Boundary value problems for nonlinear elliptic equations with a Hardy potential HABILITATION THESIS Phuoc-Tai Nguyen Department of Mathematics and Statistics, Faculty of Science, Masaryk University Contents Abstract ix Chapter 1. Introduction 1 1.1. Overview on boundary value problems 1 1.1.1. Function settings 3 1.1.2. Involvement of measures 5 1.1.3. Measure frameworks 6 1.2. Elliptic equations with a Hardy potential 11 1.2.1. The role of Hardy potentials 11 1.2.2. The role of measures 13 1.2.3. Introduction of main problems 13 1.2.4. Brief description of our contributions 15 1.3. Detailed statement of main results 19 1.3.1. Ingredients 19 1.3.2. Notions of /x-boundary trace 22 1.3.3. Linear equations 24 1.3.4. Absorption term 26 1.3.5. Source term 27 1.3.6. Gradient-dependent nonlinearities 30 1.4. Related and open problems 36 Bibliography 39 Chapter 2. Moderate solutions of semilinear elliptic equations with a Hardy potential 47 Chapter 3. Semilinear elliptic equations with a Hardy potential and a subcritical source term 69 Chapter 4. Semilinear elliptic equations and systems with Hardy potentials 99 Chapter 5. Elliptic equations with a Hardy potential and a gradient-dependent nonlinearity 143 vii Abstract The habilitation thesis is devoted to recent developments on boundary value problems for nonlinear elliptic equations with a Hardy potential in a measure framework. The presence of the Hardy potential which is singular on the boundary of the domain under consideration and the involvement of measures in the analysis yield substantial difficulties and lead to disclose the novelty of the research. New aspects are displayed not only on employed methods but also on observed novel phenomena. The thesis consists of five chapters. The first chapter addresses the main topics covered in the thesis and presents our contributions, which are collected from our recent works, including results on the existence, nonexistence, uniqueness, a priori estimates and qualitative properties of solutions, a full characterization of isolated boundary singularities, removable singularities. The major features of the problems under investigation depend essentially on the expression of the nonlinear term in equations. Therefore, typical models are successively considered throughout the last four chapters in order to reveal different phenomena. In particular, chapter 2 deals with absorption nonlinear terms and chapter 3 treats source nonlinear terms. Chapter 4 is devoted to an extension of results in the previous chapters to more general equations and systems. Finally chapter 5 focuses on the case where nonlinear terms depend on both solutions and their gradient. ix CHAPTER 1 Introduction Section 1.1 of this chapter is devoted to an overview on boundary value problems for linear and nonlinear elliptic equations involving the classical Laplace operator in function settings and in measure frameworks. We also discuss the role of measures and point out essential differences between the linear case and nonlinear case in measure frameworks. In Section 1.2, we give the motivation for the study of singular operators which are the Laplace operator perturbed by Hardy potentials. Then we address the main problems involving these operators in this thesis. The interaction between Hardy potentials and measures leads to interesting features of the problems and reveals new phenomena. We present briefly our main contributions, including results extracted from our joint paper with Moshe Marcus [106], with Kon-stantinos Gkikas [78, 80] and on our single-author work [119], as well as accompanying comments and comparisons with previous in the literature. This may help the reader to grasp the main results more easily and to follow the subsequent sections more smoothly. The detailed statements of these results are provided in Section 1.3 of the chapter for the convenience of the reader. Finally, in Section 1.4, we discuss related and open problems which have recently attracted a great deal of attention. The thesis is not a self-contained text despite of our effort to make it accessible to researchers and students with different backgrounds. We assume that the reader is familiar with basic notions in functional analysis and measure theory which can be found in standard textbooks, for instance [34, 64, 63, 1, 76]. However, at some places, relevant concepts and ideas from these fields are recalled and explained in order the make the exposition of the main results clearer. 1.1. Overview on boundary value problems In this section, we first list basic notations that are used frequently throughout the thesis. The reader is referred to the standard textbooks [1, 34, 63, 76, 128, 116, 139, 140] for more properties of these notations. Then we recall well known results for boundary value problems for linear and nonlinear equations involving the classical Laplace operator. These results, as well as the proofs, can be found in excellent references [116, 138, 139]. Basic notations. • Assume $1 is a domain (namely a connected, open nonempty subset) in M>N (N > 1). Let C(fl) be the space of continuous functions on Q. We denote by Ck{9) the space of functions k times continuously differentiable on £1 (for integer k > 1) and C°°(0,) = Dk>iCk(£l). Let CC{£1) be the space l 2 1. introduction of continuous functions on $1 with compact support in Q. Put C^(fl) = Ck(Q) n CC(Q) and Cc°°(ft) = c°°(n) n CC(Q). • Let 0 G C($l) be a positive weight function. Denote by LK(S1, 0), 1 < k < oo) the weighted Lebesgue space of functions v satisfying \v\KK(n,4>) := G LK(ft,0) : l/v G LK(Vl,4>) for every |/3| < m}. This space is endowed with the norm \/3\<-m We denote by H1^,^) = W1,2(fl, (/)). When 0=1, these spaces become the usual Sobolev spaces Wm,K(tt). We denote by W^K(Cl) the space of functions v such that v G Wm,K(tt') for any compact subset $1' C $1. • A Borel measure on $1 is called a Radon measure if it is bounded on compact sets of $1. Let 9Jt($l, 0) be the space of Radon measures r on $1 satisfying J^0d|r| < oo and 9Jt+($l,0) be the positive cone of 9Jt(Sl,0). Denote by 9Jt(9$l) the space of bounded Radon measures on d£l and by 9Jt+(cWl) the positive cone of 9Jt(9Sl). The space 9Jt($l,0) and the space 9Jt(9$l) are respectively endowed with the norms 0 -1, we have the spaces LK(Q,5e), Wm>K(Q,5e), Wl(Q,5e) and L*(n,5e). • A sequence {$ln} is a C2 exhaustion of $1 if {$ln} is uniformly of class C2 and for every n, Qn C ftn+i and Un$ln = Q. • Throughout the thesis, c,ci,C2,C,Ci,C denote positive constants which may vary from line to line. We write C = C(a,b) to emphasize the dependence of C on the data a, b. • The notation / ~ h means that there exist positive constants c\, C2 such that c±h < f < C2h. • For any a, b G M, we write a A b = min{a, b} and a V b = max{a, b}. • For re > 1, we denote by re' the conjugate exponent, i.e. re' = -^f- • For a set E in l", denote by \E the indicator function of E. • For x G M.N, denote by 5X the Dirac measure concentrated at x. 1.1. overview on boundary value problems 3 • Denote by dS the surface element on dtt. • For z G dtt, denote by nz the outer unit normal vector at z. We denote by ^ the derivative in the outer normal direction on d£l. • The gradient of u is Vu = (^-,355^-)• 1.1.1. Function settings. Nonlinear elliptic equations have been one of the most developed subject in the area of partial differential equations (PDEs) not only because of their great interest to other fields within mathematics such as calculus of variations, harmonic analysis, measure theory, differential geometry, fluid dynamics, probability theory, but also because of their applications in physics, engineering, and other applied scientific disciplines. The simplest second order PDE is the Laplace equation -Au = 0 infi (1.1.1) where Q is a domain in the Euclidean space (2 < N G N) and A denotes the Laplace operator (or Laplacian) defined by Au = X^i f^r- In (1.1.1) and throughout the present thesis, we write the Laplace operator with 'minus sign' because the operator —A is positive. In the context of this habilitation thesis, unless otherwise stated, $1 is a C2 bounded domain (see the definition of C2 domains in Gilbarg and Trugdinger [76]). A function u £ C2($l) satisfying equation (1.1.1) is called harmonic. Roughly speaking, a boundary value problem for (1.1.1) is a problem of finding an harmonic function u in $1 which satisfies certain auxiliary boundary conditions on some part of the boundary d£l in some sense. There is a huge literature on boundary value problems for (1.1.1), and for more general elliptic equations, in which one of the earliest well-known works is the Dirichlet problem which asks if we can find an harmonic function u in $1 with a prescribed boundary value u = h on d£l, where h is a given function defined on dtt. The history of such Dirichlet problem is remarkable and led to an extensive development of methods in PDEs in function settings (see a survey by Brezis and Browder [35]). Another important equation is the Poisson equation which arises in many varied physical situations -Au = t inn, (1.1.2) where r is a given datum. The Dirichlet problem associated the Poisson equation is f — Au = t in $1, [ u = v on oil, where visa given boundary datum. It is classical that if the data r and v are smooth enough then problem (1.1.3) admits a unique classical solution u G C2($l) n C($l) (see for example [76, Theorem 4.3]) and hence the equation and the boundary value condition in (1.1.3) are understood in the pointwise sense. By multiplying the equation in (1.1.3) by 0 £ Cq($1), where C2(n) = {0 G C2(n) : 0 = 0 on dfl} and then using the integration by parts, we obtain the formula - f uA4>dx= f tdx- f v^-dS, (1-1-4) Jn Jn Jan 9n 4 1. introduction where n is the outer normal unit vector to díl, ^ denotes the derivative in the outer normal direction on díl and dS denotes the surface element on díl. When r and v are not regular, one faces a difficulty stemming from the fact that (1.1.3) does not admit any classical solution. However we observe that functions satisfying (1.1.4) may exist, which leads to the definition of weak solutions. More precisely, Brezis [33] defined weak solutions to (1.1.3) with integrable data as follows: Assume r £ L1($l, 5) and v £ L1(9$l), a function u is a weak solution of (1.1.3) if u £ V-{VL) and u satisfies (1.1.4) for all 0 £ C$(íí). Here 5 is the distance function to the boundary díl. It is known, by the classical approximation method, that for any r £ L1(íž,á) and £ L1(9$l), there exists a unique weak solution u of (1.1.3) (see [116, Proposition 1.1.3]). Let Gn : $1 x $1 \ {(x,x) : x £ il} —>■ M+ be the Green kernel (or Green function) associated to the operator —A and : il x díl —> M+ be the Poisson kernel associated to —A, i.e. dG^1 Pn(x,y) = -^—(x,y) Vxeíl,yedíl. an More properties and sharp estimates of Green kernel and Poisson kernel can be found in [132, 63, 31, 128, 139]. For any r £ L1(íž,á) and v £ L1(a$l), the unique weak solution to (1.1.3) can be represented by u(x)= [ Gn(x,y)r(y)dy + [ Pn(x,y)u(y)dS(y). (1.1.5) Jn J on The theory of linear problem (1.1.3) forms a basis for the investigation of the Dirichlet problem i-Au + f(u)=t inn, 1 «n (1.1.6) [ u = v on d\l, where / : M —> M is a given function, r and v are given data. The equation in (1.1.6) consists of the linear part Au and the nonlinear part f(u). This problem has been studied by many authors in various function settings (see, for example, Brezis and Strauss [33, 40], Marcus and Veron [116], Quittner and Souplet [127, 128] and Drábek and Milota [51] and references therein). Weak solutions of problem (1.1.6) are defined Brezis and Strauss as follows: Assume r £ L1($l, 5) and v £ L1(a$l), a function u is a weak solution of (1.1.6) if u £ L1($l), f(u) £ L1($l, 5) and u satisfies - [ uA(f)dx+ [ f(u) M is a continuous, nondecreasing function with /(O) = 0 (in this case it is called absorption nonlinear term), the solvability of (1.1.6) for any r £ L1($l, 5) and v £ L1(a$l) is essentially due to Brezis and Strauss [40] and was demonstrated by Marcus and Veron in [116, Proposition 2.1.2]. It is worth emphasizing that this result holds true for a quite large class of absorption terms since it does not require any additional condition on 1.1. overview on boundary value problems 5 /. Therefore, in this regard, linear problem (1.1.3) and nonlinear problem (1.1.6) in L1 setting share a similarity. 1.1.2. Involvement of measures. Thomas-Fermi equation. A motivation for the study of semilinear elliptic equations in measure frameworks stems from the Thomas-Fermi theory (see Lieb [95] and Benilan and Brezis[20]). The theory was invented by L. H. Thomas and E. Fermi in order to describe the electron density g(x), i6l3 and the ground state energy for a system (e.g. a molecule) consisting of k nuclei of charges m-i > 0 and fixed locations A-i G M3 (1 < i < k) and £ electrons. The Thomas-Fermi energy functional for the system is £(g) = - [ gldx- [ Vgdx + - ! [ e(f-' dxdy + U 5 .7r3 Jr3 2 JR3 JE3 \x - y\ on A := {g > 0 : g e L1^3) nLf(M3), / gdx = £}, where V(x) = Yli=imi\x ~ and U(x) = ^ij'=im«mjl^« — ^ is noticed that g i—> £{g) is convex. The Thomas-Fermi energy is defined by £TF := inf £{g). (1.1.8) eeA The Euler-Lagrange equation, which is also call the Thomas-Fermi equation, is gii=(u-\) + . (1.1.9) where —A is called the Lagrange multiplier or chemical potential and u{x) = V{x) - [ r^-dy. y It is known that there is a minimizer g for (1.1.8) if and only if £ < M : = Yli=imi- The minimizer is unique, denoted by gTF, and satisfies (1.1.9) for some A > 0. Conversely, any positive solution of (1.1.9) is a minimizer of (1.1.8). In the neutral case, i.e. £ = M, one has g > 0 and A = 0, 2 therefore (1.1.9) becomes £3 = u. By applying A on both sides, one obtains a semilinear equation k - Au + 4iru'i = 4iry^mi5Ai (1.1.10) =1 where bAi denotes the Dirac measure concentrated at A{. It can be seen that the left-hand side of this equation consists of a linear part Au and a 3 nonlinear part which is expressed by «2, while the right-hand side is the sum of Dirac measures concentrated at the points A{. Interior measure data. Motivated by the investigation on equation (1.1.10), Benilan and Brezis [20] considered a more general equation - Au+\u\p-1u = t mü (1.1.11) where Q is a domain in M.N, p > 1 and r is a Radon measure on Q. Equation (1.1.10) is a particular case of (1.1.11) with N = 3 andp = |. The analysis of this equation reveals that the theory in L1 setting previously established by Brezis and Strauss [40] cannot be easily extended to a measure framework 6 1. introduction and new striking phenomena appear due to the nonlinear nature of the problem. Therefore, dealing with a measure framework would provide a deep insight about the problem. In particular, Benilan and Brezis [20] showed that the value jjr^ is a critical exponent for the existence of equation (1.1.11) with zero Dirichlet boundary condition u = 0. More precisely, they proved that if 1 < p < then the problem has a unique solution, while if p > jjr^ then there is no solution with fi = Sa for any Many developments have been achieved since the work of Benilan and Brezis [20], including Veron [136] and Brezis and Oswald [39] for a complete classification of solutions with an isolated singularity, Vazquez and Veron [134, 135] for more general nonlinearities, Brezis and Veron [41] for removable isolated singularities, Baras and Pierre [14, 15] for removability results in terms of Bessel capacities, Vazquez and Veron [133] and Friedman and Veron [70] for isolated singularities of quasilinear equations. Boundary measure data. Similar problems with boundary measures have been also studied with important motivation coming from the probability theory. Boundary value problems with measure data for linear and semilin-ear equations are respectively related to Markov processes called diffusions and super diffusions. A diffusion is a model of a random motion of a single particle and is characterized by a second order elliptic differential operator, including the Laplacian. A superdiffusion, which describes a random evolution of a cloud of particles, is closely related to semilinear equations. For further discussions about the importance of measure boundary data in the study of linear and semilinear equations in connection with diffusions and super diffusions, the reader is referred to excellent books of Dynkin [57, 58]. The role of boundary measures can be seen in particular from the representation theorem for harmonic function. More precisely, given a positive harmonic function in $1, by Herglotz-Doob theorem [116, theorem 1.4.1], there exists a unique measure v £ 9Jt+(cWl) such that u(x)= [ Pn(x,y)du(y). (1.1.12) holds. Such measure v is called boundary measure of u and it is attained as the limit of the Sobolev trace of the solution u in each surface parallel to dfl. More precisely, let {$ln} be a C2 exhaustion of $1 and denote by u\oQn the Sobolev boundary trace of u on d£ln. Then there exists a nonnegative bounded Radon measure v on d£l independent of the choice of the exhaustion such that the sequence of measures {u\gQndS} converges weakly to v. The above result shows that in order to completely characterize the boundary behavior of harmonic functions, it is insufficient to deal only with function settings and hence measures have to be involved in the analysis. 1.1.3. Measure frameworks. Linear equations. The results for linear equations in function settings can be extended to measure frameworks in which the definition of weak solutions to (1.1.3) is modified as follows: 1.1. overview on boundary value problems 7 Assume r £ and v £ 9Jt(9$l). A function u of (1.1.3) is a solution of (1.1.3) if - f u/\4>dx= [ 4>cLt- [ V0eC$(n). (1.1.13) Jn Jn Jan "n As explained in L1 setting, for any (f) £ Cq(S1), one has |0| < cc), and hence the first term on the right-hand side of (1.1.13) is finite. This also explains why 9Jl($l,5) is the largest possible measure space one can work on. The second term on the right-hand side is finite because §r- is bounded. & on It is known that, for any r £ 9Jl($l,5) and v £ 9Jt(9$l), there exists a unique solution of (1.1.3) and the solution can be represented by the Green kernel acting on r and the Poisson kernel acting on v (see [116, Theorem 1.2.2]), i.e. u(x)= [ Gn(x,y)dr(y) + [ Pn(x,y)du(y). (1.1.14) Jn Jan In particular, given a measure v £ 9Jt(cWl), the harmonic solution with boundary condition u = u is given by u(x)= / Pn(x,y)du(y). (1.1.15) Jan Absorption nonlinearities. Over the last decades, boundary value problems for semilinear equations in measure frameworks have been intensively investigated both in probabilistic approaches and analytic methods with the aim of bringing into light and describing several aspects of nonlinear phenomena. The pioneering work on the Dirichlet problem with measure boundary data for semilinear elliptic equations with an absorption term -Au + f(u)=0 infi, (1.1.16) where / : M —> M is a nondecreasing, continuous function with /(0) = 0, is due to Gmira and Veron [82]. They introduced the definition of weak solutions of (-Au+f(u)=0 inn, no (1.1.17) [ u = v on oil, in spirit of Brezis [33] as follows: Assume v G 9Jt(9$l). A function u is a weak solution of (1.1.17) if u £ L1($l), f(u) G L1($l,5) and u satisfies - f uA4>dx+ [ f(u)4>dx = - [ ^-dv V0 £ Cq(TI). (1.1.18) Jn Jn Jdn "n The notion of weak solutions is well defined. Indeed, for any \ < c5 and f(u) £ L1($l,5). It can be also seen that the term on the right-hand side of (1.1.18) is also finite because ^ is bounded on d£l and v is a bounded Radon measure on an. A highlighting feature is that, in contrast to the L1 case where the existence holds true for every L1 boundary datum, the Dirichlet problem 8 1. introduction (1.1.17) is not solvable for every measure datum in general. More precisely, Gmira and Veron [82] showed that if / satisfies d -i N+l (/(*) + Ifi-mr^—idt < oo, (i.i.i9) 1 then problem (1.1.17) has a unique weak solution. This result was demonstrated by employing a classical approximation method, in combination with Marcinkiewicz estimates on Green kernel and Martin kernel and Vitali convergence theorem. Condition (1.1.19) is sharp and the value jprj appearing in (1.1.19) is a critical exponent for the existence of a solution to (1.1.17). When f(u) = \u\p~1u with p > 1, equation (1.1.16) becomes -Au + \u\p~1u = 0 infi, (1.1.20) and condition (1.1.19) is interpreted as 1 < p < ^-j, which is the subcritical range. Hence in this range, for any v G 9Jt(cWl), problem ( -Au+\u\p~1u = 0 inn, I u = v on ou, admits a unique weak solution in this case. In the supercritical range, namely P > ]^T> ^ was shown (see [82]) that there is no weak solution of (1.1.21) if v is a Dirac measure concentrated at a point on d£l. Furthermore, when P > ]^T> anv nonnegative solution u £ C(ft\ {0}) of (1.1.20) vanishing on cWl\{0} is identically zero, i.e. isolated boundary singularities are removable. The topic has reached to its full flowering through a series of celebrated papers by Marcus and Veron [109, 111, 112, 113, 114, 115, 99, 116] and many other works (see for examples [37, 30, 120, 29, 22] and references therein). Taking into account the construction of the boundary measure of positive harmonic functions, Marcus and Veron introduced a notion of boundary trace [116, Definition 1.3.6] in order to describe the boundary behavior of solutions to equation (1.1.16). Definition 1.1 (m-boundary trace). Let u e W^(Q) for some re > 1. We say that u possesses an m-boundary trace on d£l if there exists a bounded Radon measure v on d£l such that, for every C2 exhaustion {$ln} of $1 and every tp £ C($l), lim / u\dQnipdS = / tpdv. >dnn Jdn Here u\qqu denotes the Sobolev trace of u on dCln. The m-boundary trace of u is denoted by ti{u). It is known from [116, Proposition 1.3.7] that every positive harmonic function u in $1 admits a positive m-boundary trace v G 9Jt(9$l), which in fact coincides the boundary measure given by (1.1.15). A first step to study the notion of m-boundary trace is to deal with moderate solutions of (1.1.16), namely weak solutions of (1.1.16) which are bounded by positive harmonic functions (see [116, Definition 3.1.1]). The equivalence between the notion of m-boundary trace and the concept of moderate solutions was established by Marcus and Veron [116, Theorem 3.1.2]. Moreover, these notions are equivalent to the condition f(u) G L1($l,5). 1.1. overview on boundary value problems 9 Afterwards, Marcus and Veron generalized the notion of m-boundary trace to a notion called rough boundary trace and pointed out that, under some additional assumptions on /, every positive solution of (1.1.16) possesses a rough boundary trace given by a positive outer regular Borel measure on d£l (see [115, Theorem 1.1] or [116, Theorem 3.1.8, Theorem 3.1.12 and Definition 3.1.14]). Conversely, they showed that the solvability for (1.1.17) holds true when visa positive outer regular Borel measure which may be infinite on subsets of dtt (see [116, Theorem 3.3.1]). In particular, if f(u) = \u\p~1u then, in the subcritical range p < jprj, for any positive outer regular Borel measure v, problem (1.1.21) admits a unique positive solution (see [115, Theorem 1.6]). Also, in the subcritical range, Marcus and Veron [111, 115] characterized completely boundary isolated singularities of nonnegative solutions of (1.1.20). It means that if u G C($1\{0}) is a non-negative solution of (1.1.20) vanishing on d£l \ {0}, then either u = u^, the solution of (1.1.21) with v = kSo for some k > 0 (weakly singular solutions), or u = lim/^oo (strongly singular solution). The supercritical case is more challenging and was treated by many authors using various techniques. The removability result due to Gmira-Veron has been significantly extended, either by using probabilistic approach by Le Gall [92], [93], Dynkin and Kuznetsov [59], [60], under the restriction jf^j < P < 2, or by employing purely analytic methods by Marcus and Veron [112, 113, 114] in the whole range < p. The key ingredient in analyzing the problem is the Bessel capacity C2 , in (N — l)-dimension, where p' = Among the most interesting results, it is worth mention- ing that problem (1.1.21) is solvable with u £ 9Jt+(9$l) if and only if u is absolutely continuous with respect to the C2 ,-capacity. Furthermore, if _ p,p E C d£l is compact and u G C($l \ E) is a solution of (1.1.20) vanishing on d£l \ E, then u is necessary zero if and only if C2 V/(E) = 0. A complete p characterization of positive solutions of (1.1.20) has been developed by Mse-lati [117] when p = 2, by Dynkin [57, 58] for ^ < p < 2, and finally by Marcus [99] for the whole supercritical range p > jf^j- Source nonlinearities. An important PDE with a source term is the Lane-Endem equation -Au = up infi, (1.1.22) where p > 1, which was named after astrophysicists Jonathan Homer Lane and Robert Emden. This equation was introduced in 1869 by Home Lane [89] in the study of the temperature and the density of mass on the surface of the Sun and has received special attention because it can be used to describes polytropes in hydrostatic equilibrium as simple models of a star. A systematic survey on the this equation from the physical and mathematical point of view was presented in [42, 17]. A great number of remarkable works have been carried out in various directions by many mathematicians, including Lions [96], Brezis and Nirenberg [38], Gidas, Ni and Nirenberg [73], Gidas and Spruck [74, 75], Baras and Pierre [14, 15] and Kalton and Verbitsky [86], Polacik, Quittner and Souplet [125], Quittner and Souplet [127, 128]. 10 1. introduction It is worth mentioning that a universal pointwise estimate for nonneg-ative solutions of (1.1.22) was obtained by Polacik, Quittner and Souplet [125, Theorem 2.3] for 1 < p < $±| by using a rescaling argument, a key doubling property and Liouville-type results. The Dirichlet problem with measure boundary data -Am = up in n, «o (LL23) u = v on oil, was first studied by Bidault-Veron and Vivier in [31] where estimates involving classical Green and Poisson kernels for the Laplacian were established to obtain an existence result in the subcritical range, i.e. 1 < p < Afterwards, Bidaut-Veron and Yarur [32] reconsidered this type of problem in a more general setting and gave a necessary and sufficient condition for the existence of solutions. Chen et al. [43] investigated the Dirichlet problem with a more general source term by using the Schauder fixed point theorem in combination with weighted Marcinkiewicz estimates. Recently, Bidaut-Veron et al. [28] provided new criteria expressed in terms of boundary capacities for the existence of weak solutions to problem (1.1.23). A remarkable feature of the Dirichlet problem (1.1.23) is that, not only the value of exponent p, but also the total mass of the boundary datum v plays an important role in deriving the existence and non-existence result. More precisely, when 1 < p < there exists a threshold value p* such that problem (1.1.23) admits a solution if |M|g#(<9n) < P*> and no solution if IMk(dn) > P* (see [31]). It is worth mentioning that existence results for scalar equations were extended to the Lane-Emden system -Am = vp + r in n, -Av = up + f inn, (1.1.24) u = v, v = v on d£l, where r, f are measures in n and u,u are measures on d£l. Various existence and non-existence results, as well as a priori estimates, for solutions of (1.1.24) were established in [32]. See also the celebrated paper of Polacik, Quittner and Souplet [125] and the books of Quittner and Souplet [127, 128] for related results. Gradient-dependent nonlinearities. Equations with a nonlinear term depending on the gradient of solutions (or the convection) arise in various models of optimal stochastic control with state constrains. A typical equation is - Au + \Vu\q = 0 inn, (1.1.25) with q > 1, which is also a particular case of Hamilton-Jacobi-Bellman equations. Existence, a priori estimates and qualitative properties of solutions to this equation, as well as more general class of equations, were discussed in Lions [97] and Lasry and Lions [90]. The Dirichlet problem with measure data -Am+ f(|Vd) = 0 inn, ' (1.1.26) u = v on oil 1.2. elliptic equations with a hardy potential 11 where / : M+ —> M+ is a continuous, nondecreasing function with /(0) = 0, was first studied in our joint work with Veron [120]. Under the subcriticality integral condition on / given by , , 1 N+l /(£)£ N~dt < +oo, we obtained the existence of a positive solution to (1.1.26). When / is of power type, namely /(|Vu|) = |Vu|9 with 1 < q < 2, the subcriticality integral condition reads as 1 < q < ^f^. It was showed that is the critical exponent for the solvability of (1.1.26) and a complete description of the structure of solutions with an isolated singularity on dCl was provided. Moreover, the existence of a solution to the Dirichlet problem with boundary datum given by a Borel measure in the subcritical case, i.e. 1 < q < and a removability result in the supercritical case, i.e. < q < 2, were also established. These results were then extended in our joint paper with Marcus [105] to the Dirichlet problem for a much more intricate equations with a nonlinear term depending on both solutions and their gradient f -Au+ f(u,\Vu\) = 0 in CI, ao (1-1.27) [ u = v on oil Two model cases f(u, |Vu|) = up + |Vu|9 and f(u, |Vu|) = up\X7u\q were carefully studied in [105]. It is worth mentioning that equations with a gradient-dependent nonlinear term have been studied in various directions. We refer to Ghergu and Rädulescu [71, 72] for singular elliptic equations with convection term and zero Dirichlet condition, Alarcön, Garcia-Meliän [5, 6] for Keller-Osserman type estimates and Liouville type theorems, Aghajani, Cowan and Lui [2, 3, 4] for singular solutions, Bidaut-Veron, Garcia-Huidobro and Veron [25, 26, 27] for a priori estimates on singular solutions. 1.2. Elliptic equations with a Hardy potential In this section, we first explain the role of Hardy potentials and measures in the investigation. Then we address the main problems regarding elliptic equations with a Hardy potential in the thesis and depict briefly the main results collected from our papers [106, 119, 78, 79, 80]. The detailed statements of these results will be provided in Section 1.3. 1.2.1. The role of Hardy potentials. Schrödinger operators of the form Ly = A + V, where V is a potential, have been intensively investigated by numerous mathematicians because of their applications in non-relativistic quantum mechanics, geometry, spectral and scattering theory, and integrable systems (see for example [129, 131]). The behavior of the potential V has a significant effect on properties of Ly such as spectral properties and the existence of the associated Green kernel and Martin kernel. The Coulombian potential V(x) = /igt, appears in the Thomas- Fermi-Dirac-von Weizsäcker theory [18, 19]. The case where V is an inverse square potential, i.e. V(x) = fi\x\~2, is called Leray-Hardy potential and has been studied in connection with semilinear elliptic equations by Guerch 12 1. introduction and Veron [83], Cirstea [47], Dupaigne [55], Davila and Dupaigne [49, 50], Du and Wei [53], Chen and Veron [44]. When singular potentials, which blow up on the boundary dtt, were involved, in most of the cases, they were assumed to satisfy, e.g. |V(a;)| < c5(x)2~e for some e > 0 small or more generally 5V G L1($l) (see [141, 122, 123]), which exclude the case where V behaves like 5~2 on certain part of $1. The case where \V(x)\ < CS{x)-2 (1.2.1) is of great interest and a theory of linear Schrodinger equation Lyu = 0 on manifolds was successfully and systematically developed by Ancona [7, 8]. The particular potential V(x) = fi5(x)~2 has received special attention and it is called Hardy potential because of the close link to the Hardy inequality Cff(fl) f ^rdx < I \X7ip\2dx Vy> G ^(fi), (1.2.2) Jn °2 Jn where CuiP) denotes the best constant in Hardy inequality (also called Hardy constant) given by , ^ fa \y^>\2dx CHP):= inf f . (1.2.3) In spirit of [69], Hardy inequality (1.2.2) can be interpreted as a form of uncertainty principle. This means if the function tp in (1.2.2) is localized close to the boundary (i.e. the term on the left-hand side of (1.2.2) is large) then its momentum becomes large (i.e. the left-hand side of (1.2.2) is large). The exponent —2 appearing in the power 5~2 plays a crucial role because it keeps inequality (1.2.2) scaling invariant. Inequality (1.2.2) in one dimensional case with Hardy constant Ch{$1) = ^ was discovered by Hardy [84, 85]. Moreover, he pointed out that the Hardy constant is not attained. This inequality was then extended to Lip-schitz domains in WN by Necas [118], Opic and Kufner [121] and was revisited by Brezis and Marcus [36]. It is classical that CjjiP) £ (0, |] and CjjiP) = ^ if is convex (see [103, Theorem 11]) or if — A5 > 0 in the sense of distributions (see [16, Theorem A]). Moreover, the infimum in (1.2.3) is achieved if and only if CjjiP) < \- For fi G M, denote by the Laplace operator perturbed by a Hardy potential LM:=A + ^, (1.2.4) where S(x) = dist(x, dtt). Notice that the Hardy potential blows up on d£l. The energy functional associated to the operator is given by E^) = \J^(\Vv\2-^V2)dx, 0, a, (3 G M, 1.2. elliptic equations with a hardy potential 15 does not solve (1.2.6). Consequently, standard arguments relying on the scaling invariance property are not applicable. • The boundary data are given by measures on the boundary díl, which makes solutions (if exist) significantly less regular. Therefore, the standard approximation arguments are invalid or may be valid only under some additional conditions on the nonlinear term g. The interaction of the above features yields substantial difficulties and leads to disclose new types of results. Therefore a new approach with subtle analysis is required in the investigation. 1.2.4. Brief description of our contributions. This subsection serves as a summary of the main results in the thesis and provides a comparison with previous results. This might help the reader grasp the gist of the main results. The reader who is interested in the detailed statements and the proofs is referred to Section 1.3 and Chapters 2-5. Boundary trace problem. We consider the boundary trace problem for equation (1.2.6). To this purpose, we first investigate the boundary behavior of L ^-harmonic functions in il, i.e. solutions of the equation - L^u = 0 in il, (1.2.7) which in turn indicates possible boundary behavior of solutions to corresponding semilinear equations. Based on that, we introduce a notion of normalized fi-boundary trace expressed by a bounded measure on the boundary (see Definition 1.3). This notion is new (compared with the notion of m-boundary trace when fi = 0 in Definition 1.1) because it depicts clearly how a function admitting a normalized /x-boundary trace behaves on every surface parallel to the boundary díl. Moreover, it is pointed out that this notion is equivalent to the concept of moderate solutions of (1.2.6), namely solutions which are dominated by a positive L^-harmonic function. In parallel, another notion of boundary trace defined by mean of weak convergence of measures is introduced by Gkikas and Veron [77]. We then show that these notions of boundary trace are equivalent (see subsection 1.3.2). Furthermore, it is worth mentioning that any positive solution (not necessarily moderate) admits a boundary trace given by a Borel measure on díl which may be infinite on compact subsets of díl (see [77, Theorem F]). Dirichlet problem. We investigate the Dirichlet problem f -Lliu±g(u,\Vu\) = 0 'mil, 1 an (1.2.8) [ u = v on d\l, where v is a given measure on díl, 6 (0, |] and g is a nondecreasing function with respect to u and/or \SJu\. Since the analysis depends essentially on the expression of the nonlinear term, typical models are considered separately, for which we showed that if the nonlinear term g does not grow 'too fast' with respect to u and/or \Vu\, then for any bounded Radon measure v on díl, problem (1.2.8) possesses a solution. Absorption case. Let us illustrate the above-mentioned fact by considering the absorption case, namely the equation (1.2.8) with plus sign. The 16 1. introduction following exponent is deeply involved in the analysis a:=^(l + yi^). (1.2.9) This exponent comes from the construction of a special solution of (1.2.7) in the half space M.+ which is singular at the origin 0 and vanishes on \{0} (Actually this solution plays a similar role as the Poisson kernel of —A in the half space). Notice that, since // G (0, a G [^, 1). When g : M —> M is a nondecreasing, continuous function depending only on solutions u with g{0) = 0, we show that the condition (9(t) + |ff(-t)|)t_1_pMt < oo, (1.2.10) Jl with N + a , . Pu ■=-, (1.2.11 is a sufficient condition for the existence of (1.2.8) (see Theorem 1.10 and [77, Theorem 3.3]). A typical model is g{u) = with p > 1 and (1.2.10) is satisfied if and only if 1 < p < p^. For this model, p^ is called a critical exponent for (1.2.8). We say that p is in the subcritical range if p < p^i, otherwise we say that p is in the supercritical range. When g : M+ —> M+ is a nondecreasing, continuous function depending only on |Vu| with g(0) = 0, we show that the condition /oo g^t^^dt < oo, (1.2.12) with ;= W^-v is a sufficient condition for the existence of (1.2.8) (see [79, Theorem B]). A typical model is g(|Vu|) = |Vu|9 with 1 < q < 2 and (1.2.12) is satisfied if and only if 1 < q < q^. For this model, q^ is called a critical exponent for (1.2.8). We say that q is in the subcritical range if q < q^, otherwise we say that q is in the supercritical range. When g : M+ x M+ —> M+ is a nondecreasing, continuous function depending on both solutions u and the gradient | Vu| with g(0, 0) = 0, we point out that (see [80, Theorem 1.3]) the sufficient condition for the existence of (1.2.8) can be nicely expressed by a combination of condition (1.2.10) and condition (1.2.12) as J g(t,t'»>)t-1-p>1dt < oo. (1.2.14) There are two typical models that are worth highlighting. • The first one is g(u, |Vu|) = up + |Vu|9 with p > 1 and 1 < q < 2 and (1.2.14) is satisfied if 1 < P < Pfi and 1 < q < qß. (1.2.15) For this model, we say that (p, q) is in the subcritical range if (1.2.15) holds, otherwise we say that (p, q) is in the supercritical range. 1.2. elliptic equations with a hardy potential 17 • The second model is g(u, |Vu|) = up\SJu\q with p>0,0 1 and (1.2.14) is satisfied if (N+ a-2)p + (N+ a- l)q < N + a. (1.2.16) For this model, we say that (p, q) is in the subcritical range if (1.2.16) holds, otherwise we say that (p, q) is in the supercritical range. Sharp solvability results in the typical models can be summarized in the following table 1. Absorption case Existence, uniqueness in subcritical range Non-existence in supercritical range g(u) = P < Pp P > Pp. g(\Vu\) = \Vu\q q Pp. or qp, < q < 2 g(u, \Vu\) = v?\Vu\q (N + a-2)p + (N + a - l)q < N + a (N + a-2)p + (N + a - 2)p > N + a Table 1: Absorption case The above table shows that, in the subcritical range, for any bounded boundary measure v, problem (1.2.8) with plus sign admits a unique solution. Moreover, this solution is bounded from above by the solution to the corresponding linear problem f —L,,u = 0 in $1, *o (1-2-17) [ u = v on oil. The proof of the existence result relies on the approximation method, making use of sub and super solutions theorem, delicate estimates of Green kernel and Martin kernel in weak Lebesgue spaces and Vitali convergence theorem. The condition (1.2.14) allows to show that the sequence of approximate nonlinear terms is convergent. The existence result for the case when g depends on both u and |Vu| is new, even in the case fi = 0. The uniqueness in case the nonlinear term depends only on u is based on Kato type inequalities which are achieved due to the monotonicity. The question of uniqueness has remained open for a while, even when fi = 0, in case the gradient |Vu| is involved in the analysis due to the lack of monotonicity. In [105], we obtain the uniqueness for the case g(u, |Vu|) = up + |Vu|9 and fi = 0. One of the main thrusts of our thesis is to obtain the uniqueness in the case g(u, \Vu\) = up\Vu\q and 0 < /x < | in which the interplay between up and |Vu|9 drastically complicates the situation. Our existence and uniqueness results cover well known results in case fi = 0 and provide a full understanding in the case 0 < /x < |. Also in the subcritical range, we provide a complete classification of solutions of (1.2.6) with an isolated boundary singularity. In particular, we show that there are actually two types of solutions with an isolated boundary singularity at a point y £ d£l: the weakly singular solutions, i.e. solutions uk8y °f (1-2.8) with plus sign and v = kSy with k > 0 and Sy being the Dirac -'-The cases g(u) = \u\p^1u and g(\X7u\) = |V«|9 are particular cases of g(u, |Vm|) = up\X7u\q with q = 0 and p = 0 respectively, however we include these cases in the table for the sake of completeness. 18 1. introduction mass concentrated at y, and the strongly singular solution «00,5 which is the limit UooSy = ^mk^foouk8y (see Theorem 1.17 and Theorem 1.18). In the supercritical range, we prove that boundary singularities are removable. It means if E is a compact subset of dtt which has a zero capacity in certain sense and u G C($l \ E) is a nonnegative solution of (1.2.6) with plus sign such that u vanishes on d£l \ E, then u is identically zero (see Theorem 1.19 and Theorem 1.20). Source case. Next we consider the source case, i.e. equation (1.2.6) with minus sign. Phenomena occurring in this case are sharply different from those in the absorption case. A striking distinction is that the existence for (1.2.8) holds if the norm of v is small and does not hold if the norm of v is large (see Theorem 1.12 and Theorem 1.21). Moreover, in contrast to the absorption case, in the source case, solutions (if exist) are bounded from below by the solution of (1.2.17). The method used to prove the existence in the source case is different from that in the absorption case due to the nature of the nonlinear term. In fact, when g(u) = up, we use the sup and super solutions method combined with 3-G estimates to show the existence of the minimal solution. This can be done thanks to the monotonicity of the nonlinear term. In a more general case, or in case g depends also on the gradient |Vu|, this method is invalid (due to the lack of monotonicity), hence we employ the Schauder fixed point theorem to show the existence under the smallness assumption on the boundary data. The existence result in the typical models are summarized in the following table 2 in which |M|g#(<9n) denotes the norm of the boundary measure datum v. Source case Existence Non-existence in subcritical range in supercritical range g(u) = up p < and P < Pp, \H\m(dn) > P* IMIsm(drc) ^ P* or p > p^ g{\Vu\) = \Vu\q q < qß and IMk(öü) sma11 Not known yet g(u,\Vu\) = up + \\7u\q p < Pfi and q < q^ and IMk(öü) sma11 Not known yet g(u,\Vu\) = Up\Vu\i {N + a-2)p + (N + a - l)q < N + a and imiot(öü) sma11 Not known yet Table 2: Source case From Table 2, we see that, in the source case, non-existence result holds for the model g(u) = up if the norm of v is large enough and has not been known yet in other typical models. We obtain various necessary and sufficient conditions in terms of sharp estimates on Green kernel and Martin kernel (see Theorem 1.13 for the case g{u) = up). We also establish criteria expressed in terms of capacities for the 2The cases g(u) = \u\p~1u and g(\Vu\) = \Vu\q are particular cases of g(u, |Vm|) = up\X7u\q with q = 0 and p = 0 respectively, however we include these cases in the table for the sake of completeness. 1.3. detailed statement of main results 19 existence of (1.2.8) (see Theorem 1.14 for the case g(u) = up and Theorem 1.22 for the case g depends on u and |Vu|). Our results for the case where g depends only on u extend those in [31, 32, 28] for fi = 0, and the results for the case where g depends on |Vu| are new even for fi = 0. 1.3. Detailed statement of main results In this section, we first discuss important ingredients for the investigation. Then we provide the detailed statement of our recent results on the boundary value problems for linear and nonlinear equations involving operator which were established in our papers [106, 119, 78, 79, 80]. 1.3.1. Ingredients. Main ingredients in the study of boundary value problems with measure data for equations (1.2.7) and (1.2.6) include eigen pair of — and the Green kernel and the Martin kernel of — L^. First eigenvalue and eigen]'unction. The eigenvalue problem associated to — is /n(|Vy|2-£y2)ds -V := mf -7.-jj-. (1.3.1) If /x < j then by [36, Remark 3.2], problem (1.3.1) admits a positive minimizer p^ in Hq(£1) and hence A^ is the first eigenvalue of — in Hq(£1). The function p^, normalized by jn(p2l/52)dx = 1, is the corresponding positive eigenfunction and satisfies -Lf^Pfj, = Xf^Pfj, in fl. Moreover, by [66] (see also [108, Lemmas 5,1, 5.2] and [50, Lemma 7] for an alternative proof), p^ ~ Sa in $1, where a is given in (1.2.9). It is noted that, since // G (0, |], a £ [^, 1). If /x = j then there is no minimizer of (1.3.1) in Hq(Q), but there exists a nonnegative function pi 6 H}(Q) such that —Lipi = Aiwi in $1 in the 4 wc 4 4 4 4 sense of distributions. Again by [66], pi ~ 5^ in $1. We observe from (1.2.3) and (1.3.1) that AM > 0 if // < CH(ty, AM = 0 if /x = Ch(Q) < \, while A^ < 0 when // > C#(f2). It is not known if A^ > 0 when /x = Ch(Q) = \- However, if $1 is convex or if — A5 > 0 in the sense of distributions - in these cases CH{9) = \- then Ai > 0 (see [36, Thereom II]) and [16, Theorem A with k = 1 and p = 2]). Green kernel and Martin kernel. The positivity of the first eigenvalue A^ plays a crucial role in derivation of the existence of Green kernel and Martin kernel. From the above observation, we see that this property does not hold for arbitrary // £ (0, ^]. Therefore, in order to go further in the study of the Green kernel and Martin kernel, we assume that /i£(0,|] and AM > 0. (1.3.2) Notice that this assumption (1.3.2) is fulfilled when fi 6 (0, C#(f2)). Throughout the thesis, unless otherwise stated, we assume that (1.3.2) holds. 20 1. introduction We say that u is L^-harmonic (resp. L^-subharmonic, L^-superharmonic) in fl if u G L\ (Q) and — L^u = 0 (resp. — L^u < 0, —L^u > 0) in the sense of distributions in $1, namely - / uL^tpdx = (resp. <, >) 0, Vtp G Cc°°(ft). Under assumption (1.3.2), tp^ is a positive L^-superharmonic function in $1. Therefore, a classical result of Ancona [8] and a result of Gkikas and Veron [77, Section 2] (for the case ^ = \) imply that for every y G dil, there exists a positive L^-harmonic function in $1 which vanishes on dil \ {y} and is unique up to a constant. This function is denoted by K^(-,y) with normalization KJ}(xQ,y) = 1 where xq G $1 is a fixed reference point. The function (x,y) i—> Kj}(x,y), (x,y) G $1 x all, is called the L^-Martin kernel in $1 relative to xq. We emphasize that the role of the Martin kernel is similar to that of the Poisson kernel in case fi = 0. However, unlike the Poisson kernel which has a finite mass, the Martin kernel KJ}(, y) may have zero or infinite mass at y. In particular, if fi G (0, C#(f2)), then the mass of Kj}(-,y) at y is zero and hence the Poisson kernel does not exist in this case. Furthermore, by [8] and [77, Theorem 2.33] (for // = |), there is a one-to-one correspondence between the set of positive L^-harmonic functions and the set of positive bounded Radon measures on dil. More precisely, we have: Theorem 1.2 (Representation Theorem). For every v G dJl+(dfl) the function K>](*):=/ Kj}(x,y)du(y) Vx G $1 (1.3.3) Jan is L^-harmonic in $1. Conversely, for every positive L^-harmonic function u in Q there exists a unique measure v G 9Jt+(a$l) such that u = K^[v] in a The measure v such that u = K^[u] is called the L^-boundary measure of u. If fi = 0, v becomes the m-boundary trace of u (see the definition of m-boundary trace in Definition 1.1). However, when fi G (0, C#(f2)), it can be proved that, for every v G 9Jt+(a$l), the m-boundary trace of is zero. Let G^ be the Green kernel for the operator — in Q x Q defined by poo G^{x,y) = / H^x,y,t)dt Jo where is the heat kernel associated with — L^. By [66, Theorem 4.11], for every x,y G CI, x ^ y, G%(x, y) ~ \x - y\2-N (l A 5(x)^(y)a|x - y\-2a) . (1.3.4) This estimate leads to the observation that a measure r G 9Jt+($l, Sa) if and only if G^[r] is finite a.e. in $1 (see [106, page 9]) where G^ is the Green 1.3. detailed statement of main results 21 operator acting on r defined by ,nri(z) == / GnJx,y)dr(y). t From the following relation between Green kernel and Martin kernel K%{x, y) = lim ^fX,Z\ Vx G ft, y G dSl, and estimate (1.3.4), we deduce Kj}{x,y) ~ 5{x)a\x -y\2-N-2a Vx G Q, y G dfl. (1.3.5) Estimates in weak Lebesgue spaces. Let us first recall the definition of weak Lebesgue spaces (or Marcinkiewicz spaces). See the paper of Benilan, Brezis and Crandall [21] for more details of these spaces. Let re > 1, re' = and 0 be a positive weight function. We set L£(ft,0) := {u G LJoc(n) : |M|Ls,(n,0) < °o} where ||m||lk (£},(/>) := inf "|c G [0, oo] : J \u\. The space L£,($1,0) is called the weak Lebesgue space with exponent re (or Marcinkiewicz space) with the norm \\-\\l*(Slrf)- The subscript w in the notation stands for 'weak'. The relation between the Lebesgue space norm and the weak Lebesgue space norm is given in [21, Lemma A.2(h)]. In particular, for any 1 < r < re < oo and u G LJoc(Q), there exists C(r, re) > 0 such that for any Borel subset E of $1 \u\rdx^J 1-2 We notice that LK(ft,0) C L£(n,]|| _jv+^_ ]W = / V^(x,z)dz/(z). J an Estimates (1.3.8) and (1.3.9) indicate that the value n+^°Li would be a critical exponent for semilinear equations with the nonlinear term depending on the gradient of solutions. 1.3.2. Notions of /x-boundary trace. One of the first attempt in the study of boundary value problems for linear and nonlinear equations involving operator was carried out by Bandle, Moroz and Reichel [11] who investigated L^-sub and superharmonic functions and obtained the existence and nonexistence of large solutions, i.e. solutions blowing up on the boundary Further research related to large solutions is due to Bandle and Pozio [12, 13] and Du and Wei [52]. However, there are other types of solutions which may not singular on the whole boundary dtt. Therefore, in order to characterize the boundary behavior of such solutions to equations with a Hardy potential, we need a new tool. Normalized boundary trace. An interesting observation emerging from estimate (1.3.5) and (1.3.3) is that there exists /3q > 0 small enough such that for any v g 9Jt+(9$l) and any (3 g (0,/3q), there holds v]dS ~ /31 a\\iy\\m(dn), where dS denotes the surface element on := {x g $1 : S(x) = /3}. This, together with the fact a < 1, implies that the m-boundary trace (see Definition 1.1) of is zero for any v g 9Jt+(9$l). Therefore, the notion of m-boundary trace (see Definition 1.1) is no longer an appropriate tool to describe the boundary behavior of Taking into account of the Representation theorem (see Theorem 1.2), we see that this notion does not play a role in the study of L^-harmonic functions. Therefore, we introduce a new notion (see [106, Definition 1.2]) as follows: Definition 1.3. Assume 0 < /x < C#(f2). A function u g W^(fl) with some r > 1 possesses a normalized fi-boundary trace if there exists a measure v g 9Jt(5f2) such that lim^-1 I \u-K^[u]\dS = 0, (1.3.10) iS/3' MLJI where dS denotes the surface element on E^. The normalized /x-boundary trace will be denoted by tr*(u). 1.3. detailed statement of main results 23 The terminology 'normalized' comes from the term /3a_1 in (1.3.10). Roughly speaking, the difference between the function u and on has to be normalized by the weight /3a_1 = 5{x)a^1 for x G Tip so that it tends to zero. The weight /3a_1 is also 'prepositional' to the volume of the surface Yip. The term on the left-hand side of (1.3.10) can be understood as the 'average' of the difference between u and near the boundary. This notion is well-defined, i.e. if v and u' satisfy (1.3.10) then v = v' (see the remark after [106, Definition 1.2]). The condition u G W^(£l) ensures the meaning of u on Yp as the Sobolev trace of u on Yip. The restriction 0 < /x < C#(f2) in [106] is imposed to guarantee that the first eigenfunction tp^ of — is a positive L^-superharmonic in $1. The notion was extended to the range /x < | by Marcus and Moroz in [104] due the fact that there exist local L^-superharmonic functions in a neighborhood of dCl. The notion is new even in the case /x = 0 and enables us to measure how close a function is in comparison with the Martin kernel near the boundary. An important feature of this notion is that it allows to derive tr* (G^[r]) = 0 for any r G 971(17, 5a) and tr* (K%[i/]) = v for any v G Tt(dQ). Boundary trace denned in a dynamic way. In parallel, Gkikas and Veron [77] introduced another notion of boundary trace which is defined using the weak convergence of measures. Let D (e $1 and xq G D be a fixed reference point. If h G C{dD) then the following problem -Luu = 0 in D, u = h on oD, admits a unique solution which allows to define the L^-harmonic measure ujxj» on dD by u(x0) = [ h(y)duxD°(y). (1.3.12) J dD Let {fln} be a C2 exhaustion of Q. For each n, let us^ be the L^"-harmonic measure on dttn. Definition 1.4. Let /x G (0, \}. We say that a function u possesses a fi-boundary trace if there exists a measure v G 9Jt(5f2) such that for any C2 exhaustion of $1, there holds lim I <(mdu>f° = I <\>dv V0 G C(Tl). (1.3.13) n^°° Jdn„ n Jan The /x-boundary trace of u is denoted by tr^(u) and we write tr^(u) = ia The advantage of this notion is that it does not require to determine the normalization factor in the definition, however it does not provide information on the boundary behavior of functions near d£l. Equivalence of the notions of /x-boundary trace. We show (see [78]) that the normalized /x-boundary trace in Definition 1.3 and the /x-boundary trace in Definition 1.4 are actually equivalent. This is achieved thanks to the following result (see Section 2, especially Proposition 2.5 and Proposition 24 1. introduction 2.6 in [77]) tr;(G"[r]) = tr^M) = 0 Vr G Wl{Sl, 5° L[i\^vfj, L' J/ ^{J-K^fj, [ Thanks to this result, these notions can be used interchangeably. In the sequel, we employ the notion of /x-boundary trace given in Definition 1.4 in the study of linear equations and nonlinear equations. 1.3.3. Linear equations. In this subsection, we consider nonhomoge-neous linear equations of the form - L^u = t in fl (1.3.14) with r G DJl(tt,5a). The boundary value problem for (1.3.14) with fi-boundary trace is formulated as -L^u = t in $1 , , (1.3.15) trM(u) = u, where v G dJl(dfl). Definition 1.5. Assume r G dJl(fl,5a) and v G dJl(dfl). (i) A function u is a solution of (1.3.14) if u G Lj (fl) and u satisfies (1.3.14) in the sense of distributions in $1, namely / uLfl(f)dx= J (pdr V0 G Cc°°(^)-Jn Jn (ii) A function u is a solution of (1.3.15) if u is a solution of (1.3.14) and trM(u) = v. Our main results regarding this problem provides a full understanding of equation (1.3.14) and problem (1.3.15), as listed below. Theorem 1.6. (i) For any v G Wi(dfl), the function u = K^[v] is the unique solution of problem (1.3.15) with r = 0. If u is a nonnegative L^-harmonic function and tr^(u) = 0 then u = 0. (ii) For any r G 9Jt(f2, Sa), the function u = G^[T] is the unique solution of (1.3.15) with v = 0. In particular, tr^(G^[r]) = 0. (Hi) Letu be a positive L^-subharmonic function. Ifu is dominated by an L^-superharmonic function then L^u G 9Jt+($l, Sa) and u has a pi-boundary trace. In this case tr^(u) = 0 if and only if u = 0. (iv) Let u be a positive L^-superharmonic function. Then there exist v G Wi+(dtt) and r G dx± I g(u,\Vu\)4>)dx = - I K^ujL^dx V0 G A^($l). Jn Jn Jn (1.3.21) The answer to this question is positive. We will discuss it the typical models successively. 26 1. introduction 1.3.4. Absorption term. In case g(u, |Vu|) = with p > 1, and the plus sign occurs in (1.2.6), equation (1.2.6) and problem (1.3.19) become - Ltiu+\u\p~1u = 0 inn (1.3.22) and f -Luu + \u\p^1u = 0 inn 11 (1.3.23) [ trM(u) = u, where i/ £ 9Jt(<9n). As in the the case fi = 0, the first step to analyze the /x-boundary trace of solutions to (1.3.22) is to deal with moderate solutions of (1.3.22). Definition 1.8. A function u is a moderate solution of (1.3.22) if \u\ < v where v is a positive L^-harmonic function. Our next theorem describes the relation between the concept of fi-boundary trace and the notion of moderate solutions of (1.3.22). Theorem 1.9. Assume p > 1 and letu be a positive solution of (1.3.22). Then the following statements are equivalent. (i) u is a L^-moderate solution of (1.3.22). (ii) u admits a pi-boundary trace v £ 9Jl+(9n). It means u is a solution of (1.3.23). (Hi) u £ Lp(n, Sa) and (1.3.20) holds with g{u, |Vu|) = up and the plus sign, where v = tr^(u). Furthermore, a positive function u is a solution of (1.3.23) if and only if u £ Lp(n,cP) and (1.3.21) holds with g{u, |Vu|) = up and the plus sign. This theorem is a combination of [106, Proof of Theorem A] and [77, Proof of Proposition 3.5] and covers the previous results for the case fi = 0 in [116, Section 2.1]. The proof is based on Theorem 1.6, Representation Theorem 1.2 and an approximation process. A remarkable distinction in the study of nonlinear problem (1.3.23) in comparison with that of linear problem (1.3.15) is that problem (1.3.23) is solvable for any v £ DJl(dtt) only if the nonlinear term does not grow 'too fast'. More precisely, we show that the exponent p^ given in (1.2.11) is a critical exponent for the existence of solutions to (1.3.23) in the sense that if 1 < p < p/j, then problem (1.3.23) has a unique solution for every measure v £ 9Jl+(9n) while, if p > p^ then the problem has no solution if v is a Dirac measure. This is reflected in the following theorem. Theorem 1.10. Assume p > 1. (i) Existence, uniqueness and stability. If 1 < p < p^ then for every v £ 9Jl(9n) (1.3.23) admits a unique positive solution. (i) Non-existence. If p > p^ then for every k > 0 and y £ d£l, there is no positive solution of (1.3.23) with fi-boundary trace kSy, where 8y denotes the Dirac measure concentrated at y. It is noticed that for any v £ Lx(9n), problem (1.3.23) admits a unique solution. We proved Theorem 1.10 for fi £ (0, C#(n)) in connection with normalized /i-boundary trace (see [106, Theorem E and Theorem F]). The 1.3. detailed statement of main results 27 results were extended by Marcus and Moroz [104] to the case /x < ^ for arbitrary domains, without any requirement on the positivity of the first eigenvalue A^ due to the key observation that there exists a positive local L^-superharmonic function in the whole range // < ^. In parallel, Gkikas and Veron considered the semilinear equations with a class of more general absorption terms g{u) where g : M —> M is a continuous, nondecreasing function with g(0) = 0, under assumption (1.3.2). They pointed out in [77, Theorem 3.3] that, under the condition /oo (g(t) + {gi-Wt-^dt < oo, (1.3.24) for every v G 9Jt(5f2) there exists a unique weak solution of (1.3.19) with nonlinear term g(u). Clearly, when g{u) = with p > 1, condition (1.3.24) is translated as 1 < p < p^ and the existence result by Gkikas and Veron covers statement (i) of Theorem 1.10. In particular, the result asserts that for any k > 0, there exists a unique solution Uk$ of (1.3.23) with v = kSy. It was also showed that the function u^s '■= limfc^oo uk8 1S a solution to the equation in (1.3.22) which vanishes on d£l \ {y} and admits a strong singularity at y. When p > p^, they provided a necessary and sufficient condition expressed in terms of Besov capacities for the existence of a solution to (1.3.23), which includes statement (ii) of Theorem 1.10 as a concrete case. 1.3.5. Source term. We are also interested in semilinear elliptic equations with a source term — L^u = g{u) in $1 (1.3.25) and the associated boundary value problem {—Ltlu = q(u) in Q M (1.3.26) trM(u) = v where g : M+ —> M+ is a continuous, nondecreasing function with g(0) = 0. When dealing with (1.3.25) and (1.3.26), one encounters the following difficulties. The first one stems from the lack of a universal upper bound for solutions of (1.3.25). The second difficulty is that is a subsolution of (1.3.26) and therefore it is no longer an upper bound, but a lower bound for solutions of (1.3.26). We also show that weak solutions of (1.3.26) can be represented as in (1.3.20), namely they can be decomposed as the sum of two terms: the action of Green operator on the nonlinearity and the action of the Martin kernel on the boundary datum. Equivalently, it also means that weak solutions satisfy an integral formulation weakfor-ugradu (See [119, Theorem A] for /x £ (0,Cjy($1)) and [78, Proposition A] for /x 6 (0,^] and more general results involving also interior measure data). A counterpart of Theorem 1.9 is also obtained for (1.3.25) and (1.3.26) (see [77, Proposition A]). Based on this, together with weak Lebesgue space estimate on Green kernel and Martin kernel and the Schauder fixed point 28 1. introduction theorem, we show that the value p^ given in (1.2.11) is also a critical exponent for the existence of (1.3.26), as stated below (see [119, Theorem H and Theorem I]). Theorem 1.11. (i) Subcritical case. Assume that and 0 < g(t) < AttPl + 9 for all t £ [0,1] with some px > 1, Ax > 0, 9 > 0. There exist 9q > 0 and po > 0 such that for any 9 £ (0, 9q) and v £ 9Jt+(cWl) such that |M|g#(<9n) < PO; problem (1.3.26) admits a nonnegative weak solution. (ii) Sublinear case. Assume that for some p2 £ (0,1], A2 > 0 and 9 > 0. In (1.3.28), if P2 = 1 we assume in addition that A2 is small enough. Then for any v £ 9Jt+(9$l), (1.3.26) admits a nonnegative weak solution. When g(u) = up, we obtain a deeper analysis of the existence and nonexistence phenomena. Indeed, we prove (see [119, Theorem D and Theorem G]) the existence of a threshold value for the existence of solutions to where p > 0 is a parameter and v £ 9Jt+(9$l) with H^Ugjj^n) = 1- Theorem 1.12. Letp > 1 and v £ 9Jt+(<9$l) with \v\m(dtt) = 1- I. Subcritical case: p £ (l,p^). There exists p* £ (0, 00) such that the followings hold. (i) If p £ (0, p*] then problem (1.3.29) admits a minimal positive weak solution u . (ii) If p > p* then (1.3.29) does not admit any positive solution. II. Supercritical case: p > p^. For every p > 0 and y £ d£l, there is no positive weak solution of (1.3.29) with v = 5y, where 8y is the Dirac mass concentrated at y £ d£l. Theorem 1.12 shows a sharp difference between the absorption case and the source case. More clearly, in the source case, the existence depends not only on the concentration of the boundary datum but also on its norm. It was proved later on in our recent paper [23] that the multiplicity occurs when p £ (0,p*) and the uniqueness happens when p = p*. We also established various necessary and sufficient conditions in terms of estimates on Green kernel (see [78, Theorem B]) for the existence for the Dirichlet problem with interior measure data and boundary measure data 0 < g(t) < A2tP2 +9 Vt > 0 Existence results are stated in the following theorem. 1.3. detailed statement of main results 29 Theorem 1.13. Let r G 9Jt+(ft, 5a), 1/ e 9Jt+(9ft) and p>0. (i) Assume 0 < p < p^. Then there exists a constant C > 0 such that G%[K%[v]p] < CK^[u] a.e. in ft, (1.3.31) G^[G^[r]p] < CG^[t] a.e. in ft. (1.3.32) (ii) If (1.3.31) and (1.3.32) hold then problem (1.3.30) admits a weak solution u satisfying G^[o-t] + K%\pv] 0 and p > 0 small enough if p > 1, for any a > 0 and p > 0 if 0 < p < 1. fivj If p > 1 and (1.3.30) admits a weak solution then (1.3.31) and (1.3.32) hold with constant C = fvj Assume 0 < p < p^. Then there exists a constant C > 0 suc/i t/iai /or any weak solution u of (1.3.30) there holds G^[o-t] + K%\pv] p^, we make use of interior capacities and boundary capacities which are recalled below. For 0 < 9 < /3 < AT, set Nd,p{x,y) :=--rj^g-jT- g, V(x,y) G ft x ft,x / y, \x — y|iV p max{|x — y\, o{x), o{y)\u (1.3.35) Nfl,^[r](x) := l_Ng^{x,y)dr{y), Vr G 9Jt+(ft). (1.3.36) in For a > -1, 0 < 6 < (3 < N and s > 1, define Cap| by Cap|e/3iS(S) :=inf ^J_5acf>s dx : cf> > 0, Ng^[5acf>] > xb| , (1-3-37) for any Borel set E C ft. Here xe denotes the indicator function of E. Next we recall the capacity Capf^ introduced in [28] which is used to deal with boundary measures. Let 6 G (0, N — 1) and denote by Bg the Bessel kernel in I"-1 with order 9. For s > 1, define CapBg:S{F) := inf { I Ssdy : 0 > 0, Bg * 0 > xf) (1-3.38) for any Borel set F C l"-1. Since ft is a bounded smooth domain in M.N, there exist open sets 0\, ...,Om in M.N, diffeomorphisms Tj : Oi —> -Bi(O) and compact sets K\,Km in 9ft such that (i) Ki C Oi, 1 < i < m and 9ft C U™^, (ii) r,(0, n 9ft) = Bi(o) n = o}, r,(0, n ft) = Bi(o) n {xN > o}, (iii) For any x G Ojflft, there exists y G Ojn9ft such that S(x) = \x — y\. We then define the Cap^—capacity of a compact set F C 9ft by m Cap^(F) := £ Cap^C^F n ^)), (1.3.39) j=i where Tj(F D = fi(F D #i) x {x^ = 0}. 30 1. introduction Let a > -1, 0 < 9 < (3 < N and s > 1 and assume that -1 + s'(l + 9 -(3) < a < — 1 + s'(N + 9-/3). Then the above capacities are equivalent Cap^iS(£) « Cap™g+sp_ls(E) for any Borel E C 0ft. The interested reader is referred to in [78, Section 3.3] for more properties of of such capacities. Using these capacities, we give a criteria for the existence of solutions. Theorem 1.14. Let r e DPT+(0, 5") and v G Wl+(d£l). Assume p > 1. Then the following statements are equivalent. (i) There exists C > 0 such that the following inequalities hold j Sadr 0 and p > 0 small enough. We remark that capacities are a very useful tool which provides a finer topology than Borel measures. When 1 < p < p^, we have iff CapfcS'^l) > 0 and nrf Capf?a+T,p,(M) > 0, hence the Theorem 1.14 covers Theorem 1.13 (i)-(iii). It is worth mentioning that we also extend existence results for scalar equations to systems of the form — L^u = eg{v) + ot in $1, — L^v = e g(u) + of in 17, (1.3.42) trM(u) = pi/, trM(u) = pi> where 5 and g are nondecreasing, continuous functions in M with g(0) = g(0) = 0, e = ±1, a > 0, 0, p > 0, p > 0. The reader is referred to our paper [78] for various existence results for (1.3.42). 1.3.6. Gradient-dependent nonlinearities. In this subsection, we consider the Dirichlet problem for equation (1.2.6) with g : M+ x M+ —> M+ being nondecreasing and locally Lipschitz in its two variables with g(0, 0) = 0. We recall that the nonlinearity g(u, |Vu|) is called absorption (resp. source) if the plus sign (resp. minus sign) appears in (1.2.6). Two prototype models to keep in mind are g(u, |Vu|) = up + \X7u\q and g(u, |Vu|) = up\X7u\q. First we are interested in the Dirichlet problem in the absorption case {—L,,u + q(u, \Vu\) =0 in $1, ' , (1-3-43) trM(u) = v. Weak solutions of (1.3.43) are defined in Definition 1.7. The case where g depends only on |Vu| was studied in our joint paper with Gkikas [79] where we showed that the value q^ given in (1.2.13) is a critical value for the existence of (1.3.43). Several results were established for 1.3. detailed statement of main results 31 the Dirichlet problem, including uniqueness and full description of isolated boundary trace in the case 1 < q < and a removability result in the case 9M < q < 2. Coming back to problem (1.3.43), the existence of a weak solution holds under an integral condition on g (see [80, Theorem 1.3]). Theorem 1.15. Assume g satisfies (1.2.14). Then for any v e Wi+(dfl), •problem (1.3.43) admits a positive weak solution 0 < u < in $1. The proof is a highly nontrivial adaptation of that in the case where g depends only on u or |Vu|, relying on sub and super solutions method, the Schauder fixed point theorem and the Vitali convergence theorem. Two typical models are g(u, |Vu|) = up+\X7u\q and g(u, |Vu|) = up\X7u\q. The subcritical range and supercritical range for (p, q) are defined in (1.2.15) and (1.2.16). Next we show that the uniqueness holds in the cases g(u, |Vu|) = up + |Vn|9 and g(u, |Vu|) = up\X7u\q. As a matter of fact, the uniqueness is a direct consequence of the following comparison principle (see [80, Theorem 1.5 and Theorem B.l]). This result is novel even in the case fi = 0. Theorem 1.16. Assume g(u,\X7u\) = up + |Vn|9 with p,q satisfying (1.2.15) or g(u, |Vu|) = up\SJu\q with p, q satisfying (1.2.16) and q > 1. Let V{ g 9Jt+(9Sl), i = 1,2, and U{ be a nonnegative solution of (1.3.43) with v = V{. If v\ < i/2 then u\ < u2 in CI. The proof is based on a regularity result, the maximum principle, estimates on the gradient of subsolutions of a nonhomogeneous linear equation. Assume the origin 0 g dCl and let c>o be the Dirac measure concentrated at 0. It is known from Theorem 1.15 that for any k > 0, there exists a unique solution Uq k of (1.3.43) with v = kb§. It is natural to ask what the lim/^oo Uq k could be. The answer is given in the next theorem where a complete description of isolated singularities at 0 is established. We first consider the case g(u, |Vu|) = up + |Vn|9. In this case, set (1.3.44) 2-9 Theorem 1.17. Assume g(u, |Vu|) = up + |Vn|9 with p and q satisfying (1.2.15). I. Weak singularity. For any k > 0, let u^ be the solution of (1.3.43) with v = kSo. Then there exists a constant c = c(N,fi, $1) > 0 such that u%k(x) < ck5(x)a\x\2-N-2a Vx £ Q (1.3.45) and Moreover V«ök(x)\ < ckS(x)a-1\x\2-N-2a Vx g Q. (1.3.46) lim "f . = k. (1.3.47) Q3x^0 Kj}(x, 0) Furthermore the mapping k i—> UQk is increasing. 32 1. introduction limfc- m„,. Then u, II. Strong singularity. Put solution of {—L^u + g(u, |Vu|) =0 in Q, u = 0 ondQ\{0}. Then there exists a constant c = c(N, fi,p,q,il) > 0 such that __2 __2 c~15(x)a\x\ mp^-1 < Uq^x) < c5(x)a\x\ '"p.-?-1 \fx G il, o.cxd a Moreover lim !!3i->0 T = cr £ 6 , o.cxd v (1.3.48) (1.3.49) (1.3.50) (1.3.51) locally uniformly on upper hemisphere 1 = n S^. Here = {x = (xi,...,xtv) = (x',xn) : x^v > 0} and S1^^1 is the unit sphere in 1^. T/ie function uj is the unique positive solution of -fiV-l 0 LO = 0 m 54 on aS*. iV-l (1.3.52) where 2 \2 4v. v,q sp + m s2 + \Z(' m, + 2-N p,q %-f^l'p,q if p < if p = if p> (s,0 G t>Af x (1.3.53) The above theorem shows that there is a competition between two terms up and |VuK In particular, if p > then up is the dominant term, z q otherwise |Vu|9 is the dominant term. Moreover, it is observed that the equation is not scaling invariant, unless p = Isolated boundary singularities in the case g(u, |Vu|) = up\SJu\q are depicted in the next theorem (see [80, Theorem 1.6]). We notice that, unlike the case of sum, in this case of product, the equation is scaling invariant and the blowup rate is explicitly determined by the exponent p+g_1. Theorem 1.18. Assume g(u, |Vu|) = up\X7u\q with q > 1 and p and q satisfying (1.2.16). I. Weak singularity. For any k > 0, let u^k be the solution of (1.3.43) with v = kSo. Then (1.3.45)- (1.3.47) hold. Furthermore the mapping k i—> Uq k is increasing. II. Strong singularity. Put lim m?,. Then u, n o.cxd is a solution of (1.3.48). There exists a constant c = c(N, fi,p,q,Q) > 0 such 1.3. detailed statement of main results 33 that c-15{x)a\x\"P^~a < uoi00(x) < cS(x)a\x\~^^~a Vx G £1, (1.3.54) \Vuo,oo(x)\ - c5(x)a-1\x\~^^~a Vx G Q. (1.3.55) Moreover 2-q „ lim l^lp+9_1^o oo(x) = (1.3.56) !)3i->0 ' * - a t * + locally uniformly on upper hemisphere S^1 = n SN~1, where uj is the unique solution of problem (1.3.52) with r a' , V o 2-g ( 2p + q C^uj = Auj +---^w, £Npq = —■-- —■-- - AT (eN-a)z '™ p + q-l\p + q-l 9. J(s,0 = sp ( (-1-A_)2S2 + |£|2j 2 (S,0 G R+ x RN. [1.3.57) In the supercritical range, an important ingredient in the study is Bessel capacities. First we recall below some notations concerning Besov spaces and Bessel spaces (see, e.g., [1, 98, 130]). For a > 0, 1 < re < oo, we denote by Wcr,K(Rd) the Sobolev space over R . If a is not an integer the Besov space Ba,K(Rd) coincides with Wa,K(Rd). When a is an integer we denote Ax,yf '■= f{x + y) + f{x — y) — 2f{x). The Besov space is defined by B1,K(Rd) : = j/ G LK(Rd) : G LK(Rd x Md)| , with norm bi.« := (+ ff l4rd^dxdy \y\ Then we have Bm>K(Rd) : = {/ G Wm-^K(Rd) : DeJ G B^K(Rd) W G Nd, \6\ = m - l} , with norm =m—1 'xtf 12/1 For s G M, the Bessel kernel of order s is defined by Gs(£) = T 1(1 + |.|2)_2 J-"(£), where J7 is the Fourier transform of moderate distributions in Rd. The Bessel space LSjK(Rd) is defined by LS:K(Rd) :={f = Gs*g:g£L«(Rd)}, with norm >,k '■= = \\G-S * N-l — ct,q'' 34 1. introduction It is known that if 1 < k < oo and s > 0, LS:K(Rd) = Ws>K(M.d) if s G N and LS:K(Rd) = Bs>K(M.d) if s £ N, always with equivalent norms. The Bessel capacity is defined for compact subsets K C Rd by C%{K) := inf{||/||£.iK, / G S'(Rd), f > XK}. (1.3.58) It is extended to open sets and then Borel sets by the fact that it is an outer measure. Let v G 9Jt+(9$l). We say that v is absolutely continuous with respect to the Bessel capacity C|?^ if Cf*K(E) = 0 v{E) = 0 for all Borel set E. A necessary condition expressed by Bessel capacities for the existence of a solution to (1.3.43) is obtained from [80, Theorem B.5 and Theorem 1.7]. Theorem 1.19. Let v G Wi+(dfl) and assume that problem (1.3.43) has a nonnegative solution u G C2($l). I. Assume g(u, |Vu|) = vP + |Vu|9 with (p,q) is in the supercritical range. (i) If p > Pfi then v is absolutely continuous with respect to C^Nx+a ,. (ii) If < q < 2 then the followings occur. (a) If q ^ a + 1 then u is absolutely continuous with respect to C^+i i (b) If q = a + 1 then for any e G (0, min{a + 1, ^+\^a — (1 — a)}) then v is absolutely continuous with respect to C^N 1 a+1. II. Assume g(u, |Vu|) = up\X7u\q with (p,q) is in the supercritical range. (i) If 9 / a + 1 then v is absolutely continuous with respect to the capacity ^1*^+ a+1_« (p+q)r ^ere (P + denotes the conjugate exponent ofp + q. (ii) If q = a + 1 then for any e G (0, min{a + 1, ^+11^a — (1 — a)}) then v is absolutely continuous with respect to * E (P+a+iy Define the weight function W by [5{x)l-a if/iPfi, pi & or (ii) CgT^ (E)=0ifq^ 1, q > 1, k > 0. Then there exists po > 0 depending on N, p,Cl,k,p,q such that for any p g (0,po), problem (1.3.63) admits a positive weak solution u > pK^u] in $1. It is easy to see that if g{u, |Vu|) = up + |Vu|9 or g{u, |Vu|) = up\SJu\q then (1.3.64) holds. Therefore Theorem 1.21 holds true for these typical models. Sufficient conditions for the existence for the Dirichlet problem in two typical models with (p, q) being in supercritical range are expressed in terms of capacities (see (1.3.37) and (1.3.39)). We note that the capacities used for this case are different from the Bessel capacities employed in the absorption case (see (1.3.58)). 36 1. introduction Theorem 1.22. I. Assume g(u,\Vu\) = up + \Vu\q with p > 1 and < q < ■ Assume one of the following conditions holds: (i) There exists a constant C > 0 such that for every Borel set E c dft. u{E) < Cmin{Capf ^ (E), Cap™ ^ (£)}. (L3.65) (iij There exists a positive constant C > 0 N2a,2[^(p+1)N2a,2[i/]p] < CN2a,2M < oo ffl.e. in n , ' (1.3.66) n2a-i,i[5{a-1)q+an2a-i,Mq\ < CN2a-i,iM < oo a.e. in ft. Then there exists po = po(N, fi,p,q,C,fl) > 0 such that for any p 6 (0, po), problem (1.3.63) admits a weak solution u. II. Assume g(u, |Vn|) = np|Vn|9 with p > 0, q > 0, p + q > 1 and q < i+a+(i a)p ^ Assume one oi £/je following conditions holds. (i) There exists a constant C > 0 such that v(E) < CCapf^ a+i-Vi{p]qy(E) for every Borel set E c dVl. (1.3.67) Here {p + q)' denotes the conjugate exponent of p + q. (ii) There exists a positive constant C > 0 such that n2a-i,i[5ap+{a-1)q+an2a-i,MP+q\ < CNaa-i.iM < oo a.e. in ft. (1.3.68) Then there exists po = po(N, p,p,q,C,ft) > 0 such that for any p £ (0, po), problem (1.3.63) admits a weak solution u. 1.4. Related and open problems We notice that because of the broadness of the topics regarding boundary value problems for nonlinear equations with a Hardy potential, the results presented above are a modest contribution to the recent developments and are due to the our interest. The topics have received much attention and many new results have been recently published. Some related interesting results and open problems are listed below. Multplicity and uniqueness. As pointed out before in Theorem 1.12, when 1 < p < p^, there exists a threshold value p* > 0 such that problem (1.3.29) admits a minimal positive solution for p 6 (0,p*] and admits no positive weak solution for p > p*. In [23], we carried out a deeper analysis on (1.3.29) and proved the multiplicity for p £ (0,p*) and the uniqueness for p = p*, which complements the results in [119]. More precisely, the structure of the solution set of problem (1.3.29) is described as follows. Subcritical case: p 6 (l,p^). There exists p* £ (0, oo) such that the follow-ings hold. (i) If p G (0,p*) then problem (1.3.26) admits two positive solutions, including the minimal positive solution. (ii) If p = p* then problem (1.3.26) admits a unique positive solution. (iii) If p > p* then (1.3.29) does not admit any positive solution. 1.4. related and open problems 37 Supercritical case: p > p^. For every p > 0 and z £ there is no positive weak solution of (1.3.29) with v = Sy, where Sy is the Dirac mass concentrated at y £ díl. The multiplicity for systems were also derived in [23] for measure data with small total mass. However, this result provides partial understanding of solution set of systems and needs to be improved. Schrodinger equations with potential blowing up on boundary. Let $1 C 1^ (N > 3) be a C2 bounded domain and F C díl be a k dimensional C2 submanifold, 0 < k < N — 1. Denote 5(x) := dist(x, díl), 5f(x) : = dist(x,F) and put V(x) = VF{x) := 5F(x)-2, xeil. Boundary value problems for semilinear equations with Schrodinger operator Ly := A + fiV, /i e K, in the special cases k = N and k = 0 have been extensively investigated. In particular, the case k = N — 1, F = díl and V(x) = 5(x)~2 is well understood, as shown in this thesis, while the case k = 0, F = {0} with the origin 0 £ dQ and V(x) = \x\~2 was treated by Chen and Veron [45]. It is also worth mentioning that the case of more strongly singular potential V(x) = 8(x)~a with a > 2 was considered by Du and Wei in [54]. The case 1 < k < N — 2 had remained open until our work in [107]. In fact, we considered the case where F is a C2 submanifold of dimension 0 < k < N — 2 without boundary and established the solvability for solutions to the equation — Lyu+g(u) = 0 in $1 with prescribed boundary data. The reader is referred to the work of Fall and Mahmoudi [65] for Hardy type estimates and estimate of the first eigenfunction. Recently, Marcus published papers [100, 101] in which he considered the potential V such that |V(a;)| < a5(x)~2 for all x £ $1 and under additional conditions, he obtained estimates on Green kernel, Martin kernel, as well as sub and super harmonic functions. Moreover, large solutions of semilinear equations are studied in [102]. Schrodinger equations with potential blowing up on a subset of the domain Another interesting case is that V(x) =^(x)-2, (1.4.1) where S is a compact, C2 submanifold in $1 with dimension k with 0 < k < N — 2 and 5%(x) = dist(x, S). The special case S = {0} C $1 was treated by Guerch and Veron [83], Chen and Veron [44], Chen and Zhou [46], Cirstea [47] and references therein. 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CHAPTER 2 Moderate solutions of semilinear elliptic equations with a Hardy potential This chapter is based on our join paper with Moshe Marcus [106] on boundary value problems for semilinear elliptic equations with an absorption term and a Hardy potential. In this chapter, we introduce a notion of normalized boundary trace and develop a theory of linear equations, which in turn provides a basis for the study of semilinear equations. 47 48 2. elliptic equations with a hardy potential Available online at www.sciencedirect.com ScienceDirect annai.es de uinstitut henri poikcare ELSEVIER analyse non linealre Ann. I. H. Poincaré -AN 34 (2017) 69-88 www.elsevier.com/locate/anihpc Moderate solutions of semilinear elliptic equations with Hardy potential Moshe Marcusa*, Phuoc-Tai Nguyen a,b a Department of Mathematics, Technion, Haifa 32000, Israel k Departamento de Matemätica, Pontificia Universidad Catölica de Chile, Santiago, Chile Received 21 July 2014; received in revised form 14 September 2015; accepted 22 October 2015 Available online 28 October 2015 Abstract Let Q, be abounded smooth domain inK . We study positive solutions of equation (E) —L^u+ul = 0 in Q where Lfl = A + j^, 0 < /t, q > 1 and S(x) = dist(;c, dQ). A positive solution of (E) is moderate if it is dominated by an l^-harmonic function. If /t < c//(f2) (the Hardy constant for Q) every positive l^-harmonic function can be represented in terms of a finite measure on dQ via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary trace by which we obtain a complete classification of the positive moderate solutions of (E) in the subcritical case, 1 < q < q^,c- (The critical value depends only on N and /t.) For q > q^c there exists no moderate solution with an isolated singularity on the boundary. The normalized boundary trace and associated boundary value problems are also discussed in detail for the linear operator L^. These results form the basis for the study of the nonlinear problem. © 2015 Elsevier Masson SAS. All rights reserved. MSC: 35J60; 35J75; 35J10 Keywords: Hardy potential; Martin kernel; Moderate solutions; Normalized boundary trace; Critical exponent; Removable singularities 1. Introduction In this paper, we investigate boundary value problem with measure data for the following equation in a C2 bounded domain £2, where q > 1, /x e K and S(x) = dist(x, 3£2). This problem is naturally linked to the theory of linear SchrSdinger equations —Lvu = 0 where L v := A + V and the potential V satisfies | V\ < c<5-2. Such equations have been studied in numerous papers (see [1,2] and the references therein). Corresponding author. E-mail addresses: marcusm@math.technion.ac.il (M. Marcus), nguyenphuoctai.hcmup@gmail.com (R-T. Nguyen). http://dx.doi.Org/10.1016/j.anihpc.2015.10.001 0294-1449/© 2015 Elsevier Masson SAS. All rights reserved. -Au--i-U +Uq =0 (l.i) 2. elliptic equations with a hardy potential 49 70 M. Marcus, P.-T. Nguyen/Ann. I. H. Poincare -AN34 (2017) 69-88 Put LM:=A + |j. (1.2) A solution u e Lj (Q.) of the equation —Ltlu = 0 is called an L ^-harmonic function. Similarly, if —Ltlu > 0 or — Ltlu < 0 we say that u is L ^-superharmonic or L^-subharmonic respectively. If /x = 0 we shall just use the terms harmonic, superharmonic, subharmonic. Some problems involving equations (1.1) and (1.2) with /x < 1/4 were studied by Bandle, Moroz and Reichel [4]. They derived estimates of local LM-subharmonic and superharmonic functions and applied these results to study conditions for existence or nonexistence of large solutions of (1.1). They also showed that the classical Keller-Osserman estimate [14,24] remains valid for (1.1). The condition /x < | is related to Hardy's inequality. Denote by Ch(&) the best constant in Hardy's inequality, i.e., L \Vu\2dx CH(0)= inf f . (1.3) Hi(V) Jn(u/S)2dx By Marcus, Mizel and Pinchover [17], Ch(&) e (0, ^] and Ch(&) = \ when Q, is convex. Furthermore the infimum is achieved if and only if Ch(&) < 1 /4. By Brezis and Marcus [7], for every /x < 1/4 there exists a unique number XMti such that inf fa(\Vu\2 - X^iu2)dx H^n) fa(u/S)2dx and the infimum is achieved. Thus is an eigenvalue of —LM and, by [7, Lemma 2.1], it is a simple eigenvalue. We denote by (ptl<\ the corresponding positive eigenfunction normalized by fQ( 0. Consequently, in this case, (p^x is a positive supersolution of — Ltl. This fact and a classical result of Ancona [2] imply that for every y e 3£2, there exists a positive L^-harmonic function in Q, which vanishes on dQ, \ [y] and is unique up to a constant. Denote this function by K^(-, y), normalized by setting it equal to 1 at a fixed reference point xq e £2. The function (x, y) i-> K^(x, y), (x, y) e Si x 3£2, is the L^-Martin kernel in Q, relative to xq. Further, by [2]: Representation Theorem. For every v e 9J?+(3£2) the function K°M(x):=J K°(x,y)dv(y) Vx e n (1.4) is L^-harmonic, i.e., LMK^[y] = 0. Conversely, for every positive L^-harmonic function u there exists a unique measure v e Wi+(dQ.) such that u = K^[v]. This theorem implies that - in the present case - the L^-Martin boundary of Q, coincides with the Euclidean boundary. (For the general definition of Martin boundary see, e.g. [1]. However this notion will not be used here beyond the representation theorem stated above.) The measure v such that u = K^[v] is called the L^-boundary measure of u. If /x = 0, v is equivalent to the classical measure boundary trace of u (see Definition 1.1). But if 0 < /x < Ch(&), it can be shown that, for every v e 9J?+(3£2), the measure boundary trace of K^[v] is zero (see Corollary 2.11 below). In the case /x = 0, the boundary value problem -Au + \u\q~1u = 0 inSi u = v on d£i (1.5) 50 2. elliptic equations with a hardy potential M. Marcus, P.-T. Nguyen/Ann. I. H. Poincare - AN 34 (2017) 69-88 71 where q > 1 and v is either a finite measure or a positive (possibly unbounded) measure, has been studied by numerous authors. Following Brezis [6], if v is a finite measure, a weak solution of (1.5) is defined as follows: u is a solution of the problem if u and S\u\q are integrable in Si and (-uAt + \u\q~1ut)dx = - f—dv VfeC2(£2) (1.6) J 3n where n is the outer unit normal on dSi. Brezis proved that, if a solution exists then it is unique. Gmira and Veron [13] showed that there exists a critical exponent, qc := j^rj", such that if 1 < q < qc, (1.6) has a weak solution for every finite measure v but, if q > qc there exists no positive solution with isolated point singularity. Marcus and Veron [20] proved that every positive solution of the equation -Au+uq=0 (1.7) possesses a boundary trace given by a positive measure v, not necessarily bounded. In the subcritical case the blow-up set of the trace is a closed set. Furthermore they showed that, in this case, for every such positive measure v, the boundary value problem (1.5) has a unique solution. In the case q = 2, N = 2 this result was previously proved by Le Gall [15] using a probabilistic definition of the boundary trace. In the supercritical case the problem turned out to be much more challenging. It was studied by several authors using various techniques. The problem was studied by Le Gall, Dynkin, Kuznetsov, Mselati a.o. employing mainly probabilistic methods. Consequently the results applied only to 1 < q < 2. In parallel it was studied by Marcus and Veron employing purely analytic methods that were not subject to the restriction q < 2. A complete classification of the positive solutions of (1.5) in terms of their behavior at the boundary was provided by Mselati [18] for q = 2, by Dynkin [11] for qc < q < 2 and finally by Marcus [16] for every q > qc. For details and related results we refer the reader to [23,22,21,3,10] and the references therein. In the case of equation (1.1) one is faced by the problem that, according to the classical definition of measure boundary trace, every positive L^-harmonic function has measure boundary trace zero. Therefore, in order to classify the positive solutions of (1.1) in terms of their behavior at the boundary, it is necessary to introduce a different notion of trace. As in the study of (1.7), we first consider the question of boundary trace for positive L^-harmonic or superharmonic functions. We recall the classical definition of measure boundary trace. Definition 1.1. (i) A sequence {£)„} is a C2 exhaustion of Si if for every n, Dn is of class C2, Dn c D„+i and UnDn = Si. If the domains are uniformly of class C2 we say that {£)„} is a uniform C2 exhaustion. (ii) Let u e (Si) for some p > 1. We say that u possesses a measure boundary trace on dSi if there exists a finite measure v on dSi such that, for every uniform C2 exhaustion {£)„} and every

0, denote Sip = {x e Si: S(x) p], Hp = {x e Si: S(x) = p}. Put "±:=21±\U/X- (1-8) It can be shown (see Corollary 2.11 below) that the classical measure boundary trace of K^[v] is zero but there exist constants C\, C2 such that, for every v e 9Jl(dSi), 2. elliptic equations with a hardy potential 51 72 M. Marcus, P.-T. Nguyen/Ann. I. H. Poincare -AN34 (2017) 69-88 1 P" Ci \\v\\maa) <—J K%[v](x)dSx < C2 \\v\\mm) (1.9) 2s for all p e (0, Po) where Pq > 0 depends only on £2. In view of this we introduce the following definition of trace. Definition 1.2. A positive function u possesses a normalized boundary trace if there exists a measure v e 97t+(3£2) such that 1 fi-^b p lim — / \u - E?[v]\dSx = 0. (1.10) The normalized boundary trace will be denoted by tr*(u). Remark. The notion of normalized boundary trace is well defined. Indeed, suppose that v and v' satisfy (1.10). Put v = (K^[y — y'])+ then v is a nonnegative L^-subharmonic function, v < K[v + y'] and tr*(u) = 0. By Proposition 2.14, v = 0, i.e., K^[y — y'] < 0. By interchanging the roles of v and y', we deduce that K%[v' - v] < 0. Thus y = y . Denote by the Green function of — Ltl in Q, and, for every positive Radon measure r in Q,, put G£[t](jc):= f G°(x,y)dr(y). Denote by fMf(Si), f a positive Borel function in £2, the space of Radon measures r on ^ satisfying fQfd\x\ < oo and by 97t^(£2) the positive cone of this space. If x is a positive measure such that G^[r](x) < oo for some x e £2 then r e SWj«+ (£2) and G^[r] is finite everywhere in £2. The underlying reason for this is the behavior of the Green function at the boundary: for every p > 0 there exists cp such that c-fil&{x)a+ < G^(x, y) < Cf,8(x)a+ Vx e fy/2, J e »/s- For details see Section 2.2 below. We begin with the study of the linear boundary value problem, —Ltlu = x mO, tr*(u) = v, (1.11) where v e 97t+(3£2) and r e 97tJ<+ (£2). As usual we look for solutions u e Lj (Q.) and the equation is understood in the sense of distributions. The representation theorem implies that if r = 0 the problem has a unique solution, We list below our main results regarding this problem. Proposition I. (i) If u is a non-negative L^-harmonic function and tr*(w) = 0 then u = 0. (ii) If x e 9JtJ<+ (Q.) then G^[r] has normalized trace zero. Thus G^[r] is a solution o/(l.ll) with y = 0. (iii) Let u be a positive L^-subharmonic function. If u is dominated by an L^-superharmonic function then L^u e 97tJ,+ (Q.) and u has a normalized boundary trace. In this case tr*(w) = 0 if and only if u = 0. (iv) Let u be a positive L^-superharmonic function. Then there exist v e 97t+(3£2) and x e 9JtJ<+ (Q.) such that k = G£[t] + K£[i;]. (1.12) In particular, u is an L ^-potential (i.e., u does not dominate any positive L ^-harmonic function) if and only if tt*(u) = 0. (v) For every v e SW+(3^) and x e 5W+a+ (£2), problem (1.11) has a unique solution. The solution is given by (1.12). 52 2. elliptic equations with a hardy potential M. Marcus, P.-T. Nguyen/Ann. i. h. Poincare - AN 34 (2017) 69-88 73 Next we study the nonlinear boundary value problem, —L^u + uq = 0 in £2 tr*(u) = v (1.13) where v eSW+(3^). Definition 1.3. (i) A positive solution of (1.1) is L^-moderate if it is dominated by an L^-harmonic function, (ii) A positive function u e Lqloc(Q,) is a (weak) solution o/(1.13) if it satisfies the equation (in the sense of distributions) and has normalized boundary trace v. Definition 1.4. Put X(Q.) = Re C2(£2): Sa-L^t; e L°°(£2), 1. Theorem A. Assume that 0 < /x < C//(£2), q > 1. Let u be a positive solution o/(l.l). Then the following statements are equivalent: (i) u is L^-moderate. (ii) u admits a normalized boundary trace v e 91t+(3£2). In other words, u is a solution o/(1.13). (iii) u + Gf1[ui] = Kf1[v] (1.14) where v = tr*(w). Furthermore, a positive function u is a solution o/(1.13) if and only ifu/8a- e L1 (£2), Sa+uq eL'(S2) ami {-uL^ + uqi;)dx = -JK^[v]L^dx Vf e X(£2). (1.15) £2 £2 Theorem B.Ass«mc 0 < /x < Ch(&), q > 1. I. UNIQUENESS. For every v e 91t+(3£2), rfiere ex/sta at most one positive solution o/(1.13). II. MONOTONICITY. Assume v,- e 91t+(3£2), i = 1,2. Let uVi be the unique solution o/(1.13) with v replaced by v,-, i = 1, 2. vi < v2 rfjerc Myj < u„2. III. A-PRIORI ESTIMATE. There exists a positive constant c = c{N, fi, £1) such that every positive solution u o/(1.13) satisfies, IMIl1 (£2)+ H"IIl« (£2) 1. If v e 91t+(3£2) ami K^[v] e L*a+(£2) f/ie« rfiere ex/sta a unique solution of the boundary value problem (1.13). Corollary CI. For every positive function f e L1 (3 £2) (1.13) with v = f admits a unique positive solution. Theorem D. Assume 0 < /x < C//(£2), q > 1. w is a positive solution o/(1.13) then u(x) lim —=-= 1 non-tangentially, v-a.e. on dQ,. (1-17) x~*y K"[v](jc) 2. elliptic equations with a hardy potential 53 74 M. Marcus, P.-T. Nguyen/Ann. I. H. Poincaré -AN34 (2017) 69-88 Let N + ce+ Qac-= -—■ (1-18) In the next two results we show, among other things, that qllc is the critical exponent for (1.13). This means that, if 1 < q < q^^ then problem (1.13) has a unique solution for every measure v e 97t+(3£2) but, if q > q^^ then the problem has no solution for some measures v, e.g. Dirac measure. In Theorem e we consider the subcritical case 1 < q < qllc and in Theorem F the supercritical case. Theorem E. Assume 0 < /x < C#(£2) and 1 < q < qtl,c- Then: 1. EXISTENCE AND UNIQUENESS. For every v e 97t+(3£2) (1.13) admits a unique positive solution uv. II. STABILITY. If {vn} is a sequence of measures in 97t+(3£2) weakly convergent to v e 97t+(3£2) then uVn —>• uv in Lls-a_ (£2) and in L^a+ (£2). III. LOCAL BEHAVIOR. Let v = k&y, where k > 0 and Sy is the Dirac measure concentrated at y e 3Í2. Then, under the assumptions of Theorem E, the unique solution o/(1.13), denoted by Uksy, satisfies UlrX (x) lim „yV =k. (1.19) Remark. Note that in part III we have 'uniform convergence' not just 'non-tangential convergence' as in Theorem D. Theorem F. Assume 0 < /x < C//(£2) and q > q^^- Then for every k > 0 and y e 3Í2, there is no positive solution of (1.1) with normalized boundary trace k&y. In the first part of the paper we study properties of positive L^-harmonic functions and the boundary value problem (1.11). In the second part, these results are applied to a study of the corresponding boundary value problem for the nonlinear equation (1.1). These results yield a complete classification of the positive moderate solutions of (1.1) in the subcritical case. They also provide a framework for the study of positive solutions of (1.1) that may blow up at some parts of the boundary. The existence of such solutions in the subcritical case has been studied (by different methods) in [5]. The boundary trace for positive non-moderate solutions and corresponding boundary value problems will be treated in a forthcoming paper. The main ingredients used in this paper are: the Representation Theorem previously stated and other basic results of potential theory (see [1]), a sharp estimate of the Green kernel of — Ltl due to Filippas, Moschini and Tertikas [9], estimates for convolutions in weak Lp spaces (see [23, Section 2.3.2]) and the comparison principle obtained in [4]. 2. The linear equation Throughout this paper we assume that 0 < /x < C//(£2). 2.1. Some potential theoretic results We denote by 9Jlj«(í2), a e K, the space of Radon measures r on £1 satisfying J Sa(x)d\r\ < oo and by ÍDtJ,^) the positive cone of 9Jtj«(£2). When a = 0, we use the notation 5t)t(£2) and 5t)t+(£2). We also denote by SDT(3Í2) the space of finite Radon measures on 3Í2 and by 5t)t+(3£2) the positive cone of 97t(9£2). Let D be a C2 domain such that D S^D, u) a.e. in D. 54 2. ELLIPTIC EQUATIONS WITH A HARDY POTENTIAL M. Marcus, P.-T. Nguyen/Ann. I. H. Poincare - AN 34 (2017) 69-88 75 Proof. Since u is L^-superharmonic in £2, there exists r e 97t+(£2) such that —L^u = x. Let v be the solution of "L"U=n [nD,n (2-2) u = 0 on 3D and put w = §M(D, u). Then iu > 0 and u\o = v + w > v. □ Lemma 2.2. Lef u be a nonnegative L^-superharmonic and {Dn} be a C2 exhaustion ofQ,. Then u := lim §„ (D„, u) n^oo exists and is the largest L^-harmonic function dominated by u. Proof. By Lemma 2.1, §M (/)„,«) < w|d„, hence the sequence {§M(Dn,w)} is decreasing. Consequently, u exists and is an LM-harmonic function dominated by u. Next, if v is an LM-harmonic function dominated by u then v < §M(Z)„, u) for every n e N. Letting « —>• oo yields v G^(x,xq) behaves like the first eigenfunction (Pn,\(x) near the boundary, i.e., G^(-,xq) ~ (p^x in Sip, (0 < P<8(x0)). By [19, Lemmas 5,1, 5.2] (see also [8, Lemma 7] for an alternative proof) c-lS(xf+ < < c&(xf+. (2.4) Thus, if 0

a}) 0. (2.9) The function Xf is called the distribution function of / (relative to r). For p > 1, denote LPv(Si; x) = {f Borel measurable : &wpapXf(a; x) < oo} a>0 and \\f\\*Jp(n.r, = ^VapXf(a-x))lp. (2.10) LW(Q,T) a>Q This expression is not a norm, but for p > 1, it is equivalent to the norm if \f\dx 1 H/IIlp(S2t) = SUP " :t)Cfi,ffl measurable , 0 < x(co) \. (2.11) More precisely, x(co)l/P' 56 2. elliptic equations with a hardy potential M. Marcus, P.-T. Nguyen/Ann. I. H. Poincare - AN 34 (2017) 69-88 77 ll/lll£Wr,^ll/llL£(n;r,<^rill/III£(n;r)- (2-12) Notice that, for every a > — 1, LPJ(£l;8adx)czL'sa(£i) Vre[l,p). For every ig3S2, denote by nx the outward unit normal vector to 3 £2 at x. The following is a well-known geometric property of C2 domains. Proposition 2.7. There exists Pq > 0 such that (i) For every point x e &p0, there exists a unique point ax e dQ, such that \x — ax\ = <5(x). This implies x = ax - S(x)nax. _ _ (ii) The mappings x \-> S(x) and x \-> ax belong to C2(£2p0) and Cl(£2p0) respectively. Furthermore, limx^ffW V<5(x) = -nx. By combining (2.6), (2.7) and [23, Lemma 2.3.2], we obtain Proposition 2.8. There exist constants c,- depending only on N, i±, P,Q. such that, ||G"[t]|| a±e. -l (2.13) lG°[r]\ _s±L. -2a_, (2.14) ||K"rv1|| -l. (2.15) Proof. We assume that r is positive; otherwise we replace r by |r|. We consider r as a positive measure in MN by extending r by zero outside of £2. For a e (0, N), denote Ta(x) = \x\a~N. By [23, inequality (2.3.17)], l|ra*r|| m max{-l,-a} (2.16) where c = c(iV, a, /S, diam(Q.)). By (2.6), g£(jc, y) < cmin{r2(x - y), Sixf+Hyf+r^M ~ y)}. Hence, by (2.16), |G"[r]| n±i -1, |G"[r]| n+fs -2a_. Next we extend v by zero outside 3 £2 and observe that, by (2.7), K^(x, y) < cFi+a_(x — y). Hence K^[v] < cFi+a_ * y and by (2.16), IKMl N+l> -c llri+«- *yll ^ Sc\\v\\m{dn) vp >-1. a Corollary 2.9. Let p >-l. (i) If{vn} C 9Jt+(3£2) converges weakly to v e 97t+(3£2) f/ie« {K^[v„]} converges to K^[v] m LPp{£l) for every p such that 1 < p < N^_^a ■ (ii) If {xn} C 97t+(£2) converges weakly (relative to Cq(Q,)) to x e 97t+(£2) f/ie« {G^[r„]} converges to G^[r] m LPp{£i) for every p such that 1 < p < -^rf • Proof. We prove the first statement. The second is proved in a similar way. 2. elliptic equations with a hardy potential 57 78 M. Marcus, P.-T. Nguyen/Ann. I. H. Poincare -AN34 (2017) 69-88 Since K^(x,.) e C(dSi) for every x e Si, {K^[y„]} converges to K^[v] every where in Si. By Holder inequality and (2.15), we deduce that {(K^[v„])p} is equi-integrable w.r.t. S^dx for any 1 < p < N^^a ■ By Vitali's theorem, K%[vn]^K%[v]mLPsl)(Si). a 2.3. Estimates related to the normalized trace Proposition 2.10. There exist positive constants C\, C2 such that, for every fi e (0, /So), CifJa- < J K%{x,y)dSx 0, 1 Jim„t7- / K^(x,y)dSx=0 VyedSi. (2.18) For ro fixed, the rate of convergence is independent of y. Proof. By (2.7), f K?1{x,y)dSx 0. Denote v := G^[r]. We start with the proof of (2.23). By Fubini's theorem and (2.6), jvdSx• 0. Put Vi = G^[r,]. Then, for 0 < fJ < fJi/2, vldSx[hp]< F°<>[Vl3Df)] e C2(Q.) n C(^) and G^ > 0 in ^. Similarly 5 is a proper regularization of <5 relative to xq if <5(x) = <5(x) for x e ^(x^* ^ £ C2(£2) and <5 > 0 in £2. Remark. Using (2.6) and (2.4), it is easily verified that the functions /if e X(£2). In the proofs of the next two propositions we use the following construction. Let D c= £2 be a C2 domain. The Green function for —L^'mD is denoted by G^. (To avoid misunderstanding we point out that, in the formula defining LM, <5(x) denotes, as before, the distance from x to dQ,, not to 3D.) Given xq e £2 we construct a family of functions ~ Dr i ~ ~ Dr Dr Q{xq) = {GM : 0 < p < ^P(xq)} such that, for each f3, GM is a proper regularization of G^ {xq, ■) in Dp and Q{xq) has the following properties: . For every p e (0, ±0(jco)), G°" e C2^), g£> > 0 and g£"(x) = G°f (x0, x) for xeDp\ Z>fco). • The sequences {G^} and {L^G^"} converge to G^ and L^G^ respectively, as ft —>• 0, a.e. in £2. II ~ ~ ^ II • GM + |LMGM | < Mxo where Mxo is a positive constant independent of p. II \\L°°(D/i) Q(xq) will be called a uniform regularization of {G^}. For any function h e C2(dQ,), we say that h is an admissible extension of h relative to xo in £2 if h(x) = h(a(x)) for x e ^(jo) and h e C2(£2). 2. ELLIPTIC EQUATIONS WITH A HARDY POTENTIAL 61 82 M. Marcus, P.-T. Nguyen/Ann. I. H. Poincare -AN34 (2017) 69-88 2.5. Nonhomogeneous linear equations Here we discuss the boundary value problem (1.11) in Q,. Lemma 2.17. Let u e Lj (£1) be a positive solution (in the sense of distributions) of equation (2.27) in Q. where x is a non-negative Radon measure. Ifxe ms«+(Q.) then i^[r]L^dx = j t,dx Vf e X(£2). (2.28) Proof. We may assume that r is positive. By Proposition 2.12, tr*(G^[r]) = 0. Therefore, given e > 0, there exists i p = p(e) < iftj such that. P Let Hp Sip „M[r]d5x G^(x, y) is L^-harmonic in Sip. By local elliptic estimates, for every £ e Hp, sup IV.G^x.y)!^^-1 sup G°(x,y). By Harnack's inequality, sup G°(x,y)• LllG^(z, x) pointwise and the sequence {L^G^"} is bounded by a constant Mz. We observe that U e L1 (Si); in fact by assumption u e L1g_a_ (Si) and therefore, by Proposition 2.8, U e L1g_a_ (Si). Consequently, by (2.38), U(z,) = - j U(x)L^(z„x)dx. S2 Since G^(z, ■) e X(Si), by (2.37) the right hand side vanishes. Thus U vanishes in Si, i.e., u satisfies (2.35). □ Corollary 2.19. Let u be a positive Ltl superharmonic function. Then there exist v e 9}l+(dSi) and x e 97tJ<+(£2) such that (1.12) holds. Proof. By the Riesz decomposition theorem u can be written in the form u = up + Uh where up is an LM-potential and Uh is a non-negative L^-harmonic function. Therefore there exists v e 9Jl+(dSi) such that u/, = K^[v]. Since up is an LM-potential there exists a positive Radon measure r such that up = G^[r] (see e.g. [1, Theorem 12]). This necessarily implies that r e SWj«+ (Si). □ Proposition 2.20. Let w be a non-negative L ^-subharmonic function. If w has a normalized boundary trace then it is dominated by an L ^-harmonic function. Proof. There exist a positive Radon measure r in Si and a measure v e 9Jl+(dSi) such that —L/1w = —x inSi, tr*(w) = v. Let up be the solution of —L/1u = —xp in Dp, u = K^[v] on Hp where xp := xxDp- Then, up+G^[xp] = K^[v]. Letting /i^Owe obtain, G£[r] < K°[v]. Hence r e 97tJ,+ (Si) and consequently w + G%[x] = E%[v]. a (2.39) 64 2. ELLIPTIC EQUATIONS WITH A HARDY POTENTIAL M. Marcus, P.-T. Nguyen/Ann. I. H. Poincare - AN 34 (2017) 69-88 85 3. The nonlinear equation In this section, we consider the nonlinear equation -L^u + uq=0 (3.1) in Si with 0 < /x < Cfi(£i) and q > 1. Proof of Theorem A. Since u is a positive solution of (1.1), u is L^-subharmonic. Assuming (i), u is dominated by an LM-harmonic function. Therefore, by Proposition 2.14, (i) ==> (ii) and u e Lqa+(Si). On the other hand, by Proposition 2.20 (ii) ==> (i). As mentioned above, (i) implies that u e Lqa+(Si) and that there exists v e W^a+(dSi) such that tr*(u) = v. Therefore, by Theorem 2.18, (1.14) is a consequence of (2.37). Thus (i) ==> (iii). Finally, the implication (iii) ==> (i) is obvious. It remains to prove the last assertion. If u is a positive solution of (1.13) then, by (iii), u e Lqa+(Si) and (1.15) follows from Theorem 2.18. Conversely, assume that Sa+uq, u/Sa- eL'(fi) and (1.15) holds. Then, by (1.15) with f e C™(Si), u is a solution of (1.1). Taking f/ = G^[/] where / e Cc(Si) and / > 0 we obtain J(K"[v] - u)f dx = j uqi;f dx < co. This implies u < K^[y], i.e., u is LM-moderate. Therefore by (i), u is a solution of (1.13). □ Proof of Theorem B. Uniqueness. Let u\ and 112 be two positive solutions of (1.13). Then v := (u\ — 112)+ is a subsolution of (1.1) and therefore an LM-subharmonic function. Furthermore, by (iii) in Theorem A, «i,«2 e Lqa+(Si) and v < G^[uq + uq] =: v. Obviously v is LM superharmonic and tr*(u) = 0. Therefore, by Proposition 2.14, v = 0. Thus u\ < 112 and similarly 112 < u\. Monotonicity. As before, v := (u\ — 112)+ is LM-subharmonic and it is dominated by an LM-superharmonic function. Since vi < V2, tr*(u) = 0. Hence by Proposition 2.14, v = 0. A-priori estimate. Suppose that u is a positive solution of (1.13). Then (1.15) with f = G^[l] implies (1.16). (Recall thatG^fl] ~<5a+.) □ For the proof of the next theorem we need Lemma 3.1. Let D (<= Si be a C2 domain and q > 1. If h is a positive function in Ll(dD) then there exists a unique solution of the boundary value problem, —Ltlu + uq = 0 in D u = h on 3D. (3.2) Proof. First assume that h is bounded. Let denote the Poisson kernel of — Ltl in D and put uq := F^[h]. Thus uq is bounded. We show that there exists a non-increasing sequence of positive functions {«„[", dominated by uq, such that un is the solution of the boundary value problem, fi — Av + vq = — u„_i in D S2 v = h on 30 n = l,2,... (3.3) As usual <5 denotes the distance to 3Si, not to 3D. For n = 1, uq is a supersolution of the problem and, obviously v = 0 is a subsolution. Consequently there exists a unique solution u\. By induction, for n > 1, 2. ELLIPTIC EQUATIONS WITH A HARDY POTENTIAL 65 86 M. Marcus, P.-T. Nguyen/Ann. I. H. Poincare -AN34 (2017) 69-88 „ /X /X -Aw„_l = ^2U"-2 > ^2Un-l- Thus u = un-\ is a supersolution of (3.3) and it is bounded. It follows that there exists 0 < un < w„_i such that a /X — Aun + u„ = -r:un-\ in D, un = h on 3D. <52 As the sequence is monotone we conclude that u = \imun is a solution of (3.2). If h e L1 (3D), we approximate it by a monotone increasing sequence of non-negative bounded functions {hk}. If Vk is the solution of (3.2) with h replaced by then {v^} increases (by the comparison principle [4, Lemma 3.2]) and y = lim Vk is a solution of (3.2). Uniqueness follows by the comparison principle. □ Proof of Theorem C. Put uq := K^[v] and hp := moLx^- Let up be the solution of (3.2) with h replaced by hp, P e (0, Po). Since uq is a supersolution of (1.1) it follows that {up} decreases as f3 1 0. Therefore u := lim^ow^ is a solution of (1.1). We claim that tr*(u) = v. Indeed, up+G?/[ul] = F?/[hp] = u0. (3.4) Furthermore, in Dp, up «]. Hence, by (3.4), u + G£[u*] = u0 = k£[v]. By Proposition 2.12, tr*(u) = v. By Theorem B the solution is unique. □ Proof of Corollary CI. By the previous theorem, if v = f where / is a positive bounded function then (1.13) has a solution. If 0 < / e L'(£2) then it is the limit of an increasing sequence of such functions. Therefore, once again problem (1.13) with v = f has a solution. Proof of Theorem D. Put v = K^[y] - u. By the comparison principle v > 0. Clearly v is L^-superharmonic in Q. and, by definition tr*(u) = 0. By Proposition I(iv) v is an Ltl potential. Consequently, by Theorem 2.6, v(x) lim —t:— = 0 non-tangentially, v a.e. on 3 £2. This implies (1.17). □ Proof of Theorem E. By Proposition 2.8, specifically inequality (2.15), K^[v] e L^a+(£i) for every q e (1, q^.c) and y e 97t+(3£2). Therefore the first assertion of the theorem is a consequence of Theorem C. We turn to the proof of stability. Put vn = K^[y„]. By Proposition 2.8, [vn] is bounded in L^a+(£i) for every q e (l,^iC) and in LP_a_{Q.) for every p e (1, )• ^n addition vn —>• y pointwise in £2. This implies that {v%8a+} and {u„/<5a-} are uniformly integrable in £2. Since uV)! < vn it follows that this conclusion applies also to {uvJ. By the extension of the Keller-Osserman inequality due to [4], the sequence {uVn} is uniformly bounded in every compact subset of £2. Therefore, by a standard argument, we can extract a subsequence, still denoted by {uVn} that converges pointwise to a solution u of (1.1). In view of the uniform convergence mentioned above we conclude that uVn^u in Lga+ (£1) and in (£2). By Theorem A, "v,+G>«,] = K>#1]. 66 2. ELLIPTIC EQUATIONS WITH A HARDY POTENTIAL M. Marcus, P.-T. Nguyen/Ann. I. H. Poincare - AN 34 (2017) 69-88 87 In view of the previous observations, passing to the limit as n —>• oo, we obtain, u + G£[u«] = K£[v]. Again by Theorem A it follows that u is the (unique) solution of (1.13). Because of the uniqueness we conclude that the entire sequence {uVn} (not just a subsequence) converges to u as stated in assertion II. of the theorem. Finally we prove assertion III. By Theorem A «tt,+G>^] = y). (3.5) Combining (2.7), (2.6) and the fact uksy < kK^{-, y), we obtain G"[mL ](x) GnUKn( y)4~\(x) kS> < i1 I1 y JW < ckq\x _ y]N+a+-q(N-\-a-) _ K°{x,y) ~ K°(x,y) Since 1 < q < gM>c, it follows that lim o -= 0- ■i^)' K°°(x, y) Therefore, by (3.5), we obtain (1.19). □ Proof of Theorem F. Letye3£2.By negation, assume that there exists a positive solution u of (1.13) with v = kSy for some k > 0. By Theorem A,u< fcK^(., y) and u eIj,+ (52). Let y e (0, 1) and denote Cy(y) = [x e £2: y |x — y| < <5(x)}. By Theorem D, ,. u(x) hm ieCr(j),MJ^(ij) This implies that there exist positive numbers ro, c such that u(x)>cK°(x,y) VxeCK(y)nBro(y). (3.6) By (2.7), JY--=fcy(j)nB«(y)&2(X'y»q8Ma+dx > c'y^+D fCy(y)nBro(y) \x - yr+-«<"-i-«->d*. Since ^ > 5M>C the last integral is divergent. But (3.6) and the fact that u e L^a+ (£1) imply that Jy < oo. We reached a contradiction. □ Conflict of interest statement No conflict of interest. Acknowledgements This research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, through grant 91/10. Part of this research was carried out, by the first author, during a visit at the Isaac Newton Institute, Cambridge as part of the FRB program. He wishes to thank the institute for providing a pleasant and stimulating atmosphere. The second author was also partially supported by a Technion fellowship. The authors wish to thank Professor Pinchover for many useful discussions. 2. ELLIPTIC EQUATIONS WITH A HARDY POTENTIAL 67 88 M. Marcus, P.-T. Nguyen/Ann. I. H. Poincaré -AN34 (2017) 69-88 References [1] A. Ancona, Theorié du potentiel sur les graphes et les variétés, in: Ecole ďété de Probabilités de Saint-Flour XVIII-1988, in: Springer Lecture Notes in Math., vol. 1427, 1990, pp. 1-112. [2] A. Ancona, Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. Math. (2) 125 (1987) 495-536. [3] A. Ancona, M. Marcus, Positive solutions of a class of semilinear equations with absorption and Schrodinger equations, J. Math. Pures Appl. 104(2015) 587-618. [4] C. Bandle, V. Moroz, W. Reichel, Boundary blowup type sub-solutions to semilinear elliptic equations with Hardy potential, J. Lond. Math. Soc. 2 (2008) 503-523. [5] C. Bandle, M. Marcus, V. Moroz, Boundary singularities of solutions to elliptic equations with Hardy potential, preprint. [6] H. Brezis, Une equation semilinéaire avec conditions aux limites dans Z.1, unpublished note, 1972. [7] H. Brezis, M. Marcus, Hardy inequalities revisited, Ann. Sc. Norm. Super. Pisa, CI. Sci. (4) 25 (1997) 217-237. [8] J. Dávila, L. Dupaigne, Hardy-type inequalities, J. Eur. Math. Soc. 6 (2004) 335-365. [9] S. Filippas, L. Moschini, A. Tertikas, Sharp two-sided heat kernel estimates for critical Schrodinger operators on bounded domains, Commun. Math. Phys. 273 (2007) 237-281. [10] E.B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations, Amer. Math. Soc, Providence, RI, 2002. [11] E.B. Dynkin, Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations, Amer. Math. Soc, Providence, RI, 2004. [12] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, second edition, Springer, Berlin, 1983. [13] A. Gmira, L. Veron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J. 64 (1991) 271-324. [14] J.B. Keller, On solutions of Au = /(«), Commun. Pure Appl. Math. 10 (1957) 503-510. [15] J.F Le Gall, The Brownian snake and solutions of Au = u^ in a domain, Probab. Theory Relat. Fields 102 (1995) 393^132. [16] M.Marcus, Complete classification of the positive solutions of-Au + uq = 0, J. Anal. Math. 117(2012) 187-220. [17] M. Marcus, V.J. Mizel, Y. Pinchover, On the best constant for Hardy's inequality in , Trans. Am. Math. Soc. 350 (1998) 3237-3255. [18] B. Mselati, Classification and probabilistic representation of the positive solutions of a semilinear elliptic equations, Mem. Am. Math. Soc 168 (798) (2004). [19] M. Marcus, I. Shafrir, An eigenvalue problem related to Hardy's Lp inequality, Ann. Sc. Norm. Super. Pisa, CI. Sci. 29 (2000) 581-604. [20] M. Marcus, L. Veron, The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Arch. Ration. Mech. Anal. 144(1998) 201-231. [21] M. Marcus, L. Veron, Removable singularities and boundary trace, J. Math. Pures Appl. 80 (2001) 879-900. [22] M. Marcus, L. Veron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion, J. Eur. Math. Soc. 6 (2004) 483-527. [23] M. Marcus, L. Veron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter Series in Nonlinear Analysis and Applications, 2013. [24] R. Osserman, On the inequality Au >/(«), Pac. J. Math. 7 (1957) 1641-1647. CHAPTER 3 Semilinear elliptic equations with a Hardy potential and a subcritical source term This chapter is based on the paper [119]. In this chapter, we discuss semilinear elliptic equations with a source term and a Hardy potential. Various necessary and sufficient conditions for the existence of solutions to the corresponding Dirichlet problem are obtained. 69 70 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Calc. Var. (2017) 56:44 DOI 10.1007/s00526-017-l 144-6 Calculus of Variations CrossMark Semilinear elliptic equations with Hardy potential and subcritical source term Phuoc-Tai Nguyen1 Received: 10 October 2015 / Accepted: 20 February 2017 / Published online: 17 March 2017 © Springer-Verlag Berlin Heidelberg 2017 Abstract Let £2 be a smooth bounded domain in M.N (N > 2) and S(x) := dist (x, 9 £2). Assume [i e R+, v is a nonnegative finite measure on 9 £2 and g e C(£2 x R+). We study positive solutions of fi —Au--ttii = g(x, u) in £2, tr (w) = v. (P) Sz Here tr* (w) denotes the normalized boundary trace of u which was recently introduced by Marcus and Nguyen (Ann Inst H Poincare Anal Non Lineaire, 34, 69-88, 2017). We focus on the case 0 < [i < Ch (£2) (the Hardy constant for £2) and provide qualitative properties of positive solutions of (P). When g(x, u) = uq with q > 0, we prove that there is a critical value q* (depending only on N, fi) for (P) in the sense that if q < q* then (P) possesses a solution under a smallness assumption on v, but if q > q* this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where g is subcritical [see (1.28)]. We also investigate the case where g is linear or sublinear and give an existence result for (P). Mathematics Subject Classification 35J60 • 35J75 • 35J10 Contents 1 Introduction ............................................... 2 2 Preliminaries............................................... 8 2.1 WeakL^ spaces........................................... 8 2.2 Green and Martin kernels...................................... 8 Communicated by F. Helein. Phuoc-Tai Nguyen nguyenphuoctai.hcmup@gmail.com; phnguyen@mat.puc.cl Departamento de Matematicas, Pontificia Universidad Catolica de Chile, Vicuna Mackenna 4860, Santiago, Chile Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 71 44 Page 2 of 28 P.-T. Nguyen 2.3 Some results on linear equations.................................. 9 3 Nonlinear equations with source term.................................. 10 3.1 Properties of weak solutions .................................... 10 3.2 Nondecreasing source........................................ 11 4 Power source............................................... 13 4.1 Subcritical case........................................... 13 4.2 Supercritical case.......................................... 21 5 More general source........................................... 22 5.1 Subcriticality ............................................ 22 5.2 Sublinearity............................................. 26 References.................................................. 28 1 Introduction This paper concerns a study of weak solutions of semilinear elliptic equations with Hardy potential and source term [i — Au--7ru=g(x,u) in £2 (1.1) Sz where £2 is a C2 bounded domain in M.N(N > 2), fi > 0, S(x) := dist(x, 9£2) and g e C(£2 x R+). Henceforth, we will use the notations LjL := A + ^ and (g o w)(x) := g(x, u{x)). Definition 1.1 (i) A function u is an L^-harmonic function (resp. L^-subharmonic, LjL-superharmonic) in £2 if u e L\oz(^i) and —LjLu = 0 (resp. — LjAu < 0, —LjAu > 0) in the sense of distributions in £2. (ii) A function u is called a nonnegative weak solution (resp. subsolution, supersolution) of (1.1) if u > 0, u € L\QC(Sl), goue Lloc(£2) and —L^u = g o u (resp. — L^u < g o u, —L^u > g o u) in the sense of distributions in £2. Boundary value problem with measures for (1.1) with [i = 0 and job = uq, i.e. the problem — Au = uq in £2, u = v on 3£2 (1.2) was first considered by Bidault-Veron and Vivier [7]. They established estimates involving classical Green and Poisson kernels for — A and applied these estimates to obtain an existence result in the subcritical case, i.e. 1 < q < qc := ^-rj-- Then Bidaut-Veron and Yarur [9] reconsidered this type of problem in a more general setting and provided a necessary and sufficient condition for the existence of a solution of (1.2). Chen et al. [12] investigated (1.1) with [i = 0 and g satisfying a subcriticality condition. Their approach makes use of Schauder fixed point theorem, essentially based on estimates related to weighted Marcinkiewicz spaces. Recently, Bidaut-Veron et al. [8] provided new criteria for the existence of weak solutions of problem (1.2) and extended those results to the case where A is replaced by LjL. When )i/0, the study of (1.1) relies strongly on the investigation of the linear equation -L[lu = Q in £2. (1.3) Springer 72 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Semilinear elliptic equations with Hardy potential and... Page 3 of 28 44 Equation (1.3) with fi < 0, and more generally Schrodinger equations — Aw + V(x)w = 0 where V is a nonnegative potential, was studied by Ancona [1,2], Marcus [18], Ancona and Marcus [3] and by Veron and Yarar [25]. The case [i > 0 was considered by Bandle et al. [4-6], Marcus and Nguyen [20], Gkikas and Veron [15] and by Marcus and Moroz [19] in connection with the Hardy constant Ch (£2) which is given by C„(£2) = inf f . (1.4) It is well known (see [11,21]) that C#(£2) e (0, ±] and C#(£2) = \ when £2 is convex. Moreover the infimum is achieved if and only if Ch (£2) < 1 /4. Let (/> > 0 in £2 and p > 1, we denote by LP(S2; ). When = 1, we use the usual notations 9Jt(£2) and 9Jt+(£2). We also denote by 911(3 £2) the space of finite measures on 3 £2 and by 9Jt+(3£2) the nonnegative cone of 911(3 £2). Let Gtl and KjL be the Green and the Martin kernels for — LjL in £2 respectively (see [20] for more detail). Denote by and the associated operators defined by Put M(x) = f G„(x, y)dr(y), Vr e 9Jt(£2), (1.5) [v](x) = Kilix,z)dviz), Vve9Jt(3£2). (1.6) i±yr^i or± :=---. (1.7) Let be the first eigenvalue of — LjL in £2 and denote by 0 and by [13] (see also [22]), there exists a constant c\ > 0 such that c^l&a+ < cp^i < ciSa+ in £2. (1.8) For j3 > 0, put £2/; = {x e £2 : 5(x) < f3}, Dp = {x e £2 : 5(x) > f3}, Y,p = {x e £2 : <5(x) = ,S}. When dealing with boundary value problem associated to (1.1) with [i > 0 one encounters the following difficulties: - The first one is due to the fact that every positive LjL -harmonic function has classical measure boundary trace zero (see [20, Corollary 2.11]). Therefore, the classical notion of boundary trace no longer plays an important role in describing the boundary behavior of L^-harmonic function or solutions of (1.1). - The second one stems from the invalidity of the classical Keller-Osserman estimate, as well as the lack of a universal upper bound for solutions of (1.1). Moreover, contrast to the case of nonnegative absorption nonlinearity, K^[v] with v e 9Jt+(3£2) is a subsolution of — L/1u = gou in £2 (1.9) and therefore it is no longer a natural upper bound for solutions of (1.9). Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 73 44 Page 4 of 28 P.-T. Nguyen In order to overcome the first difficulty, we shall employ the notion of normalized boundary trace which is defined as follows: Definition 1.2 A function u possesses a normalized boundary trace if there exists a measure v € 2Jt(3ft) such that lim p-a- I \u(x) - KM[v](x)| 0 when [i e (0, C//(£2)) and hence (p^\ is a positive L^-superharmonic function in £2. This fact, together with a classical result of Ancona [2], guarantees the validity of Representation theorem (see [20]). The notion of normalized boundary trace turned out to be appropriate to investigate the problem -L^u + uq=0 inn, tr*(w) = v. (1.11) More precisely, when [i e (0, C//(£2)), they showed that there exists a critical exponent q*=q*(N,VL) := J[+a+ 0- (1-12) N + a+ — 2 for (1.11). This means that if 1 < q < q*, for every positive finite boundary measure v on 9£2, (1.11) admits a unique positive solution, while if q > q* there exists no positive solution of (1.11) with v beingaDirac measure. Stability result was also discussed in the case 1 < q < q*. Problem (1.11) with uq replaced by amore general nonlinearity f{u) was then investigated by Gkikas and Veron [15] in a slightly different setting. When f(u) = \u\i-lu, they provided a necessary and sufficient condition in terms of Besov capacity for solving (1.11) in the supercritical case, i.e. q > q*. Because of the second difficulty, we mainly deal with the minimal solution of (1.9) which possesses several exploitable properties. This solution is constructed due to sub-supersolutions theorem in Sect. 3. Observe that K^[v] with v e 9Jt+(3£2) is a subsolution of (1.9); hence in order to prove the existence of a minimal solution of (1.9), it is sufficient to find a supersolution of (1.9) which dominates K^[v], Throughout the present paper, we assume that [i e (0, C//(£2)). We now introduce the definition of solutions of — L/1u=gou infi, tr*(w) = v. (1-13) Definition 1.3 (i) A nonnegative function u is called a (weak) solution of (1.13) if u is a solution of (1.1) and has normalized boundary trace v. (ii) Put A function f e X(£2) is called an admissible test function for (1.13). Notice that 0, C3 > 0. 0 < r\ < r2 < 00 and a function I : R+ —>■ R+ such that g(x,rs) <£(r)g(x,s) Vs > 0, r > 0,x 6 R, (1.16) 1(1 + c2C3r'1£(r)) < c2 Vr€ (n, r2), (1.17) &„lg o (K^[v])] < C3KM[v] a.e. in Si. (1.18) 1. existence. For any q € (n, r2) the problem — Ljlu = gou inS2, tr*(u)=gv (1-19) admits a minimal positive weak solution u_QV in the sense that ifv is a positive weak solution of (1.19) then u_QV < v in £1. 2. Estimates. There exists a positive constant c4 = c4(c2, c3, £, q) such that eKM[v] < ugv < c4gKf£[v] a.e. in £2. (1.20) 3. Nontangential convergence. For v-a.e. point z e 3fi, there holds M.„v(x) lim- = q non-tangentially. (1-21) x^z KM[v](x) Remark When g(x, u) = uq with q > 1, c4 can be chosen independently of q. In the next results, we focus on the pure power case, namely the problem —Ljlu = uq in £2, tr*(w) = v (Dv) where q > 0 and v e 9Jt+(9£2). We shall establish some estimates related Green and Martin operators and a necessary condition for the existence of solutions of (Dv) in the case q > 1. Theorem C Let q > 0 and v € 9Jl+(d£2). Then there exists a positive constant c5 = cs(N, (i,q, £2) such that G^lvf] < c5 ||v||g,n) K„[v] a.e in S2. (1.22) Furthermore, if q > 1 and problem (Dv) admits a positive weak solution then there holds G^lvf] < —L-K^tv] a.e. in £2. (1.23) Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 75 44 Page 6 of 28 P.-T. Nguyen Remark It is worth mentioning that when fi = 0, (1.22) and (1.23) were obtained by Bidaut-Veron and Vivier [7]. When LjL is replaced by a uniformly elliptic differential operator of second order with bounded Holder-continuous coefficients, (1.23) is relevant to [17, Theorem 7.6]. Recently, (1.23) with an inexplicit multiplier on the right hand-side were proved by Veron et al. in [8, Theorem 4.1]. In this paper we employ the method in [7] to prove (1.22) for q > 0 and apply the idea in [7,9,10] to point out that the multiplier on the right hand-side of (1.23) can be explicitly chosen as ^tj- When q = 1, estimate (1.22) becomes GfiP&fTv]] < csIK^v] with c5 = c$(N, fi, £2), which can be regarded as the limiting case of (1.23). The next results reveal that q* is a critical exponent for (Dv). More precisely, in the subcritical case, namely 1 < q < q*, (Dv) admits a solution under a smallness assumption on the boundary datum, while in the supercritical case, i.e. q > q*, this problem possesses no solution with isolated boundary singularity. For z € 3£2, we denote by Sz the Dirac measure concentrated at z. Existence and nonexistence results when 0 0, consider the problem —Ljlu = uq in £2, tr*(w) = qv. (Dev) 1. Case: q e (1, q*). There is a threshold value q* e R+ for (Dev) such that the following holds. (i) If q € (0, q*] then problem (Dev) admits a minimal positive weak solution ugv. Moreover, if q e (0, q*), ugv satisfies (1.20) and (1.21). In addition, if {q„} is a nonde-creasing sequence converging to q* then {u_QnV\ converges to uQ*v in L1 (£2; 5~a-) and in Li(Sl;8a+). (ii) If q > q* then there exists no positive weak solution of (Dev). 2. Case: q e (0, 1). For every q > 0 problem (Dev) admits a minimal solution ugv which satisfies satisfies (1.20) and (1.21). Moreover, lime^00u_ev = oo a.e. in £2. For any 1 7^ q e (0, q*), if v = Sz with z € 3£2 then there holds H-oX (x) lim ^j— = q. (1.24) x^z K^(x, z) Remark Note that in the absorption case [namely equation (1.11)], if 1 < q < q*, there are two types of solution with isolated boundary singularity: the weakly singular solutions waz (the solution of (1.11) withv = qSz) and the strongly singular solution Wco,z- Actually, u^>z is the limit of the sequence wez as q —» 00. This limiting process can not be executed in the source case since (De$) admits no solution if q > q* due to Theorem D. We next give a stability result. Theorem E Let q € (0,g*),g 7^ 1 and {vn} is a sequence of measures in 9Jt+(3£2) which converges weakly to v e Wl+(dQ,). If q > 1, assume in addition that SUP llvjgjton) < q*. (1.25) n For each n, let uVn be a positive weak solution of (DVn). Then, up to a subsequence, {uVn} converges to a positive weak solution uv of (Dv) in L1 (£2; 8~a-) and in Lq (£2; Sa+). Springer 76 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Semilinear elliptic equations with Hardy potential and... Page 7 of 28 44 An existence and stability result in the case q = 1 is stated in the following theorem in which is the first eigenvalue of — LjL in Q.. Theorem F Let v e 9Jt+(9£2). For k > 0, consider the problem —L/1u = Ku inS2, tr*(w) = v. There exists a number k* € (0, A^j] such that the following holds. (i) Ific € (0, k*) then problem (E* ) admits a minimal positive weak solution u_K . Moreover, u^ v satisfies (1.21). Assume {v„} is a sequence of measures in 9Jt+(3£2) which converges weakly to v € DJl+(d£2) and for each n denote by wKVn a positive weak solution of(E* ). Then, up to a subsequence, {uKVn\converges to a positive weak solution wKjV of (E*) in L1 (£2; 5~a-). (ii) If k > k* then (E*) admits no positive weak solution. Furthermore, problem (E^'1) admits no positive weak solution. Remark It is notified by the referee that k* = A^j. The way to prove it is to note that —LjL — k admits the Green function G^jK for any k < fi and then to prove a modification of Proposition 2.4 for G^jK (see [24] for the existence of the Green function G^jK). The weaker statement k* < A^i in the present paper is essentially in order to simplify the proofs and to streamline the exposition. In the supercritical case, i.e. q > q*, there is no solution with an isolated boundary singularity. Theorem G Assume q > q*. Then for every q > 0 and z € d£2, there is no positive weak solution of —LjLu = uq in S2, tr*(w) = qSz. (^e^) Here Sz denotes the Dirac measure concentrated at z. Remark Interesting removability results for (Dv) in terms of capacities in the supercritical case was provided in [8], In the next two theorems, we consider the case (g o u)(x) = S(x)y g(u(x)) where y > — 1 — a+ and g : R+ —» R+ is nondecreasing and continuous. In this framework, the critical exponent for (1.1) is * *r\t ^ N + a+ + y <7v = qY{N, fi, y) := —--• (1-26) r r N + a+ — 2 Clearly q* = q*. Theorem H gives an existence result for the problem — Lllu = Svg(u) in £2, tr*(u)=gv. (1-27) Theorem H Let v e 9Jt+(9£2) with IMI^g^ = 1. Assume that s~l~q*g(s)ds < +oo, (1.28) g(s) < Aisqi +6, Ws e [0, 1] for some qi > 1, A: > 0, 6 > 0. (1.29) Then there exist 0q > 0 and qq > 0 depending on N, [i, y, Aq, A\ andq\ such that for every 9 € (0, 0q) and q € (0, qq) problem (1.27) admits a weak solution u > £?K^[v] in £2. Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 77 44 Page 8 of 28 P.-T. Nguyen Remark If g satisfies (1.28) we say that g is subcritical with respect to y. The case where g is linear or sublinear is treated in the following theorem. Theorem I Let v e Tl+(dS2) with || v Hot(3£2) = 1- Assume that g(s) < A2sq2+9, Ws>0 (1.30) for some q2 € (0, 1], A2 > 0 and 6 > 0. In (1.30), if q2 = 1 we assume in addition that A2 is small enough. Then for any q > 0, (1.27) admits a weak solution u > £?K^[v]. Remark In Theorem I, when q2 < 1, the smallness assumption on 6 is not required. The plan of the paper is as follows. In Sect. 2 we give results concerning Green and Martin kernels and boundary value problem for linear equations with Hardy potential. Theorems A and B are proved in Sect. 3. It is noteworthy that main ingredients in proving Theorem A are: a generalization of Herglotz-Doob theorem to LjL -superharmonic functions and the theory of Schrodinger linear equations. Theorem B is established using a sub-supersolutions theorem. The proof of Theorems C-G are presented in Sect. 4. Finally, in Sect. 5 the existence result in the case of more general source terms (Theorems H and I) is obtained due to the Schauder fixed point theorem and estimates in weak Lp spaces. 2 Preliminaries Throughout this paper we assume that 0 < [i < Ch(S2). 2.1 Weak Lp spaces We denote by LP,(SI; r), 1 < p < 00, r e 9Jt+(£2), the weak Lp space (or Marcinkiewicz space) (see [23]). When r = Sadx, for simplicity, we use the notation LP,(SI; Sa). Notice that, for every a > — 1, L? (SI; 8adx) C Lr{Q.\ 8a), Vre[l,p). Ifw € Lpw(£l; &a)(a > -l)then f fdxs] wy 2.2 Green and Martin kernels Let be the Green kernel for the operator — LjL in Q. x Q. and denote by the associated operator defined by (1.5). It was shown in [20] that for every r e 9Jt(£2; <5a+), |G^[r]| < oc a.e. in Q.. Denote by KjL the Martin kernel for — LjL in Q. and by the Martin operator defined by (1.6). In what follows the notation / ~ g means: there is a constant c > 0 such that c^1 f < g < cf in the domain of the two functions. By [14, Theorem 4.11] and [20] (see also [15]), G,Ax,y) ~ min{|x - y\2~N , S(x)a+S(y)a+ \x - y\2a-~N} Vx, y e SI, x # y, (2.2) ^(x,z) ~5(x)a+|x -z\2a-~N Vx &Sl,z&dSl. (2.3) Springer 78 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Semilinear elliptic equations with Hardy potential and... Page 9 of 28 44 The following estimates can be found in [16, Proposition 2.4] Proposition 2.1 (i) Let € ( — N+2a+ -2' TT^)' Tnen there exists a constant c~i = cj(N, [i, P, £2) such that |G,Jr]| N+e — 1. Then there exists a constant cs = cg(iV, [i, ft, £2) such that ||IMv]|| N+e 0 such that for every point x € £lpQ, there exists a unique point ax € 9£2 such that x = ax — S(x)nax. The mappings x \-> S(x) and x \-> ax belong to C2(£fyj0) and C^(£lp0) respectively. Moreover, limx^CT(x) V<5(x) = — nax. For D <£ £2, let G® and Kf? be the Green and Poisson kernels of —L^'mD respectively. Denote by and K.® the corresponding Green and Poisson operators in D. We prove below main properties of solutions of (1.13). Proof of Theorem A (i) (ii). Assume u is a positive weak solution of (1.13). Put r = g o u and for j3 e (0, j3o) denote rp := t\op and kp := u | . Consider the boundary value problem This problem admits a unique solution vp (the uniqueness is derived from [5, Lemma 2.1] since [i < C#(£2)). Therefore vp = u\pp. We have lim - x^z w2 (x) 0 non-tangentially. — L^u = g o u (3.1) Lßv = Tß in Dp, v = Xß on Tiß. u\Dß=Vß=Gllß[Tß-] + Kllß[*.ßl It follows that JDß Letting ß —» 0, we get (3.2) Springer 80 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Semilinear elliptic equations with Hardy potential and... Page 11 of 28 44 Fix a point xq € £2 such that u(xq) < oo. Keeping in mind that G^(xo, y) > cXQS(y)a+ for every y e £2, we deduce from (3.2) that gou e L1^; Sa+). Thanks to Proposition 2.4 (v) , we obtain (1.14). (ii) (i). Assume u is a function such that g o u e Ll(£2; <5"+) and (1.14) holds. By Proposition 2.4 (i) — L^K^Iv] = 0, which implies that u is a solution of (3.1). On the other hand, since g o u e L1^; Sa+), we deduce from Proposition 2.4 (ii) that tr *(GjAg o w]) = 0. Consequently, tr *(w) = tr *(KAt[v]) = v. (i) (iii). Assume u is apositive solution of (1.13). From the implication (i) (ii), we deduce that u e L1^; 8~a-) and g o w e L1^; <5a+). Hence, by Proposition 2.4 (vi) , u satisfies (1.15). (iii) ==> (i). This implication follows directly from Proposition 2.4 (vi). □ 3.2 Nondecreasing source We start with an existence result for (3.1) in presence of sub and super solutions. Theorem 3.2 Let g € C(£2 x R+), g(x, r) be nondecreasing with respect to r for any x € £2. Assume that there exist a subsolution V\ and a supersolution V2 of (3.1) such that 0 < V\ < V2 in £2. Then there exists a solution u of (3.1) which satisfies V\ < u < V2 in £2. Moreover, ifV\ = K^[v] for some v € Wl+(d£2) and g o V2 € L1 (£2; Sa+) then there exists a minimal positive weak solution uv 0/(1.13) in the sense that uv < v in £2 for every positive weak solution v of (1.13). Lemma 3.3 Let D <<= £2, f e Ll(D), f > 0 and 77 e L1(3D), 77 > 0. Then there exists a unique solution of —L^u = f in D, u = 77 on dD. (3.3) Proof We start with the case f e L2 (D) and r\ = 0. Let us consider the functional J(V) -=\f (l^l2 " ^2v2) dx ~ f fvdx over the space Hq(D). Since fi < C#(£2), by Hardy inequality and the variational method, one can show that the problem min^i^ J(v) admits a solution v e Hq(D). The minimizer 11 is the unique weak solution of (3.3). If / € Ll+(D) then we can approximate it by an increasing sequence {fm} C L^°(D). Let vm be the solution of (3.3) with 77 = 0 and / replaced by fm. By comparison principle [5, Lemma 2.1], {vm} increases and therefore v := hmm^oo vm is a solution of (3.3) with (7 = 0. We next consider the case 77 e L1(dD). Let v be a solution of (3.3) with 77 = 0 then is a solution of (3.3). The uniqueness follows from the comparison principle. □ Proof of Theorem 3.2 Put uq '■= Vi and 77^ := Vils^ for j3 e (0, /3o). For n > 1, consider the problem — L^u = g o w„_i in Dp, u = r\f>j on dDp. (3.4) For each n > 1, by Lemma 3.3 there exists a unique solution u^n of (3.4). Moreover, since g(x,r) is nondecreasing with respect to r for every x e £2, by applying the comparison principle, we deduce that Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 81 44 Page 12 of 28 P.-T. Nguyen V\ < up,n < u^n+i < v2 in Dp. Therefore up := lhrin^oo upt„ is a solution of (3.1) in Dp which satisfies V\ < up < V2 in Dp. Moreover, up = G^[goup] + F^[np]. (3.5) ForO < j3' < j3 < ySo, by the comparison principle, upt\ < up<\ in Dp. By the monotonicity assumption on g, it follows that upt„ < upi n in Dp for every n > 1. Therefore V\ < up < up< < V2 in Dp and hence u := lim^o up is a solution of (3.1) in £2 satisfying V\ < w < V2. In the case V\ = K^[v], formulation (3.5) becomes up=G^lgoup]+K„lv]. (3.6) Put uv := lim^o up. Since 0 K^[v] and g o v £ L1^; <5"+); consequently uv < v in £2. □ Proof of Theorem B We first notice that since g o (K^ [v]) e L|oc(fi) andG/Lt[go(IK/Lt[v])] < 00, it follows that g o (K^[v]) e L1^; 8a+) due to a similar argument as in the proof of Theorem A. It is easy to see that K^[v] is a subsolution of (3.1). For q e (r\, r2), we look for a supersolution v of the form « = eKp[v] + C2G,ko(eK,[v])] (3.7) where c2 is the constant in (1.17). By (1.16) and (1.18), we obtain v < e(l +C2C3q-1£(q))Ki1[v] in£2. The monotonicity property of g implies gov 3 = ß\(DiUD2), and := f lx-yl^-Nly_zl(2-a+-N)q min\\x - y\a+, \y - z\a+}dy, i = 1, 2, 3. For every y e T>\, \x — z\ < 2\y — z\, therefore h < cn\x -zf-«+-N^ [ \x - y\1+a-Ndy < c'12\x - z|l+«--(jv-h.+ -2)?_ (44) For every y e T>2, |x — z\ < 2|x — y\, hence h < c13\x - z\1+a-N f \y- z\{2-^-N)qdy Jv2 < c'13|x - z|i+"--(jv+«+-2)?. (4 5) Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 83 44 Page 14 of 28 P.-T. Nguyen For every y e D3, \y — z\ < 3|x — y\, therefore h < C14 [ \y~ z\l+<*--N-(N+<*+-Vidy < c'14|x - z|l-H»--(JV+a+-2),_ (4 6) Combining (4.3)-(4.6), we obtain (•,*)*](*) < cii(c'12 + c'13 +C'14)5(x)a+|x _Z|1-H»—(JV+«+-2),_ (47) Estimate (4.2) follows straightforward from (2.3) and (4.7). □ Proposition 4.2 Assume 0 < q < q* and v is a positive finite measure on d£2. Then IK/*[v] € Lq(£2\ Sa+) and there exists a constant C15 = c\s(N, [i, q, £2) such that G^[K^[v]*] 1. From (2.5) and the fact that Lqm (£2; Sa+) C Lq(S2\ Sa+), we deduce that KM[v] e L?(fi; 5a+). It follows from (1.6) and Jensen's inequality that Kß[v](x)q < / Kß(x,z)qdv(z) fora.e.xeß. Jan Consequently, G^K^vfKx) < /" /" G^(x,y)^(y,z)*dv(z)dy. Jan '3£2 JQ By Lemma 4.1, since N + a+ — (N + a+ — 2)q > 0, G^K^vfK*) < cio f \x - z\N+a+-{N+a+-2)qK„(x, z)dv(z) JdQ. 0, v e 9Jt+(3£2), v ?k 0 and cp e C^fO, oo)) fee a concave, nondecreasing function such that (p(l) > 0 and (/>' is bounded. Let cp be a positive function in LJgc(£2) such that —L^cp > f. Then (4.9) / in the weak sense. (4.10) Springer 84 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Semilinear elliptic equations with Hardy potential and... Page 15 of 28 44 Proof Since -L^cp > f > 0, by Proposition 2.4, there exist r e DJt+(£2; Sa+) and X e Wi+idtt) such that

0. Due to [10, Lemma 5.3], It follows that Consequently, (-A(p„) + (-Lß(p„) + n} and {^r„} are uniformly integrable with respect to 8a-dx. In view of Vitali theorem {(pn} and {iffn} converge to

-t" Due to Fatou lemma, by sending n —» oo in (4.12), we obtain (4.9) and (4.10). □ Theorem 4.4 Let q > \andv s9Jt+(9£2),v ^ 0. Ifproblem (Dv) admits a positive weak solution then l t[v] in £2. (4.14) Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 85 44 Page 16 of 28 P.-T. Nguyen Proof Let u a positive weak solution of (Dv); then by Theorem A, uq e Ll(£l\ Sa+) and (1.14) holds. Consequently K^[v]? e L1^; Sa+). Now applying Lemma 4.3 with / = uq ,

1. - 1 1 ifO\ then there exists a positive number qq > 0 depending on n, jJL,q,£l such that for every q € (0, qq) problem (DgV) admits a minimal weak solution u_^v. (ii) If q € (0, 1) then for every q > 0 problem (DgV) admits a minimal weak solution u_QV. For any 1 ^ q € (0, q*), u_QV satisfies (1.20) and (1.21). Proof We shall apply Theorem B to deduce the existence of a solution of (Dev). One can verify that the functions g(x,s) = sq and l(s) = sq with q > 0 satisfy (1.16). From Proposition 4.2 we deduce that condition (1.18) is fulfilled with the constant c\s. For such g and £, condition (1.17) is valid if one can find a positive constant cjs such that I + enseal-1 <4_ (4.16) If q > 1 then there exist qq = Qo(q, cis) and cjs = c\%(q) such that (4.16) holds true for every q e (0, qq). If q < 1 then for every q e [1, oo) one can choose cjs = cig(cis) large enough such that (4.16) holds. If q < 1 then for every q e (0, 1) one can choose cjg = ci8(o, g, c15) large enough such that (4.16) holds. Hence, by Theorem B, there exists a minimal solution uQV of (Dev) which satisfies (1.20) and (1.21). □ Lemma 4.6 Let 0 < q ^ 1 and v € 9Jt+(9£2). Then there is a constant c19 = ci9(iV, [i, q, £i) such that ifu is a solution of(Dv) IImHl1^-"-) + IKIL1^;^) - cw(1 + IMIjOTOn))- (4-17) Springer 86 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Semilinear elliptic equations with Hardy potential and... Page 17 of 28 44 Proof Indeed, by taking f = cp^j (the first eigenfunction of—L^) in the formulation satisfied by u, we obtain / u(Pß,idx = / uq 1. By Young inequality, we get / utp^idx < (2k^i) 1 / uq(p^idx + (2k^i)i-1 / cp^idx. (4.19) Ja Ja Ja By (4.18) and (4.19), we obtain / uq(p^idx + 2k^i / K^vty^idx < (Zk^i)^ / cp^idx. (4.20) Ja Ja Ja Since the second term on the left hand-side of (4.20) is nonnegative, we deduce by (1.8) that q ll«lll»(n:*"+) < cf(2Vi)*11 jja+dx ^ c20- (4-21) On the other hand, we derive from (1.14), (2.4) and (2.5) that INIl1(£2;<5-"-) - c2i(||«9|Li(n.3«+) + llvllajton)). (4.22) Combining (4.21) and (4.22), we obtain (4.17). Case 2: q e (0, 1). By Young inequality, we have / uqa Ja Ja Therefore Wuhl(a-s«+) < c22(l + IMIaxon)). (4.23) Combining (4.22) and (4.23) leads to (4.17). □ Theorem 4.7 Assume q € (l,q*) andv € 9Jt+(3£2) with II v || jxrt(as2") = 1- Then there exists a threshold value q* € WL+for (DgV) such that the following holds. (i) If q € (0, q*] then (DgV) admits a minimal weak solution u_^v. If q € (0, q*) then u_^v satisfies (1.20) and (1.21). Moreover {u_gV} is an increasing sequence which converges, as q —>■ q*, to the minimal positive weak solution u_g*v of (Dg*v) in L1 (£2; 8~a-) and inL1(Sl;8a+). (ii) If q > q* then there exists no positive weak solution of(DgV). Proof Put a := {q > 0 : (Dev) admits a weak solution} and q* := sup A By Proposition 4.5, (Dev) admits a solution for q > 0 small, therefore a ^ 0. Moreover, from Theorem 4.4, we deduce that q* is finite. We shall show that (0, q*) C a. To this purpose, we have to show that if 0 < q < q' and a 3 q' < q* then q e a. Since q' e a, due to Theorem 4.4, there exists a minimal Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 87 44 Page 18 of 28 P.-T. Nguyen weak solution uQ,v of (Dgiv) which is greater than gK^[v]. By Theorem 3.2, (Dgv) admits a minimal weak solution uQV, i.e. q e a. Next we prove that q* e a, namely problem (De*v) admits a weak solution. Let {q„} be an increasing sequence converging to q*. For each n, let uQnV be a weak solution of (DQnV). Then uQnV e L1 (S2; S~a-) n Lq(£2; Sa+) and it satisfies - / uenVL,^dx= [ uq Sdx-Qn [ K^ML^dx Vf e (4.24) JQ. JQ. JQ. It follows from Lemma 4.6 that the sequence {uqnV} is uniformly bounded in L1^; <5"+) and hence by local regularity for elliptic equations there exists a subsequence, still denoted by the same notation, such that {uenV} converges a.e. to a function ue*v. From Theorem A, there holds (4.25) Thanks to Proposition 2.1, {uenV} is uniformly bounded in Lqx (£2; S~a-) and in Lqi (£2; 8a+) where 1 < q\ < and q < q2 < q*■ We invoke Holder inequality to infer that {ugnV} and {ugnV} are uniformly integrable with respect to 8~a-dx and Sa+dx respectively. As a consequence, {ugnV} converges to ug*v in L1 (£2; 5~a-) and {uqnV} converges to uqQ*v in l}(Q.\ Sa+). Letting n oo in (4.24) implies / ue*vL,^dx= [ uqi;dx-Q* I KpWLrfdx Vf e (4. JQ. JQ. JQ. 26) We infer from Theorem A that solution of (Dg*v). Notice that, in light of Theorem 3.2 and the above argument, one can prove that {ugv} is an increasing sequence converging to the minimal solution Ug*v of (Dg*v) in L1^; 5~a-) and mLq(£2;Sa+). We next show that for each q e (0, q*), there exists a minimal weak solution ugv of (Dev) which satisfies (1.20). Take q' = et,e and let uQv, f = uq^,v and (s) We get Therefore s(l+e^_1) 7 s + t[e'v]0 t[e'v] if* > l, if 0 < s < 1 Un> with e t[e'v]0 Q'V [e'v; 7-1 is a supersolution of (4.1). Moreover *I> > gK^[v]. By Theorem 3.2 there exists a minimal weak solution u_QV of (Dev) such that t[v] inß. Springer 88 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Semilinear elliptic equations with Hardy potential and... Page 19 of 28 44 This implies fiKjjM < iLgV < e_^Te'KA[[v] inn. _ __i_ Therefore we get (1.20) with 04 = q~ q's i~l. Finally, (1.21) can be obtained by a similar argument as in the proof of Theorem B. □ Proof of Theorem D Part(l) follows from Theorem4.7. Part (2) follows from Proposition 4.5 (ii). If v = qSz, by (1.14) and (1.20), we obtain K^(x,z) K^(x,z) Since q < q*,it follows from Lemma 4.1 that ,. G^[^(-,z)*](x) lim —-—-- = 0. x^z K,j,(x,z) Thus, by (4.27), we conclude (1.24). □ Proof of Proposition E If q > 1, assumption (1.25) guarantees the existence of a solution uVn of (DVn). Moreover, since {v„} converges weakly to v, it follows that II v||jxrt(as2") — 6*-Due to Lemma 4.6, the sequence {uVn} is uniformly bounded in Lq (£2; 8a+). Employing a similar argument as in the proof of Theorem 4.7, we obtain the convergence in L1 (£2; 8~a-) and mLi(Sl;8a+). If q € (0, 1), due to Lemma 4.6, we obtain the convergence in L1 (£2; S~a~). □ We next consider the case q = 1. Lemma 4.8 Let k > 0 and u be a positive solution of — L^u = ku in Q.. (4.28) Then u satisfies the Harnack inequality; i.e. for every a € (0, 1) and x € £2, sup u < C24 inf u (4.29) B(x,aS(x)) B(x,aS(x)) where C24 = C24(N, [i, q, £2). Proof Equation (4.28) can be written as follows -Ah=^ + kJm infi. (4.30) Take arbitrarily a e (0, 1) and xo e £2. Put d := ^-S(xo) and M := maxB(X0,d) u. Put yo:=d~lxo, S2d:=d~lS2, 8d(y) := dist (y, d£2d) with y € £2d. We define vd(y):=M-lu(dy), Vy € nd. Clearly, max£(-y0ji) vd = 1 and due to (4.30) we deduce that is a solution of -Avd = Vvd in£2d. (4.31) Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 89 44 Page 20 of 28 P.-T. Nguyen where SdiyV One can find a positive constant C25 such that V(y) < C2s8d(y)~2 for every y e £2d. Notice that B(yo, 1) C £2d and for every y e B(yo, 1), there holds 1 — a sd(y) > t—■ 1 + a Hence 0 < V < C26 in B(yo, 1) where C26 = C26(a, fi). By applying Harnack inequality, we deduce that there is a constant C2i = C2j(a, fi, N, £2) such that sup Vd < C2i inf vd. Thus we obtain (4.29). Proof of Theorem F Put k0 = min{l, ||G^[l]||LI(n)}- Claim 1 For any k € (0, icq) there exists a minimal solution u^ v of(E*). Fix q e (1, q*) such that k < (q |G^[1] f^oo^x)-1 and let g* be the threshold value for (Dev). Putg = llvllgjton) > 0. We first assume that g e (0, g*) and let u_v be the minimal weak solution of (Dv). By Young inequality, we get «v + 1 > Kv + ~ > K(M.V +Gm[1]) in £2. It follows that -L^+GjAU) >k(uv+ G„[l]) in £2. Therefore uv + G^[l] is a super solution of the equation — Ljlu = icu in £2. (4.32) Clearly K^[v] is a subsolution of (4.32). By Theorem 3.2 there is a minimal weak solution uK v of (E*) which satisfies K^[v] < uK v < u_v + GM[1] in £2. Since GM[1] < ci6KM[v], we infer that u^ v satisfies (1.20) and (1.21). If Q > Q* then there exists m > 0 such that g/m e (0, g*). Let « be the minimal weak solution of (EV). Put uKV= muK v_ then by the linearity, we deduce that uK is the minimal weak solution of (E*) with satisfies (1.20) and (1.21). Claim 2 There exists a number k* € (0, A^i ] such that the following holds. (i) If k € (0, k*) then (E*) admits a solution; (ii) If k > k* then (E*) admits no solution. Springer 90 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Semilinear elliptic equations with Hardy potential and... Page 21 of 28 44 Put 23 := {k > 0 : admits a weak solution} and denote k* := sup 23. We shall show that (0, k*) C B. Take k' e B and let uKiv be the minimal solution of (E* ). For any k e (0, k'), ukiv and K^[v] are respectively super and sub solutions of such that I&^lv] < wK'jV. Then by Theorem 3.2 there exists a minimal solution v of satisfying I&^lv] < wK v < uKiv in £2. Hence k e 23. By Lemma 4.8, G^ [w^. y] satisfies local Harnack inequality. Hence, we deduce from Theorem 2.5 that, for v-a.e. z € 3&>, there holds ,. G^u ](x) hm-:-= 0. v](x) Consequently, (1.21) remains valid with uQV replaced by v. Nowletv € (3 £2), k e 23 and denote by wKjV a solution of (E*). Then by Theorem A, — / uKVhjL^dx = k j uKV^dx — / IK^[v]L^f(ix Vf € Jo. Jo. Jo. Taking f = ^j, we obtain / uK^vq>^idx = k / uK^vq>^idx + k^i / K^M^jdx, (4.33) in which implies that k < A^j. Consequently, k* < A^j. We show that A^j ^ 23 by contradiction. Indeed, suppose that there exists v e 9Jt+(3£2) such that the problem — Ljlu = Xjliu in £2, tr*(w) = v (4.34) admits a weak solution u. Take 1 -k)_1 ||v||OT(an). By a similar argument as in the proof of Theorem 4.7, we deduce that, up to a subsequence, {uk,v„} converges to a solution wKjV of (£„) in L1^, Remark (i) If k > 0 small then v satisfies (1.20). Moreover, if v = qSz with q > 0, z € 3 £2 then uK gS satisfies (1.24). (ii) It is notified by the referee that k * = A^ \. This can be obtained by noticing that —L^—k admits the Green function G^jK for any k < fi and then by proving a modification of Proposition 2.4 for G^jK (see [24] for the existence of the Green function G^jK). The weaker statement k* < A^j in the present paper is essentially in order to simplify the proofs and to streamline the exposition. 4.2 Supercritical case Proof of Theorem G This theorem is a consequence of a more general result established in [8]. We present below a simple proof for the special case treated here. Suppose by contradiction that for some q > 0 and z e 3£2 there exists a positive weak solution u of (Desz). Then by Theorem A, u e Lq{Q.\ <5"+) and u > qK^-, z). This, along with (2.3), implies Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 91 44 Page 22 of 28 P.-T. Nguyen / S(x)a+u(x)qdx > J S'xf+K^'x^fdx f S(x)a+i9+1)\x -y\(2a--N)qdx >[ S(x)a+(q+1)\x-y\(2a--N)qdx. > c30 Fix ro > 0 such that x:\x\_\\x-y\\Ax^:&{x)>_\\x-y\ Then f 8(x)a+u(x)qdx >c'30 f \x-y\a+-(N-l-a-)qdx. (4.35) Jo. Je Since q > q*, the integral on the right hand-side of (4.35) is divergent, which in turn implies that u £ Lq (S2; <5"+). Thus we get a contradiction. □ Remark Interesting removability result in the supercritical case can be found in [8]. 5 More general source In this section, we assume that (g o u)(x) = S(x)Y g(u(x)) where y > — 1 — a+ and g : R+ —» R+ is nondecreasing and continuous. Theorems H and I are obtained by using the method in [12]. 5.1 Subcriticality Let {gn} be a sequence of C1 nonnegative functions defined on R+ such that gn(O) = g(0), gn < gn+1 < g, supgn = n and lim \\gn - g\\L°° (R } = 0. (5.1) Put y '■= min{a+ + y, —aJ\ > —1. (5.2) In preparation for proving Theorem H, we establish the following lemma. Lemma 5.1 Let v e 9Jt+(9£2) with II v||jxrt(as2") = 1 and {gn} C C^R-i-) be a sequence satisfying (5.1). Assume (1.28) and (1.29) are satisfied. Then there exist X,0q > 0 and q0 > 0 depending on Ao, A\, N, (i, y and q\ such that for every 0 € (0, 0q) and q € (0, qq) the following problem -Lllv = sygn(v + QKll[v]) inQ, tr»=0 (5.3) admits a positive weak solution vn € (£2; Sa++V) Pi Lqi (£2; 5~a-) satisfying llVnhlh^^y) + llVnhq>(^sf)-L (5A) Springer 92 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Semilinear elliptic equations with Hardy potential and... Page 23 of 28 44 Proof We shall use Schauder fixed point theorem to show the existence of a positive weak solution of (5.3). For n e N, define the operator S„ by Sn(v) := G„[8ygn(v + qK„[v])] Wv e L*.(£2). (5.5) Set itfi(«):= \\v\\ q* + VveLqJ(n;Sa++y), M2(v)-=\\v\\LiHa;sn VueL"(S2;5'?), (5.6) M(d) := Mi(v) +M2(v) Vd e L^f (£2; <5a++K) n L?1 (£2; 8?). Step 1: To estimate L1^; <5"++>0-norm of g„(u + gK^fv]) for u e L^(£2; 8a++Y) n For A > 0, set AA := {x e £2 : v + gl^[v] > A} and a(k) := fA 8a++Ydx. We write \\g„(v + eKAt[v])||Ll(n.3„++y) = / g„(u + eKft[v])50'++,' 0, a(s) < \\v + QK,j,[v]\\qh s~qy _ * < c3i(Mi(v) + c32Q)qyg„(l) +c3i(Mi(v) +c32Q)qy J s qydg„(s) < C3l(Mi(u) + C32Q)qy£j 'g„(.£j) + C3iq*(Mi(.v) + c32Q)qy I s 1 qygn(s)ds. By virtue of (5.8), letting j —» oo yields /DO S'1^ gn(s)ds < c33A0Mi(v)qy + c33A0Qqy (5.11) where c33 = c33(N, fi, y, £2). To handle the remaining term//, without lost of generality, we assume q\ e (1, NN_^la )■ Since g satisfies condition (1.29) and g„ < g, it follows that g„ satisfies this condition too. Hence II 1 and q\ > 1, there exist go > 0 and 9$ > 0 such that for any g e (0, go) and 9 € (0, #0) the equation C37A0Xqy + c38A1A'71 + c409 + C39de = X admits a largest root X > 0. Therefore, M(v) < X =^ M(S„(u)) < X. (5.18) Step 3: We apply Schauder fixed point theorem to our setting. Set 0 :={(/> € L\{ST) : M((/>) < X}. Clearly, O is a convex subset of Lx(^2). We shall show that O is a closed subset of l}(^2). Indeed, let {4>m} be a sequence in 0 converging to cp in L1^). Obviously, (/> > 0. We can extract a subsequence, still denoted by { a.e. in £2. Consequently, by Fatou's lemma, M,((/>) < liminfm^ooM,^,,,) for = 1, 2. It follows that M((/>) < X. So e 0 and therefore 0 is a closed subset of L1 (£2). In light of (5.13) and (5.18), S„ is well-defined on O and S„(0) C O. We observe that S„ is continuous. Indeed, if (pm —» (f> as m —» 00 in L1 (£2) then g„ (4>m + Q^Avl) -+ g„( + eKM[v]) as m 00 in L\S2; 8a++Y). By (2.4), S„(m). Hence {A^rm} is uniformly bounded in LP(G) for every compact subset G C £2. Therefore {^m} is uniformly bounded in Wl'p(G). Consequently, there exists a subsequence, still denoted by {^m}, and a function ^ such that iffm —» ^ a.e. in £2. By dominated convergence theorem, \jfm —» \jr in L1 (£2). Thus S„ is compact. By Schauder fixed point theorem there is a function vn e L}+(£2) such that S„(i>„) = i>„ and M(vn) < X where X is independent of n. Due to Proposition 2.4, tr *(vn) = 0 and v„ is a nonnegative solution of (5.3). Moreover, there holds - f vnL^dx = f 8*gn(vn + QKpMKdx Vf eX(fi). (5.19) Jo. Jo. □ Proof of Theorem H Let 9 e (0, #0) and g e (0, go)- For each n, set w„ = v„ + glK^fv] where v„ is the solution constructed in Lemma 5.1. Then tr *(w„) = gv and - / u„L^dx = j 8vgn(u„)i;dx-g [ K^vJL^dx V£ e X(£2). (5.20) Since {i>„} C 0, the sequence {gn(vn + glK^fv])} is uniformly bounded in L1^; 8a++Y) and the sequence {^vn} is uniformly bounded in Lqi (G) for every compact subset G C £2. As a consequence, {Ai>„} is uniformly bounded in L1^). By regularity result for elliptic Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 95 44 Page 26 of 28 P.-T. Nguyen equations, there exists a subsequence, still denoted by {v„}, and a function v such that v„ —» v a.e. in £2. Therefore u„ —» u a.e. in £2 with u = v + gK^v] and g„(u„) —» g{u) a.e. in £2. We show that w„ —» w in L1 (£2; 8~a-). Since {vn} is uniformly bounded in Lqi (£2; 8y), by (2.5), we derive that {u„} is uniformly bounded in Lqi (£2; 8~a-). Due to Holder inequality, («„} is uniformly integrable with respect to 8~a- dx. We invoke Vitali's convergence theorem to derive that un —» u in L1^; <5~a~). We next prove that g„(un) g(u) in L1 (£2; Sa++y). For X > 0 and n e N set Bnx := ji 6 : u„ > 1} and fc„(A) := JB ^ <5a++)/(ix. For any Borel set E C £2, f gn{un)8a++ydx = f gn{un)8a++ydx + f gn{un)8a++y'dx Je JEnBn^ JEr\Bcn k < [ g„(un)8a++Ydx + 0k f 8a++Ydx JBn,k Je 0, there exists X > 0 such that the right hand-side of (5.22) is smaller than |. Fix such X and put rj = j^-- Then, by (5.21), f 8{xf++Ydx < r, /" g„(M„)5(x)^ +>^x Therefore the sequence lg„(u„)} is uniformly integrable with respect to Sa++Ydx. Due to Vitali convergence theorem, we deduce that g„{u„) g(u) in Lr(£2; Sa++y). Finally, by sending n —» oo in each term of (5.20) we obtain - / uL^dx = / SYg(u)i;dx-Q / K^L^dx Vf € X(fi). (5.23) JO. JO. JO. By Theorem A, u is a nonnegative weak solution of (1.27). □ 5.2 Sublinearity In this subsection we deal with the case where g is sublinear. Lemma 5.2 Let v e 9Jt+(3£2) with ||v||jrj|(3S2) = 1 flw<^ {.?«} <^ C,1(M+) be a sequence satisfying (5.1). Assume (1.30) is satisfied. Then for every q > 0 problem (5.3) admits a nonnegative solution vn satisfying \M\LHa;S9)<^ (5-24) where y is as in (5.2) and X depends on A2, q2, N, [i and y. Springer 96 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM Semilinear elliptic equations with Hardy potential and... Page 27 of 28 44 Proof The proof is similar to that of Lemma 5.1, also based on Schauder fixed point theorem. So we point out only the main modifications. Let S„ be the operator defined in (5.5). Fix « 6 (l, NNiIa_ ) and put N(v) := \\v\\Lq3(n.sf) Vd €L«(£2;a>>). Combining (2.4), (2.5) and (1.30) leads to N(S„(v)) 0 then N($n(v)) < c44a2kq2 + c43(qq2 + 9). Consider the following algebraic equation c44a2kq2 + c43 (gq2 +9)=k. (5.25) If q2 < 1 then for any q > 0 (5.25) admits a unique positive root k. If q2 = 1 then for a2 small such that C44 A2 < 1 and q > 0 Eq. (5.25) admits a unique positive root k. Therefore, N(v) < k =^ JV(S„(i;)) < X. (5.26) By proceeding as in the proof of Lemma 5.1, one can prove that S„ is a continuous, compact operator from the closed, convex set O := {v € L+(£2) : N(v) < k} into itself. Thus by appealing to Schauder fixed point theorem, we see that there exists a function vn e Ll+ (£2) such that S„ (vn) = vn and N(vn) < k with k being independent of n. By Proposition 2.4, tr *(v„) = 0 and v„ is a nonnegative solution of (5.3). Moreover (5.19) holds. □ Proof of Theorem I Let v„ be the solution of (5.3) constructed in Lemma 5.2. Put u„ = v„ + qK^Iv] then u„ satisfies (5.20). By a similar argument as in the proof of Theorem H, there exists a subsequence, still denoted by {u„} and a function u such that u„ —» u a.e. in S2. Since {v„} C O, it follows that {v„} is uniformly bounded in Lq3(S2; 8~a-), so is {w„}. By Holder inequality, {w„} is uniformly integrable in L1^; 8~a-). Due to (1.30), {gn(un)} is uniformly integrable in L1^; 8a++y). Vitali convergence theorem implies that un —» u in Ll(£l; 8~a~) and gn(un) g(u) in L1^; Sa++y). Letting n —» 00 in (5.20), we conclude that u is a positive solution of (1.27). □ Acknowledgements This research was supported by Fondecyt Grant 3160207. The author is grateful to Dr. Q. H. Nguyen for many useful discussions. The author would like to thank the anonymous referee for a careful reading of the manuscript and helpful comments. Springer 3. EQUATIONS WITH A HARDY POTENTIAL AND A SOURCE TERM 97 44 Page 28 of 28 P.-T. Nguyen References 1. Ancona, A.: Theorié du potentiel sur les graphes et les variétés. In Ecole ďété de Probabilités de Saint-Flour XVIII-1988. Springer Lecture Notes in Mathematics, vol. 1427, pp. 1-112 (1990) 2. Ancona, A.: Negatively curved manifolds, elliptic operators and the Martin boundary. Ann. Math. (2) 125, 495-536 (1987) 3. Ancona, A., Marcus, M.: Positive solutions of a class of semilinear equations with absorption and schrodinger equations. J. Math. 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Pisa. CI. Sci. 29, 581-604 (2000) 23. Marcus, M., Veron, L.: Nonlinear second order elliptic equations involving measures. In: De Gruyter Series in Nonlinear Analysis and Applications, vol. 21. De Gruyter, Berlin (2014) 24. Pinchover, Y., Tintarev, K.: A ground state alternative for singular Schrodinger operators. J. Funct. Anal. 230, 65-77 (2006) 25. Veron, L., Yarur, C: Boundary value problems with measures for elliptic equations with singular potentials. J. Funct. Anal. 262, 733-772 (2012) Springer CHAPTER 4 Semilinear elliptic equations and systems with Hardy potentials This chapter, which is based on a collaboration with Gkikas [78], is a continuation of our study on semilinear equations with a Hardy potential. We offer a unified approach and go further in the analysis of the boundary value problems with both interior and boundary measure data. We also extend several existence results for semilinear equations to systems. 99 100 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS ELSEVIER Available online at www.sciencedirect.com ScienceDirect J. Differential Equations 266 (2019) 833-875 Journal of Differential Equations www.elsevier.com/locate/jde On the existence of weak solutions of semilinear elliptic equations and systems with Hardy potentials Konstantinos T. Gkikasa, Phuoc-Tai Nguyenb* a Centro de Modelamiento Matemdtico (UMI2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile b Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic Received 4 May 2018 Available online 30 July 2018 Abstract Let Q, C RN (N > 3) be a bounded C2 domain and S(x) = dist(x, dQ). Put LM = A + jfr with fi > 0. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to —LfJlu=up + T in Q, u = v on dQ, where /x > 0, p > 0, r and v are measures on Q and dQ respectively. We then establish existence results for the system — Lfj,u = e vp + x mQ,, — LfJlv = €up + x in Q, u = v, v = v on 3Q, where e = ±1, p > 0, p > 0, x and f are measures on Q, v and v are measures on dQ. We also deal with elliptic systems where the nonlinearities are more general. ©2018 Elsevier Inc. All rights reserved. MSC: 35J10; 35J57; 35J61; 35J75; 35R05 Corresponding author. E-mail addresses: kgkikas@dim.uchile.cl (K.T. Gkikas), ptnguyen@math.muni.cz, nguyenphuoctai.hcmup@gmail.com (R-T. Nguyen). https://doi.Org/10.1016/j.jde.2018.07.060 0022-0396/© 2018 Elsevier Inc. All rights reserved. 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 101 834 K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 Keywords: Hardy potential; Semilinear equations; Elliptic systems; Boundary trace Contents 1. Introduction ........................................................834 2. Preliminaries........................................................843 2.1. Green kernel and Martin kernel ......................................843 2.2. Boundary trace..................................................846 3. The scalar problem....................................................850 3.1. Concavity properties and Green properties...............................850 3.2. New Green properties.............................................855 3.3. Capacities and existence results ......................................856 3.4. Boundary value problem...........................................858 4. Elliptic systems: the power case...........................................860 5. General nonlinearities..................................................864 5.1. Absorption case.................................................864 5.2. Source case: subcriticality..........................................867 5.3. Source case: sublinearity...........................................873 5.4. Source case: subcriticality and sublinearity...............................874 References..............................................................875 1. Introduction Let Q C RN (N > 3) be a bounded C2 domain, S(x) = dist (x, dtt) and g e C(M). Put := A + ||. In the present paper we study semilinear problems with Hardy potential of the form L/j,u — g(u) + x infi, (1.1) where fi > 0, r is a Radon measure on £2. The boundary value problem with measures for (1.1) without Hardy potential and with power absorption nonlinearity, i.e. \i — 0, r — 0, g(u) — — \u\p~1u, p > 1, was well understood in the literature, starting with a work by Gmira and Veron [10]. It was proved that there is the critical exponent p* :— -^jy in the sense that if p € (1, p*) then there is a unique weak solution for every finite measure v on £2, while if p e [p*, oo) there exists no solution with a boundary isolated singularity. Marcus and Veron [15,16] studied this problem by introducing a notion of boundary trace, providing a complete description of isolated singularities in the subcritical case, i.e. 1 < p < p*, and giving a removability result in the supercritical case, i.e. p> p*. The solvability for boundary value problem for (1.1) without Hardy potential and with power source term, namely fi — 0, r — 0, g(u) — up, p > 1, was studied by Bidaut-Veron and Vivier [4] in connection with sharp estimates of the Green operator and the Poisson operator associated to (—A) in n. They proved that, in the subcritical case 1 < p < p*, the problem admits a solution if and only if the total mass of the boundary datum v is sufficiently small. Afterwards, Bidaut-Veron and Yarur [6] reconsidered this type of problem in a more general setting and provided a necessary and sufficient condition for the existence of solutions. Recently, Bidaut-Veron 102 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 835 et al. [5] provided new criteria for the existence of solutions with p > 1 in terms of the capacity associated to the Besov spaces. Let 4> > 0 in !T2 and p > 1, we denote by LP(Q,\ <))) the space of all function w on Q satisfying fa \v\p ) the space of Radon measures r on £2 satisfying f^(j)d\T\ < oo and by 9Jl+(£2; <))) the nonnegative cone of 91?(!T2; (/>)■ When 0 in Q in the sense of distributions, the first eigenvalue of in £2 is positive, i.e. fQ(\Vcp\2-£cp2)dx V:= inf —-?—Tl->0. (1.3) For /I € (0, |], denote by a the following fundamental exponent 1 2(l+v/l-4/i). (1.4) Notice that j < a < 1. The eigenfunction cp^ associated to with the normalization f^cp^/S^dx — 1 satisfies c~1Sa < cp^ < cSa for some constant c > 0 (see [7]). In relation to Hardy constant, Bandle et al. [3] classified large solutions of the linear equation —L^u — 0 in £2, (1.5) and of the associated nonlinear equation with power absorption -L^u + u" = 0 in £2. (1.6) In [14], Marcus and P.-T. Nguyen studied boundary value problem for (1.5) and (1.6) with fi e (0, Ch) in measure framework by introducing a notion of normalized boundary trace which is defined as follows: Definition 1.1. A function u € LJoc(£2) possesses a normalized boundary trace if there exists a measure v e 911(3£2) such that Hin^""1 j \u-K^[v]\dS = 0. (1.7) The normalized boundary trace is denoted by tr*(w). 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 103 836 K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 The restriction p e (0, Ch) in [14] is due to the fact that in this case is weakly coercive in Hq (!T2) and consequently by a result of Ancona [2, Remark p. 523] there is a (1 — 1) correspondence between 9Jt+(3!T2) and the class of positive harmonic functions, namely any positive harmonic function u can be written in a unique way under the form u — KM[v] for some v eWi+(dtt). The notion of normalized boundary trace was proved [14] to be an appropriate generalization of the classical boundary trace to the setting of Hardy potentials, giving a characterization of moderate solutions of (1.6). In addition, it was showed in [14] that there exists the critical exponent N + a p^=nT^2 (L8) such that if p e (1, p^) then there exists a unique solution of (1.6) with tr*(u) — v for every finite measure v on 3!T2, while if p > p^ there is no solution of (1.6) with an isolated boundary singularity. Marcus and Moroz [13] then extended the notion of normalized boundary trace to the case p < \ and employed it to investigate (1.6). When p — ^,L/x is no longer weakly coercive and hence Ancona's result cannot be applied. However, Gkikas and Veron [9] observed that if the first eigenvalue of—Li is positive then the kernel Ki (•, y) with pole at y € dQ is unique up 4 4 to a multiplication and any positive L i harmonic function u admits such a representation. Based on that observation, they considered the boundary value problem with measures for (1.6), fully classifying isolated singularities in the subcritical case p e (1, p^) and providing removability result in the supercritical case p > p^. A main ingredient in [9] is the notion of boundary trace which is defined in a dynamic way and is recalled below. Let Dgfi and xq e D. If h e C(3D) then the following problem I—LnU =0 in D, (1-9) u — h on 3D, admits a unique solution which allows to define the L^-harmonic measure ojXp on 3D by «(*o) = j h{y)dof£{y). (1.10) 3D A sequence of domains {!T2„} is called a smooth exhaustion of £2 if 3!T2„ € C , !T2„ C £2n+i, U„!T2„ = £2 and T-LN~l {dQ,n) —>■ T-LN~l {dQ,). For each n, let cox£ be the L^"-harmonic measure on 3!T2„. Definition 1.2. A function u possesses a boundary trace if there exists a measure v e 9Jt(3!T2) such that for any smooth exhaustion {!T2„} of £2, lim / tudü)x° = tdv VteC(Q.). (1.11) n—>oo " The boundary trace of u is denoted by tr (u). 104 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 837 It is worthy mentioning that in Definition 1.2, /i is allowed to belong to the range (0, |] provided X^ > 0. In parallel, semilinear equations with Hardy potential and source term -Lllu = up in £2 (1.12) were treated by Bidaut-Veron et al. [5] and by P.-T. Nguyen [18] and a fairly complete description of the profile of solutions to (1.12) was obtained in subcritical case p < (see [18]) and in supercritical case p > p^ (see [5]). Our first contribution is to show that the notion of normalized boundary trace given in Definition 1.1 is equivalent to that in Definition 1.2 by examining tr(G^[r]) — tr*(G^[r]) and tr (K^ [v]) — tr * (K^ [v]). This enables to establish important results for the boundary value problem for linear equations (see Proposition 2.13) which in turn forms a basic to study the boundary value problem for I-LuU = g(u) + r in £2, (1.13) tr(w) — v. When dealing with (1.13), one encounters the following difficulties. The first one is due to the presence of the Hardy potential in the linear part of the equations. More precisely, since the singularity of the potential at the boundary is too strong, some important tools such as Hopf's lemma, the classical notion of boundary trace, are invalid, and therefore the system cannot be handled via classical elliptic PDEs methods. The second one comes from the interplay between the nonlinearity, the Hardy potential and measure data. The interaction between the difficulties generates an intricate dynamics both in £2 and near 3 £2 and leads to disclose new type of results. Convention. Throughout the paper, unless otherwise stated, we assume that /x e (0, ^] and the first eigenvalue X^ of —L^ in Q is positive. We emphasize that if pu € (0, Ch) then X^ > 0. Definition 1.3. (i) The space of test functions is defined as XM(£2):=R €H/oe(£2):5-«f eH^^S2"), S~aL^ e L°°(S2)}. (1.14) (ii) Let (r, v) € Tt(Q, Sa) x Tt(dQ). We say that u is a weak solution of (1.13) if u € Ll(Q; Sa), g(u) € Ll(Q; Sa) and - juL^i;dx = j g(u)^dx + j t,dx-jK^[v]L^dx Vf€XM(£2). (1.15) Q Q Q Q Main properties of solutions of (1.13) are established in the following proposition. Proposition A. Let t € 9Jt(£2; Sa) and v € 9Jt(3£2). The following statements are equivalent. (i) u is a weak solution o/(1.13). (ii) g(u) eLl(Q;Sa) and « = G^[g(«)] + G^[r]+K^[v]. (1.16) (iii) u € LJoc(£2), g(u) € LJoc(£2), u is a distributional solution of (1.1) and tr (u) — v. 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 105 838 K.T. Gkikas, P.-T. Nguyen /J. Differential Equations 266 (2019) 833-875 This allows to establish necessary and sufficient conditions for the existence of a weak solution of I—Lflu = up+crr infi, tr(u) — qv. Theorem B. Let t € Tt+(Q; 8a), v e Wi+(dtt) and p > 0. (i) Assume 0 < p < p^. Then there exists a constant C > 0 such that (1.17) (1.18) (1.19) G4KA[vf] 0 such that G^[G^[t]"]v] 0 and q > 0 small enough if p > 1, /or any er > 0 a« 0 ifO < p < I. (iv) If p > 1 a« 0 .ywc/z that for any weak solution u of (1.17) there holds t[err] +Kß[Qv] v] + Sa) a.e. in Q,. (1.21) In order to study (1.17) in the supercritical case, i.e. p > p^, we make use of the capacities introduced in [5] which is recalled below. For 0 < 6 < fi < N, set Ne,ß(x, y) := 1 \x - y\N~ß max{|x - y\, S(x), S(y)}9 Ne,ß[r](x) := j Ne,ß(x, y)dx Vr € Wi+(£2) V(x, yjefixSl,!^, (1.22) (1.23) For a > — 1, 0 < ö < jß < iV and s > 1, define Cap^e s by CaP%^^ :=inf j 8a>0, Ne,^[5fl0]>X£ for any Borel set E c£2. For 6 e (0, N — I) and s > 0, let Capjj^ be the capacity defined in [5 (1.24) Definition 1.1]. Notice that if 0s > N - 1 then Cap^({z}) > 0 for every z e d£2 106 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 839 Theorem C.Let x e 9Jl+(£2; Sa) and v e 9Jl+(d£2). Assume p > 1. Then the following statements are equivalent. (i) There exists C > 0 such that the following inequalities hold VBorelEcti, (1.25) VBorelFcdtt. (1.26) (ii) There exists a positive constant C such that (1.18) and (1.19) hold. (iii) Problem (1.17) has a positive weak solution for a > 0 and q > 0 small enough. Remark. When x — 0, Theorem C covers Theorem B (i), (iii) due to the fact that Cap^ a+1 ,({z}) > c > 0 for every z € 3£2 if 1 < p < p/x. Also if 1 < p < p/x then (see Lemma 3.10) which implies the statements (ii) and (iii) in Theorem B. The next goal of the present paper is the study of weak solutions of semilinear elliptic system involving Hardy potential -Llxu = g(v) + r in £2, —L^v — giu) + x in £2, (1.27) tr(u) — v, tr(i;) = v where x, r € SDC(£2; Sa), v,ve SDC(3£2), g, g € C(M). Definition 1.4. A pair (u, v) is called a weak solution of (1.27) if u € Ll(Q; Sa), veLl(Q; Sa), g(u) € Z/(£2; 5"), € L1^; 5") and - j uL^t; dx - j vL^ dx A counterpart of Proposition A in the case of systems is the following: Proposition D. Let x, x € 9Jt(£2; Sa) and v, v € 9Jt(3£2). Then the following statements are equivalent. J8«dx q'-=p——r- (1.31) We first treat the system — Lßu — vp + ax inSl, — Lßv = üp+äx inS2, tr(u) = Qv, ü(v) — qv, (1.32) where p > 0, p > 0, x, r e Wl(S2; Sa) and v, v e Wl(dS2). The next theorem provides a sufficient condition for the existence of solutions of (1.32). Theorem E. Let p > 0, p > 0, x, x e Wl+(S2; Sa) and v,ve Wl+(dS2). Assume pp^l,q< p^, G^[r] + K^[v + v] € Lp(Sl, Sa). Then system (1.32) admits a weak solution (u, v) for a > 0 and cr > 0 small if pp > 1, for any a > 0 and cr > 0 if pp < 1. Moreover u « Gß[(Gu.[a>\ + Ku.[v])P] + GM[r] + Kß[v] where the similarity constants depend on N, p, p, ß, Sl,a,ä,x,x and (1.33) (1.34) A new criterion for the existence of (1.32), expressed in terms of the capacities Cap^ s and Cap^, is stated in the following result. 108 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 841 Theorem F. Let p > 1, p > 1, r, r € SOT1-(£2; Sa) and v, v e 9Jl+(d£2). Assume there exists C > 0 such that max Sadi < Cmin{Cap^,(F), Cap^, (£)}, VF C £2, (1.35) maxMF), v(F)} 0, cr>0, q > 0, q > 0 small enough. There exists C > 0 such that 5M[or] + KM[ev] < w < C(GM[ar + cfr] -^[cfr] + KM[§v] < i; < C(Gß[ar + cfr] (1.37) Finally, we deal with elliptic systems with more general nonlinearities — L^u — e g{v) + err in Q, — L^v — e g(u) + err in £2, tr(M) = g>v, tr(i;) = £v on 3£2 (1.38) where g and g are nondecreasing, continuous functions in M, e — ±1, cr > 0, a > 0, g> > 0, £ > 0. We shall treat successively the cases e — — 1 and e — I. For any function /, define oo A/ := Js-l-^\f(s)- f(-s)\ds (1.39) with /jm defined in (1.8). Theorem G.Let e — — 1 a«if o,o,q,q be positive numbers, r, r € SD?(£2; 5") a« 1, a\ > 0, b\ > 0 .ywc/z f/zaf IgWI <«iM?1 Vsef-1,1], \8(s)\ <«iM?1 +bi Vse[-l,l]. (1.40) (1.41) Then (1.38) admits a weak solution for b\, a, cr, q, q small enough. 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 109 842 K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 II. SUBLINEARITY. Assume that there exist q\ > 1, q2 € (0, 1], a2 > 0 and b2 > 0 .ywc/z f/zaf !M|v|] + G^[|r|] € Lqi (Q; Sa~l) and \g{s)\ 0 z's .yma// f/ze« (1.38) admits a weak solution for any a > 0, a > 0, g > 0, g > 0. (b) If q\q2 < 1 f/ze« (1.38) admits a weak solution for any a > 0, a > 0, q > 0, q > 0. III. SUBCRITICALITY AND SUBLINEARITY. Assume that Ag < oo. /« addition, assume that there exist a\ > 0, a2 > 0, fei > 0, &2 > 0, q\ € (1, /j^), 92 € (0, 1], such that (1.40) and (1.43) hold. (a) If q\q2 > 1 then (1.38) admits a weak solution for b\, b2, a, a, q, q small enough. (b) If q2Pfi — 1 and a2 is mall enough then (1.38) admits a weak solution for any a > 0, cr > 0, q > 0, q > 0. (c) If q2pfl < 1 then (1.38) admits a weak solution for every for any a > 0, a > 0, q > 0, q > 0. Remark about elliptic equations and systems with weights. We emphasize that Theorems B and C can be extended to the case of equations with weights of the form -Lilu = d>vup +ox in £2, (1.44) and Theorems E-H can be extended to the case of systems with weights of the form I— LnU — e Sr g(v) + ax inQ,, (1.45) — L^v — € Syg(u) + ox mQ., by using similar arguments. However, in order to avoid the complication of the proofs, we state and prove the results without weights. The paper is organized as follows. In Section 2 we investigate properties of the boundary trace defined in Definition 1.2 and prove Propositions A and D. Theorems B and C are proved in Section 3 due to estimates on Green kernel, Martin kernel and the capacities Capj^"^, and Cap3S „,, . In Section 4 sufficient conditions for the existence of weak solutions to elliptic systems with power source terms (1.32) (Theorems E and F) are obtained by combining the method in [6] and the capacity approach. Finally, in Section 5, we establish existence results for elliptic systems with more general nonlinearities (Theorems G and H) due to Schauder fixed point theorem. Notations. Throughout this paper, C, c, c',... denotes positive constants which may vary from one appearance to another. The notation A B means c~l B < A < cB for some constant c > 1 depending on some structural constant. 110 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 843 Acknowledgments. The first author has been supported by FONDECYT Grant 3140567 and by Millenium Nucleus CAPDE NC130017. The second author was supported by FONDECYT Grant 3160207. The authors would like to thank the referee for a careful reading of the manuscript and useful comments. 2. Preliminaries 2.1. Green kernel and Martin kernel Denote by L^(£2; r), 1 < p < oo, r € DJi+(£2), the weak Lp space (or Marcinkiewicz space) (see [17]). When r — 8sdx, for simplicity, we use the notation L^(£2; 8s). Notice that, for every s > -1, LP(Q;8s)cLr(Q;8s) Wre[l,p). (2.1) Moreover for any u € L^(Q,\ 8s) (y > -1), \\u\>X) Let G^j and K^f be respectively the Green kernel and Martin kernel of — in Q, (see [14,9]) for more details). We recall that G%(x,y)^mm{\x-y\2-N,8(x)a8(y)a\x-y\2-N-2a} Vx,ye£2,x^y, (2.3) K^(x,y)^8(x)a\x-y\2~N~2a Vxe£2,yed£2. (2.4) Finally, we denote by GM and KM be the corresponding Green operator and Martin operator (see [14,9]), namely GjArKx)^ j GlAx,y)dr(y), Vr eWi(£2), (2.5) K^[v](x)^ j K^(x,z)dv(z), Vv€9rt(9£2). (2.6) Let us recall a result from [4] which will be useful in the sequel. Proposition 2.1. ([4, Lemma 2.4]) Let co be a nonnegative bounded Radon measure in D — Q or dQ and r\ € C(Q) be a positive weight function. Let H be a continuous nonnegative function on ((ij)eQxD: x ^ y}. For any A. > 0 we set Ax(y) '■— {x € Q \ {y}: H(x,y)>\} and mx(y)'■— J r\(x)dx. Ax(y) 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 111 844 K.T. Gkikas, P.-T. Nguyen /J. Differential Equations 266 (2019) 833-875 Suppose that there exist C > 0 and k > 1 such that mx (y) < CX k for every X > 0. Then the operator M[co](x):= j H(x,y)dco(y) D belongs to LW{Q,\ rj) and i™ilL*(n:,)<(l- Ck k-1 )co{D). By combining (2.3), (2.4) and the above Lemma we have the following result. Lemma 2.2. Let y € ^— N"£t_2i F-l)- Then there exists C — C(N, p, y, Q.) > 0 such that sup JV+y LS+a 2 (Q-SV) < c. (2.7) Proof. Let £ € Q,. We will apply Proposition 2.1 with D — Q,, r) — 8y with y > —l,co — <5"<%, where is the Dirac measure concentrated at £, and H{x,y)-. Then Gp(x, y) S(yr Gß(x, y) 8(y)« S(y)adS^y) = Gß(x,^. From (2.3), there exists C — C(N, p, Q,) such that, for every (x, y) e Q, x !T2, x ^ y, 2-N-a Gß(x,y)CX\x - y\N+2a~2 (2.8) (2.9) (2.10) (2.11) (2.12) 112 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 845 We consider two cases: y > 0 and — 1 < y < 0. Case 1: y > 0. Due to (2.11) and (2.12) we have (y j\x-y\2~NJ dx (£2) =8(%)a,by Proposition 2.1, we get ||GM(-,£)|| J±±y — N"^_2- By arguing similarly as in Case 1, we get (2.7). □ Lemma 2.3. Let y > —1. Then there exists C — C(N, /i, y, Q) > 0 such that SUp H^G, N+y < C. Proof. Let £ € 3!T2. We will apply Proposition 2.1 with D — 3£2, r\ — d>v with y > —1 and a> = <5^. The rest of the proof can be proceeded as in the proof of Lemma 2.2 and we omit it. □ In view of (2.1), Lemma 2.2 and Lemma 2.3, one can obtain easily the following proposition (see also [14,18]). Proposition 2.4. (i) Let y € (— N"2a-2> W^i^- ^nen there exists a constant c — c(N, jjl, y, £2) such that G/*[r] N+y — 1. 77ze« there exists a constant c — c(N, /i, y, Q) such that ||K^[v]|| _N±y ■ oo, ensures j UL uuu lim / udúL>„ — 0, namely tr (u) — 0. In the general case, the result follows from the linearity property of the problem. □ The next result shows that the boundary trace of L^ harmonic function can be achieved in a dynamic way. Proposition 2.6. [9, Proposition 2.34] Let xq e £2\ and p e SCÍ(3Í2). Put v(x) := j K^(x, y)dv(y), dQ then for every f € C(Q), lim j tvarný = j i;dv. (2.17) dQ.n dQ. 114 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 847 Also we have the following representation formula for harmonic functions. Proposition 2.7. [9, Theorem 2.33] Let u be a positive L^ harmonic in Q,. Then u € Ll(Q,\ Sa) and there exists a unique Radon measure v on dQ such that j Kß(x, y)dv( u(x) = J Kll(x,y)dv(y). (2.18) dQ In the following proposition, we study the boundary trace of L^ subharmonic functions. Proposition 2.8. Let w be a nonnegative L^ subharmonic function. If w is dominated by an L^ superharmonic function then L^w € $R+(Q; Sa) and w has a boundary trace v € 9Jt(3£2). In addition, if tr (w) — 0 then w — 0. Proof. By proceeding as in the proof of [14, Proposition 2.14] and using Proposition 2.7, we obtain the desired result. □ Proposition 2.9. Let w be a nonnegative L^ subharmonic function. If w has a boundary trace then it is dominated by an L^ harmonic function. Proof. The proof is similar to that of Proposition 2.20 in [14]. For the sake of convenience we give it below. Let {!T2„} be as in the proof of Proposition 2.5 and fix xq e £2i. For any x e !T2, set Unix)— j wdoofc, , 3Q„ then u„ is L^jn harmonic function with boundary trace w. Furthermore, by the maximum principle we have that w < un in !T2„. Let v e 9Jt(3!T2) be such that lim / twdcag = / tdv Vr € C(£2). (2.19) Then dQy, dQ unixo) — j wdci)XQ —> j dv. dQy, dQ We infer from Harnack inequality that {un} is locally uniformly bounded and hence there exists an L^ harmonic function u such that un —>■ u locally uniformly in £2. By Proposition 2.8, there exists a nonnegative measure r € 9Jt+(!T2; Sa) such that w = -Gll[t] + Kll[v]. 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 115 848 K.T. Gkikas, P.-T. Nguyen /J. Differential Equations 266 (2019) 833-875 On the other hand, w = -g£"[r] + u„ -gM[r] + u, locally uniformly in !T2. Thus we can deduce that u — kM[v] and the result follows. □ Proposition 2.10. Let u be a nonnegative L^ superharmonic function. Then there exist v € Wi+(dtt) and x € Tt+(Q; 8a) such that u=Gll[t]+Kll[v]. Proof. Let Q,n and &>^° be as in the proof of Proposition 2.5. Since u is L^ superharmonic function there exists a nonnegative Radon measure in Q, such that —LfjM — x in Q in the sense of distributions. Note that u is the unique solution of I-LnW — x in Q„ (2.20) w — u on dQ„. Therefore u = G%n[r] + K%n[u]. (2.21) Set wn — k^"[w]. Since r > 0, by the above quality, we have 0 < w„(x) < u(x). Thus by the Harnack inequality, wn —>■ w locally uniformly in !T2. Furthermore, w is an L^ harmonic function in !T2 and by Proposition 2.18 there exists v e 9Jt+(9 !T2) such that w^K^v]. (2.22) Now since G^jn t G^ as n —>■ oo, we deduce from (2.21) and (2.22) that u = g£» [r] + k^" [u] gM[r] + k^fv]. Since GIJL(x,y)>c(x,fi,N)S(y)a, we can easily prove that r € 9Jt+(!T2; 8a) which completes the proof. □ The above results enable to study the boundary value problem for the linear equation {—LnU — x in Q,, (2.23) tc(u) — v. 116 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 849 Definition 2.11. Let (r, v) € Tt(Q; Sa) x Wi(dQ). We say that u is a weak solution of (2.23) if u € Ll(Q; Sa) and -j uL^i;dx = j t,dx- j K^[v]L^dx Vf €XM(S2), (2.24) Proposition 2.12. For any (t, v) € 9Jt(£2; 5") x 9Jt(3£2) f/zere exwf* a unique weak solution of (2.23). Moreover « = G^[r]+K4v], (2.25) INIlHJW) - c(llrll2rt(£2;<5«) + IIvllaK(3q))- (2.26) In addition, for any f € X^(£2), f > 0, \u\L^dx< j £ sign(u)dr-j K^[\v\]L^dx, (2.27) a«• (ii). By an argument similar to that of the proof of [18, Theorem B], we deduce that (ii) -<=>• (iii). □ For f3 > 0, put Qß := {x € Q : S(x) < ß}, Dß := {x € Q : S(x) > ß}, := {x € Q : S(x) = ß}. (2.29) 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 117 850 K.T. Gkikas, P.-T. Nguyen /J. Differential Equations 266 (2019) 833-875 Lemma 2.14. There exists (3* > 0 such that for every point x e £2pt, there exists a unique point ax € d£2 such that x — ax — 5(x)nCTj. The mappings x h-> S(x) and x h-> ax belong to C2(£2pt) and Cl(£2pt) respectively. Moreover, \imx^a(x) VS(x) — — nax. Proof of Proposition D. (iii) =>• (ii). Assume (u, v) is a distribution solution of (1.30). Put oj :— g(v) and denote cop :— o)\d^, xf> '■— X\D(, and Xp := u\-£p for f3 e (0,/J*). Consider the boundary value problem —L^w—cop + tp in Dp, w—Xp on ~Ep. This problem admits a unique solution wp (see [9]). Therefore wp — u\d^- We have u \Dp = up = G^ [cop] + G^ [rp] + [Xp] where G^ and are respectively Green kernel and Poisson kernel of — in Dp. It follows that / G/(;y)g(v(y))dy Letting (3 0, we get / Gß(-, y)g(v(y))dy < oo. (2.30) Fix a point xq e £2. Keeping in mind that GM(xo, y) & S(y)a for every y e £2pt, we deduce from (2.30) that g(v) e Ll(£2; Sa). Similarly, one can show that g(u) eL1(£2; Sa). Thanks to Proposition 2.13 (v), we obtain (1.29). (ii) =>• (iii). Assume u and v are functions such that g(u) e Ll(Q.;&a), g(v) e Ll(Q.;&a) and (1.29) holds. By Proposition 2.13 (i) L^K^lv] — L^K^lv] — 0, which implies that (u, v) is a solution of (1.30). On the other hand, since g(u) e Ll(Q.;&a) and g(v) e Ll(Q.;&a), we deduce from Proposition 2.13 (ii) that tr(G/j,[g(u)]) — tr(Gfj,[g(v)]) — 0. Consequently, tr (u) = tr (KM[v]) = v and tr (v) = tr (KM[v]) = v. (iii) =>• (i). Assume (u, v) is a positive solution of (1.30) in the sense of distributions. From the implication (iii) =>• (ii), we deduce that u e Ll(Q; Sa), v e Ll(Q; Sa), g(u) e Ll(Q; Sa) and g(v) eL1^; Sa). Hence, by Proposition 2.13, (1.28) holds for every e XM(£2). (i) =>• (iii). This implication follows straightforward from Proposition 2.13. □ 3. The scalar problem 3.1. Concavity properties and Green properties Here we give some concavity lemmas that will be employed in the sequel. 118 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 851 Proposition 3.1. Let cp € Ll (Q; Sa), cp > 0 and t e Tt+(Q; Sa). Set w :— GfjXq) + r] and ijr — G^[r]. Let

0. Then '(w/f) ip-n > 0 for any « > «o for some «o € N, we have by straightforward calculations . wn \ I wn wn i wn \ i wn -A tfr„0(-i) = (-Atfr„) 0(-p) - + (-Atu„)0'(-p) * 2 fn^'i-1) . IV, V . fn Now note that, since 0' > 0, we have , wn , wn ( ikn wn (-Au)n) (/)'(—) -A^„ - Mtt + M-^r + \t) + 0'(f) > 0 for any t > 1, implies wn wn , wn \ , wn (-Atfr„) - + (-Atu„)0'(-p) V V« ^« fn J fn > (-Atfr„) 0(-p) - + 0'(-p) + 0'(-p) -/^ + + ^» V V« ^« fn fn J fn \ Sz Sz <52 V fn fn fn fn J fn \ Sz Sz fl wn . wn SZ fn fn Thus we have proved -Lß f„{^)L^dx> f '{^) 0 such that (1.19) holds. Proof. First we assume that p > 1. By (2.13) we have that QfJi[x]p e L1^; 8a). We write GjAy, z) G^[r](y) = j GjAy,z)dx(z)^ j 8(z)° Q -8(zfdx(z), thus '/ S(zf (- drl(y)p 0 and x € $R+(Q; Sa). Assume that there exists a positive constant C such that (1.19) holds. Then problem (3.5) admits a weak solution u satisfying G^lerr] < u < CGAax\ a-e- in ^ (3-6) with another constant C > 0, for any a > 0 small enough if p > 1, for any a > 0 if p < 1. Proof. We adapt the idea in the proof of [6, Theorem 3.4]. Put w := AG^Ierr] where A > 0 will be determined later. By (1.19), !Awp + err] < (CApap~l + l)GM[err] in £2. 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 121 854 K.T. Gkikas, P.-T. Nguyen /J. Differential Equations 266 (2019) 833-875 Therefore we deduce that w > Gtl[wp + err] as long as CApap~l + 1 < A. (3.7) If p > 1 then (3.7) holds if we choose A > 1 and then choose a > 0 small enough. If p e (0, 1) then (3.7) holds if we choose a > 0 arbitrary and then choose A > 0 large enough. Next put uq := G^lerr] and un+\ :— G^uf, + err]. It is clear that {un} is increasing and un < w in !T2 for all «. Since (1.19) holds, wp € L1^; 5"). Consequently, by monotone convergence theorem, there exists a function u € LP(Q,; Sa) such that —>■ mp in Z.1 (!T2; 5"). It is easy to see that u is a solution of (3.5) satisfying (3.6). □ Estimate (1.19) is also a necessary condition for the existence of weak solution of (3.5). Proposition 3.5. Let p > 1, a > 0 and x € $R+(Q; Sa). Assume that problem (3.5) admits a weak solution. Then (1.19) holds with C — Proof. We adapt the argument used in the proof of [6, Proposition 3.5]. Assume (3.5) has a solution u e LP(S2; Sa) and assume a — 1. By applying Proposition 3.1 with cp replaced by up and with we get (1.19) with C = □ (l-s1~p)/(p-l) ifs>l, s — 1 if s < 1, Proposition 3.6. Let 0 < p < /j^, a > 0 a«l. Set u\ '.— u GM[r] — Gfl[up], hence u—u \ + GAt[r]. Since m € LP(S2; Sa) (by assumption), it follows that u + x e 9Jl+(£2; Sa), therefore by (2.13), u e LS(S2; Sa), for all 1 < s < p^. Thus there exists kr, > 1 such that mp € Z>(n; 5"). 122 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 855 Let p < s < Pfj,. By Holder inequality we obtain MlW*oS=G/i[MP]WtoS= f [ G^\(y)au(yydy\ -C/(^t^)S5(yrM(^' This, joint with Lemma 2.2, yields j u^yfv&iyfdySC j u(y)^S(y)a j ( k0 k0P,Ma f (Gß(x,y)YKxfdxdy■ oo, by [17, Lemma 2.3.2] we have for 1 < s < p^ un8(y)ady + j \x - y\2-a~N u^&iyf \Q Q / 0. This, together with the inequality GM[1] < CSa, implies (3.8). □ 3.2. New Green properties Lemma 3.7. Let 0 < p < p^, x e 9Jt+(!T2; Sa). Let s be such that max (0, p — pß + l) < s < 1. (3.9) 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 123 856 K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 Then there exists a constant C > 0 such that G/JG/JrF] < CGM[r]s a.e. in £2. (3.10) Proof. First we assume that p > 1. In view of the proof of Lemma 3.3, we have G^[G^[tF](jc) 0, 0 < 0 < fi < N and s > 1, let Ne,(s, ^e,(s and Cap^ s be defined as in (1.22), (1.23) and (1.24) respectively. In this section we recall some results in [5, Section 2]. We recall below the definition of the capacity associated to Ng^ (see [11]). 124 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 857 Definition 3.8. Let a > 0, 0 < 6 < ß < N and s > 1. Define Capj^ s by CapaNeßJE) :=inf j8a>0, N0,ß[8a]>XE for any Borel set E C £2. Clearly we have CapV^(£)=inf -aO-l),AS 0s dy: 0>O, Ne,ß[]>XE for any Borel set E C £2. Furthermore we have by [1, Theorem 2.5.1] i (Cap£M s(£))7 =inf {<«(£): «€OT+(£2), ||NM[«w]||Li,(jj;4fl) < l} , (3.17) for any compact set £ C £2 where s' is the conjugate exponent of s. Using [5, Theorem 2.6], we obtain easily the following result. Proposition 3.9. Let p > 1, a > 0 a« 0 such that the following inequality hold E for any Borel E C Q. 2. There exists a constant C > 0 such that (1.19) holds. 3. Problem (3.5) has a positive weak solution for a > 0 small enough. Proof. First we note that G(x, y) = 8{x)a8{y)aN2a,2{x, y), Vx, y € £2, x # y. Thus the inequality 1, q > 0 and v e 9Jl+(3f2). Then, the following statements are equivalent. 1. There exists C > 0 such that the following inequality holds v{F) 0 such that (1.18) holds. 3. Problem (3.19) has a positive weak solution for q > 0 small enough. Lemma 3.12. Let v € 9Jt+(3£2) and 0 < p < p^. Then there exists a constant C > 0 such that (1.18) /zo/*. 126 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 859 Proof. We first assume that 1 < p < p/x. Let £ € 3!T2; we have 8^ (F) < cCapdQ l+a r(F) for every F C 3!T2 where c is independent of By Proposition 3.11, (1.18) holds with v replaced by 8^ and with the constant C independent of By taking integral over £ € 3!T2, we get (1.18). Next, if p e (0, 1], we choose s > 1 such that 1 < ps < p^. By Young's inequality, 0, q > 0 and v e 9Jl+(d£2). (i) Assume there exists a constant C > 0 such that (1.18) holds. Then problem (3.19) admits a weak solution u satisfying IK^Jgv] < u < CK^^v] a.e. in Q, (3.21) with another constant C > 0, for any q > 0 .yma// enough if p > 1, /or any g > 0 rf p € (0, 1). (ii) Assume p > 1 a« 0 such that for any weak solution u of (3.19) there holds ^[Qv] 0, a > 0, q > 0 and x e Tt+(Q; 8a) and v e Wi+(dtt). 7/(1.18) and (1.19) hold then problem (1.17) admits a weak solution u satisfying (1.20) for a > 0 and q > 0 small enough if p > I, for any a > 0 and q > 0 ifO < p < 1. Furthermore ifO < p < p^ there exists a constant C > 0 such that for any weak solution u of (1.17) estimate (1.21) holds. Proof. We adapt the argument in the proof of [4, Theorem 3.13]. Put v :—u — KM[gv] then v satisfies —Luv — (v + Ku[pv])p + ax in !T2, (3.23) tr(t;) =0. Consider the following problem —LnW — Cr>wp + cp(Ku[pv])p + ax in !T2, (3.24) tr(wj) = 0 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 127 860 K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 where cp := max{l, 2^"1}. Since (1.18) holds, it follows that KM[vF € L\S2; sa). Since (1.19) holds, we infer from Proposition 3.4 that problem (3.24) admits a weak solution w for a > 0 and q > 0 small enough if p > 1, for any a > 0 and g> > 0 if 0 < p < 1. Notice that wj is a supersolution of (3.24), we infer that there is a weak solution v of (3.23) satisfying v < w a.e. in £2. By Proposition 3.4 and (1.18), we get w < cG^K^vf + • (ii) =>• (iii) follow from Proposition 3.11, Proposition 3.9 and Proposition 3.14. We will show that (iii) =>• (ii). Since (1.17) has a weak solution for a > 0 small and q > 0 small, it follows that (3.5) admits a solution for a > 0 small and (3.19) admits a solution for q > 0 small. Due to Proposition 3.11 and Proposition 3.9, we derive (1.19) and (1.18). This completes the proof. □ 4. Elliptic systems: the power case Let /I € (0, |]. In this section, we deal with system (1.32). We recall that p^ is defined in (1.31) and + 1 - P + l 1'-=P^r—, q:=P——• p+1 p+1 Without loss of generality, we can assume that 0 < p < p. Then p\. Put tfj, := p{p- Pfj, + 1)-Notice that if q < then t^ < q < . Lemma 4.1. Let p > 0, p > 0 and x € $R+(Q; sa). Assume q < p^. Then for any t € (max(0, tp), p], there exists a positive constant c — c(N, p, p, /i, t, r) (independent of x if p > I) such that G/JG/JrFF^cG/Jr]'. (4.1) In particular, G^[G^[t]"F 0, p > 0, r, r € 9Jl+(£2; 5") a«if v, v € 9Jl+(3£2). Asswrae that there exist positive functions U € Lp (Q; Sa) and V € LP(Q; Sa) such that U > GM[(V + KM[qv])p] + GM[ar], V > GM [(£/ + KM [qv])p] + GM [or r] r'« £2. 77ze« f/zere exists a weak solution (u, v) of (1.32) such that G^t] + KAQv] v in L1^), «« up in L1 (£2; 5"), u£ up in L1 (£2; 5"). Moreover w < t/ and v < V in £2. By letting « oo in (4.6), we obtain r]+KML5v], (4.7) i^[crr]+K^[Qv]. Thus (w, v) is a weak solution of (1.32) and satisfies (4.5). □ Proof of Theorem E. We first show that the following system has weak a solution — L^w — (w + KfJi[Qv])p + ax in £2, - Lllw = {w + Kll[Qv]y+ax in Q, (4.8) tr(w) =tr(t>) = 0. 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 862 K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 Fix ůi > 0, (i — 1, 2, 3, 4) and set * := G^r + K^vYY + K^vý + ů4r. For ze € (0, 1], put i i i_ a:—Kfů\, a:—KŮ4, q-.— kpůj,, ~q :—k fi>ů^. Then from the assumption, we deduce that * € 9Jt+(!T2; 8"). By Lemma 4.1, Gll[Gll[Gllmpý] 0 will be determined later on. We have (U + K^[qv])P +ax < c ^[(a3KG^m + K^[qv])pY + G^[axý + K^Qvý} + ax where c — c(p, p). It follows that G;A(U + K^[QV]ý + ax\ 1 then we can choose A > 0 large enough and then choose k > 0 small enough (depending on A) such that (4.14) holds. If pp < 1 then for any k > 0 there exists A large enough such that (4.14) holds. For such A and k, we obtain G4(£/ + K/J(?v]y5 + orT]< V. By Lemma 4.2, there exists a weak solution (w, w) of (4.8) for a > 0, a > 0, v > 0, v > 0 small if pp > 1, for any er > 0, cr > 0, v > 0, v > 0 if pp < 1. Moreover, (w, w) satisfies w^G^o}], (4.15) w « GM[(GM[«] + K^[v])"] + GM[r] (4.16) where C = C(iV, /5, p, Q,, a, a, r, r). Next put m := w + K^Iqv] and v := uj + KM[^v] then (w, i>) is a weak solution of (1.32). Moreover (1.33) and (1.34) follow directly from (4.15) and (4.16). □ Proof of Theorem F. Put r* := maxjr, r} and v* := maxjv, v}. Fix # > 0, ■& > 0 and for k e i - ~ -L (0, 1], put cr = £ = p and a — KtJ, g = pp . Set r#:=#r + #r and v# := #v + hv then r# < {§ + il)x* and v# < {§ + il)v*. Put V :— A(G/it[/cr#] + KM[/cv#]) where A > 0 will be determined later on and put U :— Gfj,[(y + ^[i[qv])P + err]. We have t/P+OTT 0 and k > 0 such that C(Appkpp~1 + 1) < A. (4.21) Since pp > 1, one can choose A large enough and then choose k > 0 small enough such that (4.21) holds. For such A and k, we have G^[Up+ar]+K^[Qv]< V. By Lemma 4.2, there exists a weak solution (u, v) of (1.32) which satisfies (1.37). □ 5. General nonlinearities 5.1. Absorption case In this section we treat system (1.38) with e — —1. We recall that Ag and Ag are defined in (1.39). Proof of Theorem G. Step 1: We claim that J g(K^[\v\] + G^[\a\])Sadx + j g(K^[\v\] + G^[\r\])Sadx 0, set Ax := {x e Q : KM[|v|] + GM[|r|] > A} and a (A.) := fA) Sadx. We write ||5(K^[|v|] + G^[|T|])||Ll(n.4a) = j g(KlA\v\l + GlA\r\l)Sadx Ai + y'g(K/Lt[|v|] + G/t[|r|])5adx < J g(KM[|v|] + GM[|r|])5adx + g(l) J 5 0, «C*> < C7 (IIKmCIvIIH^^^ + l|G/4[|?|]||^Mcn:a„))*-*'- < C7,-"m (5.3) where C = C(N, /x, £2, y, II vllart(a^) > ll*llatt(fl;a«))- Thus CO CO «(l)g(l) + y fl(s)^(s) < C + j"1"^^ < C^Ag. (5.4) l l By combining the above estimates we obtain ||g(KM[|v|] + GM[|r|])||Ll(S2.^) 1, since g, g satisfy (5.1) there exist functions uk and vk satisfying - L^Uk +g(vk-i) — x in Q,, - L^vk+g(uk) = x in £2, (5.5) tt(uk) = v, tx(vk) — v. Moreover (5.6) uk + ^Agivk-i)] = GM[r] + KM[v], wjt + Gp[g(uk)] = GM[r] + KM[v]. Since g, g > 0, it follows that, for every k > 1, K/Jv] + GM[r] - GM[g(KM[v] + GM[r])] < uk < K^fv] + GM[r] = «0 and K^[v] + GM[r] - GM[g(KM[v] + GM[r])] < Wjt < K^[v] + GM[r] in Q,. Now, suppose that for some k>l,uk< uk-\. Since g and g are nondecreasing, we deduce that vk = K/Jv] + GM[r] - G^[g(«it)] > K^fv] + GM[r] - G^fo-i)] = ujt-i, Mjt+i = K^[v] + GM[r] - GM[g(^)] < KM[v] + GM[r] - GM[g(^_i)] = Mjt. This means that {vk} is nondecreasing and {uk} is nonincreasing. Hence, there exist u and t> such that uk \u and ^ f t> in £2 and IMv] + GM[r] - GM[g(KM[v] + GM[r])] < « < K^[v] + GM[r], K^[v] + GM[r] - GM[g(KM[v] + GM[r])] < v < K^[v] + GM[r]. Since g and g are continuous and nondecreasing, we infer from monotone convergence theorem and (5.1) that g(vk) —>■ g(v) in L1^; Sa) and g(uk) —>■ §(m) in L1^! As a consequence, <&Li[g(uk)] G^[g(u)] a.e. in £2, Gi,[g(vk)l^GAg(v)] a.e.in£2. By letting —>■ oo in (5.6), we obtain the desired result. □ 134 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 867 5.2. Source case: subcriticality In this section for simplicity we consider system (1.38) with e — 1. Assume that g(Q) — g(Q) — 0. In preparation for proving Theorem H, we establish the following lemma: Lemma 5.1. Assume e — 1, g and g are bounded, nondecreasing and continuous functions in M. Let x, x € 9Jt(£2; Sa) and v, v € 9Jt(9£2). Assume there exist a\ > 0, b\ > 0 and q\ > 1 such that (1.40) and (1.41) are satisfied. Then there exist b* > 0 and > 0 depending on N, [i, Q y, y, Ag, Ag, a\, q\ such that the following holds. For every b\ € (0, b.*) and a, a, q, q € (0, £*) the system - Lflu = g(v + QKfl[v] + aGfl[x]) in -LfJ,v = g(u+eKfJ,[v] + aGfJ,[x]) intl, (5.8) tr(w) — tr(v) — 0 admits a weak solution (u, v) satisfying (5.9) Proof. Without loss of generality, we assume that ||r||g^(Q.^a) — ll?ll«r/t(S2;«a) — IMIaK(3Q) — ||v||«r/t(3j2) — 1. We shall use Schauder fixed point theorem to show the existence of positive weak solutions of (5.8). Define S(w) := Gß[g(w + QKß[v] + ctG/Jt])], §(w) := Gß[g(u) + eKß[v] + oGß[x])], Vw € L^Q). (5.10) Set Mi(wj) := \\w\\LPf(Q.&a), Vwj € L^(S2; Sa), Mi(w) := l|w||lpm(n.,5a), Vwj € L^(S2; 5"), M2(w) := ||w||L?i(n;^-i), Vwj € L?1(^; <5"_1), M(wj) := Mi (w) + M2(w), Vwj € L^(S2; 5") n L?1 (£2; 5""1), M(wj) := Mi (w) + M2(w), Vwj € L^(S2; 5") n L?1 (£2; 5""1). Step 1: Upper bound for g(w + qK^v] + ctGm[t]) in L1^;^) with wj € L^(£2; 8a) n (fi;^"1). For A > 0, set fi^ := {x e Q : \w\ + £KM[|v|] + crGM[|r|] > A} and fe(A) := fa 8adx. We write 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 135 868 K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 \\g(w + QK^[v]+5Gll[T])\\LHa.sa)< jg(\w\+QK^[\v\]+aG^[\r\])Sadx Bi + J g(\w\+QKIJ,[\v\\+5Gll[\T\])Sadx - j g(-M-QK^[\v\]-aG^[\r\])Sadx (5'U Bi - j g(-\w\-eKll[\v\]-aGll[\r\l)Sadx -. I + II + III + IV. We first estimate /. Since g e C(M+) is nondecreasing, one gets I = b(l)g(l) + j b(s)dg(s). l Since g is bounded, there exists an increasing sequence of real positive number {£j} such that limf^oo and lim Cp»g(lj) = 0. (5.12) Observe that j b(s)dg(s)= Urn J b(s)dg(s). On the other hand, by (2.2) one gets, for every s > 0, a(s)< ||M+eK4|v|]+CTG4|T|]||^»(a.s-p» ■ oo yields I (1> nV-I)- Since S satisfies condition (1.40), it follows that (5.15) max{II,IV] Kfl[v] + 3r'Gfl[i])\\Li(Q.sa), which implies M2(S(«;)) < CAgMi(ffl)p" +aiCM2(w)qi +biC + Cd~Qj. (5.19) Consequently, M(§(u))) 1 and q\ > 1, there exist > 0 and b* > 0 depending on N, [i, !T2, Ag, Ag, a\, q\ such that for any q, q e (0, g*) and b\ e (0, fe*) there exist X* > 0 and X* > 0 such that )5(A*)=X* and fj(X*) = X*. Here A* and X* depend on N, [i, £2, Ag, Ag, a\, q\. Therefore, M(w) < A* =>• M(S(wj)) < X* (5.22) M(wj) < X* =>• M(S(wj)) < A*. Step 3: To apply Schauder fixed point theorem. For w\, w2 € Z.1^), put T(Wl,«;2):=(S(w2),S(wi)), (5.23) V := {(«9, 0 and (p > 0. We can extract a subsequence, still denoted by the same notation, such that (cpm, (pm) —>■ (p) a.e. in £2. Consequently, by Fatou's lemma, Mi (cp) < liminf Mi ( and therefore T> is a closed subset of Z/(£2) xt'(S). Clearly, T is well defined in T>. For (w, w) e T>, we get M(w) < X* and M(«j) < X*, hence M(S(wj)) < X* and M(§(«j)) < V It follows that T(V) C V. We observe that T is continuous. Indeed, if wm —>■ w and «jm —>■ u) as m —>■ oo in L1 (£2) then since g, g € C(M) n L°°(M), it follows that g(wm + qK^v] + orG^fr]) g(w + §KM[v] + orG^fr]) in L> (£2; 5"), and g(Wm + eKM[v] + aGM[r]) £(u, + eKM[v] + aGM[r]) in L1^; 5") as m —>■ oo. By (2.13), §(u)m) —>■ S(ui) and S(«;m) —>■ §(«;) as m ^ oo in L1^). Thus T(wm, ujm) —>■ T(iu, uj) in Ll(Q,) x L1^). We next show that T is a compact operator. Let {(wm, wn)} cT> and for each m > 1, put ^rm — §(«jm) and \j/m — S(itim). Hence {Aip-m} and {Ai/fm} are uniformly bounded in LP(G) for every subset G g £2. Therefore {^ml is uniformly bounded in W1,P(G). Consequently, there exists a subsequence, still denoted by the same notation, and functions i/r, ijr such that (fffl, i/rm) —> (i/f, i/f) a.e. in £2. By dominated convergence theorem, (i/rm, i/rm) —>■ (i/r, i/r) in L1^) x L1^). Thus T is compact. By Schauder fixed point theorem there is (u, v) e T> such that T(u,v) — (u,v). □ Proof of Theorem H.I. Let {gn} and {gn} be the sequences of continuous, nondecreasing functions defined on M such that g„(0) = g(0), \g„\< \gn+i\< \g\, sup|g„| M g„(0) = g(0), |g„| < |g„+i| < sup|g„| M Due to Lemma 5.1, there exist A.*, X*, > 0 and g„ > 0 depending on N, fi, £2, Ag, Ag, a\, qi such that for every b\ e (0, fe*), q, q e (0, g*) and n > 1 there exists a solution (w„, uj„) € T> of -L^Wn =g„(wn +QKfJi[v] + aGfJi[i]) in £2, -L^xbn =g„(wn +£>KM[v] + erGM[r]) in £2, (5.25) tr(w„) = tr(uj„) =0. = «and hm ||g„ - g||L«> (M) = 0, = «and lim ||g„ - g||Loo (M) = 0. (5.24) 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 139 872 K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 For each n, set un—wn+ £KM[v] + erG^r] and vn — wn+ gK^v] + cfG^fr]. Then — j unL^4>dx — j g„(v„)dx + a j dx V0€XM(£2), (5.26) Q Q Q Q - j vnL^4>dx — j gn (un)4>dx + a J (pdt — q j [vM^dx V0eX^(£2). (5.27) Q Q Q Q Since {(wn, wn)} C T> and the fact that Agn < Ag, we obtain from (5.16) that \\gn(vn)\\Li{Q.sa) < CAgXl11 +a\CXqi + b*C + dQt (5.28) l, Hence the sequence {g{vn)} is uniformly bounded in L (£2; Sa). Since {(«;„, «)„)} C 2?, the se-w„] and {£ quence {||wj„} and {||«j„} are uniformly bounded in Lqi(G) for every subset G g £2. As a consequence, {Auj„} and {Auj„} are uniformly bounded in L1(G) for every subset G g £2. By regularity results for elliptic equations, there exist subsequences, still denoted by the same notations, and functions w and w such that (wn, wn) —>■ (w, w) a.e. in £2. Therefore («„, t>„) —> (u, v) a.e. in £2 with u — w + qK^Iv] + ffG^fr] and u — w + £>KM[v] + cfG^fr]. Moreover ign{un), gn(v„)) ~+ (g(u), g(v)) a.e. in £2. We show that u„ —>■ u in L1^; 5"). Since {wn} is uniformly bounded in Lqi{Q,\ <5"_1), by (2.14), we derive that {un} is uniformly bounded in Lqi (£2; 5"). Due to Holder inequality, {un} is uniformly integrable with respect to Sadx. We invoke Vitali convergence theorem to derive that un —> u in L1 (£2; 5"). Similarly, one can prove that v„ —>■ i> in L1^; 5"). We next prove that g«(u«) —>■ g(v) in L1^; 5"). For X > 0 and n € N set fi„^ := {x € £2 : |t>„ | > A} and bn (X) := JB ^ Sadx. For any Borel set E C £2, (5.29) J g„(v„)Sadx = y g„(i;„)5". Er\BcnX < j gn(v„)Sadx + mg,x j Sadx Bn,>. e oo ■ oo. Take arbitrarily e > 0, there exists X > 0 such that the right hand-side of (5.30) is smaller than |. Fix such X and put r\ — 2me ^. Then, by (5.29), 140 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 873 j S(x)adx < x] =>■ j g„(vn)S(x)adx < s. E E Therefore the sequence {g„(vn)} is uniformly integrable with respect to Sadx. Due to Vitali convergence theorem, we deduce that g„(v„) —>■ g(v) in L1^; Sa). By sending n —>■ oo in each term of (5.26) we obtain —j uL^cpdx — j g(v)■ g(M) m £ (£2; By letting « —>■ oo in (5.27), we get —j vL^cpdx — j g(u)) is a solution of (1.27). □ 5.3. Source case: sublinearity We next deal with the case where g and g are sublinear. ProofofTheoremH.il. The proof is similar to that of Lemma 5.1, also based on Schauder fixed point theorem. So we point out only the main modifications. Let § and § be the operators defined in (5.10). Put Ni(«;):= IMIL?i(n;a«-i), Vwj € Lqi (£2; Sa~l), N2(w) := IMIlH^0 Combining (2.13), (2.14) and (1.42) leads to On the other hand Define Then N2(S(w)) < a2CNi (w)qx + C(eqi + Stqx + b2) Ni (§(«;)) < a2CN2(w)q2 + C(eq2 + aqi + b2) §1 (X) := a2CXqi + C(Qqi + 5qi + b2), := a2CXq2 + C(Qq2 + aq2 + b2). N2(S(u/))<£i(Ni(uO) and Ni(§(w)) <&(N2(uO). 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS 141 874 K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 If qiq2 < 1 then we can find X\ and X2 such that ^i(X\) — X2 and §2(^2) — X\. Thus if Ni(«;) < X\ then N2(S(w)) < X2 and if N2(w) < X2 then Ni(§(«;)) < X\. If qiq2 — 1 and «2 small enough we can find, X\ and A 2 such that §1 (A 1) = A 2 and §2(^2) — M ■ The rest of the proof can be proceeded as in the proof of Lemma 5.1 and the proof of Theorem H.I. and we omit it. □ 5.4. Source case: subcriticality and sublinearity Proof of Theorem H.III. Set N(wj) := IMIL?i(n;a«-i), Vwj € Lqi(£2; &a~l). By an argument similar to the proof of Lemma 5.1 and Theorem H.II, we get N(S(uO) < CAgMi(u))p» + aiCM2(w)qi +biC + d^a. On the other hand M(S(wj)) < a2CN(w)q2 + C(eqi + aqi + b2). Set li(X) :=CAgXp» +aiCXqi +biC + d~Qj, l2(X) := a2CXq2 + C(Qq2 + aq2 + b2). Then N(S(uO) < li (M(wj)) and M(S(uj)) < |2(N(wj)). We consider there cases. Case 1: <7i#2 > 1- Since > qi, it follows that 7^(72 > 1- Therefore there exist b* > 0 and > 0 such that for b\, b2 € (0, b*) and q e (0, g*) one can find Ai > 0 and X2 > 0 satisfying ti(M) = *2 and |2(X2) = M. (5.33) Case 2: /j^^ = 1. In this case, there exist a* > 0 such that if «2 € (0, a*) then for every q > 0 and £ > 0 one can find X\ > 0 and A 2 > 0 satisfying (5.33). Case 3: Pfj,q2 < 1- In this case for every q > 0 and q > 0 one can find X\ > 0 and A 2 > 0 such that (5.33) holds. Hence, in any case, M(w) < M => N(S(uO) < |i = A.2 N(iu) < X2 => M(S(iu)) < |2(^2) = The rest of the proof can be proceeded as in the proof of Lemma 5.1 and the proof of Theorem H.II. and we omit it. □ 142 4. EQUATIONS AND SYSTEMS WITH HARDY POTENTIALS K.T. Gkikas, P.-T. Nguyen/J. Differential Equations 266 (2019) 833-875 875 References [1] D.R. Adams, L.I. Heberg, Function Spaces and Potential Theory, Grandlehren der Mathematischen Wisenschaften, vol. 31, Springer-Verlag, 1999. [2] A. 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Blow-Up, Global Existence and Steady States, Birkhauser Verlag, Basel, 2007, xii+584 pp. CHAPTER 5 Elliptic equations with a Hardy potential and a gradient-dependent nonlinearity An investigation on semilinear equations with a Hardy potential and a gradient-dependent nonlinear term is presented in this chapter, which is based on a joint work with Gkikas [80]. We establish sharp existence and uniqueness results and obtain a complete description of isolated singularities in subcritical range. We also show that singularities are removable in the supercritical range. 143 144 DE GRUYTER 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY Adv. Nonlinear Stud. 2020; aop Research Article Konstantinos T. Gkikas and Phuoc-Tai Nguyen* Elliptic Equations with Hardy Potential and Gradient-Dependent Nonlinearity https://doi.org/10.1515/ans-2020-2073 Received September 27, 2019; revised January 8, 2020; accepted January 10, 2020 Abstract: Let Cl c KN (N > 3) be a C2 bounded domain, and let S be the distance to dCl. We study equations (E+), -Luu ± g(u, \Vu\) = 0 in Cl, where Lu = A + J^, pi e (0, |] and g: R x R+ R+ is nondecreasing and locally Lipschitz in its two variables with g(0,0) = 0. We prove that, under some subcritical growth assumption on g, equation (E+) with boundary condition u = v admits a solution for any nonnegative bounded measure on dCl, while equation (£_) with boundary condition u = v admits a solution provided that the total mass of v is small. Then we analyze the model case g(s, t) = \s\ptq and obtain a uniqueness result, which is even new with pi = 0. We also describe isolated singularities of positive solutions to (E+) and establish a removability result in terms of Bessel capacities. Various existence results are obtained for (£_). Finally, we discuss existence, uniqueness and removability results for (E±) in the case g(s, t) = \s\p + tq. Keywords: Hardy Potential, Singular Solutions, Boundary Trace, Uniqueness, Critical Exponent, Gradient Term, Isolated Singularities MSC 2010: 35J10, 35J25, 35J60, 35J66, 35J75 Communicated by: Julian Lopez Gomez Dedicated to Laurent Veron on his 70th birthday. 1 Introduction and Main Results Let Cl be a C2 bounded domain in M.N (JV > 3), pi € (0, ^] and <5(x) = <5q(x) := dist(x, dCl). In this paper, we investigate the boundary value problem with measure data for equation -Lpu ±g(u, \Vu\) = 0 inQ, (E±) where 1^ = := A + andg: Rx R+ R+ is nondecreasing and locally Lipschitz in its two variables with g(0,0) = 0. The term is called Hardy potential since it is related to the Hardy inequality. The nonlinearity g(u, |Vu|) is called absorption (resp. source) if the "plus sign" (resp. "minus sign") appears in (£+). One prototype model to keep in mind is g(u, | Vu|) = |up|Vu|9. 1.1 Background and Main Contributions The boundary value problem for (E+) without Hardy potential, i.e. pi = 0, has received substantial attention over the last decades, starting from the pioneering work of Brezis [10]. In particular, Brezis proved that, for ^Corresponding author: Phuoc-Tai Nguyen, Department of Mathematics and Statistics, Masaryk University, 611 37 Brno, Czech Republic, e-mail: ptnguyen@math.muni.cz Konstantinos T. Gkikas, Department of Mathematics, National and Kapodistrian University of Athens, 15784 Athens, Greece, e-mail: kugkikas@gmail.com Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 145 2 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER every prescribed I1 boundary datum, the semilinear equation with absorption term -Au + g(u) = 0 in a (1.1) admits a unique solution. Afterwards, equation (1.1) in measure framework was first considered by Gmira and Veron in [18] where they showed that boundary value problem for (1.1) is not always solvable for every measure boundary datum. Because of its applications in many areas, equation (1.1) with g(u, | Vu|) = |uP_1u has been intensively studied in many works, among them is the celebrated series of papers of Marcus and Veron [26-28]. These results were then extended to the equation with gradient-dependent absorption term -Au + g(u, |Vu|) = 0 inQ. We refer to [32] for the case when g depends only on Vu and to [24, 30] for the case when g depends on both u and Vu. Equation (£_) with pi = 0, i.e. - Au -g(u, |Vu|) = 0 inQ, (1.2) has been studied in various directions. Necessary and sufficient criteria in terms of capacities for the existence of a solution with measure boundary data were obtained in [9]. Singular solutions of (1.2) with g(u, |Vu|) = |Vu|? in a perturbation of the ball was studied in [2]. Recently, Bidaut-Veron, Garcia-Huidobro and Veron have established a priori estimates for solutions of (1.2) with g(u, |Vu|) = |up|Vu|9 org(u, |Vu|) = \u\p + M|Vu|« (see [7, 8]). The case with Hardy potential has been intensively studied over the last decade. See e.g. Bandle, Moroz and Reichel [5], Bandle, Marcus and Moroz [4], Marcus and Nguyen [25], Gkikas and Veron [17], Marcus and Moroz [23], Nguyen [31], Gkikas and Nguyen [16]. In the aforementioned papers, the best constant in the Hardy inequality L\V(p\2dx CH(Q):= inf fJ° , (1.3) is deeply involved in the analysis. It is well known that Ch(Q) e (0, |] and Ch(Q) = ^ if Q is convex (see [22, Theorem 11]) or if -A<5 > 0 in the sense of distributions (see [6, Theorem A]). Moreover, the infimum in (1.3) is achieved if and only if Ch(Q) < \. Moreover, Brezis and Marcus [11, Remark 3.2] proved that, for any pi< \, the eigenvalue problem p€Hj(Q)\{0} Jn (p2 dx admits a minimizer (pu in Hq(Q), and hence Au is the first eigenvalue of -Lu in Hq(Q). Moreover, -L^ip^ = A^ip^ in Q. WhenU = \, there is no minimizer of (1.4) inffJ(Q), but there exists anonnegative function ipi e Hl0C(Q) such that -I =Ai(pi in Q in the sense of distributions. We see from (1.3) and (1.4) that Au > 0 if pi < CH(Q), Au = 0 if pi = CH(Q) < \, while Au < 0 when pi > Cff(Q). It is not known if Au > 0 when pi = Ch(Q) = \. However, if Q is convex or if -A<5 > 0 in the sense of distributions (in these cases Ch(Q) = \), then A' > 0 (see [11, Theorem II] and [6, Theorem A] with k = 1 and p = 2). Throughout the present paper, we assume that pi € (0,1] and Au > 0. (1.5) This assumption implies the validity of the representation theorem which states that every positive Inharmonic function u in Q (i.e. u is a solution of L^u = 0 in Q in the sense of distributions) can be uniquely represented in the form u = Kj,[v] for some positive measure v € 9Jt+(dQ) (the space of positive bounded measure on dQ), where denotes the Martin operator (see Subsection 2.2 for more details). The representation theorem is derived from Ancona [3] (see also [25, page 70]) in the case pi < Ch(Q) and was proved by Gkikas and Veron [17, Theorem 2.33] in the case pi = \ and Ai > 0. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 146 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K.T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 3 In order to investigate the boundary behavior of I^-harmonic functions, Gkikas and Veron [17] introduced a notion of boundary trace in a dynamic way which is recalled below. Let D D° on dD by u(x0)= jh(y)dcoxD°(y). dD A sequence of domains {Cln} is called a smooth exhaustion of Cl if dCln e C2, Cln c Cln+i, \JnCl„ = Cl and M^-^dQ,,) -» ,HN-1(da). For each n, let be the I°" -harmonic measure on dQ„. Definition 1.1. Let e (0, ^]. We say that a function u possesses a boundary trace if there exists a measure v € 9Jt(dQ) (the space of bounded measure on dCi) such that, for any smooth exhaustion {Cln} of Cl, it holds lirn^ | 0u dw*° = | 4> dv for all 0 € C(Cl). The boundary trace of u is denoted by tr(u), and we write tr(u) = v. In [17, Proposition 2.34], Gkikas and Veron proved that if tr(Kj,[v]) = v for every v € 9Jt(dQ). This fact and the representation theorem allow to characterize Lu -harmonic functions in terms of their boundary behavior. It was shown in [15] that, when pi e (0, Ch(CI)), the notion of boundary trace in Definition 1.1 coincides with the notion of normalized boundary trace introduced by Marcus and Nguyen in [25, Definition 1.2]. This notion was employed to formulate the boundary value problem -Luu±\u\p-1u = 0 inQ, " (1.6) tr(u) = v. A complete description of the structure of positive solutions to (1.6) with "plus sign" was established in [17, 25], and various existence results for (1.6) with "minus sign" were given in [15, 31] in connection to the critical exponent N + a , 1 |l , . Px:=N7^2 wltha:=- + ^--^. (1.7) In particular, it was proved that, when 1 < p < pu, equation (1.6) with "plus sign" admits a unique solution for every finite measure v € W'(dCl), while the existence phenomenon occurs for (1.6) with "minus sign" only with boundary measure of small total mass. When p > pu, the nonexistence phenomenon happens, i.e. equations (E+) do not admit any solution with an isolated singularity. Related results were obtained in [4, 5, 23] and references therein. Very recently, a thorough study of the boundary value problem -Luu + \Vu\q = 0 inQ, U (1.8) tr(u) = v was carried out in [16], revealing that the value N + a q" := N + a-1 is a critical exponent for the solvability of (1.8). This means that if 1 < q < qu, then, for every v € 9Jt+(dQ), there is a unique solution of (1.8); otherwise, if qu < q < 2, singularities are removable. Motivated by the aforementioned works, in the present paper, we aim to investigate related issues for (£+). Main features of a boundary value problem for (E+) with measure are Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 147 4 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER the presence of the Hardy potential which blowups strongly at the boundary, the dependence of the nonlinearity on both solution and its gradient, rough data which cause the invalidity of some classical results. The interplay between the features leads to new essential difficulties, hence complicates drastically the analysis and produces interestingly new phenomena. Our contributions are the following. We establish the existence of weak solutions of (E±) with prescribed boundary trace v under sharp assumption on g. In particular, by using the standard approximation method, combined with the estimates of the Green kernel and the Martin kernel as well as their gradient [15], the sub- and supersolutions theorem and the Vitali convergence theorem, we show that, for every measure v € 9Jl+(dQ), equation (£+) admits a solution. Unlike the absorption case, thanks to the Schauder fixed point theorem, we can construct a solution to (E+) under the smallness assumption on the total mass of the boundary datum. We prove the comparison principle for (£+), which in turn implies the uniqueness. This result, which is obtained by developing the method in [16, 24] and the theory of linear equations [15, 17, 25], is new, even in the case without Hardy potential. We show sharp a priori estimates for singular solutions of (£+). This allows to study solutions with an isolated singularity. As a matter of fact, we show that there are two types of solutions with an isolated singularity of (£+): weakly singular ones and strongly singular one. Moreover, the strongly singular solution can be obtained as the limit of the weakly singular solutions. It is interesting that this phenomenon does not occur for (£_). The interaction of up and \Vu\q is a source of difficulties, which requires a delicate analysis and heavy computations. We demonstrate removability results of singularities in terms of capacities. The absorption case and source case are treated differently using different types of capacities (see [9,16]). Our results cover and refine most of the aforementioned works in the literature and provide a full understanding of equations with Hardy potential and gradient-dependent nonlinearity. 1.2 Main Results First we are concerned with a boundary value problem for equations with absorption term of the form -Luu + g(u,\Vu\) = 0 inQ, tr(u) = v. Before stating the main results, let us give the definition of weak solutions of (P+). Definition 1.2. Let v € 9Jt(dQ). A function u is called a weak solution of (P+) if uel^Q.fi"), g(u,|Vu|)€l1(Q,5a) and -jul^f dx+ |g(u, \Vu\X dx = - ^KpMLpt; dx for all f e xu(Ci), □ □ □ where the space of test function Xj,(Q) is defined by X„(Q) := {<" € Hlc(Q) : S~a( e H\a, S2a), S~aL^ e I°°(Q)}. (1.9) We notice that this definition is inspired by the definition in [17, Section 3.2]. For more properties of the space of test functions Xj,(Q), we refer to [17]. Our first result is the existence of a weak solution of (P+) under an integral condition on g. Theorem 1.3 (Existence). Assumegsatisfies 00 Ag := g(s,sv )s~1~p" ds < oo. (1.10) 1 Then, for any v € 9Jl+(dQ), (P+) admits a positive weak solution 0 < u < Ku[v] in Cl. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 148 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K.T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 5 Remark 1.4. We remark the following. (i) Ifg(s, f) = \s\pt? fors e R, f e B.+,p,q > Q,p + q > 1, thengsatisfies (1.10) if (N + a- 2)p + (JV + a- l)q < N + a. (1.11) (ii) Ifg(s, f) = \s\p + tq fors € R, f € R+,p > 1, 1, thengsatisfies (1.10) if 1 < P < P]i and 1 < 9 < (1-12) It is worth noticing that this theorem is established by developing the sub- and supersolutions method in [16], in combination with the Schauder fixed point theorem and the Vitali convergence theorem. It seems infeasible to obtain the uniqueness in case of general nonlinearity; however, when g(u,\Vu\) = W\Vu\\ we are able to prove the comparison principle, which in turn implies the uniqueness. The method is delicate, relying on a regularity result (see Proposition 4.1), maximum principle (see Lemma 4.2), estimates on the gradient of subsolutions of a nonhomogeneous linear equation (see Lemma 4.4). We emphasize that this result is new even in the case without Hardy potential, i.e. pi = 0. Theorem 1.5 (Comparison Principle). Assumeg(u, |Vu|) = |up|Vu|9 withq > 1 andp and q satisfying(1.11). Let V( € 9Jl+(dQ), i = 1,2, and let U( be a nonnegative solution of'(P+) with v = Vi.lfvi < Vj, then Ui < Uj in Q. Assume 0 € dQ, and denote by <5o the Dirac measure concentrated at 0. A complete picture of isolated singularities concentrated at 0 is depicted in the next theorem. Theorem 1.6. Assumeg(u, |Vu|) = |uP|Vu|« with q > 1 andp and q satisfying (1.11). (I) Weak singularity. For any k> 0,letUg k be the solution of -Luu + g(u,\Vu\) = 0 in Q (1.13) tr(u) = kSo. Then there exists a constant c = c(N,pi,Q) > 0 such that u^ k(x) < ck5(x)a\x\2~N~2a for every x e Qand |Vu°k(x)| < cfc5(x)a-1|x|2-JV-2a for all x € Q. Moreover, Um = k. (1.14) 03*-y KJ/(x, 0) Furthermore, the mapping k<-^> Ugkis increasing. (II) Strong singularity. Putu^^ : = lim;MCO u^ k. Then u^^is a solution of (1.15) -Luu + g(u,\Vu\) = 0 inQ, u = 0 ondQ\{0}. There exists a constant c = c(JV, pi, p, q, Q) > 0 such that c-15(x)a|xr^"a < Uq.coW s cS(xf\x\~^~" for all x e Q, |Vug|0O(x)| < c5(x)a-1|xr^"a for all x e Q. Moreover, lim \x\£fru$00(x) = aj(o), (1.16) Qbx—>0 locally uniformly on the upper hemisphere S+_1 = R^ n SN~1, where ti> is the unique solution of problem (4.35). Here m% = {x = (xu ... ,xN) = (x',xN) : xN > 0}, and S"'1 is the unit sphere in Rw. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 149 6 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER Let us discuss briefly the proof of Theorem 1.6. The main ingredients in the proof of convergence (1.14) are the estimates on the Green kernel (2.1) and the Martin kernel (2.2) and condition (1.11). From the monotonicity of the sequence {Ug k], universal estimate (4.15) and a standard argument, we deduce that Ug ^ is a solution of (1.15). The proof of convergence (1.16) relies strongly on the similarity transformation Te (see (4.19)) and the study of problem (4.35) in the upper hemisphere S+-1. The existence and uniqueness result for (4.35) is provided in Section 4.4. When g(u, |Vu|) = |up|Vu|9 with q > 1, in order to deal with a wider range of p and q (i.e. p and q may not satisfy (1.11)), we make use of Bessel capacities (see Section 5). A necessary condition for the existence of a solution to (P+) and a removability result are stated in the following theorems. Theorem 1.7 (Absolute Continuity). Assumeg(u, |Vu|) = |uP|Vu|« with p>0, ll and (JV + a - 2)p + (JV + a - l)q > JV + a. Let v € 9Jl+(dQ) and assume that problem (P+) has a nonnegative solution u e C2(Q). (i) If q # a + 1, then v is absolutely continuous with respect to C" a, (p+qy, i.e. v(K) = 0/or any Borel set K c dQ, such that C" a, m-, lj)+qy (K) = 0. Here (p + q)' denotes the conjugate exponent ofp + q. (ii) If q = a + 1, then, for any £ e (0, minja + 1, '■N~^a - (1 - a)}), v is absolutely continuous with respect to C" gl , (p+a+1y.Here the capacity CfK is defined in Section 5. Put W(x) : 4 (1.17) <5(x)i|ln<5(x)| il]x = \. We note that, by [17, Propositions 2.17, 2.18], for any h e C(dQ), there exists a unique Inharmonic function uh e C(Q) n I1(Q, 8") such that lim ¥rrT=h(£) forall^edQ. (1.18) xeQ,x->f W(X) In addition, tr(Uf,) = hoi*0, where Xn € Q, is a fixed reference point and is the I^-harmonic measure in Q, (see [17, Subsection 2.3] for further details). It is worth mentioning that (1.18) can be viewed as the boundary condition in the case with Hardy potential. If pi = 0, then a = 1 and W(x) = 1, in which case (1.18) becomes the boundary condition in the classical sense. The following result provides a removability result for solutions with "zero boundary condition" on dQ\K(see (1.20)). Theorem 1.8 (Removability). Assume p >0,1 < q < 2,p + q > 1 and (JV + a - 2)p + (JV + a - l)q > JV + a. Let K c dQ.be compact such that (0 Cfa^i(p+q)l(K) = Oifqta + lor (ii) cfa1+!Z.,ip+a+1y(K) = Qforsomes e (0, minfa + 1, ^£ _ (1 _ a)}) ifq = a + 1. Then any nonnegative solution u e C2(Q) n C(Q \ K) of -Luu + \u\p\Vu\q = 0 in CI (1.19) such that lim = 0 for all 11 dO.\K (1.20) xeQ,x->f W(X) is identically zero. Next we deal with the boundary value problem for equations with source term of the form -Luu-g(u,\Vu\) = 0 inQ, tr(u) = v. Weak solutions of (PZ) are defined similarly to Definition 1.2. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 150 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K.T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 7 Phenomena occurring in this case are different from those in the case of absorption nonlinearity. This is reflected in Theorem 1.9 which ensures the existence of a weak solution under a smallness assumption on the total mass of the boundary data. In order to make the statement clear and lucid, we rewrite equation (P*) as -Luu-g(u,\Vu\) = 0 inQ, tr(u) = gv, where g is a positive parameter and v € 9Jl+(dQ) with ||v||rm(dQ) = 1. Theorem 1.9 (Existence Result for (P6V) in Subcritical Case). Let v e 9Jl+(dQ) with ||v||rr«(dQ) = 1. Assume g satisfies (1.10) and g(as, bt) < k(ap + bq)g(s, f) for all (a, b, s, t) e R+, (1.21) for some p > 1, q > 1, k > 0. Then there exists go > 0 depending on N,pi, a, Ag, k,p, q such that, for any g € (0, go), problem (P£v) admits a positive weak solution u > qKu [v] in a. This result is established by combining an idea in [34] and the Schauder fixed point theorem. Remark 1.10. It is easy to see that if g(s, f) = \s\ptq or g(s, f) = \s\p + f«, then (1.21) holds. The next result provides sufficient conditions for the existence of a solution to (P?v) with g( u, |Vu|) = | u P | Vu |9 in terms of capacities. See the definition of the capacities Capdn and N2a-i,i in Section 7. Theorem 1.11 (Existence Result for (P£v)). Assume that g(u, |Vu|) = \u\p\Vu\qwithp >0,q>0,p + q>l and q < 1+a+i^-a>p m Assume one of the following conditions holds. (i) There exists a constant C > 0 such that v(E) < CCapdn „+1 „ , (E) for every Borel set E c da. Here (p + q)' denotes the conjugate exponent of p + q. (ii) There exists a positive constant C > 0 such that Wia-i.i [<5ap+(a-1)«+aN2a_1,1 [v]p+l] < CK2a.1A [v] < co a.e. in a. (1.22) Then there exists go = Qo(N, pi,p,q,C,a) > Osuch that, for any q € (0, go), problem (P£v) admits a weak solution u satisfying \u\ < C'<5aN2a-i,i[ev], |Vu| < C'fi^Nza-i.iIev] ina, (1.23) where C' = C'(N,pi, Q) is a positive constant. Organization of the paper. In Section 2, we recall main properties of the first eigenvalue and the corresponding eigenfunction of -Lu in a and collect estimates on the Green kernel and the Martin kernel, as well as their gradient. In Section 3, we prove Theorem 1.3, and in Section 4, we demonstrate Theorem 1.5 and Theorem 1.6. In Section 5, we give the proof of Theorem 1.7 and Theorem 1.8. Section 6 is devoted to the proof of Theorem 1.9, and in Section 7, the proof of Theorem 1.11 is provided. In Appendix A, we construct a barrier in the case g(u, |Vu|) = |uP|Vu|*. Finally, in Appendix B, we discuss the case g(u, |Vu|) = \u\p + \Vu\q and state main results without proofs since the arguments are similar to those in the case g(u, | Vu|) = up\Vu\q. 1.3 Notations We list below some notations that we use frequently in the present paper. For

0, denote by L"(a, 1) the space of functions v satisfying JqM*^ dx < co. We denote by E1(a, 1, we denote by k' the conjugate exponent. Throughout the paper, c,ct, Cj, C, Ct, C' denote positive constants which may vary from line to line. We write C = C(a, b) to emphasize the dependence of C on the data a, b. The notation/ = h means that there exist positive constants c\, such that cth < f < Cjh. Denote by xe the indicator function of a set E. For z € dQ, denote by nz the outer unit normal vector at z. 2 Preliminaries We recall that, throughout the paper, we assume that pi e (0, j] and Au > 0. 2.1 Eigenvalue and Eigenfunction We recall important facts of the eigenvalue Au of -Lu and the associated eigenfunction (pu which can be found in [13, 14]. If 0 < pi < |, then the minimizer (pu e Hq(Q) of (1.4) exists and satisfies (pu = 5", where a is defined in (1.7). If pi = \, there is no minimizer of (1.4) in ffJ(Q), but there exists a nonnegative function (pi € H^QC(Q) such that ip^. ~ 5? and it satisfies -I i^j = Av(p i in Q in the sense of distributions. In addition, we have <5" € Hq(Q, 5). 2.2 Green Kernel and Martin Kernel Denote by andif^ the Green kernel and the Martin kernel of-L,, in Q respectively (see [17, 25]). The Green operator and the Martin operator are defined as follows: G°[r](x) := ^G®(x,y)dT(y) foreveryr e 9Jt(Q, <5"), Q K°[v](x) := | K$(x,z)dv(z) for every v € 9Jt(dQ). By [14, Theorem 4.11], it holds Gu(x,y) = min{|x-y|2-JV,5(x)a5(y)a|x-y|2-JV-2a} for every x,y e Q, x # y. (2.1) Since (1.5) holds, by [3] and [17, Proposition 2.29], the Martin kernel Ku exists. Moreover, it holds (see [25, (2.7), page 76] for pi < CH(Q) and [17, Theorem 2.30] for ]x = \) Ku(x, y) = S(x)a\x - y\2-N~2a for every x € Q, y e dQ. (2.2) For estimates on the Green kernel and the Martin kernel of a more general Schrodinger operator, we refer to [21]. Next we recall estimates of Green kernel and Martin kernel in weak L* spaces. Let r € 9Jt+(Q). For k > 1, k' = and u e Ij10C(Q, r), we set llullc(n,r) := infjc € [0, co] : dr < cQ dr^j' for any Borel set E c qJ e e Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 152 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K.T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 9 and I* (0, t) := {u € 14(0, T) : \\u\\LUn,T) < co}. IJ(0, r) is called weak V space (or Marcinkiewicz space with exponent jc) with quasi-norm || • Hi* (n,r)-See [29] for more details. Notice that, for every s > -1, L* (0, <5S) c Lr(Q, <5S) for every r e [1,jc). (2.3) Moreover, for any u e 14(0, 5s) (s > -1), | 8s dx0. (2.4) 1i"i>A| Proposition 2.1 ([15, Proposition 2.4]). The following statements hold. (i) Let y € (-JV+2n_2- spi)- There exists a constant c = c(JV, ^, y, O) such that 11^1^11^(0,«,) < c||T||m(Q,s.) for all t € OT(Q, 5"). (2.5) (ii) lef y > -1. Then there exists a constant c = c(N,p, y, O) such that HIK^[v]||LsS^(a,fil.) < cllvlla^oa) forallveWKdCl). (2.6) Proposition 2.2 ([16, Proposition A]). The following statements hold. (i) lef 8 € [0, a] and y e [0, jfrj-). Then there exists a positive constant c = c(JV, p., 8, y, O) such that l|VG^[|T|]||i^i(Q,sy) 0. Then there exists a positive constant c = c(N,p, y, O) such that IIVlMvlllli^Q.sy) < c||v||a«(dQ) forallv e OT(dQ), where VKu[v](x) = JdQ VxK^(x,2r) dv(z). 2.3 Linear Equations The Green kernel and the Martin kernel play an important role in the study of the boundary value problem for the linear equation -Luu = t in O, ■ U (2.8) tr(u) = v. Definition 2.3. Assume (r, v) € 9Jt(Q, 5") x 9Jt(dQ). We say that u is a weak solution of (2.8) if u € I1(Q,5a) and -|ul^fdx= jfdr-|K^[v]I^<"dx for all <" € X^(Q), □ □ □ where X^(0) is defined in (1.9). Proposition 2.4 ([15, Proposition 2.11]). Assume that (r, v) € 9Jl(0, 8") x 9Jl(dO). JTjeri u is a weak solution of (2.8) i/arid only ifu = Gu[t] + Ku[v] in O. Moreover, there exists a constant C = C(N,p, O) > 0 such that ll"llii(Q,«») S C(||T||fr«(Q,s») + l|v||fr«(dQ)). For D 0 such fhaf ||u||Li(D,sD) < c(||^||mi(n,«I)) + llnllmi(dD))-Finally, we will need the following classical properties of C2 domains. Proposition 2.6 ([29]). There exists a positive constant f>o such that 5 e C2(Q^0). Moreover, for anyx e Q^0, there exists a unique %x e dQ such that (a) <5(x) = |x - £f | and = -V<5(x) = -|*~|'|, where denotes the outer unit normal vector at %x £ dQ, (b) x(s) := x + sV<5(x) e and <5(x(s)) = |x(s) - fx| = <5(x) + s for any 0 < s < 4/So - <5(x). 3 Nonlinear Equations with Subcritical Absorption In this section, we establish the existence of a positive solution of (P+). The approach is based on a combination of the idea in [20], estimates on the Green kernel, the Martin kernel, their gradient and the Vitali convergence theorem. Proof of Theorem 1.3. We divide the proof into three steps. Step 1. In this step, we assume that M := sup |g(s, f)l < +co. (3.1) Let D cc Q be a smooth open domain, and consider the equation -L„v + g(v + K„[v],|V(v + K„[v])|) = 0 infl. (3.2) First we note that Ui = 0 is a supersolution of (3.2) and Uj = -Ku [v] is a solution of (3.2). Let 0 ifO (1 - u)S(x)-2 for allx € D, by standard elliptic estimates, there exists a constant Cj = C1(N, pi, do,a, D) > 0 such that sup|A[u](x)| < d(M + ||v||o«(dQ)) =: Ax. Also, by (3.3) and standard elliptic estimates, there exists a positive constant C2 = C2(N,pi, do,a, D) such that is continuous and compact. Now set x := € W1'1(D) : ||^1Im?i,~(d) < Ai + A2}. Then x is a closed, convex subset of IV1,1 (D) and A(x) c x. Thus we can apply the Schauder fixed point theorem to obtain the existence ofafunctionv € x such that A[v] = v. This means v is a weak solution of (3.6). By the standard elliptic theory, we can easily deduce that v, u2 e C2(D) n C(D). Moreover, it can be seen that v < 0. Now we allege that v > u2 in D by employing an argument of contradiction. Suppose by contradiction that there exists Xo € D such that infX£D(v(x) - U2M) = v(xo) - < 0. Then we have Vv(xo) = Vu2(xo), -A(v - u2)(x0) < 0, T[v](x0) = T[u2](x0) = u2(x0). But which is clearly a contradiction. Therefore, v > u2 in D. As a consequence, T(v) = v, and therefore v is a solution of (3 A). Step 2. In this step, we still assume that (3.1) holds. Let {£!„} be a smooth exhaustion of Cl, and let vn be the solution of (3.4) in D = Cln (constructed in Step 1) satisfying (3.5). Then there exists a constant C = C(N,pi,Q) > 0 such that |v„(x)| < G„[xo„g(v„ + K„[v], |V(v„ + K„[v])|)](x) < CMS(xf for allx € Q„. This implies that there exists a subsequence, still denoted by {v„}, such that v„ —» v in (Cl) and v satisfies -I^v + g(v + K^[v],|V(v + K^[v])|) = 0 inQ, tr(v) = 0. Furthermore, -Kj,[v] < v < 0 for all x € Cl. Setting u = v + Kj,[v], then u is a solution of (P+) satisfying 0 < u < Kp [v] in Cl. Step 3. In this step, we drop condition (3.1). Set gn := min(g, n), and let un be a nonnegative solution (the existence of un is guaranteed in Step 2) of sup|VA[u](x)| < C2(M + ||v||o«(dQ)) =: A2. By using an argument similar to the proof of [16, Theorem B, Step 1], we can show that A: W1A(D) -» W1A(D) -A(v - u2)(x0) = -(dcfn " ^<5(x0)"2)(v(x0) - u2(x0)) -g(v(x0) + K^[v](x0), |Vv(x0) + VK„[v](x0)|) + g(u2(x0) + K^[v](x0), |Vu2(x0) + VK^[v](x0)|) > 0, -Luu„+gn(u„,\Vun\) = 0 inQ, tr(u„) = v satisfying 0 < u„ < Kj,[v] in Cl. (3.7) Then un satisfies (3.8) u„ + Gplgn(um |Vu„|)] =K^[v]. (3.9) Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 155 12 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER Choosing ( = q>ii, where (p^ is an eigenfunction associated to the first eigenvalue of -L^, by (3.8), we have A? ^\u„\(pu dx + |g„(u„,\Vu„\)(pf, dx 0 and any function w, set A™ := {x e Q : |w(x)| > A}, aw(A) := | Sa dx, K B™ := {x e Q : |Vw(x)| > A^}, bw(A) := | Sa dx, ^ ^ CJ'^Aj'nB* cw(A) := | 5" dx. c; Then, for A > 0 and n e N, put A„,A=A^, a„(A) = a""(A), B„,A = B^, b„(A)=b""(A), CM = C^, c„(A) = c""(A). For any Borel set E c Q, ^g„(u„,\Vu„\)Sadx= | g„(u„,\Vu„\)Sadx+ | g„(u„, \Vu„\)Sadx E £nC„iA EnAcn AnB„iA + | gn(un,\Vun\)Sadx+ | g„(u„,|Vu„|)<5adx co < g(s,s^)s~1~p" ds + g(A,A^)^8a dx. (3.13) A £ Note that the first term on the right-hand side of (3.13) tends to 0 as A —» co. Therefore, for any e > 0, there exists A > 0 such that the first term on the right-hand side of (3.13) is smaller than f. Fix such A, and put £ 2max{g(A, A"), 1} Then, by (3.13), ^6adx jg„(u„,|Vu„|)<5adx<£. E E Therefore, the sequence {g„(un, |Vu„|)} is equi-integrable in 5"). Thus, by invoking the Vitali convergence theorem, we derive (3.11). From (3.7), we deduce thatO < un —> u inI1(Q, 5"). Therefore, letting n —> coin (3.8), we deduce that u is a weak solution of (P+). □ Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 156 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 13 4 Absorption g(u, |Vw|) = Vul*7: Subcritical Case In this section, we assume g(u, \Vu\) = \u\p\Vu\q withp > 0, q > 0, p + q > 1. We recall that (see Remark 1.4) g satisfies (1.10) if and only if (1.11) holds. Moreover, g satisfies (1.21). Therefore, by Theorem 1.3, for any v € m+(dQ), the problem -Luu + \u\p\Vu\q = 0 inQ, (4.1) tr(v) = v admits a positive weak solution. Next we prove the following regularity result. Proposition 4.1. Assumep > 0 and 0 < q < jpj. Ifu is a nonnegative solution of -Lvu + \u\p\Vu\q = 0 in Q, (4.2) then u € C2(Q). Proof. LetD cc Q be a smooth open domain. Since u is a nonnegative solution of (4.1), by(2.10), we can easily obtain 0 < u(x) < K^[v](x) < Co for allx € D. Consequently, |up|Vu|9 < CpD\Vu\q inD. Hence, by invoking [16, Lemma 4.2], we can derive the desired result. □ 4.1 Comparison Principle Lemma 4.2. Let u e C2(Q) be a nonnegative solution of (4.2). If there exists Xn € Q such that u(xn) = 0, then u = 0. Proof. By Young's inequality, |uP|Vu|? < \u\p+q + \Vu\p+q in Q. As a consequence, u satisfies - Luu + \u\p+q + \Vu\p+q > 0 inQ. (4.3) Nowseta(x) := \Vu(x)\p+q-2Vu(x) and b(x) := |u(x)P+«_1. Let jS € (0, jS0) be small enough such that x0 € Dp. Sinceu € C2(Q), there exists a constant such that supxeDp|a(x)| + supxeDp b(x) < Cp. From (4.3), we deduce -Au + a • Vu + bu > ^u > 0 in Dp. Since b(x) > 0, by the maximum principle, u cannot achieve a nonpositive minimum in Dp. Thus the result follows straightforward. □ Next we state the comparison principle for (4.2). Lemma 4.3. Let p > 0, q > 1 and D c Q. We assume that ut,u2 e C2(D) are respectively nonnegative subsolu-tion and positive supersolution of (4.2) in D such that limsup^i-^ 1. xeD mx) By (4.4), we deduce that there exists Xn € D such that Ul(X0) Ux(x) —— = sup —— = m. "2(X0) x€D "2(X) Let r > 0 be such that B(xn, r) c D. Then we see that - A(m"V -u2) + (m~1u1f\m~1Vu1\q -up2\Vu2\q < J^(m~V -u2)<0 inB(x0, (4.5) Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 157 14 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER Now note that (m-1u1f\m-1Vu1\q - u\ \Vu2\q = (m-1u1)p\m-1Vu1\q - (m-1u1f\Vu2\q + (m-1u1f\Vu2\q - up\Vu2\q = a(x)(m"1Vu1 - Vu2) + ^(xXm"1^! - u2), (4.6) where , , -i ^mr^u^ -\Vu2\q, _!„ „ , ax)= (m 'uif-—r---—^-(m 1Vui-Vu2) and „/ (m~1u1)p - uP \ b(x) := \Vu2\q[--^-2- > 0. \ m~1Ui -u2 I Since Ui,u2 e C2(D), u2(x) > 0 for any x € D and g > 1, there exists a positive constant C > 0 such that sup |a(x)| + sup £>(x) < C. «eB(Xo.j) X€£(x0,;) Combining (4.5) and (4.6), we have -Atm-1^ - u2) + a • V(m~1u1 - u2) + ^(m-1^ - u2) < 0 in B^xn, Hence, by the maximum principle, m_1ui - u2 cannot achieve a nonnegative maximum in B(xn, ■§). This is a contradiction. Thus ut 0 satisfies -1)1 v < t tr(v) = 0. Then, for any 1 < k < q^, there exists a constant c = c(N, Q,pi) such that ||Vv||l«(q,s») < c||r||a)i(n,«-). Proof of Theorem 1.5. Since u,- is a nonnegative solution of (4.1), lUiHVuil9 € 5"), i = 1,2. Moreover, from Propositions 2.1 and 2.2, we deduce that ll"illyi(Q,8») + l|VUillL!i(n,«») < CidllUiPlVUil^lliHQ,^) + ||Vi||fr«(dQ)) for any 1 < pi < pv, 1 < 0 in Q. In addition, by Proposition 4.1, u,- € C2(Q) for i = 1,2. Finally, by the representation formula, we have Ui + G^[|UiP|VUi|«] =K^[v,-], i = 1,2. LetO < £ < l.Then (£Ui - u2)+ < (G¥[\u2\p Vu2\q] - £G„[|uiHVui|«])+ < G„[\\u2\p\Vu2\q - |£UiHVui|'|] =: v, which implies tr((£Ui - u2)+) < tr(v) = 0. Hence tr((£Ui - u2)+) = 0. Note that eu\ is a subsolution of (4.2). Also, since u,- € C2(Q) and u2 > 0 in q, it follows that, for small enough p > 0, := sup — < co. x£dp u2 Without loss of generality, we assume that Cp > 1. Set £p = < 1. Then £^Uj - u2 < 0 in D^. Moreover, in view of the proof of Lemma 4.3, we derive that s«u1 - u2 < 0 inD». (4.7) Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 158 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 15 Put Ep := {x € Q : £pui - u2 > 0}. Due to Kato's inequality [29], we get - Lu(£pUl - u2)+ < {up2\Vu2\" - (eu-,f\eVu^)xEf < ((£Ul)P(|Vu2|' - \eVu^))XEr (4.8) By (4.7), we derive that Ep c Qp. Let k > 1 and maxjl N + a 1, small enough such that (£/i,Ui)^5adx) M (|Vu1r? + |Vu2r?)5adx) <-. ^* ^* By the above inequality and (4.9), we obtain V(£p,u! - u2)+ = 0 => (£p,u! - u2)+ = c, for some constant c, > 0, and since (£pt ut - u2)+ = 0 on Dpt, we have c, = 0, namely £pt ut < u2 in Q. As a consequence, Ui(x) Ui(x) Ui(x) _j sup —— = sup —— = sup —— = £g > 1. (4.10) xen u2(x) xeDp< u2(x) X€dDft u2(x) P- This implies the existence of x, e dDpt such that (£ftUi-U2)(x») = 0. (4.11) Next we take p < p,; then £^ < £pt. On the other hand, we infer from (4.10) that £p > £pt and hence £p = £pt. Therefore, (4.11) contradicts (4.7). The proof is complete. □ Lemma 4.5. Letp > 0,1 < q < jpy andp + q > l.Ifuis a nonnegative solution of (4.2), then u(x) < c8(x)~£& + Mp0 for all x € Q, (4.12) |Vu(x)| < C'5(x)"5^ forallxeQ, (4.13) where Mp0 := supofc u, C = C(N,pi, q,p, p0,Mp0) andC = C\N,pi,q,p,p0,Mp0). Proof The proof is similar to that of [16, Lemma 4.6], and hence we omit it. □ Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 159 16 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER 4.2 Isolated Singularities In this section, we assume the origin 0 € dCl and study the behavior near 0 of solutions of (4.2) which vanish on dCl \ {0}. We first establish pointwise a priori estimates for solutions with an isolated singularity at 0, as well as their gradient. Proposition 4.6. Assume 0 € dCl,p >0,q>l,p + q> 1 and p and q satisfy (1.11). Let u be a positive solution of (4.2) in Cl such that lim t77tt = 0 for all $ edQ\{0}, (4.14) xza,x-( W(x) locally uniformly in dCl \ {0}. Here W is defined in (1.17). Then there exists a constant C = C(N, pi, q, p, Cl) > 0 such that u(x) < CS(x)a\x\~^~" forallxeCl, (4.15) |Vu(x)| < C8(x)a-1\x\~'i^~a for all x e Cl. Proof. We split the proof into two steps. Step 1. Let Po be the constant in Proposition 2.6. Let x; € dCl be such that |x;| > y|, dCl c b(o, JJb(x,-, =: A for somen € N. Notice that there exists a constant £o = £o([>o) > 0 such that dist(dyl, dCl) > £q. Let w; be the function constructed in Proposition A.l in B(x,-, y|) for R = yg, i = 1,..., n. Then, by the maximum principle (see [17, Propositions 2.13 and 2.14]), we have u(x) < w;(x) for allx € b(xi, ^-7), i = 1,..., n. 16 As a consequence, there is a constant Co = Cq(JV, pi, q,p,Cl, ßo) > 1 such that u(x) < C0 for all x € |J ß(x,-, || Set v(x):=d(|x|- Po \ where C\ > 0 will be chosen later. We will show that v(x) > u(x) for every x € Cl \ A. Indeed, by a direct computation, we can show that there is a constant Cj > 0 such that, for all x € Cl \ A, _Av = -±^-({N - DIxKlxl - - ^±i-(|x| - q+p-l\ \ 4/ p+q-l\ 4/ / (p + q - l)2 V 4 / |Vv|* = C?(^-W|x|-^y^, (4.17) p + q-1, 2p+q \«1 I'^Hi £ Ci£o (SUPM)2 |x| - ^ . (4.18) Gathering estimates (4.16)-(4.18) leads to, for Ci = Ci(JV, pi,p, q, Pa, Cl) > 0 large enough, 1 »m id A 1 AjV^T „ (2-g)(l+p) _2 2 2-q -Luv + vP\W\q>_{\x\--) [-Cl ^ + g_i)2 -Cl£o2(sup|x|)2 + C^ *(FT?_, >0 forallxeQ\S. Moreover, we can choose Ci = Ci(Co,N,pi, q,p, Cl,[>o) large enough such that lim supx_a(Q\^)(u-v) < 0. By Lemma 4.3, we deduce that u < v in Cl \ A, which implies that u < C2 in Dp0 for some positive constant C2 = C2(N,pi, q,p, Cl,Po)- Thus, by Lemma 4.5, there exists C3 = C}(Cl, N,pi, q,p,Po) > 0 such that 2-g u(x) < CjS(x) forallxeQ. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 160 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 17 Step 2. For£ > 0, put Te[u](x) := e£fcu(ex), x€Qř:=r1Q. (4.19) Let £ € dCl \ {0}, and put d = d(£) := We assume that d < 1. Denote Ud := Td[u]. Then Ud is a solution of (4.2) in Cld = jQ. Let Rq = where /Jo is the constant in Proposition 2.6. Then the solution wo mentioned in Proposition A.l satisfies Ud(y) < w^2!s.(y) for all y € B2io (£) n Qd. Thus Ud is bounded above in B2£o(£) n Qd by a constant C > 0 depending only on N,p.,p, q and the C2 characteristic of Cld (see [29] for the definition of the C2 characteristic of Cl). As d < 1, a C2 characteristic of Q is also a C2 characteristic of Qd; therefore, the constant C can be taken to be independent of We note here that the constant Rq e (0,1) depends on the C2 characteristic of Cl. The rest of the proof can proceed similarly to the proof of [16, Proposition E], and we omit it. □ 4.3 Weak Singularities Proof of Theorem 1.6. We use the same idea as in the proof of [16, Theorem F]. Let u = Ugk be the positive solution of (1.13). By Theorem 1.3 and Lemma 4.2, 0 < u < kK^i •, 0) in Cl. Moreover, u + G^IVul9] =kKy(-,Q). (4.20) This and (2.2) imply that u(x) < kKu(x, 0) < ckS(x)a\x\2-N-2a for allx € Cl. (4.21) By proceeding as in the proof of (4.13), we obtain |Vu(x)| < ci:5(x)a-1|x|2-JV-2a forallxeQ. (4.22) It follows from (2.1), (4.21) and (4.22) that Gu[uP\Vun(x) < ckP+" J 8(y)aP+(a-1)«Gu(x,y)\y\(2-N-2a)(P+«) dy. (4.23) n Case 1: a + ap + (a - l)q > 0. By the assumption and (2.1), we have Gy[ď\Vll\1](x) < ckP+«8(xf ||x_y|2-JV-2a|y|a-(JV+í,-2)p-(JV+í,-l)9dy_ n Since p and q satisfy (1.11), it follows that j|x - y|2-JV-2a|y|a-(JV+a-2)p-(JV+a-l)9 dy < c|x|2-a-(JV+a-2)p-(JV-l+a)9_ n Combining (4.24), (4.25) and (2.2) yields Gy[ď\Vu\l](x) < ckP^xf^-^-^P-^-^K^x, 0). As a consequence, .. G¥[uP\Vu\9](x) hm „ ,---= 0. (4.26) IxHO K„(x,0) Case 2: -1 + a < a + ap + (a - l)q < 0. By (4.23) and (2.1), we have GuluP\Vu\"](x) < ckP*i J 8(y)aP+(a-1)«Fu(x,y)\y\(2-N-2a)(P+«) dy, a where Fu(x,y) := |x-y|2-Jvmin{l,5(x)a5(y)a|x-yr2a} forallx.y € Cl, x # y. (4.27) Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM (4.24) (4.25) 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 161 18 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER LetjS € (0,j30) be such that <5 € C2(Qp). We consider the cut-off function 0 € C°° (Hi") such that 0 <

(P+«>0(y) dy D +| 5(y)aP+(a-1)«F^(x, y)|y|-(Jv+2a-2)(p+9)(1 _ ^(y)) dy_ (4_28) n We first deal with the first term on the right-hand side of (4.28). By the definition of

) such that | SiyfP^-^F^x, y)|y|-(tf+2«-2)(p+«)(1 _ 0(y)) dy < CS(x)a. (4.29) n Now we deal with the second term on the right-hand side of (4.28). Let /J € (0, |) be such that |x - y | > rn > 0 for any y € Q.^ and some rn > 0. Let £ > 0 be such that (JV + a - 2)p + (N + a - l)q = N + a - £, and let £ e (0, £) be such that ap + (a - l)q + 1 - £ > 0. Then, by (4.27), we have | SiyfP^-^F^X, y)|y|-(JV+2a-2)(p+9) ds{y) < g^a^-N-la f 5(y)£ |y|-]V+l+(£-£) ds(y)_ Note that, by the choice of £, N - 2 - N + 1 + (£ - i) > -1, which implies sup Uy\-N+1+ic-l>dS(y)(P+«>0(y) dy = ap + (a1_1)q + 1 I V(5(y)^+C-1>«+1)V5(y)F,(x,y)|yr(JV+2a-2>(P+*0(y) dy < CS(x)a ||X - y|-(JV+2a-2)|y|-(JV+a-2)i'-(JV+a-l)9+a r\y a + CS(x)" ||x_y|-(JV+a-l)|y|-(JV+a-2)p-(JV+a-l)9+l dy a =: M(x) + N(x). (4.32) Since 03 and p and q satisfy (1.11), we infer from (2.2) that max{M(x), N(x)} < C\x\N+a-W+a-2)P-W+a-1)iKy(x, 0)_ (433) Combining (4.23), (4.28), (4.29), (4.32) and (4.33) implies that there exists a positive constant C = C(N,pi,p,q, Q) > 0 such that G„[iP|Vu|«](x) < CJt«|x|JV+a-(JV+a-2)P-(JV+a-1)«^(x,0) forallx € Q. (4.34) Since p and q satisfy (1.11), we deduce (4.26) from (4.34). Thus, from (4.26) and (4.20), we obtain (1.14). Finally, the monotonicity comes from the comparison principle. □ 4.4 Strong Singularities Let S"'1 be the unit sphere in B.N and R+ = {x = (xj,..., Xjy) = (x', Xjy) : Xjy > 0}. We denote by x = (r, a) € IR+ x S*-1 with r = |x| and a = r_1x the spherical coordinates in M.N, and we recall the representation . N~1 l.i Vu = ure + -V u, Au = urr +-ur + -^A u, r r rl where V' denotes the covariant derivative on SN~1 identified with the tangential derivative and A' is the Laplace-Beltrami operator on S*-1. We look for a particular positive solution of -Luu + \u\p\Vu\q = 0 inR^, u = 0 on dTB.^ \ {0} = R*-1 \ {0}, under the separable form u(x) = u(r, a) = r e+i-1 m(a) (r, a) e (0, co) x S" . It follows from a straightforward computation that &i > 0 satisfies - £jv,moj +/(&), V'&i) = 0 inS+~ h) = 0 on dS' JV-l (4.35) Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 163 20 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER where L„oi :=b!a>+ U w, eN,p,q := 2 " q ( 2p + q - n), F (eN-o)2 p+q-l\p+q-l I J(s, 0 := «"(( ^!J2s2 + l^l2)5. (s- r\ Let ku be the first eigenvalue of -Lu in S+_1 and o(cf)I2 = 1 -o5 = 7^o5IV'0ol2 + — 0o5 = T^r? + »4*o- Taking into account that jci - ^1 = ~i> fr°m tne above equalities, we obtain (4.37). Denote Y^Sjf"1) := {0 € ^(S"-1) : 0-0 6 H^S?"1, 02a)}. It is asserted below that condition (1.11) is sharp for the existence of a positive solution of (4.35). Theorem 4.7. Assume p > 0, q > 0 and p + q > 1. (i) // (1.11) does not hold, then there exists no positive solution of (4.35). (ii) // (1.11) holds and q > 1, fhert problem (4.35) admits a unique positive solution ti> e Y^(S+_1). Moreover, there exists a positive constant C = C(N, pi,p,q) such that aWzC-ZZ^Y^W) for alio ^St-\ JV-l \V'(u(a)\ < Cfaio)*? for all a e Proo/. (i) By multiplying (4.35) by 0,,, we obtain (Kv-tN,p,q) | W0„d<7 + | /(w, v'w))0„d<7 = O. (4.38) s»-i s»-i Note that the second term on the left-hand side of (4.38) is nonnegative. Thus, if (1.11) does not hold, or equivalently £s,p,q s ku, then no positive solution of (4.35) exists. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 164 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 21 (ii) The proof is split into two steps. Step 1: Existence. Set Yi Oft I Then the function ZJ = yil + lv>«l2)2 > [yi(^-%M) + «m+?10S = o. In the above estimates, we note that (1.11) implies -^pi > a-Let an € (a, 1) be such that JV + aQ 9a0°, where ]x0 = \-(a0- \)2 < ]i. We allege that there exists a positive constant y2 = yi(N, q, p., pin) < yi such that the function &) = Y2 1, by (4.36) and (4.37), we have - Lum - eN.p.qU + /(w, V'aj) 2-, ^ 2-g ' p + q-1, < (y2(Ho - H) + xr^)0o°"2 + (X2(^0 - + y^K-^-^f - al\2yQ° < 0, provided Y2 is small enough. Notice that we can choose Y2 oi locally uniformly in C^S* 1) and &i satisfies -L^w - £jv,moi + K<>>, Voj) = OinS+ 1. Furthermore, by (4.39), we have &)( 0 such that u,(x) < C,xJ for all x € B(xn, \). Therefore, in view of the proof of (4.40), there exists a positive constant Co such that wt(a) < Co4>0(a)a for all a e S^-1, i = 1,2, (4.41) |V'w,-((7)| < Co0o(cr)'""1 for all a e S^-1, i = 1,2. (4.42) Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 165 22 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER Set bt := infc>i{c : c&jj > h>2, a e St} < 00. Without loss of generality, we may assume that bt0 > 1 for some fo € (0,1); thus, by (4.41), we have lt, h2 = |V&i(|, fci = (j)^1)&)1 and k2 = |V'oii| and keeping in mind estimates (4.41) and (4.42), we obtain - ( — )PJ(o>t, V'wt) +/(&>!, VV) < C(q, Co)0o,'+(?"1)('""1)(|w( - Wil + \V'(ait - o>i)\). (4.46) V (lit ' Now set Vt := ip~1(h)t - &)i)+.By (4.44), (4.46) and the definition of ip, we can easily deduce the existence of a positive constant C = C(N, pi, q, Co) such that -div'(^vv()+m-w) < c(4>rq{a-1)+a\r1^t - woi+0r(?"1)(a"1)+2aiv'(r1(^( - ^»1). Now, since ipVt e Y^(S+_1) and Vt(a) < 0 for any a e S(, multiplying the above inequality by (Vt)+ and integrating over S+_1, we get ||V'(^)+|2t/)2 dS(a) + J W)2(-.C„!/i) dS(u) s, s, < c( J 0^+9(a-1)+a(^)2 dS(rj) + J 0^+(«-1)(a-1)+2a|V'(^)+|(^)+ dS(rj)). (4.47) s, s, By the definition of t/i and (4.43), we have j|V'(Vy+|2t/)2 dS(a) + J W)2(-£„i/<) dS(a) s, s, > i ||V'(^)+|202a dS(o) + |(^)202a+£-2 dS(rj) - ^1 |(^)20^a dS(rj). (4.48) s, s, s, Here we note that if £ < 1 - a, then q < 2 < 2~°~£. This leads to 2-a-£-g(l-a) > 0 and 4 - 2a - £ - 2q(l - a) > 0. (4.49) By Young's inequality, we deduce that C J 0^+(«-1)(a-1)+2a|V'(^)+|(^)+ dS(a) 4 J *o"l v'M)+l2 dS(a) + C J 02^2<«-i><«-i>+2«(1/()2 ds(a)> (4_50) s, s, where C is the constant in (4.47) and C = C(N, pi, p, q). Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 166 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 23 Gathering (4.47), (4.48) and (4.50) yields i J0oa|V'(Vy+|2dS(<7) S' < JLJ± | 4>l™-2(Vt)l dS(o) + Cj ^(^Ha-l)^ + 02ap+2(,-l)(a-l)+2a + fla^l ds((j) s, s, < I 4>la+£~2{^^- + C1(f2+aP-a-£-«(1-a) + f4+2«P-2«-£-2?0-«) + f2-£)^()2 dS((T), S, where Ci = C(iV, ^, p, q). By (4.49) and the above inequality, we can rind a positive constant ti = h(N,q,p,£, Co) such that i | 0ga|V'(^fl)+|2 dS(a) < 0, which implies (Vy+ = 0 inS(l since (Vtl)+ = 0 on {a e S+_1 : 0n(<7) = fih Hence b~t^a>2 < ii>\ for all a e S(l. Thus we have proved that £>f, = inf {c : c&)! > h>2, a € S(,} = inf {c : ca>i > (1)2, a e S*-1}. This means that (a>i - a>tl)(o) > 0 for any a e S+_1 and ^1(^0) - ^fi(^o) = 0 for some (To £ §(l. (4.51) But -£^(&)! - a>tl) - £jv,9(&)i - h)tl) + J((i>i, V'oji) -/(oifj, V&)(l) > 0, which implies -Al(h)1 -cotl)+ ](d>i, V'&ij) -/(&)(, V'&ij) + /(&)(, V'&ij) -/(&)(;, V'&)(l) > 0. By the above inequality, the fact that min(&)!, &i() > 0 in Sii and the mean value theorem, there exists A > 0 such that _ -AVi -o)t,)+ ^(V&Ji - vW) + A(&jj -wf ) > 0 inSa, where s and £are functions of <7 € Si such that ^g-r^ € I°°(Si). By the maximum principle, oil - ait, cannot achieve a nonpositive minimum in Sn \ dSti, which clearly contradicts (4.51). The result follows by exchanging the role of a>i and a>2- □ 5 Absorption g(u, |Vw|) = Vul*7: Supercritical Case Let us recall the following result in [16, 25]. Proposition 5.1. Lefv e 9Jt+(dQ), and let Po be the constant in Proposition 2.6. Then the following inequalities hold: sup 0<£< ip P"-1 [ K„[v] dS < C(p0, a, Q)||v||m(dQ) ifp 0,1 < q < 2, and /ef u € C2(Q) be a nonnegative solution of (4.1). (i) i/g # a + 1, then thereexists a constant Pi = Pi(N,p,p, q, Q) > Osuch that J S"-9uP+9 dx < CQ sauP\Vu\" dx + 1} (5.1) □ □ where C depends only on N, p, p, q, Q and supXf; (K,, [v] Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 167 24 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER (ii) Ifq = a + 1, then, for any £ > 0 small enough, thereexists a constant P\ = f>i(N,pi,p, Cl, £) > Osuch that | 5£-1uP+a+1 dx < c(| SauP\Vu\a+1 dx + l), (5.2) where C depends only on N, p.,p, Cl, £ and supj (KJ1[v])p+',+1. Proof Since u is a nonnegative solution of (4.1) we have uf | Vu| € I«(Q, 5"). Let ft € (0, ft), where ft is the constant in Proposition 2.6. (i) First we assume that q > 1, q ± a+ 1, and let y + -1. Then, for p e (0, ft), | <5>V+' dx = (y + l)"1 | V8Y+1V8uP+i dx Dp\Dh Df,\Dh = (y '+1^+«dx<2C|a + l-V+«dx. Consequently, by Young's inequality, we can find a constant Cj = Ci(N,p.,p, q, Cl) such that Cly+ir1 | 5>'+1^+«-1|Vu|dx = C|y + ir1 j S^1^"1"? uf |Vu| dx Dp\DH Df,\Dh V+«dxa-l,we obtain | 8V+1uP+« dS < Cp^-1 | u dS < Cp^-1 | K„[v] dS -> 0 as £ -> 0. Therefore, by letting /J —» 0 in (5.3), we obtain | 8VuP+« dx < C2( | <5y+,V|Vu|« dx + #+1 sup(K„[v]y+«). (5.4) By the dominated convergence theorem, we can send y —> a - q in (5.4) to obtain | s"-«uP+«dx< C2( | <5au^|Vu|'dx + ^"?+1sup(K^[v])P+'). This implies (5.1). The proof of (5.2) follows by arguments similar to the proof of (5.1) (with y = £ - 1) with some modifications, and we omit it. □ Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 168 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential 25 We recall below some notations concerning the Besov space (see e.g. [1, 33]). For a > 0, 1 < k < co, we denote by W°,K(Rd) the Sobolev space over Rd. If a is not an integer, the Besov space B"'K(B.d) coincides with W'K(Rd). When a is an integer, we denote Ax,y/ := f(x + y) + /(x - y) - 2/(x) and B1,K(WL ) := / € L"(Rd) with norm Then with norm Bm'*(]Rd) := {/ € W m-l,K,md Ax.y/ lyl1+; |A*.y/T \y\K+d € L"(Rd x Rd) j dxdy) (Rd) : D"J € B1'K(]Ra) for all 6 e Nd, |0| = m - 1} I^Xy/T .1,11 lyr - dx dy) These spaces are fundamental because they are stable under the real interpolation method developed by Lions and Petree. For s e R, we denned the Bessel kernel of order s by Gs(£) = 3r_1(l + | • |2)~i3r(4), where J is the Fourier transform of moderate distributions in Rd. The Bessel space Is,K(Rd) is denned by LsATS-") :={f = Gs*g:ge L"(B.d)} with norm ||/1|L„ := ||g||L, = ||G_S */Ileitis known that if 1 < k < coands > 0, Is,K(Rd) = Ws'"(M.d) if s € N, and Is,K(Rd) = Bs'"(M.d) if s $ N, always with equivalent norms. The Bessel capacity is denned for compact subsets K c Rd by cfK(i<) := inf{||/l|^, / € S'(Rd), / It is extended to open sets and then Borel sets by the fact that it is an outer measure. Proof of Theorem 1.7. Let £ > 0, and let u e C2(Q) be the solution of (4.1). Put S = da. If « € I°°(dQ) nB^^^'^'OQ), we denote by H := H[n] the solution of 4^ + = 0 in (0, co) x da, ds H(0,-) = n onda. Let h € C°°(]R+) be such that 0 < h < 1, h' < 0, h = 1 on [0, h = 0 on [jS0, co]. The lifting we consider is expressed by H[n](S2,o(x)M5) ifx€%, 0 ifx€D«n, J?[«](x) : (5.5) with x = (5, fj) = (<5(x), (t(x)). Case 1: q + a + 1. Set £ = Oand ( = ip^Rlrffi*^ , where (pu is the eigenfunction associated to the first eigenvalue Ajj oi-Ljj in a (see Subsection 2.1). By proceeding as in the proof of [17, Lemma 3.8, (3.46)], we deduce that there exists Co = Co(JV, pi, a, ||v||rr«(dQ)) such that a\ (p+q)' c c ndvj < \up\Vu\q(dx + Au\u(dx - (p + ,)'( J u^>--f dx) A (J I[iz]<™>' dx)5*7 (5.6) where I[«] := (2^*p5I"'+'|V^-VJ?[«]|+^ ^~"+'|Ai?MI). Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 169 26 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER Following the arguments of the proof of [17, Lemma 3.9, (3.48)], we can obtain | Lln]^' dx < cIIizII^^IIizIIb.-^ W(M). (5.7) We infer from (5.1) that | u^if^dx < Qr^L) | <5<"- N + a, if C"-«+^.Q>+?>'(K) = 0' then there exists a sequence {nn} in Cg(dQ) with the following properties: 0 0 as rt ■ This implies that v(K) = 0. Thus v is absolutely continuous with respect to C""a' Case 2: q = a + 1. LetO < £ < a + 1 and f = ^J?[^]^+a+1' . Proceeding as in the proof of (5.6), we can prove * (p+a+l)' ndv) < up\Vu\a+1(dx + Au u(dx a a up+a+1 dx\ a a where Using (5.2) and the ideas of the proof of (5.9), we can obtain the inequality a,(p+a+l)' ( ( ndvj < up\Vu\a+1(dx + Au u(dx dn □ □ where the constant C depends on JV, pi, p, Q. and £. The rest of the proof follows by using an argument similar to the first case. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 170 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 27 Proposition 5.3. Letu e C2(CI) be a positive solution of(1.19). If'up\Vu\ e Iq(Cl, 8"), then u possesses a boundary trace v e *XH+(dCl), i.e. u is the solution of boundary value problem (4.1) with boundary trace v. Proof If v := Gu[up\Vu\q], then v e 8") and u + v is a positive Inharmonic function. Hence we have u + v € I1(Q, 8a), and there exists a measure v € 9Jt+(dQ) such that u + v = Kj,[v]. By [16, Proposition 2.2], we obtain the result. □ Proof of Theorem 1.8. In view of the proof of [17, Proposition A.2], we can obtain the estimates |u(x)| < c8(xf dist(x, K)~^~" for all x € CI, |Vu(x)| < c8(x)a~1 dist(x, K)~^~" for all x € CI, where C depends on JV, pi, p, q, CI and supXfc u. Case 1. Assume that q + a + 1 and C^+^__1{p+qy(K) = 0. Then there exists a sequence {nn} in Cl(dCl) satisfying (5.10). In particular, there exists a decreasing sequence {0„} of relatively open subsets of dCl, containing K such that nn = 1 on 0„, and thus nn = 1 on Kn := 0„. We set n„ = l-n„ and („ = |Afi[^„]| < cminfl, 5(x)-4-Jve-(4«W)-2(dist(x,^))2} forallx e Q_ Proceeding as in the proof of [17, Theorem 3.10, (3.65)], we have (uI]1'rn + up\Vu\"'rn)dx = 0. (5.12) Using the expression of IyX„, we derive from (5.12) that | upm\Hn dx = ^(-A^Rlrln]2^1 -Hp + qyRlnnl^^'-^^-VRlrln] ■ 2(p + qyRlnnf^'^ip^nnminn] + (2(p + q)' - l)|Vi?[>2„]|2))udx (I ^+>/f„ dx)"+' (I I[^„]^+«>' dx) < c n where iW = ^"^iv^- vRin„]\ + ^+^"^|aj?[^„]| + ^+^"^|vj?[^„]|2. By proceeding as in the proof of [17, Theorem 3.10, (3.75)], we can prove j\u\p\Vu\yiJRlrln]2(P+l'>' dx < C||;z„||Bi-^W(M)(| 8a-lu"R[r[n]2l' dxf. a a The rest of the proof is similar to the proof of [16, Theorem I], and we omit it. Case 2. Assume that q = a + l and cf^+_j_^+a+1),(K) = 0 for £ as in statement (ii). Then we can obtain the desired result by combining the ideas in Case 1 of this theorem and in Case 2 of Theorem 1.7. □ Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 171 28 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER 6 Nonlinear Equations with Subcritical Source In this section, we prove Theorem 1.9. We first establish an existence result for the case when g is smooth and bounded. Lemma 6.1. Letv e m+(dQ) with ||v||m(dQ) = 1 and g e C'(Rx R+) n I°°(R x Assume (1.10) and (1.21) are satisfied. Then there exists Qo > 0 depending on N,pi,n, ag, ksuch that, for every q e (0, go), the problem -Lyv = g(v + gKy[v],\V(v + gKy[v])\) inn, tr(v) = 0 admits a positive weak solution v satisfying l|v|IC(n,«») + l|Vv|IC(n,«») < fo, where to > 0 depends on N, pi, n, ag, k,p, q. Here ag is defined in (1.10) and k,p,q areas in (1.21). Proof We shall use the Schauder fixed point theorem to show the existence of a positive weak solution of (6.1). Define the operator S by S(v) := G„[g(v + qKv[v], |V(v + qKv[v])\)], v e W^hq, Sa). Fix 1 < k < minjp, q, qy}, Qi(v) := \\v\\C(Q,s-) for v e I#(Q, 5"), Qi(v) Q3(v) Q4(v) H|Vv||C(Q,s.) for|W| eL*(Q,<5a), ; IMb(Q,8") for v € I^Q, 5"), ■■ \m\LHQ,s.) for |W| € I^Q, 5") and Q(v) := Qi(v) + Q2(v) + Q3(v) + Q4(v). Step 1: Estimate the <5a)-norm of g(v + qKu[v], |V(v + gK,,[v])|). For A > 0 and any function w, we use the notation as in (3.12). For the sake of simplicity, when w = v + qKu [v] , we drop the superscript v + qKu[v] in the above notations. For instance, we use the notations Aa and a(A) instead of A^"1"6"^ ^ and av+eK'' m (A). Then, by (2.4), we have a(A) qu > k > 1, there exists go > 0 depending on JV, pi, Cl, k, ag such that, for any q e (0, go), the equation c(tpv + tqv + 2t" + q") = t admits a largest root fn > 0 which depends on JV, pi, Cl, ag, k. Therefore, Q(v) < f0 => Q(S(v)) < f0. (6.8) Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 173 30 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER Step 3. We apply the Schauder fixed point theorem to our setting. By a standard argument, we can show that S: W1'1(Q, Sa) -» W1'1(Q, Sa) is continuous and compact. Set 0 := {£e W1'1(C1,S") : Q(u) < f0}. (6.9) Then 0 is a closed, convex subset of W1'1(C1, <5"), and by (6.8), S(0) c 0. Thus we can apply the Schauder fixed point theorem to obtain the existence of a function v e 0 such that S(v) = v. This means that v is a non-negative solution of (6.1), and hence it holds -j vL^dx = |g(v + eK^[v], \V(v + qKu[v])\)( dx for every <" e X^(Q). □ Proof of Theorem 1.9. Let {g„} be a sequence of C1 nonnegative functions defined on R2. such that g„(0,0) =g(0,0) = 0, g„ 0. As a consequence, {Avn} is uniformly bounded in I1(G). By a standard regularity result for elliptic equations, {vn} is uniformly bounded in Wx,Pl(G) for some p2 > 1. Consequently, there exists a subsequence, still denoted by {vn}, and a function v such that vn —> v a.e. in CI and W„ —» Vv a.e. in CI. Therefore, un —> u a.e. in CI, where u = v + qKu[v] andg„(u„, |Vu„|) —> g(u, |Vu|) a.e. in CI. We show that un —> u inI1(Q, <5"). Since {vn} is uniformly bounded in Lp(CI, <5"), by (2.6), we derive that {un} is uniformly bounded in LP(C1, 5"). Due to Holder's inequality, {un} is equi-integrable in 5"). We invoke Vitali's convergence theorem to derive that un —> u in 5"). Next proceeding as in the proof of (3.11), we obtain thatg„(u„, |Vu„|) —> g(u, |Vu|) in <5"). Therefore, by sending n —> co in each term of (6.10), we obtain - | uL^dx = | g(u, \Vu\)(dx - q | K^L^dx for every <" € xU(C1). □ □ □ This means u is a nonnegative weak solution of (P?v). Therefore, u = G„[g(u, |Vu|)]+eK„[v] inQ, which implies that u > qKu [v] in CI. □ 7 Nonlinear Equations with Supercritical Source 7.1 Capacities and Existence Results In this subsection, we introduce the definition of some capacities and provide related results which will be employed to prove Theorem 1.11 in the next subsection. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 174 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 31 For 0 < 6 < p < N, set Ne,p(x,y) :=--—-—,,9 forall(x,y) eQxii, x # y, (7.1) |x - y\n-p max{|x -y|, <5(x), <5(y)}B Ne,/i[r](x) := | JV9,/i(x, y) dr(y) for all r € (7.2) Q For a > -1, 0 < 8 < p < N and s > 1, define CapJ^ ,. by CapJJjop S(E) := inf jj<5a0sdx : 0 > 0, Ng,p[8aip] >xe} for any Borel set £ c Q. n Here ^£ denotes the indicator function of E. Let Z be a metric space and &i € 9Jt+(Z). Let /: Z x Z —»(0, oo] be a Borel positive kernel such that / is symmetric and /_1 satisfies a quasi-metric inequality, i.e. there is a constant C > 1 such that, for all x, y, z, e Z, 1 / 1 1 \ r(x) := {y € Z : d(x, y) < r} the open d-ball of radius r > 0 and center x. Note that this set can be empty. For h) € 9Jl+(Z), we define the potentials J[&i] and J[0, &i] by JIM(x):= j/(x,y)dw(y) and J[[0, w](x) := J/(x,y)0(y) dw(y). z z For f > 1, the capacity Capj^ in Z is defined by Capj;((E) := infj J0(x)řdúj(x) : 0 > 0, J[0, cj] >xe} for any Borel £ c Z. z Proposition 7.1 ([19]). Lefp> landT,ai € m+(Z) such that |íMdssC|5Mds> (73) 0 0 sup --r^— ds < C -7L-ds (7.4) yEBr(x)J S2 J S2 0 0 for any r > 0, x € Z, where C > 0 is a constant. Then the following statements are equivalent. (1) The equation u = J[up, &i] + 0 small. (2) For any Borel set Eel, it holds J JIte]'' < Ct(E), where te = XeT- (3) For any Borel set E c Z, if ho/ds r(E) < C CapJ* , (£). (4) The inequality J[J[t]p, &)] < CJ[r] < co holds w-a.e. We point out below that Ng^ defined in (7.2) satisfies all assumptions of J in Proposition 7.1. Proposition 7.2 ([9, Lemma 2.2]). JVg,^ is symmetric and satisfies the quasi-metric inequality. Next we give sufficient conditions for (7.3), (7 A) to hold. Proposition 7.3. lef cj = S(x)axn(x) dx with a > -1. Then (7.3) and (7A) hold. Proof. If a > 0, then the statement follows from [9, Lemma 2.3]. We now treat the case -1 < a < 0. We claim that, for any 0 < s < 8 diam(Q) and any x € Q, we have w(Bs(x)) = max{<5(x), s}"sJv. (7.5) Indeed, in order to obtain (7.5), we consider four cases. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 175 32 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER Case 1: 4s < <5(x). Then <5(x) = <5(y) for any y € Bs(x), and the proof of (7.5) can be obtained easily. 4 Case 2: s > ^2. Then <5(y) < 5s; thus | S(y)ady>Csa+N >Cmax{S(x),s}asN Bs (x)na Case 3: ^ < s < 4<5(x). Since Q is smooth, there exists r* > 0 such that [ S(y)a dy < -^-rg+JV for all r0 < — and <5(x,) < X—. (7.6) J + 1 8 ' 4 B,0(Xi)nn Set » <5(x) r0 := r* Then there exist x; € Bs(x), i = 1,..., k, such that Bs(x) c Bro(x,). We note that k does depend neither 32diam(Q) t,..., k, such that on x, nor on <5(x). Thus we have [ S(y)ady 4<5(x). Set ro := r 32diam(Q) Then the proof of (7.5) follows due to an argument similar to Case 3. The rest of the proof can proceed as in the proof of [9, Lemma 2.3], and we omit it. We recall below the definition of the capacity associated to ¥ie,p (see [19]). Definition 7.4. Let a > -1, 0 < 8 < p < N and s > 1. For any Borel set EcD, define Cap^gf s by Cap^s(E) := inf| j 8"4>s dy : 0 > 0, NM[5a0] > xe}-n Clearly, for any Borel set £ c Q, we have Cap^ S(E) = inf j | <5-a 0, Ke,pl4>] >Xe\-n Furthermore, by [1, Theorem 2.5.1], we have (CapJ^s(E)); = inf{w(E) : oi e 9JtJ(Q), IINe./jMll^.g,,) < 1} for any compact set E c Cl, where s' is the conjugate exponent of s. Thanks to Propositions 7.2 and 7.3, we can apply Proposition 7.1 to obtain the following result. Proposition 7.5. Let r € 9Jl+(Q), a > -1, 0 < 8 < p < N, p > 1. Then the following statements are equivalent. (1) For any Borel set E cQ, it holds t(E) < C CapJ^ , (E). (2) The inequality ¥Sg,p[8a¥Sg^[t\p\ < CNe,^[r] < co holds a.e. in Q. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 176 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 33 Recall the capacity Cap^ introduced in [9] which is used to deal with boundary measures. Let0 € (0,JV- 1), and denote by Tig the Bessel kernel in R*-1 with order 0. For s > 1, define CapBo |S(F) := infj j 5s dy : ip > 0, ■Bg *

Xf} for any Borel set F c R*-1. Since Q is a bounded smooth domain in M.N, there exist open sets Oi,..., Om in M.N, diffeomorphisms T(: 0 —> Bi (0) and compact sets Kt,..., Km in dQ such that (i) Ki c Oi, 1 < i < m, and dQ c ij^i Ki< (ii) r,(Oi n dQ) = Bi(0) n {xjv = 0}, r,'(Oi n Q) = B^O) n {Xjv > 0}, (iii) for any x € 0,- n Q, there exists y € 0,- n dQ such that <5(x) = |x - y|. We then define the Capj^-capacity of a compact set F c da by m Cap^(F) := £ CapB9,s(fi(Fn/?,•)), where r,-(F n K,) = fn K,) x {xjv = 0}. The following result is obtained by the same argument as in the proof of [9, Proposition 2.9]. Proposition 7.6. Let a > -1,0 < 0 < p < Nands > 1. Assume that-1 + s'(l + 0-0) < a < -1 + s'(JV + 0 - 0). TTien if ho/ds Cap^ S(E) = Cap^^j s(B) for any Borel E c da. 7.2 Caseg(u, |Vu|) = lunvul' Proof of Theorem 1.11. We see that, under the assumption on p and 0 such that if q e (0, go), then H(£) c £. IMIl? = l|v||y+»(Q,8-»+») + l|Vv||y+,(Q,sp«). (7.8) Define V the space of functions v € IV,1'1 (Q) with the norm We can see that £ c V and £ is convex and closed under the strong topology of V. Moreover, it can be justified that H is a continuous and compact operator. Therefore, by invoking the Schauder fixed point theorem, we conclude that there exists u e £ such that H[u] = u. Therefore, u is a weak solution of problem (P£v) satisfying (1.23) with C = 2Cj. □ Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY 177 34 - K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential DE GRUYTER A Barrier In this section, we will provide a barrier which plays an important role. This barrier will have the same properties as the barrier in [17, Proposition 6.1]. Let f>o be the constant in Proposition 2.6. Proposition A.l. Let Cl c KN be a C2 domain, 0 < pi < j, q > 0 and p + q > 1. Then, for any z e dCl and 0 < R < Yg, there exists a supersolution w := wz,r of (4.2) in CI n Br(z) such that w e C(C1 n Br(z)), w(x) —» co when dist(x, K) —> 0, for any compact subset K c CI n óBr(z), and w vanishes on dCl n Br(z). More precisely, _ Jc(J?2-|x-z|2)-,,5(x)'' for all y e (1 - a, a) ifO)5 ifpi = \, where b is a constant such thatb > max} 1>~+q~]'1 + y, ^j^, 1} and c = c(N, pi,p,q, b, y). Proof. The proof is similar to that of [17, Proposition 6.1] with some minor modifications, and hence we omit it. □ B Case g(u, |Vw|) = \u\p + \Vu\q In this section, we assume that g(u, | Vu|) = \u\p + |Vu|?withp > landl < q < 2. We will state main results for this case without proving since the proofs are similar, even simpler, to those for the caseg(u, |Vu|) = | u P | Vu |9. B.l Absorption Case This subsection is devoted to the study of the equation - Lnu + \u\p + \Vu\q =0 inQ. (B.l) When g(u, |Vu|) = \u\p + \Vu\q with p, q > 1, then g satisfies (1.10) if p and q satisfy (1.12). Moreover, g satisfies (1.21). Hence, if p and q satisfy (1.12), then, for any v € 9Jl+(dQ), the problem -Luu + \u\p + \Vu\q = 0 inQ, (B2) tr(v) = v admits a positive weak solution. Theorem B.l. Assumep and q satisfy (1.12). Let v,- € 9Jl+(dQ), i = 1,2, and let U( be a nonnegative solution of (B.2) with v = Vj. Ifvi < Vj, then Ui < u2 in Q. Set mp,q :=max|p, j^j}-Lemma B.2. Let p > 1 and 1 < q < Ifu is a nonnegative solution of (B.l), then 2 u(x) < CS(x) m"-i-1 forallxefl, |Vu(x)| < CSix)'1^1'1 forallxeQ. Lemma B.3. Let p and q satisfy (1.12). Assume u is a positive solution of (B.l) in Q such that (4.14) holds locally uniformly in dQ \ {0}. Then there exists a constant C = C(N, pi,p,q, Q) such that u(x) < CS(x)a\x\~^^~a forallxefl, |Vu(x)| < CS(x)a~1\x\~l:!A^~a for all x € CI. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM 178 5. EQUATIONS WITH A GRADIENT-DEPENDENT NONLINEARITY DE GRUYTER K. T. Gkikas and P.-T. Nguyen, Elliptic Equations with Hardy Potential - 35 Theorem B.4. Assumeg(u, |Vu|) = \u\p + | Vu|« withp and q satisfying (1.12). (I) Weak singularity. For any k > 0, let Ugk be the solution of (1.13). Then (1.14) holds. Furthermore, the mapping k<-^> Ugk is increasing. (II) Strong singularity. Put Uq ^ := lim/^co Ugk. ThenUg ^is a solution of (1.15). Then there exists a constant c = c(JV, pi,p, q, Cl) > 0 such that c^SixfW"^ < "o.coW s c<5(x)"|xfforallxe Cl, |Vug|0O(x)| < cS(x)a-1\x\~1^~a for all xe Cl. Moreover, \x\^u<0ico(x) = d>(o) locally uniformly on upper hemisphere S+_1 = n SN. Here Hi is the unique positive solution of - £jv,p,a&) +](<>>, V'&i) = 0 in Si'1, cj = 0 ondSl'1, where J(s, 0 : .I1 ... ._ 2 ( 2 (ejV'Cf) mp,q^mp,q I 11-) s2 + ifp<-3-, (s, j^, (s,£) € RtXl*. Theorem B.5. Let v e *XH+(dCl),p > py or qy < q < 2. Assume problem (B.2) admits a weak solution. (I) Ifp > py, then v is absolutely continuous with respect to C*"-1 (II) If qu < q < 2, then the following occurs. (i) If q + a + 1, then v is absolutely continuous with respect to _a ,. (ii) If q = a + 1, then, for any e e (0, minja + 1, ^N~^a - (1 - a)})! v is absolutely continuous with respect tocf:^. Theorem B.6. Assume p>p]lorq]lpu, P1 C^LaJK) = ° ifqu 1 and < 0 < ^a2- ^ssume °"e of the following conditions holds. (i) There exisfs a constant C > 0 such fhaf, /or every Borel set E c dCl, v(E) < CminiCap^^i^^.Cap^^,,,^)}. (ii) There exisfs a positive constant C > 0 such fhaf N2a,2[<5a(P+1)N2a,2[vF] < CN2a,2[v] < co a.e. in Cl, N2a-i,i[<5(a-1)«+aN2a-i,i[v]«] 0 such that if q e (0, go), then problem (P?v) admits a weak solution u satisfying (1.23). Acknowledgment: The authors wish to thank Professor L. Veron for useful discussions. The authors are also grateful to the anonymous referee for the valuable comments which help to improve the manuscript. Funding: P.-T. Nguyen was supported by Czech Science Foundation, project GI19-14413Y. References [i] D. R.Adams and L. I. 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Quasilinear Elliptic Singular Problems, World Scientific, Hackensack, 2017. Brought to you by | Utrecht University Library Authenticated Download Date I 3/5/20 9:22 PM