Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in R N

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1 Available online at Journal of Functional Analysis Existence and multiplicity of solutions to equations of -Laplacian type with critical exponential growth in guyen Lam a, Guozhen Lu b,a, a Department of Mathematics, Wayne State University, Detroit, MI 48202, USA b School of Mathematical Sciences, Beijing ormal University, Beijing , China Received 5 July 2011; accepted 31 October 2011 Available online 16 ovember 2011 Communicated by Fang Hua-Lin Abstract In this paper, we deal with the existence of solutions to the nonuniformly elliptic equation of the form div ax, u + Vx u 2 u = fx,u + εhx 0.1 in when f : R R behaves like expα u / 1 when u and satisfies the Ambrosetti Rabinowitz condition. In particular, in the case of -Laplacian, i.e., ax, u = u 2 u, we obtain multiplicity of weak solutions of 0.1. Moreover, we can get the nontriviality of the solution in this case when ε = 0. Finally, we show that the main results remain true if one replaces the Ambrosetti Rabinowitz condition on the nonlinearity by weaker assumptions and thus we establish the existence and multiplicity results for a wider class of nonlinearity, see Section 7 for more details Elsevier Inc. All rights reserved. Keywords: Ekeland variational principle; Mountain-pass theorem; Variational methods; Critical growth; Moser Trudinger inequality; -Laplacian; Ambrosetti Rabinowitz condition Research is partly supported by a US SF grant DMS * Corresponding author at: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. addresses: nguyenlam@wayne.edu. Lam, gzlu@math.wayne.edu G. Lu /$ see front matter 2011 Elsevier Inc. All rights reserved. doi: /j.jfa

2 . Lam, G. Lu / Journal of Functional Analysis Introduction In this paper, we consider the existence and multiplicity of nontrivial weak solution u W 1, u 0 for the nonuniformly elliptic equations of -Laplacian type of the form: div ax, u + Vx u 2 u = fx,u + εhx in 1.1 where, in addition to some more assumptions on ax,τ and f which will be specified later in Section 2, we have ax,τ c0 h0 x + h 1 x τ 1 for any τ in and a.e. x in, h 0 L / 1 and h 1 L loc R and f satisfies critical growth of exponential type such as f : R R behaves like expα u / 1 when u and when f either satisfies or does not satisfy the Ambrosetti Rabinowitz condition. A special case of our equation in the whole Euclidean space when ax, u = u 2 u has been studied extensively, both in the case = 2 the prototype equation is the Laplacian in R 2 and in the case >3in for the -Laplacian, see for example [10,2,3,26,18,14 16, 5], etc. We should mention that problems involving Laplacian in bounded domains in R 2 with critical exponential growth have been studied in [4,18,8,7,11,29], etc. and for -Laplacian in bounded domains in >2 by the authors of [2,14,26]. The problems of this type are important in many fields of sciences, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. They have been extensively studied by many authors in many different cases: bounded domains and unbounded domains, different behavior of the nonlinearity, different types of boundary conditions, etc. In particular, many works focus on the subcritical and critical growth of the nonlinearity which allows us to treat the problem variationally using general critical point theory. In the case p<, by the Sobolev embedding, the subcritical and critical growth mean that the nonlinearity cannot exceed the polynomial of degree p = p p. The case p = is special, since the corresponding Sobolev space W 1, 0 Ω is a borderline case for Sobolev embeddings: one has W 1, 0 Ω L q Ω for all q 1, but W 1, 0 Ω L Ω. So, one is led to ask if there is another kind of maximal growth in this situation. Indeed, this is the result of Pohozaev [27], Trudinger [32] and Moser [25], and is by now called the Moser Trudinger inequality: it says that if Ω is a bounded domain, then sup u W 1, 0 Ω, u L 1 1 Ω Ω e α u 1 < where α = w and w 1 is the surface area of the unit sphere in. Moreover, the constant α is sharp in the sense that if we replace α by some β>α, the above supremum is infinite. This well-known Moser Trudinger inequality has been generalized in many ways. For instance, in the case of bounded domains, Adimurthi and Sandeep proved in [3] that the following inequality

3 1134. Lam, G. Lu / Journal of Functional Analysis sup u W 1, 0 Ω, u L 1 Ω e α u 1 < holds if and only if α α + β 1 where α>0 and 0 β<. On the other hand, in the case of unbounded domains, B. Ruf when = 2 in [28] and Y.X. Li and B. Ruf when >2in [23] proved that if we replace the L -norm of u in the supremum by the standard Sobolev norm, then this supremum can still be finite under a certain condition for α. More precisely, they have proved the following: u W 1, 0 where sup exp α u / 1 S 2 α, u if α α, {, u L + u L 1 =+ if α>α, 2 S 2 α, u = k=0 α k u k/ 1. k! We should mention that for α<α when = 2, the above inequality was first proved by D. Cao in [10], and proved for >2 by Panda [26] and J.M. do Ó [14,15] and Adachi and Tanaka [1]. Recently, Adimurthi and Yang generalized the above result of Li and Ruf [23] to get the following version of the singular Trudinger Moser inequality see [5]: Lemma 1.1. For all 0 β<, 0 <αand u W 1,, there holds 1 { exp α u / 1 S 2 α, u } <. Furthermore, we have for all α 1 β α and τ>0, sup u 1,τ 1 1 { exp α u / 1 S 2 α, u } < where u 1,τ = u + τ u 1/. The inequality is sharp: for any α>1 β α, the supremum is infinity. Motivated by this Trudinger Moser inequality, do Ó [14,15] and do Ó, Medeiros and Severo [16] studied the quasilinear elliptic equations when β = 0 and Adimurthi and Yang [5] studied the singular quasilinear elliptic equations for 0 β<, both with the maximal growth on the singular nonlinear term fx,u which allows them to treat the equations variationally in a

4 . Lam, G. Lu / Journal of Functional Analysis subspace of W 1,. More precisely, they can find a nontrivial weak solution of mountainpass type to the equation with the perturbation div u 2 u + Vx u 2 u = fx,u + εhx. Moreover, they proved that when the positive parameter ε is small enough, the above equation has a weak solution with negative energy. However, it was not proved in [5] if those solutions are different or not. We should also stress that they need a small nonzero perturbation εhx in their equation to get the nontriviality of the solutions. In this paper, we will study further about the equation considered in the whole space [2,14 16,5]. More precisely, we consider the existence and multiplicity of nontrivial weak solution for the nonuniformly elliptic equations of -Laplacian type of the form: div ax, u + Vx u 2 u = fx,u + εhx 1.2 where ax,τ c0 h0 x + h 1 x τ 1 for any τ in and a.e. x in, h 0 L / 1 and h 1 L loc R. ote that the equation in [5] is a special case of our equation when ax, u = u 2 u. In fact, the elliptic equations of nonuniform type is a natural generalization of the p-laplacian equation and were studied by many authors, see [17,19,31,34,30]. As mentioned earlier, the main features of this class of equations are that they are defined in the whole and with the critical growth of the singular nonlinear term fx,u and the nonuniform nonlinear operator of p-laplacian type. In spite of a possible failure of the Palais Smale compactness condition, in this paper, we still use the mountain-pass approach for the critical growth as in [14,5,15,16] to derive a weak solution and get the nontriviality of this solution thanks to the small nonzero perturbation εhx. In the case of -Laplacian, i.e., ax, u = u 2 u, our equation is exactly the equation studied in [5]: div u 2 u + Vx u 2 u = fx,u + εhx. 1.3 Using the Radial Lemma, Schwarz symmetrization and a modified result of Lions [24] about the singular Moser Trudinger inequality, we will prove that two solutions derived in [5] are actually different. Thus as our second main result, we get the multiplicity of solutions to Eq Our existence result extends that in [5] to the nonuniformly elliptic type of equations. The multiplicity of nontrivial solutions was considered in [16] when β = 0 using a rearrangement inequality which does not hold in general. We will give a substantially different proof here and establish the multiplicity in all the singular cases 0 β<. See Remark 5.2 in Section 5 for more details.

5 1136. Lam, G. Lu / Journal of Functional Analysis Our next concern is about the existence of solution of the equation without the perturbation div u 2 u + Vx u 2 u = fx,u. 1.4 Using an approach as in [14 16], we prove that we don t even require the nonzero perturbation as in [5] to get the nontriviality of the mountain-pass type weak solution. Our main tool in this paper is critical point theory. More precisely, we will use the mountainpass theorem that is proposed by Ambrosetti and Rabinowitz in the celebrated paper [6]. Critical point theory has become one of the main tools for finding solutions to elliptic equations of variational type. We stress that to use the mountain-pass theorem, we need to verify some types of compactness for the associated Lagrange Euler functional, namely the Palais Smale condition and the Cerami condition. Or at least, we must prove the boundedness of the Palais Smale or Cerami sequence [12,13]. In almost all of works, we can easily establish this condition thanks to the Ambrosetti Rabinowitz AR condition, see f 2. However, there are many interesting examples of nonlinear terms f which do not satisfy the Ambrosetti Rabinowitz condition. Thus our next result is that we will show the main results remain true when one replaces the AR condition by weaker assumptions see Section 7. For the -Laplacian equation or polyharmonic operators in a bounded domain in, such a result of existence has been established by the authors in [20] and [22]. We mention in passing that the study of the existence and multiplicity results of nonuniformly elliptic equations of -Laplacian type are motivated by our earlier work on the Heisenberg group [21]. Our assumptions on the potential V are exactly those considered in [14 16,5], namely Vx V 0 > 0in and V 1 L 1. Very recently, Yang has established in [33] when ax, u = u 2 u the multiplicity of solutions when the nonlinear term f satisfies the Ambrosetti Rabinowitz condition and the potential V is under a stronger assumption than ours. More precisely, it is assumed in [33] that V 1 L 1 1 which implies V 1 L 1 when Vx V 0 > 0in. The stronger assumption of integrability on V 1 in [33] guarantees that the embedding E L q is compact for all 1 q<. The argument in [33], as pointed out by the author of [33], depends crucially on this compact embedding for all 1 q<. The assumption on the potential V in our paper only assures the compact embedding E L q for q. evertheless, this compact embedding for q is sufficient for us to carry out the proof of the multiplicity of solutions to Eq. 1.3 and existence of solutions to Eq. 1.4 without the perturbation term. See Proposition 5.2 and Remark 5.2 in Section 5 for more details. The paper is organized as follows: In the next section, we give the main assumptions which are used throughout this paper except the last section and our main results. In Section 3, we prove some preliminary results. Section 4 is devoted to study the existence of nontrivial solutions for the nonuniformly elliptic equations of -Laplacian type 1.2. The multiplicity of nontrivial solutions to Eq. 1.3 is investigated in Section 5. Section 6 is about the existence of nontrivial solutions to the equation without the perturbation 1.4. Finally, in Section 7 we study the results in Sections 5 and 6 without the Ambrosetti Rabinowitz AR condition. 2. Assumptions and main results Motivated by the Trudinger Moser inequality in Lemma 1.1, we consider here the maximal growth on the nonlinear term fx,uwhich allows us to treat Eq. 1.2 variationally in a subspace of W 1,. We assume that f : R R is continuous, fx,0 = 0 and f behaves like

6 . Lam, G. Lu / Journal of Functional Analysis expα u / 1 as u. More precisely, we assume the following growth conditions on the nonlinearity fx,uas in [14 16,5]: f 1 There exist constants,b 1,b 2 > 0 such that for all x, u R +, where 0 <fx,u b 1 u 1 + b 2 [ exp α0 u / 1 S 2,u ], 2 α0 k S 2,u= k! u k/ 1. k=0 f 2 There exists p>such that for all x and s>0, 0 <pfx,s= p s 0 fx,τdτ sf x, s. This is the well-known Ambrosetti Rabinowitz condition. f 3 There exist constants R 0,M 0 > 0 such that for all x and s R 0, Fx,s M 0 fx,s. Since we are interested in nonnegative weak solutions, it is convenient to define fx,u= 0 for all x, u, 0]. 2.1 Let A be a measurable function on R such that Ax, 0 = 0 and ax,τ = Ax,τ τ is a Caratheodory function on R. Assume that there are positive real numbers c 0,c 1,k 1 and two nonnegative measurable functions h 0, h 1 on such that h 1 L loc R, h 0 L / 1, h 1 x 1 for a.e. x in and the following conditions hold: A1 ax,τ c 0 h 0 x + h 1 x τ 1, τ,a.e.x, A2 c 1 τ τ 1 ax,τ ax,τ 1, τ τ 1 τ,τ 1,a.e.x, A3 0 ax,τ.τ Ax,τ τ,a.e.x, A4 Ax,τ k 0 h 1 x τ τ,a.e.x. Then A verifies the growth condition: Ax, τ c0 h0 x τ +h 1 x τ τ, a.e. x. 2.2 ext, we introduce some notations:

7 1138. Lam, G. Lu / Journal of Functional Analysis { E = u W 1, 0 R : u E = h 1 x u + } Vx u <, h 1 x u + 1 1/ k 0 Vx u, u E, { u } E λ 1 = inf u : u E \{0}. We also assume the following conditions on the potential as in [14 16,5]: V 1 V is a continuous function such that Vx V 0 > 0 for all x, we can see that E is a reflexive Banach space when endowed with the norm u E = and for all q<, with continuous embedding. Furthermore, h 1 x u + 1 1/ k 0 Vx u E W 1, L q { u } E λ 1 = inf u : u E \{0} In order to get the compactness of the embedding > 0 for any 0 β<. 2.3 E L p for all p we also assume the following conditions on the potential V : V 2 Vx as x ; or more generally, for every M>0, or V 3 The function [Vx] 1 belongs to L 1. μ { x : Vx M } <, ow, from f 1, we obtain for all x, u R, Fx,u b 3 [ exp α1 u / 1 S 2 α 1,u ]

8 . Lam, G. Lu / Journal of Functional Analysis for some constants α 1, b 3 > 0. Thus, by Lemma 1.1, we have Fx,u L 1 for all u W 1,. Define the functionals J,J ε : E R by 1 J ε u = Ax, u + Ju= 1 u + 1 Vx u Vx u Fx,u Fx,u ε hu, then the functionals J,J ε are well defined by Lemma 1.1. Moreover, J,J ε are the C 1 functional on E and u, v E, DJ ε uv = DJ uv = ax, u v+ u 2 u v+ Vx u 2 v Vx u 2 v ote that in the case of -Laplacian: Ax, τ = 1 τ, we choose fx,uv fx,uv ε. hv, We next state our main results. ax,τ = τ 2 τ, k 0 = 1, h 1x = 1. Theorem 2.1. Suppose that V 1 and V 2 or V 3 and f 1 f 2 are satisfied. Furthermore, assume that Fx,s f 4 lim sup s 0+ k 0 s <λ 1 uniformly in x. Then there exists ε 1 > 0 such that for each 0 <ε<ε 1, problem 1.2 has a nontrivial weak solution of mountain-pass type. Theorem 2.2. Suppose that V 1 and V 2 or V 3 and f 1 f 3 are satisfied. Furthermore, assume that Fx,s f 4 lim sup s 0+ s <λ 1 uniformly in x. and there exists r>0 such that f 5 lim s sf x, s exp s / 1 > 1 [ r β β eα d β/ + Cr β r β β ] β 1 > 0

9 1140. Lam, G. Lu / Journal of Functional Analysis uniformly on compact subsets of where d and C will be defined in Section 3. Then there exists ε 2 > 0, such that for each 0 <ε<ε 2, problem 1.3 has at least two nontrivial weak solutions and one of them has a negative energy. Theorem 2.3. Under the same hypotheses in Theorem 2.2, the problem without the perturbation 1.4 has a nontrivial weak solution. As we remarked earlier in the introduction, the main theorems above remain to hold when the nonlinear term f satisfies weaker assumptions than the Ambrosetti Rabinowitz condition. As a result, we then establish the existence and multiplicity of solutions when the nonlinear term is in a wider class. See Section 7 for more details. 3. Preliminary results First, we recall what we call the Radial Lemma see [9,16] which asserts: ux u ω 1 x, x R \{0} for all u W 1, radially symmetric. Using this Radial Lemma, we can prove the following two lemmas with an easy adaptation from Lemma 2.2 and Lemma 2.3 in [16] for β = 0 and Lemma4.2in[5]. Lemma 3.1. For κ>0, 0 β<and u E M with M sufficiently small and q>,we have [expκ u / 1 S 2 κ, u] u q C,κ u q E. Lemma 3.2. Let κ>0, 0 β<, u E and u E M such that M / 1 <1 β α κ, then for some p >. ext, we have the following [expκ u / 1 S 2 κ, u] u C,M,κ u p Lemma 3.3. Let {w k } W 1, Ω where Ω is a bounded open set in, w k L Ω 1. If w k w 0 weakly and almost everywhere, w k w almost everywhere, then exp{α w k / 1 } is bounded in L 1 Ω for 0 <α<1 β α 1 w L Ω 1/ 1. Proof. Using the Brezis Lieb Lemma in [9], we deduce that w k L Ω w k w L Ω w L Ω.

10 . Lam, G. Lu / Journal of Functional Analysis Thus for k large enough and δ>0 small enough: 0 <α1 + δ w k w / 1 L Ω <α 1 β. By the singular Trudinger Moser inequality on bounded domains [3], we get the conclusion. In the next two lemmas we check that the functional J ε satisfies the geometric conditions of the mountain-pass theorem. Then, we are going to use a mountain-pass theorem without a compactness condition such as the one of the PS type to prove the existence of the solution. This version of the mountain-pass theorem is a consequence of Ekeland s variational principle. Lemma 3.4. Suppose that V 1, f 1 and f 4 hold. Then there exists ε 1 > 0 such that for 0 <ε<ε 1, there exists ρ ε > 0 such that J ε u > 0 if u E = ρ ε. Furthermore, ρ ε can be chosen such that ρ ε 0 as ε 0. Proof. From f 4, there exist τ,δ > 0 such that u δ implies Fx,u k 0 λ1 τ u 3.1 for all x. Moreover, using f 1 for each q>, we can find a constant C = Cq,δ such that Fx,u C u q[ exp κ u / 1 S 2 κ, u ] 3.2 for u δ and x. From 3.1 and 3.2 we have Fx,u k 0 λ1 τ u + C u q[ exp κ u / 1 S 2 κ, u ] for all x, u R. ow,bya4, Lemma 3.2, 2.3 and the continuous embedding E L, we obtain J ε u k 0 u E k 0 λ1 τ u C u q E ε h u E k 0 1 λ 1 τ u E λ 1 C u q E ε h u E. Thus [ J ε u u E k 0 1 λ 1 τ u 1 E C u q 1 E ε h ]. 3.3 λ 1 Since τ>0and q>, we may choose ρ>0 such that k 0 1 λ 1 τ λ 1 ρ 1 Cρ q 1 > 0. Thus, if ε is sufficiently small then we can find some ρ ε > 0 such that J ε u > 0if u =ρ ε and even ρ ε 0asε 0.

11 1142. Lam, G. Lu / Journal of Functional Analysis Lemma 3.5. There exists e E with e E >ρ ε such that J ε e < inf u =ρε J ε u. Proof. Let u E \{0}, u 0 with compact support Ω = suppu. Byf 2, wehavethatfor p>, there exists a positive constant C>0 such that s 0, x Ω: Fx,s cs p d. 3.4 Then by 2.2, we get J ε tu Ct Ω h 0 x u + Ct u E Ctp Ω u p + C + εt Ω hu. Since p>,wehavej ε tu as t. Setting e = tu with t sufficiently large, we get the conclusion. ow, we define the Moser Functions which have been frequently used in the literature see, for example, [14,16,5]: m l x, r = 1 ω 1/ 1 log l 1/ if x r l, log x r if r log l 1/ l x r, 0 if x r. We then immediately have m l., r W 1,, the support of m l x, r is the ball B r, and m l x, r = 1, and m l W 1, = ! log l r + o l Then m l E 1 + max x r Vx 1! log l r + o l 1. Consider m l x, r = m l x, r/ m l E, then we can write m / 1 l x, r = ω 1/ 1 1 log l + d l for x r/l. 3.6 Using 3.5, we conclude that m l 1asl. Consequently, d l log l 0 asl, d = lim inf l d l, d max x r Vxω 1/ 1 1 2! r. 3.7

12 . Lam, G. Lu / Journal of Functional Analysis ext we will adapt the idea from J.M. do Ó s works [14,16] when no singular term is present to establish the minimax level in our case. See also [21] for a similar result on the Heisenberg group. Lemma 3.6. Suppose that V 1 and f 1 f 5 hold. Then there exists k such that { t max t 0 Fx,tm k } < 1 β Proof. Choose r>0 as in the assumption f 5 and β 0 > 0 such that where lim sf x, s exp s / 1 s β 0 > 1 [ r β β eα d β/ + Cr β r β β ] α 1. β 1, 3.8 C = lim k ζ k log k ζ 1 k C 1 e βlog n. β 0 exp [ βlog k s / 1 ζ k s ] ds > 0, ζ k = m k, Suppose, by contradiction, that for all k we get { t max t 0 Fx,tm k } 1 β α 1 where m k x = m k x, r. By 3.4, for each k there exists t k > 0 such that tk Fx,t k m k { t = max t 0 Fx,tm k }. Thus tk Fx,t k m k 1 β α 1. From Fx,u 0, we obtain t k β α

13 1144. Lam, G. Lu / Journal of Functional Analysis Since at t = t k we have d t dt Fx,tm k = 0 it follows that t k = fx,t k m k t k m k = x r t k m k fx,t k m k Using hypothesis f 5, givenτ>0 there exists R τ > 0 such that for all u R τ and x r, we have From 3.10 and 3.11, for large k, we obtain uf x, u β 0 τexp u / t k β 0 τ x r k exp t k m k / 1 Thus, setting we have = β 0 τ ω 1 β L k = log k α r β exp t / 1 k k t / 1 k ω 1/ 1 1 log k + t / 1 k d k. + t / 1 k d k log t k βlog k 1 β 0 τ ω 1 β r β exp L k. Consequently, the sequence t k is bounded. Otherwise, up to subsequences, we would have lim k L k = which leads to a contradiction. Moreover, by 3.7, 3.9 and t k β 0 τ ω 1 β r β exp [ t / 1 ] k β log k + t / 1 k d k α it follows that t k k β α Set

14 . Lam, G. Lu / Journal of Functional Analysis From 3.10 and 3.11 we have t k β 0 τ A k ={x B r : t k m k R τ } and B k = B r \ A k. x r β 0 τ B k exp t k m k / 1 + B k t k m k fx,t k m k exp t k m k / otice that m k x 0 and the characteristic functions χ Bk 1 for almost everywhere x in B r. Therefore the Lebesgue dominated convergence theorem implies B k t k m k fx,t k m k 0 and B k exp t k m k / 1 ω 1 β r β. Moreover, using that t k k β α 1 we have x r = exp t k m k / 1 x r x r/k + expα m k / 1 β/ r/k x r expα m k / 1 β/ expα m k / 1 β/ and

15 1146. Lam, G. Lu / Journal of Functional Analysis x r/k = expα m k / 1 β/ x r/k exp[α ω 1/ 1 1 log k β/ + d k α β/] = ω 1 r β k β+α d k log k β/ β k = ω 1 β r β k α ow, using the change of variable d k log k β/. x = log r s ζ k log k with ζ k = m k by straightforward computation, we have r/k x r expα m k / 1 β/ = ω 1 r β ζ k log k ζ 1 k which converges to Cω 1 r β as k where 0 exp [ βlog k s / 1 ζ k s ] ds C = lim k ζ k log k ζ 1 k 0 exp [ βlog k s / 1 ζ k s ] ds > 0. Finally, taking k in 3.13, using 3.12 and using 3.7 see [14,16], we obtain β which implies that α 1 [ ω 1 β 0 τ β r β e α d β/ + Cω 1 r β ω ] 1 β r β β 0 1 [ r β β eα d β/ + Cr β r β β ] β 1. This contradicts to 3.8, and the proof is complete.

16 . Lam, G. Lu / Journal of Functional Analysis The existence of solution for the problem 1.2 It is well known that the failure of the PS compactness condition creates difficulties in studying this class of elliptic problems involving critical growth and unbounded domains. In next several lemmas, instead of PS sequence, we will use and analyze the compactness of Cerami sequences of J ε. Lemma 4.1. Let u k E be an arbitrary Cerami sequence of J ε, i.e., J ε u k c, 1 + uk E DJ ε u k E 0 as k. Then there exists a subsequence of u k still denoted by u k and u E such that fx,u k fx,u strongly in L 1 loc R, u k x ux almost everywhere in, ax, u k ax, u weakly in L / 1 loc R, u k u weakly in E. Furthermore u is a weak solution of 1.2. For simplicity, we will only sketch the proof where includes the nonuniform terms ax, u and Ax, u. Proof of Lemma 4.1. Let v E, then we have 1 Ax, u k + Vx u k Fx,u k ε hu k k c 4.1 and DJ ε u k v = ax, u k v+ Vx u k 2 u k v τ k v E 1 + u k E fx,u k v ε hv 4.2 where τ k 0ask. Choosing v = u k in 4.2 and by A3, we get fx,u k u k + ε hu k Ax, u k Vx u k 2 u k u k E τ k 1 + u k E 0. This together with 4.1, f 2 and A4 leads to

17 1148. Lam, G. Lu / Journal of Functional Analysis and hence u k E is bounded and thus p 1 u k E C 1 + u k E fx,u k u k C, Fx,u k C. 4.3 Thanks to the assumptions on the potential V, the embedding E L q is compact for all q, by extracting a subsequence, we can assume that Thanks to Lemma 2.1 in [18], we have u k u weakly in E and for almost all x. fx,u n fx,u in L 1 loc R. 4.4 ext, up to a subsequence, we can define an energy concentration set for any fixed δ>0, { Σ δ = x : lim lim r 0 k B r x uk + u k } δ. Since u k is bounded, Σ δ must be a finite set. Adapting an argument similar to [5] we omit the details here, we can prove that for any compact set K \ Σ δ, lim k K fx,u k u k fx,uu = ext we will prove that for any compact set K \ Σ δ, lim k K u k u = It is enough to prove for any x \ Σ δ, and B r x,r \ Σ δ, there holds lim k B r/2 x u k u = For this purpose, we take φ C 0 B rx with 0 φ 1 and φ = 1onB r/2 x. Obviously φu k is a bounded sequence. Choose h = φu k and h = φu in 4.2, then we have:

18 B r x. Lam, G. Lu / Journal of Functional Analysis φ ax, u k ax, u u k u B r x + B r x ax, u k φu u k φax, u u u k + + τ k φu k E + τ k φu E ε B r x B r x φhu k u. φu k u fx,u k ote that by Holder s inequality and the compact embedding of E L Ω, we get lim k B r x Since u k u and u k u, there holds lim k B r x ax, u k φu u k = φax, u u u k = 0 and lim k B r x φhu k u = This implies that lim k B r x φu k uf x, u k = 0. So we can conclude that lim k B r x φ ax, u k ax, u u k u = 0 and hence we get 4.7 by A2. Thus we have 4.6 by a covering argument. Since Σ δ is finite, it follows that u k converges to u almost everywhere. This immediately implies, up to a subsequence, ax, u k ax, u weakly in L / 1 loc 2. Using all these facts, letting k tend to infinity in 4.2 and combining with 4.4, we obtain DJε u, v = 0 v C 0 R. This completes the proof of the lemma. ow, we are ready to prove Theorem 2.1. The existence of the solution of 1.2 follows by a standard mountain-pass procedure.

19 1150. Lam, G. Lu / Journal of Functional Analysis The proof of Theorem 2.1 Proposition 4.1. Under the assumptions V 1 and V 2 or V 3, and f 1 f 4, there exists ε 1 > 0 such that for each 0 <ε<ε 1, the problem 1.2 has a solution u M via mountain-pass theorem. Proof. For ε sufficiently small, by Lemmas 3.4 and 3.5, J ε satisfies the hypotheses of the mountain-pass theorem except possibly for the PS condition. Thus, using the mountain-pass theorem without the PS condition, we can find a sequence u k in E such that J ε u k c M > 0 and 1 + u k E DJε u k 0 where c M is the mountain-pass level of J ε. ow, by Lemma 4.1, the sequence u k converges weakly to a weak solution u M of 1.2 in E. Moreover, u M 0 since h The multiplicity results of the problem 1.3 In this section, we deal with the problem 1.3. ote that this is the special case of the problem 1.2 with Ax, τ = τ. Some preliminary lemmas in the case β = 0 were treated in [14,16]. We have included details here. The key ingredient of this section is the proof of Proposition 5.2 which is substantially different from those in [14,16]. Lemma 5.1. There exist η>0 and v E with v E = 1 such that J ε tv < 0 for all 0 <t<η. In particular, inf u E η J ε u < 0. Corollary 5.1. Under the hypotheses V 1 and f 1 f 5,ifε is sufficiently small then { t max J εtm k = max t 0 t 0 Fx,tm k t } εhm k < 1 β α 1. ote that we can conclude by inequality 3.3 and Lemma 5.1 that <c 0 = inf J ε u < u E ρ ε ext, we will prove that this infimum is achieved and generate a solution. In order to obtain convergence results, we need to improve the estimate of Lemma 3.6. Corollary 5.2. Under the hypotheses V 1 and f 1 f 5, there exist ε 2 0,ε 1 ] and u W 1, with compact support such that for all 0 <ε<ε 2, J ε tu < c β α 1 for all t 0.

20 . Lam, G. Lu / Journal of Functional Analysis Proof. It is possible to increase the infimum c 0 by reducing ε. By Lemma 3.4, ρ ε 0. ε 0 Consequently, c 0 0. Thus there exists ε 2 > 0 such that if 0 <ε<ε 2 then, by Corollary 5.1, we have max J εtm k <c β t 0 α 1. ε 0 Taking u = m k W 1,, the result follows. Lemma 5.2. If u k is a Cerami sequence for J ε at any level with β lim inf u k E < k α 1/ 5.2 then u k possesses a subsequence which converges strongly to a solution u 0 of 1.3. Proof. See Lemma 4.6 in [5] Proof of Theorem 2.2 The proof of the existence of the second solution of 1.3 follows by a minimization argument and Ekeland s variational principle. Proposition 5.1. There exists ε 2 > 0 such that for each ε with 0 <ε<ε 2,Eq.1.3 has a minimum type solution u 0 with J ε u 0 = c 0 < 0, where c 0 is defined in 5.1. Proof. Let ρ ε be as in Lemma 3.4. We can choose ε 2 > 0 sufficiently small such that β ρ ε < α 1/. Since B ρε is a complete metric space with the metric given by the norm of E, convex and the functional J ε is of class C 1 and bounded below on B ρε, by Ekeland s variational principle there exists a sequence u k in B ρε such that Observing that J ε u k c 0 = inf J ε u and DJε u k 0. u E ρ ε β u k E ρ ε < α 1/ by Lemma 5.2 it follows that there exists a subsequence of u k which converges to a solution u 0 of 1.3. Therefore, J ε u 0 = c 0 < 0.

21 1152. Lam, G. Lu / Journal of Functional Analysis Remark 5.1. By Corollary 5.2, we can conclude that 0 <c M <c β α 1. Proposition 5.2. If ε 2 > 0 is sufficiently small, then the solutions of 1.4 obtained in Propositions 4.1 and 5.1 are distinct. Remark 5.2. Before we give a proof of the proposition, we like to make some remarks. We note the following Hardy Littlewood inequality holds for nonnegative functions f and g in : fxgx f xg x where f and g are symmetric and decreasing rearrangement of f and g respectively. However, the following inequality, which has been used in [16] to derive the multiplicity of nontrivial solutions in the case of β = 0, fxgx f xg x, does not hold for all R>0 in general. Therefore, we will avoid using the symmetrization argument when we prove Fx,v k Fx,u 0. evertheless, this can be taken care by a double truncation argument in both cases of β = 0 and 0 <β<. This argument differs from those given in [14 16,33]. Using this argument, the compact embedding E L q for q is sufficient and thus establish the multiplicity of nontrivial solutions under our assumptions on the potential V. Proof. By Propositions 4.1 and 5.1, there exist sequences u k, v k in E such that and u k u 0, J ε u k c 0 < 0, DJ ε u k u k 0 v k u M, J ε v k c M > 0, DJ ε v k v k 0, v k x u M x almost everywhere in. ow, suppose by contradiction that u 0 = u M. As in the proof of Lemma 4.1 we obtain fx,v k fx,u 0 in L 1 B R for all R>0. 5.3

22 . Lam, G. Lu / Journal of Functional Analysis Moreover, by f 2, f 3 Fx,v k R 0fx,v k + M 0fx,v k so by the Generalized Lebesgue s Dominated Convergence Theorem, Fx,v k Fx,u 0 in L 1 B R. We will prove that Fx,v k Fx,u 0. It s sufficient to prove that given δ>0, there exists R>0 such that Fx,v k Fx,u 0 3δ and 3δ. To prove it, we recall the following facts from our assumptions on nonlinearity: there exists c>0 such that for all x, s R + : Fx,s c s + cf x, s, Fx,s c s + cr,ss, fx,v k v k C, Fx,v k C. 5.4 First, we will prove it for the case β>0. We have that v k >A Fx,v k c v k + c c R β v k E + c 1 A v k >A fx,v k v k fx,v k Since v k E is bounded and using 5.4, we can first choose A such that c 1 fx,v k v k A < δ for all k and then choose R such that.

23 1154. Lam, G. Lu / Journal of Functional Analysis c R β v k E <δ which thus v k >A Fx,v k 2δ. ow, note that with such A,wehavefor s A: Fx,s c s + cr,ss c s α j 0 + c s [c + c C,A s. j! s j/ 1+1 j= 1 α j 0 j! Aj/ 1+1 j= 1 ] So we get v k A Fx,v k C,A R β v k A C,A R β v k E. v k Again, note that v k E is bounded, we can choose R such that v k A Fx,v k δ. In conclusion, we can choose R>0 such that Fx,v k 3δ. Similarly, we can choose R>0 such that Fx,u 0 3δ.

24 . Lam, G. Lu / Journal of Functional Analysis ow, if β = 0, similarly, we have Fx,v k c v k >A c A v k + c v k >A v k +1 + c 1 A c A v k +1 E + c 1 A fx,v k fx,v k v k so since v k E is bounded and by 5.4, we can choose A such that Fx,v k 2δ. v k >A ext, we have Fx,v k C,A v k A v k A 2 1 C,A v k v k A v k u 0 + fx,v k v k v k A u 0. ow, using the compactness of embedding E L q, q and noticing that v k u 0, again we can choose R such that Fx,v k δ. v k A Combining all the above estimates, we have the fact that Fx,v k Fx,u 0 since δ is arbitrary and 5.3 holds. From this, we have lim v k = c M lim Vx v k + k k Fx,u 0 + ε hu

25 1156. Lam, G. Lu / Journal of Functional Analysis ow, let w k = v k v k and w 0 = u 0 lim k v k we have w k = 1 for all k and w k w 0 in D 1,, the closure of the space C 0 R endowed with the norm ϕ. In particular, w 0 1 and w k BR w 0 BR in W 1, B R for all R>0. We claim that w 0 < 1. Indeed, if w 0 = 1, then we have lim k v k = u 0 and thus v k u 0 in W 1, since v k u 0 in L q, q. So we can find g W 1, for some q such that v k x gx almost everywhere in. From assumption f 1, wehaveforsome α 1 > that fx,ss b1 s + C [ exp α 1 s / 1 S 2 α 1,s ] for all x, s R. Thus, fx,v k v k v k b 1 + C [expα 1 v k / 1 S 2 α 1,v k ] b 1 v k k + C [expα 1 g / 1 S 2 α 1,g] almost everywhere in. ow, by Lebesgue s dominated convergence theorem, fx,v k v k fx,u 0 u 0 lim =. Similarly, since u k u 0 in E,wealsohave fx,u k u k lim k = fx,u 0 u 0. ow, noting that DJ ε u k u k = u k E fx,u k u k εhu k 0 and we conclude that DJ ε v k v k = v k E fx,v k v k εhv k 0 lim k v k E = lim k u k E = u 0 E and thus J ε v k J ε u 0 = c 0 < 0 and this is a contradiction.

26 . Lam, G. Lu / Journal of Functional Analysis So w 0 < 1. Using Remark 5.1 we have and thus c M J ε u 0 < 1 < β β α 1 α [c M J ε u 0 ] 1/ 1. ow if we choose q>1 sufficiently close to 1 and set Lw = c M 1 Vx w + Fx,w + ε hw then for some δ>0, q v k / 1 β = β α v k / 1 [c M J ε u 0 ] 1/ 1 δ α Lv k 1/ 1 + o k 1 [c M J ε u 0 ] 1/ 1 δ. ote that lim Lv 1 k = c M lim Vx v k + k k Fx,u 0 + ε hu 0 + o k 1 and 1 c M lim Vx v k + k c M J ε u 0 so for k,r sufficiently large, Fx,u 0 + ε hu 0 1 w 0 q v k / 1 β α [1 w 0 δ. L ]1/ 1 B R By Lemma 3.3, note that w k w 0 almost everywhere since v k x u M x = u 0 x almost everywhere in : B R expq v k / 1 w k / 1 C. 5.6

27 1158. Lam, G. Lu / Journal of Functional Analysis By f 1 and Holder s inequality, fx,v k v k u 0 b 1 b 1 + b 2 v k 1 v k u 0 + b 2 v k 1/ B R v k u 0 q 1/q where q = q/q 1. By 5.6, we have v k u 0 exp v k / 1 v k u 0 B R 1/ expq v k / 1 w k / 1 1/q fx,v k v k u 0 C 1 v k u 0 / v k u 0 + C 2 q. /q Using the Holder inequality and the compact embedding E L q, q, we get v k u 0 = x <1 0 x <1 v k u 0 + x 1 1 1/s s ask x <1 v k u 0 1/s v k u 0 s + v k u 0 for some s>1 sufficiently close to 1. Similarly, v k u 0 q k 0. Thus we can conclude that v k 2 v k v k u 0 + Vx v k 2 v k v k u 0 0 since DJ ε v k v k u 0 0.

28 . Lam, G. Lu / Journal of Functional Analysis On the other hand, since v k u 0 u 0 2 u 0 v k u 0 0 and we have Vx u 0 2 u 0 v k u 0 0 v k u 0 + Vx v k u 0 R C 1 vk 2 v k u 0 2 u 0 vk u 0 + C 2 Vx vk 2 v k u 0 2 u 0 vk u 0 where we did use the inequality x 2 x y 2 yx y 2 2 x y. So we can conclude that v k u 0 in E. Thus J ε v k J ε u 0 = c 0 < 0. Again, this is a contradiction. The proof is thus complete. 6. The existence result to the problem 1.4 In this section, we deal with the problem 1.4. The main result of ours shows that we don t need a nonzero small perturbation in this case to guarantee the existence of a solution Proof of Theorem 2.3 It s similar to the proof of Theorems 2.1 and 2.2. We can find a sequence v k in E such that Jv k c M > 0 and 1 + v k E DJv k 0 where c M is the mountain-pass level of J. ow, by Lemma 4.1, the sequence v k converges weakly to a weak solution v of 1.4 in E. ow, suppose that v = 0. Similarly as in the proof of Proposition 5.2, we have that: Fx,v k So lim v k E = lim Jv k + k k Fx,v k = C M.

29 1160. Lam, G. Lu / Journal of Functional Analysis ote that by Lemma 3.6, we have 0 <C M < 1 β α α0 1,so Thus by f 1,wehave β lim sup v k E < k α 1/. fx,v k v k vk b 1 + b R,v k v k 2. ote that b 1 vk + b 2 R,v k v k 0 since by Lemma 3.2 and by the compact embedding E L s, s, R,v k v k v k CM, v k s 0. Moreover, x 1 v k 0 again by the compact embedding E L and vk x 1 C v k r 0 by Holder s inequality and by the compact embedding E L s, s. So we can conclude that which thus lim k v k E = lim k nontriviality of the solution. fx,v k v k 0 fx,v kv k = 0 and it s impossible. So we get the 7. Existence and multiplicity without the Ambrosetti Rabinowitz condition The main purpose of this section is to prove that all of the results of existence and multiplicity in Sections 5 and 6 hold even when the nonlinear term f does not satisfy the Ambrosetti Rabinowitz condition. It is not difficult to see that there are many interesting examples of such f which do not satisfy the Ambrosetti Rabinowitz condition, but satisfy our weaker conditions listed below. The existence of nontrivial solutions to a class of nonlinear equations of -Laplace type [20] or polyharmonic operators [22] on bounded domains in has been established when the nonlinear term satisfies the exponential growth but without satisfying the Ambrosetti Rabinowitz condition. In this section, instead of conditions f 2 and f 3, we assume that f 2 Hx,t Hx,s for all 0 <t<s, x where Hx,u= uf x, u Fx,u. f 3 There exists c>0 such that for all x, s R + : Fx,s c s + cf x, s. f 4 lim u Fx,u u = uniformly on x.

30 . Lam, G. Lu / Journal of Functional Analysis We should stress that f 1 + f 3 will imply f 3. The key to establish the results in earlier sections is to prove that the Cerami sequence [12, 13] associated to the Lagrange Euler functional is bounded. Once we will have proved this, the remaining should be the same as in previous sections. Therefore, we only include the proof of this essential ingredient in this section. Lemma 7.1. Let {u k } be an arbitrary Cerami sequence associated to the functional Iu= 1 u Fx,u such that 1 u k Fx,u k C M, 1 + uk u k 1 u k v+ Vx u k 1 u k v ε k 0, fx,u k v ε k v, where C M 0, 1 1 β α α0 1. Then {u k } is bounded up to a subsequence. Proof. Suppose that Set u k. 7.1 v k = u k u k then v k =1. We can then suppose that v k v in E up to a subsequence. We may similarly show that v k + v+ in E, where w + = max{w,0}. Thanks to the assumptions on the { potential V, the embedding E L q is compact for all q. So, we can assume that v + k x v + x a.e. in, v k + v+ in L q, q. We wish to show that v+ = 0a.e.. Indeed, if S + ={x : v + x > 0} has a positive measure, then in S +,wehave and thus by f 4 : lim k u+ k x = lim k v+ k x u k =+ Fx,u + k lim x k u + =+ a.e. in S+. k x

31 1162. Lam, G. Lu / Journal of Functional Analysis This means that lim n Fx,u + k x u + k x v + k x =+ a.e. in S and so lim inf k Fx,u + k x u + k x v + k x = However, since {u k } is the arbitrary Cerami sequence at level C M, we see that u k = C M + Fx,u + k x + o1 which implies that Fx,u + k x + and then lim inf k = lim inf k = lim inf k Fx,u + k x u + k x v + k x Fx,u + k x u k Fx,u + k x C M + Fx,u + k x + o1 = ow, note that Fx,s 0, by Fatou s lemma and 7.3 and 7.4, we get a contradiction. So v 0 a.e. which means that v + k 0inE. Let t k [0, 1] such that It k u k = max t [0,1] Itu k. For any given R 0, 1 β α 1, letε = 1 β α / 1 f 1 on, there exists C = CR > 0 such that > 0, since f has critical growth

32 . Lam, G. Lu / Journal of Functional Analysis Fx,s C s 1 + β α / 1 R + ε, s, x, s R. 7.5 Since u k,wehave R It k u k I u k u k = IRv k 7.6 and by 7.5, v k =1 and the fact that Fx,v k = Fx,v + k, we get IRv k C C C v + k v + k v + k 1 β α 1 1 β α 1 1 β α 1 R + ε, R v k + R + ε 1, v + k R1 β α, v k. 7.7 Since v k + 0inE and the embedding E Lp is compact for all p, usingthe Holder inequality, we can show easily that v + k x k 0. Also, by Lemma 1.1, R1 R β α, v k x is bounded by a universal C. Thus using 7.6 and letting k in 7.7, and then letting R [ 1 β α lim inf It ku k 1 1 β k α 1 ], we get 1 >C M. 7.8 ote that I0 = 0 and Iu k C M, we can suppose that t k 0, 1. Thus since DIt k u k t k u k = 0, tk u k = fx,t k u k t k u k. By f 2 : It k u k = tk u k = Fx,t k u k [fx,t k u k t k u k Fx,t k u k ]

33 1164. Lam, G. Lu / Journal of Functional Analysis [fx,u k u k Fx,u k ] Moreover, we have [fx,u k u k Fx,u k ] = u k + C M u k + o1 = C M + o1. which is a contraction to 7.8. This proves that {u k } is bounded in E. References [1] S. Adachi, K. Tanaka, Trudinger type inequalities in and their best exponents, Proc. Amer. Math. Soc [2] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the - Laplacian, Ann. Sc. orm. Super. Pisa [3] Adimurthi, K. Sandeep, A singular Moser Trudinger embedding and its applications, odea onlinear Differential Equations Appl [4] Adimurthi, S.L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of R 2 involving critical exponent, Ann. Sc. orm. Super. Pisa Cl. Sci [5] Adimurthi, Yunyan Yang, An interpolation of Hardy inequality and Trudinger Moser inequality in and its applications, Int. Math. Res. ot. IMR [6] Antonio Ambrosetti, Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal [7] F.V. Atkinson, L.A. Peletier, Ground states and Dirichlet problems for u = fu in R 2, Arch. Ration. Mech. Anal [8] F.V. Atkinson, L.A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations [9] Haïm Brézis, Elliott Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc [10] Daomin Cao, ontrivial solution of semilinear elliptic equation with critical exponent in R 2, Comm. Partial Differential Equations [11] L. Carleson, S.Y.A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math [12] Giovanna Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A in Italian. [13] Giovanna Cerami, On the existence of eigenvalues for a nonlinear boundary value problem, Ann. Mat. Pura Appl in Italian. [14] João Marcos do Ó, -Laplacian equations in with critical growth, Abstr. Appl. Anal [15] João Marcos do Ó, Semilinear Dirichlet problems for the n-laplacian in with nonlinearities in the critical growth range, Differential Integral Equations [16] João Marcos do Ó, Everaldo Medeiros, Uberlandio Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in R n, J. Differential Equations [17] Duong Minh Duc, guyen Thanh Vu, onuniformly elliptic equations of p-laplacian type, onlinear Anal [18] D.G. de Figueiredo, O.H. Miyagaki, B. Ruf, Elliptic equations in R 2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations [19] Giovanni Gregori, Generalized solutions for a class of non-uniformly elliptic equations in divergence form, Comm. Partial Differential Equations [20] guyen Lam, Guozhen Lu, -Laplacian equations in with subcritical and critical growth without the Ambrosetti Rabinowitz condition, arxiv:

34 . Lam, G. Lu / Journal of Functional Analysis [21] guyen Lam, Guozhen Lu, Existence and multiplicity of solutions to the nonuniformly subelliptic equations of Q-subLaplacian type with critical growth in H n, preprint, [22] guyen Lam, Guozhen Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth, Discrete Contin. Dyn. Syst., in press. [23] Yuxiang Li, Bernhard Ruf, A sharp Trudinger Moser type inequality for unbounded domains in R n, Indiana Univ. Math. J [24] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoam [25] J. Moser, A sharp form of an inequality by. Trudinger, Indiana Univ. Math. J / [26] R. Panda, ontrivial solution of a quasilinear elliptic equation with critical growth in R n, Proc. Indian Acad. Sci. Math. Sci [27] S.I. Pohozaev, On the eigenfunctions of the equation u + λf u = 0, Dokl. Akad. auk SSSR in Russian. [28] Bernhard Ruf, A sharp Trudinger Moser type inequality for unbounded domains in R 2, J. Funct. Anal [29] Mei-Chi Shaw, Eigenfunctions of the nonlinear equation u + νf x, u = 0inR 2, Pacific J. Math [30] A.S. Tersenov, On quasilinear non-uniformly elliptic equations in some non-convex domains, Comm. Partial Differential Equations [31] Hoang Quoc Toan, Quoc-Anh go, Multiplicity of weak solutions for a class of nonuniformly elliptic equations of p-laplacian type, onlinear Anal [32] eil S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech [33] Yunyan Yang, Existence of positive solutions to quasilinear equations with exponential growth in the whole Euclidean space, arxiv: v1. [34] Huang Yisheng, Yuying Zhou, Multiple solutions for a class of nonlinear elliptic problems with a p-laplacian type operator, onlinear Anal

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