Asymptotic Analysis 93 (2015) DOI /ASY IOS Press

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1 Asymptotic Analysis DOI /ASY IOS Press Combined effects of concave convex nonlinearities and indefinite potential in some elliptic problems Nikolaos S. Papageorgiou a and Vicenţiu D. Rădulescu b,c, a Department of Mathematics, National Technical University, Athens, Greece npapg@math.ntua.gr b Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia c Institute of Mathematics Simion Stoilow of the Romanian Academy, Bucharest, Romania vicentiu.radulescu@math.cnrs.fr Abstract. We consider a nonlinear Dirichlet problem driven by the p-laplacian and a reaction which exhibits the combined effects of concave that is, sublinear terms and of convex that is, superlinear terms. The concave term is indefinite and the convex term need not satisfy the usual in such cases Ambrosetti Rabinowitz condition. We prove a bifurcation-type result describing the set of positive solutions as the positive parameter λ varies. Keywords: indefinite concave term, superlinear perturbation, p-laplacian, nonlinear regularity, positive solutions 1. Introduction Let R N be a bounded domain with a C 2 -boundary. In this paper, we study the following nonlinear parametric elliptic problem p uz = ϑzuz q 1 + f z, uz, λ in, u = on, u> in, P λ where λ>and1<q<p.here p denotes the p-laplace differential operator defined by p u = div Du p 2 Du for all u W 1,p, 1 <p<. The perturbation z, x fz,x,λis a Carathéodory function that is, for all x R, z fz,x,λis measurable and for a.a. z, x fz,x,λis continuous, which exhibits p 1-superlinear growth * Corresponding author: Vicenţiu D. Rădulescu, Institute of Mathematics Simion Stoilow of the Romanian Academy, P.O. Box 1-764, 147 Bucharest, Romania. vicentiu.radulescu@math.cnrs.fr /15/$ IOS Press and the authors. All rights reserved

2 26 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems near +, without satisfying the usual in such cases unilateral Ambrosetti Rabinowitz condition ARcondition for short. So, in problem P λ we have the combined effects of a concave that is, of a p 1- sublinear nonlinearity which is expressed by the term ϑzu q 1 recall 1 <q<p and of a convex that is, of a p 1-superlinear nonlinearity, expressed by the term fz,u,λ. Hence, we are dealing with a concave convex problem. The interesting feature of our work here, is that the concave term ϑzu q 1 is indefinite, namely the weight function ϑ may change sign. Problems with combined nonlinearities, were first investigated by Ambrosetti, Brezis and Cerami [2], where p = 2 semilinear problem and the parametric reaction has the form 2N λx q 1 + x r 1 for all x, with 1 <q<2 <r<2 if 2 <N, = N 2 + if N = 1, 2. They proved bifurcation type results describing the dependence of the set of positive solutions on the parameter λ>. Their work was extended to nonlinear problems driven by the p-laplacian, by Garcia Azorero, Manfredi and Peral Alonso [8] and Guo and Zhang [1]. Problems with more general reactions, were studied by Hu and Papageorgiou [11] and Marano and Papageorgiou [14]. Problems with indefinite concave nonlinearities were investigated by de Paiva [6], Li, Wu and Zhou [12], and Papageorgiou and Rădulescu [18] only in the context of semilinear equations that is, p = 2 and with a particular reaction of the form x ϑzx q 1 + λx r 1 for all x, with ϑ L and 1 <q<2 <r<2.wealso refer to the related papers by de Figueiredo, Gossez and Ubilla [5] and Narukawa and Takajo [16]. Using variational methods based on the critical point theory, combined with suitable truncation and comparison techniques, we establish the existence, nonexistence and multiplicity of positive solutions for problem P λ as the parameter λ>varies. 2. Mathematical background Let X be a Banach space and X be its topological dual. By, we denote the duality brackets for the pair X,X.Givenϕ C 1 X, we say that ϕ satisfies the Cerami condition the C-condition for short, if the following is true: Every sequence {u n } n 1 X such that {ϕu n } n 1 R is bounded and 1 + un ϕ u n in X as n, admits a strongly convergent subsequence. This is a compactness type condition on the functional ϕ, which is needed since the ambient space X need not be locally compact since, in general X is infinite dimensional. The C-condition is the main tool in proving a deformation theorem, from which one can derive the minimax theory for the critical values of ϕ. One of the main results in this theory, is the so-called mountain pass theorem due to Ambrosetti and Rabinowitz [3], stated here is a slightly more general form see Gasinski and Papageorgiou [9]. Theorem 1. Assume that ϕ C 1 X satisfies the C-condition, u,u 1 X, u 1 u >ρ>, max { ϕu, ϕu 1 } < inf [ ϕu: u u =ρ ] = η ρ

3 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems 261 and c = inf max ϕ γt with Γ = { γ C [, 1],X } : γ = u,γ1 = u 1. γ Γ t 1 Then c η ρ and c is a critical value of ϕ. In the analysis of problem P λ, in addition to the Sobolev space W 1,p we will also use the Banach space C 1 ={u C1 : u = }. This is an ordered Banach space with positive cone C + = {u C 1 : uz forallz }. This cone has a nonempty interior given by { int C + = u C + : uz > forallz, u } n <. Here by n we denote the outward unit normal on. Let f : R R be a Carathéodory function with subcritical growth in x R, thatis, f z, x a z 1 + x r 1 for a.a. z, all x R, Np with a L + and 1 <r<p if p<n, = N p + if N p. We set F z, x = x f z, s ds and consider the C 1 -functional ϕ : W 1,p R defined by ϕ u = 1 p Du p p 1,p F z, uz dz for all u W. The next result can be found in Garcia Azero, Manfredi and Peral Alonso [8] and essentially is a consequence of the nonlinear regularity theory of Lieberman [13]. Proposition 2. Assume that u W 1,p is a local C 1 -minimizer of ϕ, that is, there exists ρ > such that ϕ u ϕu + h for all h C 1 with h C 1 ρ. Then u C 1,α for some α, 1 and it is also a local W 1,p -minimizer of ϕ, that is, there exists ρ 1 > such that ϕ u ϕu + h for all h W 1,p with h ρ 1. Hereafter by we denote the norm of W 1,p. By virtue of the Poincaré inequality, we have u = Du p for all u W 1,p.

4 262 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems Let A : W 1,p W 1,p = W 1,p 1 p + 1 p = 1 be the nonlinear map defined by Au, y = Du p 2 Du, Dy R N dz for all u, y W 1,p. The next proposition summarizes the main properties of this map see, for example, Papageorgiou and Kyritsi [17, p. 314]. Proposition 3. The map A : W 1,p W 1,p is bounded that is, maps bounded sets to bounded sets, demicontinuous, strictly monotone, hence maximal monotone too and of type S + that is, if w 1,p u n u in W and lim sup Aun, u n u, n then u n u in W 1,p as n. We recall that the Dirichlet p-laplacian p,w 1,p admits a smallest eigenvalue ˆλ 1 >. Sometimes we write ˆλ 1 > to emphasize the domain. This eigenvalue is isolated, simple with eigenfunctions of constant sign. The nonlinear regularity theory and the nonlinear maximum principle see, for example, Gasinski and Papageorgiou [9, pp ], imply that every positive eigenfunction corresponding to ˆλ 1 > belongs in int C +. Finally let us fix our notation. So, for x R,wesetx ± = max{±x,}. Thengivenu W 1,p, we set u ± = u ±.Wehave u ± W 1,p, u = u + u, u =u + + u. Given any measurable function h : R R for example, a Carathéodory function, we define N h u = h,u for all u W 1,p the Nemytski map corresponding to h. Evidently z N h uz is measurable on. By N we denote the Lebesgue measure on R N. 3. Positive solutions The hypotheses on the data of problem P λ are the following: H 1 : ϑ L and if D + ={z : ϑz > }, thend + and there exists an open set + such that + D +, + is C 2 and ess inf + ϑ = m + >. H 2 : f : R, R is a function such that for all λ >, z, x fz,x,λ is Carathéodory, fz,,λ= for a.a. z and i fz,x,λ az, λ1 + x rλ 1 for a.a. z with rλ p, p, a,λ L +, a,λ as λ + ;

5 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems 263 ii if Fz,x,λ = x fz,s,λds, then Fz,x,λ lim =+ uniformly for a.a. z x + x p and there exist τ rλ p max{ N p, 1},p, τ > q and η λ > with λ η λ nondecreasing such that fz,x,λx pfz,x,λ η λ lim inf uniformly for a.a. z ; x + x τ iii for every ρ>, there exists m ρ λ > suchthatm ρ λ + as λ +, inf [ fz,x,λ: x ρ ] = m ρ λ >, for a.a. z, allx, the map λ fz,x,λis nondecreasing and for every ξ>, for a.a. z, allx ξ, allλ > λ >, we have fz,x,λ m ξ λ with m ξ nondecreasing, m ξ λ + as λ and fz,x,λ fz,x,λ η ξ > ; iv for every ρ>, there exists ξ ρ λ > such that for a.a. z, the function x fz,x,λ+ ξ ρ λx p 1 is nondecreasing on [,ρ]. Remark 1. Since we are interested on positive solutions and all the above hypotheses concern the positive semiaxis R + =[, +, without any loss of generality, we may assume that fz,x,λ = for a.a. z, allx, all λ>. Hypothesis H 2 ii implies that for a.a. z and all λ>, the perturbation x fz,x,λ is p 1-superlinear near +. However, we do not employ the usual in such cases AR-condition. We recall that the AR-condition unilateral version, since we assume that fz,x,λ = for a.a. z, allx, all λ>, says that there exist μ = μλ > p and M = Mλ > such that a <μfz,x,λ f z, x, λx for a.a. z, allx M; b ess inf F,M,λ> see Ambrosetti and Rabinowitz [3] and Mugnai [15]. Integrating a and using b, we obtain the weaker condition c 1 x μ Fz,x,λ for a.a. z, all x M and some c 1 = c 1 λ >. Evidently this unilateral growth estimate implies the much weaker condition lim x + Fz,x,λ x p =+ uniformly for a.a. z.

6 264 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems Note that our hypothesis H 2 ii is weaker than the AR-condition. Indeed, suppose that the AR-condition holds see a and b above. We may assume that μ > rλ p max{ N, 1}. Wehave p f z, x, λx pfz,x,λ x μ = f z, x, λx μfz,x,λ x μ + μ p Fz,x,λ x μ μ p Fz,x,λ for a.a. z, all x M see a x μ μ pc 1 for a.a. z, all x M fz,x,λx pfz,x,λ = lim inf μ pc x + x μ 1 λ = η λ uniformly for a.a. z. So, hypothesis H 2 ii holds. See the examples that follow for functions which satisfy our hypothesis H 2 ii but not the AR-condition. Example 1. The following functions satisfy hypotheses H 2. For the sake of simplicity we drop the z-dependence: f 1 x, λ = λx r 1 for all x with p<r<p, f 2 x, λ = ξλx ln1 p 1 x + x + for all x px + 1 with λ ξλ strictly increasing on, +, lim λ ξλ =+, lim λ + ξλ =. Note that f 2,λdoes not satisfy the AR-condition. Let L = { λ>: problem P λ admits a positive solution }, Sλ = set of positive solutions of problem P λ. First we establish the nonemptiness and a structural property of the set L of admissible parameters. Proposition 4. If hypotheses H 1 and H 2 hold, then L, for every u Sλ we have uz > for all z and λ L implies,λ] L. Proof. We consider the following auxiliary Dirichlet problem p ez = 1 in,e =. 1 Recalling that A is maximal monotone, strictly monotone and coercive by virtue of the Poincaré inequality, we see that problem 1 has a unique solution e W 1,p \{} see, for example, Gasinski and Papageorgiou [9, p.319].actingon1 with e W 1,p, we obtain De p = e dz p = e, e.

7 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems 265 From the nonlinear regularity theory and the nonlinear maximum principle see, for example, Gasinski and Papageorgiou [9, pp ], we have e int C +. Claim 1. There exists λ> such that for all λ, λ, we can find ξ = ξλ > for which we have ξ q 1 ϑ e q 1 + a,λ 1 + ξ r 1 e r 1 <ξ p 1. Arguing by contradiction, suppose that we can find λ n such that ξ p 1 ξ q 1 ϑ e q 1 + a,λn 1 + ξ r 1 e r 1 for all ξ>, all n 1. Passing to the limit as n and using hypothesis H 2 i, we obtain ξ p q ϑ e q 1 for all ξ>, a contradiction. This proves the claim. Let u = ξe int C +.Thenwehave p uz = ξ p 1 >ϑzuz q 1 + f z, uz, λ for a.a. z 2 see Claim 1 and hypothesis H 2 i. Fix λ, λ and consider the following Carathéodory function if x<, g λ z, x = ϑzx q 1 + fz,x,λ if x uz, ϑzuz q 1 + f z, uz, λ if uz < x. We set G λ z, x = x g λz, s ds and consider the C 1 -functional ψ λ : W 1,p R defined by ψ λ u = 1 p Du p p 1,p G λ z, uz dz for all u W. From 3 it is clear that ψ λ is coercive. Also, using the Sobolev embedding theorem, we can see that ψ λ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find u λ W 1,p such that 3 ψ λ u λ = inf [ ψ λ u: u W 1,p ]. 4 Let u C + \{} with supp u +. Recall that u int C +. Hence, we have that u + > see hypothesis H 1. Therefore, we can find t, 1 small such that tu u. So, using hypothesis H 2 iii and 3, we have ψ λ tu t p p Du p p t q q ϑzu q dz.

8 266 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems Since supp u + and q<p, choosing t, 1 even smaller if necessary, we infer that ψ λ tu < = ψ λ u λ < = ψ λ see 4, hence uλ. From 4, we have ψ λ u λ = = Au λ = N gλ u λ. 5 On 5 first we act with u λ W 1,p and obtain Du p = p see 3 λ = u λ,u λ. Also, on 5 we act with u λ u + W 1,p and obtain Auλ, u λ u + = g λ z, u λ u λ u + dz = {u λ >u} = [ ϑzu q 1 + fz,u, λ ] u λ u + dz ξ p 1 u λ u + dz see Claim 1 = Au, u λ u + see 1 see 3 Duλ p 2 Du λ Du p 2 Du, Du λ Du R N dz = {uλ > u} N =, hence u λ u. So, we have proved that u λ [, u] = { u W 1,p : uz uz for a.a. z }. This fact and 3 imply that u λ Sλ. The nonlinear regularity theory see Lieberman [13], implies that u λ C + \{}. Invoking Harnack s inequality see Pucci and Serrin [19, p. 163], we infer that u λ z > forallz. Now let λ L and let μ,λ, u λ Sλ.Wehave p u λ z = ϑzu λ z q 1 + f z, u λ z, λ ϑzu λ z q 1 + f z, u λ z, μ for μ a.a. z 6 see hypothesis H2 iii.

9 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems 267 We introduce the Carathéodory function γ μ z, x defined by if x<, γ μ z, x = ϑzx q 1 + fz,x,μ if x u λ z, ϑzu λ z + f z, u λ z, μ if u λ z < x. 7 We set Γ μ z, x = x γ μz, s ds and consider the C 1 -functional τ μ : W 1,p R defined by τ μ u = 1 p Du p p Γ μ z, uz dz for all u W 1,p. From 7 it is clear that τ μ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u μ W 1,p such that τ μ u μ = inf [ τ μ u: u W 1,p ]. As before, we can show that τ μ u μ < = τ μ, hence u μ. Also, we have τ μ u μ = = Au μ = N γμ u μ. Acting with u μ W 1,p and with u μ u λ + W 1,p, as before, we show that u μ [,u λ ]= { u W 1,p : uz u λ z for a.a. z } = u μ Sμ see 7 = μ L and so,λ] L. This completes the proof. Let λ = sup L. Proposition 5. If hypotheses H 1 and H 2 hold, then λ <. Proof. Let ˆλ + 1 = ˆλ 1 + see Section 2 andtakeβ>ˆλ + 1. Claim 2. There exists λ > big such that ϑzx q 1 + fz,x,λ βx p 1 for a.a. z +, all x, all λ λ.

10 268 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems Since q<p, we can find δ> small such that ϑzx q 1 βx p 1 for a.a. z +, all x δsee hypothesis H 1. 8 Note that hypothesis H 2 ii implies that for all λ>, we have fz,x,λ lim =+ uniformly for a.a. z. x + x p 1 So, we can find M = Mλ > suchthat fz,x,λ βx p 1 for a.a. z all x M. 9 Let δ>beasin8 andλ>. By virtue of hypothesis H 2 iii, we have inf [ fz,x,λ: x δ ] = m δ λ >, with λ m δ λ nondecreasing and m δ λ + as λ +. So, we can choose λ 1bigsuch that m δ λ βm p 1 for all λ λ = fz,x,λ βx p 1 for a.a. z, all δ x M, all λ λ. 1 Combining 8, 9, 1, we conclude that Claim 2 holds. Take λ>λ and assume that λ L. Then we can find u λ Sλ such that p u λ z = ϑzu λ z q 1 + f z, u λ z, λ βu λ z q 1 for a.a. z + see Claim 2. Let û + int C + + be the principal, L p -normalized that is, û + L p + = 1 positive eigenfunction for p,w 1,p +. From Proposition 4, we know that u λ + >. So, we can find t, 1 small such that We have tû + z u λ z for all z p tû + z = ˆλ + 1 tû +z p 1 <βtû + z p 1 for a.a. z + 12 recall that ˆλ + 1 <β.

11 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems 269 We consider the following Carathéodory function βtû + z p 1 if x<tû + z, k β z, x = βx p 1 if tû + z x u λ z, z, x + R, βu λ z p 1 if u λ z < x 13 see 11. We set K β z, x = x k βz, s ds and consider the C 1 -functional χ β : W 1,p + R defined by χ β u = 1 p Du 1,p L p +,R N K β z, uz dz for all u W +. + From 13 it is clear that χ β is coercive and also it is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find ũ β W 1,p + such that χ β ũ β = inf [ χ β u: u W 1,p + ] = χ β ũ β = = Âũ β = N kβ ũ β with  = A 1,p W On 14, first we act with tû + ũ β + W 1,p +.Then Aũβ, tû + ũ β + = k β z, ũ β tû + ũ β + dz + = βtû + p 1 tû + ũ β + dz + ˆλ + 1 tû + p 1 tû + ũ β + dz + = Âtû +, tû + ũ β + see 12 = Âtû + Âũ β, tû + ũ β + = {tû + > ũ β } + N =, hence tû + ũ β in +. Next, on 14 we act with ũ β ũ λ + W 1,p + where ũ λ = u λ + recall that u λ + >. We have Âũβ, ũ β ũ λ + = k β z, ũ β ũ β ũ λ + dz + = βũ p 1 λ ũ β ũ λ + dz +

12 27 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems [ q 1 ϑzũ λ + fz,ũ λ,λ ] ũ β ũ λ + dz see Claim 2 + = Âũ β Âũ λ, ũ β ũ λ + since u λ Sλ = {ũβ > ũ λ } + N =, hence ũ β ũ λ in +. So, finally we have tû + z ũ β z u λ z for all z +. From 13 and14 it follows that p ũ β z = βũ β z p 1 for a.a. z +, ũ β + =, ũ β, a contradiction since β>ˆλ + 1 recall that every nonprincipal eigenvalue of p,w 1,p +, has nodal that is, sign changing eigenfunctions, see [9]. This means that λ λ <. In what follows, for every λ>, by ϕ λ : W 1,p R we denote the energy functional for problem P λ definedby ϕ λ u = 1 p Du p p 1 ϑzu + z q dz F z, uz, λ dz q for all u W 1,p. Evidently ϕ λ C 1 W 1,p. Proposition 6. If hypotheses H 1 and H 2 hold, then λ L. Proof. Let {λ n } n 1 L such that λ n λ as n and for every n 1, let u n Sλ n.wemay assume that ϕ λn u n < forall n Indeed, if λ< λ <λ and ũ S λ, then by virtue of hypothesis H 2 iii, we have p ũz = ϑzũz q 1 + f z, ũz, λ ϑzũz q 1 + f z, ũz, λ for a.a. z 16 see hypothesis H 2 iii. Then reasoning as in the proof of Proposition 4, we introduce the following truncation of the reaction of problem P λ : if x<, w λ z, x = ϑzx q 1 + fz,x,λ if x ũz, ϑzũz q 1 + f z, ũz, λ 17 if ũz < x.

13 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems 271 This is a Carathéodory function. We set W λ z, x = x w λz, s ds and consider the C 1 -functional ˆψ λ : W 1,p R defined by ˆψ λ u = 1 p Du p p 1,p W λ z, uz dz for all u W. Again, ˆψ λ is coercive see 17 and sequentially weakly lower semicontinuous. So, we can find u λ W 1,p such that ˆψ λ u λ = inf [ ˆ ψ λ u: u W 1,p ]. 18 As in the proof of Proposition 4, we show that ˆψ λ u λ < = ˆψ λ, hence u λ. From 18, we have ˆψ λ u λ = = Au λ = N wλ u λ. 19 On 19 first we act with u λ W 1,p and obtain u λ, u λ see17. Then we act with u λ ũ + W 1,p. We obtain Auλ, u λ ũ + = w λ z, u λ u λ ũ + dz + [ = ϑzũ q 1 + fz,ũ, λ ] u λ ũ + dz see 17 + Aũ, u λ ũ + see 16 So, we have = Au λ Aũ, u λ ũ + = {uλ > ũ} N =, hence u λ ũ. u λ [, ũ] = { u W 1,p : uz ũz for a.a. z } = u λ Sλ see 17 and ϕ λ u λ < since ϕ λ [,ũ] = ˆψ λ [,ũ]. This proves that we can always assume that 15 holds. From 15wehave Du n p p p ϑzu q n q dz pfz, u n,λ n dz < for all n 1. 2

14 272 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems Also, since u n Sλ n for all n 1, we have Au n = ϑzu q 1 n + N fλn u n, n 1, 21 where f λn z, x = fz,x,λ n.on21 we act with u n W 1,p and obtain Du n p p + ϑzu q n dz + fz,u n,λ n u n dz = for all n Adding 2 and22, we obtain [ fz,un,λ n u n pfz, u n,λ n ] p dz q 1 ϑzu q n dz for all n It is clear that in hypothesis H 2 i without any loss of generality, we may assume that λ rλ and λ a,λ are both nondecreasing in, +. Then hypotheses H 2 iii, imply that we find c 2,c 3 > such that c 3 x τ c 4 fz,x,λ n x pfz,x,λ n for a.a. z, all x, all n Using 24 in23 and recalling that τ>qsee hypothesis H 2 ii, we infer that {u n } n 1 L τ is bounded. 25 It is clear from hypothesis H 2 iii that without any loss of generality, we may assume that τ < r 1 = rλ 1 rλ n for all n 1 see hypothesis H 1 i. Suppose N p. Then we can find t, 1 such that 1 = 1 t r 1 τ + t p. Invoking the interpolation inequality see, for example, Gasinski and Papageorgiou [9, p. 95], we have u n r u n 1 t τ u n t p for all n 1 = u n r r M 1 u n tr for some M 1 >, all n see 25 and use the Sobolev embedding theorem. From hypothesis H 2 i, we have fz,x,λ n x c x r for a.a. z, all x, some c 4 >. From 22, 25 and since τ>qsee H 2 ii and ϑ L see H 1 Du n p p c un r r for some c 5 >, all n 1 c un tr for some c 6 >, all n 1 see

15 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems 273 From 26 and our hypothesis on τ see H 2 iii, it follows that tr < p. Hence from 28 we infer that {u n } n 1 W 1,p is bounded. If N = p, thensincep =+ and W 1,p is compactly embedded into L q for all q [1,, then the above argument works, if we replace p by η>r>pbig. Then again we reach the same conclusion. So, we may assume that u n w u in W 1,p and u n u in L r r = r λ as n. 29 On 21 we act with u n u W 1,p, pass to the limit as n and use 29. Then lim Aun, u n u = n = u n u in W 1,p as n 3 see Proposition 3. Passing to the limit as n in 21 and using 3, we obtain Au = ϑzu q 1 + N fλ u. 31 We need to show that u, because then u Sλ,thatisλ L. To this end, we consider the following auxiliary Dirichlet problem p uz = ϑzuz q 1 in +, u + =, u >. 32 Since q<p, a straightforward application of the direct method as before, establishes that problem 32 admits a nontrivial solution u W 1,p, u. The nonlinear regularity theory and the nonlinear maximum principle see, for example, Gasinski and Papageorgiou [9, pp ], imply that u int C +. Moreover, Theorem 2 of Diaz and Saa [7], implies that u int C + is the unique positive solution of 32. Now, let λ L and u λ Sλ. We introduce the following Carathéodory function if x<, jz,x = ϑzx q 1 if x u λ z, z, x + R, ϑzu λ z q 1 if u λ z < x. 33 Let Jz,x = x jz,sds and consider the C1 -functional J : W 1,p + R defined by Ju = 1 p Du p L p +,R N + J z, uz dz for all u W 1,p +.

16 274 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems As before, 33 implies that J is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find ũ W 1,p + such that Jũ = inf [ Ju: u W 1,p + ]. 34 Since q<pand u λ + > recall u λ int C +, as in previous similar cases, we have Jũ < = J, hence ũ. From 34, we have J ũ =, = Aũ = N j ũ. On 35 we act with ũ W 1,p + and with ũ u λ + W 1,p + here we use the fact that u λ + > and we obtain < ũz u λ z for all z +, ũ So, we have = ũ is a positive solution of 32 = ũ = u recall that u is the unique positive solution of 32 = u u λ in +. uz u n z for all z +, all n 1 = uz uz for all z + see 3 = u. Therefore, u Sλ and so λ L. Next, we look for additional positive solutions for problem P λ. To this end, we consider the following auxiliary Dirichlet problem: p uz = ϑ + zuz q 1 + f z, uz, λ in, u = on, u> in. 35 Au λ Reasoning as in the proofs of Propositions 4, 5 and 6 with + replaced by we obtain the following proposition. Proposition 7. If hypotheses H 1 and H 2 hold, then there exists λ,λ ] such that for all λ,λ ] problem Au λ has at least one positive solution ũ λ int C +.

17 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems 275 Remark 2. Note that in this case, the solution ũ λ C + \{} satisfies p ũ λ z = ϑ + zũ λ z q 1 + f z, ũ λ z, λ for a.a. z see hypotheses H 2 iii = ũ λ int C + see Gasinski and Papageorgiou [9, p. 738]. We can use Proposition 7, to produce a multiplicity result for the positive solutions of problem P λ with λ,λ ]. Proposition 8. If hypotheses H 1 and H 2 hold and λ,λ ], then problem P λ has at least two positive solutions u λ, û λ C + \{}, u λ û λ, <u λ z û λ z for all z. Proof. Let μ λ, λ. From Proposition 4, we know that λ, μ L and we can find u λ Sλ C + and û λ int C + solution of Au μ. We claim that we can have u λ ũ μ. Indeed note that p ũ μ z = ϑ + zũ μ z q 1 + f z, ũ μ z, μ ϑzũ μ z q 1 + f z, ũ μ z, λ for a.a. z 37 see hypothesis H 2 iii. We consider the following truncation of the reaction of problem P λ. if x<, ĝ λ z, x = ϑzx q 1 + fz,x,λ if x ũ μ z, ϑzũ μ z q 1 + f z, ũ μ z, λ if ũ μ z < x. This is a Carathéodory function. We set Ĝ λ z, x = x ĝλz, s ds and introduce the C 1 -functional ˆψ λ : W 1,p R defined by ˆψ λ u = 1 p Du p p Ĝ λ z, uz dz for all u W 1,p Evidently ˆψ λ is coercive see 38 and sequentially weakly lower semicontinuous. So, we can find u λ W 1,p such that ˆψ λ u λ = inf [ ˆψ λ u: u W 1,p ]. 39

18 276 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems Since q<p, as before see, for example, the proof of Proposition 4, we have ˆψ λ u λ < = ˆψ λ, hence u λ. From 39wehave ˆψ λ u λ = = Au λ = Nĝλ u λ. 4 On 4 we act with u λ W 1,p and with u λ ũ μ + W 1,p. As in the proof of Proposition 4, using this time 37, we show that u λ [, ũ μ ]= { u W 1,p : uz ũ μ z for a.a. z }. Hence u λ Sλ C + see 38 and u λ z > forallz by Harnack s inequality, see Pucci and Serrin [19, p. 163]. Therefore 36 holds. We introduce the following truncation of the reaction of problem P λ : k λ z, x = { ϑzu λ z q 1 + f z, u λ z, λ if x u λ z, ϑzx q 1 + fz,x,λ if u λ z < x. This is a Carathéodory function. We set K λ z, x = x k λz, s ds and consider the C 1 -functional τ λ : W 1,p R defined by τ λ u = 1 p Du p p 1,p K λ z, uz dz for all u W. From the proof of Proposition 6, we know that τ λ satisfies the C-condition. 42 We truncate k λ z, as follows: ˆk λ z, x = { k λ z, x if x ũ μ z, k λ z, ũμ z if ũ μ z < x This too is a Carathéodory function. We set ˆK λ z, x = x ˆk λ z, s ds and consider the C 1 -functional ˆτ λ : W 1,p R defined by ˆτ λ u = 1 p Du p p ˆK λ z, uz dz for all u W 1,p.

19 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems 277 From 43 it is clear that ˆτ λ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u λ W 1,p such that ˆτ λ u λ = inf [ ˆτ λ u: u W 1,p ] = ˆτ λ u λ = = Au λ = Nˆk λ u λ. 44 On 44 we act, first with u λ u λ W 1,p andthenwithu λ ũ μ + W 1,p. As in the proof of Proposition 5, we show that u λ [u λ, ũ μ ]= { u W 1,p : u λ z uz ũ μ z for a.a. z }. If u λ u λ, then this is the desired second positive solution of problem P λ andu λ u λ =û λ. So, we may assume that u λ = u λ.letρ = ũ μ recall that ũ int C +, see Proposition 7 andlet ξ ρ = ξ ρ μ > be as postulated by hypothesis H 2 iv for the perturbation fz,,μ.wehave p u λ z + ξ ρ u λ z p 1 = ϑzu λ z q 1 + f z, u λ z, λ + ξ ρ u λ z p 1 = ϑzu λ z q 1 + f z, u λ z, μ + ξ ρ u λ z p 1 + [ f z, u λ z, λ f z, u λ z, μ ] ϑ + zu λ z q 1 + f z, u λ z, μ + ξ ρ u λ z p 1 δ λ,λz with δ λ,λz = f z, u λ z, μ f z, u λ z, λ ϑ + zũ μ z + f z, ũ μ z, μ + ξ ρ ũ μ z p 1 see hypothesis H2 iv = p ũ μ z + ξ ρ ũ μ z p 1 for a.a. z. 45 For every K compact, we have u λ K ξ> recall u λ z > forallz, see Proposition 4. So, by hypothesis H 2 iii we have δ λ,μ K m ξ >. Then from 45 and Proposition 2.6 of Arcoya and Ruiz [4] recall that ũ μ int C +, we infer that ũ μ u λ int C We claim that u λ is a local C 1-minimizer of the functional τ λ. Indeed, if this is not the case, we can find {u n } n 1 C 1 such that u n u λ in C 1 and τ λu n <τ λ u λ for all n From 46 and47 it follows that we can find n N such that u n ũ μ for all n n = u + n ũ μ for all n n recall ũ μ int C +.

20 278 N.S. Papageorgiou and V.D. Rădulescu / Combined effects in some elliptic problems Also, we have ˆτ λ u + n = τλ u + n τλ u n <τ λ u λ for all n n which contradicts the fact that u λ = u λ is a global minimizer of ˆτ λ. Since u λ is a local C 1-minimizer of τ λ. We may assume that u λ is an isolated critical point of τ λ, or otherwise we already have a sequence of distinct positive solutions, since the critical set of τ λ is in [u λ ={u W 1,p : u λ z uz for a.a. z } see 41. Therefore, we can find ρ, 1 small such that τ λ u λ <inf [ τ λ u: u u λ =ρ ] = m λ 48 see Aizicovici, Papageorgiou and Staicu [1, Proof of Proposition 29]. Moreover, by virtue of hypothesis H 2 ii, if u int C +,then τ λ tu as t Because of 42, 48 and49, we can apply Theorem 1 the mountain pass theorem and find û λ W 1,p such that τ λ û λ = and m λ τû λ. 5 From 49, 5 and since the critical set of τ λ is in [u λ, it follows that û λ Sλ C + and û λ u λ,u λ û λ. Finally from Proposition 4,wehave <u λ z û λ z for all z. The proof is now complete. So, summarizing the situation for problem P λ, we can state the following result describing the set of positive solutions as the parameter λ>varies. Theorem 9. If hypotheses H 1 and H 2 hold, then a there exists λ > such that for all λ,λ ] problem P λ has at least one positive solution u λ C + with u λ z > for all z and for λ>λ there are no positive solutions; b there exists λ,λ ] such that for all λ,λ problem P λ has at least two positive solutions u λ, û λ C +,u λ û λ,u λ û λ,u λ z > for all z. Remark 3. It will be interesting to know if λ = λ. Also, it is not clear to us if this result can be extended to Neumann problems. A careful inspection of the proofs reveals that they fail in the Neumann case.

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