International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 595429 | https://doi.org/10.1155/2012/595429

Guowei Dai, Chunfeng Yang, "Existence and Multiplicity of Solutions for Semipositone Problems Involving p-Laplacian", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 595429, 10 pages, 2012. https://doi.org/10.1155/2012/595429

Existence and Multiplicity of Solutions for Semipositone Problems Involving p-Laplacian

Academic Editor: Peter Takac
Received22 Mar 2012
Accepted19 Apr 2012
Published19 Jul 2012

Abstract

We prove existence and multiplicity of positive solutions for semipositone problems involving ๐‘-Laplacian in a bounded smooth domain of โ„๐‘ under the cases of sublinear and superlinear nonlinearities term.

1. Introduction

In this paper, we shall study the following semipositone problem involving the ๐‘-Laplacian: ๎‚€||||โˆ’divโˆ‡๐‘ข๐‘โˆ’2๎‚โˆ‡๐‘ข=๐œ†๐‘“(๐‘ข)inฮฉ,๐‘ข=0on๐œ•ฮฉ,(1.1) where ฮฉโŠ‚โ„๐‘ is a smooth bounded domain, ๐œ† is a positive parameter, and ๐‘“โˆถ[0,+โˆž)โ†’โ„ is a continuous function satisfying the condition(F0)๐‘“(0)=โˆ’๐‘Ž<0.

Such problems are usually referred in the literature as semipositone problems. We refer the reader to [1], where Castro and Shivaji initially called them nonpositone problems, in contrast with the terminology positone problems, coined by Keller and Cohen in [2], when the nonlinearity ๐‘“ was positive and monotone.

Under the case of ๐‘โ‰ก2, a novel variational approach is presented by Costa et al. [3] to the question of existence and multiplicity of positive solutions to problem (1.1), where they consider both the sublinear and superlinear cases. The aim of this paper is to extend their results to the case of ๐‘-Laplacian. The main difficulty is in verifying (PS)๐‘-condition because of the operator โˆ’ฮ”๐‘ is not self-adjoint linear.

We define the discontinuous nonlinearity ๐‘”(๐‘ ) by ๎‚ป๐‘”(๐‘ )=0,if๐‘ โ‰ค0,๐‘“(๐‘ ),if๐‘ >0.(1.2) We shall consider the modified problem ๎‚€||||โˆ’divโˆ‡๐‘ข๐‘โˆ’2๎‚โˆ‡๐‘ข=๐œ†๐‘”(๐‘ข)inฮฉ,๐‘ข=0on๐œ•ฮฉ.(1.3) We note that the set of positive solutions of (1.1) and (1.3) do coincide.

Our main results concerning problems (1.1) and (1.3) are the following:

Theorem 1.1. Assume (F0) and the following assumptions:(F1)lim๐‘ โ†’+โˆž(๐‘“(๐‘ )/๐‘ ๐‘โˆ’1)=0 (the sublinear case);(F2)๐น(๐›ฟ)>0 for some ๐›ฟ>0, where โˆซ๐น(๐‘ข)=๐‘ข0๐‘“(๐‘ )๐‘‘๐‘ .Then, there exist 0<๐œ†0โ‰ค๐œ†โˆ— such that (1.3) has no nontrivial nonnegative solution for 0<๐œ†<๐œ†0, and has at least two nontrivial nonnegative solutions ๐‘ข๐œ†, ๐‘ฃ๐œ† for all ๐œ†>๐œ†โˆ—. Moreover, when ฮฉ is a ball ๐ต๐‘…=๐ต๐‘…(0), these two solutions are non-increasing, radially symmetric and, if ๐‘โ‰ฅ2, at least one of them is positive, hence a solution of (1.1).

Theorem 1.2. Assume (F0), (F2) and the following assumptions:(F3)lim๐‘ โ†’+โˆž(๐‘“(๐‘ )/๐‘ ๐‘โˆ’1)=+โˆž, lim๐‘ โ†’+โˆž(๐‘“(๐‘ )/๐‘ ๐‘โˆ—โˆ’2)=0 (the sublinear, subcritical case);(F4)๐œƒ๐น(๐‘ )โ‰ค๐‘“(๐‘ )๐‘ for all ๐‘ โ‰ฅ๐พ and some ๐œƒ>๐‘ (the AR-condition).Then, (1.3) has at least one nonnegative solution ๐‘ฃ๐œ† for all ๐œ†>0. Ifฮฉ=๐ต๐‘… then ๐‘ฃ๐œ† is nonincreasing, radially symmetric and one of the two alternatives occurs. (i)There exists ๐œ†โˆ—>0 such that, for all 0<๐œ†<๐œ†โˆ—, ๐‘ฃ๐œ† is a positive solution of (1.1) having negative normal derivative on ๐œ•๐ต๐‘…. (ii)For some sequence ๐œ‡๐‘›โ†’0, problem (1.1) with ๐œ†โ‰ก๐œ‡๐‘› has a positive solution ๐‘ค๐œ‡๐‘› with zero normal derivative on ๐œ•๐ต๐‘….

2. Preliminaries

We start by recalling some basic results on variational methods for locally Lipschitz functionals. Let (๐‘‹,โ€–โ‹…โ€–) be a real Banach space and ๐‘‹โˆ— is its topological dual. A function ๐‘“โˆถ๐‘‹โ†’โ„ is called locally Lipschitz if each point ๐‘ขโˆˆ๐‘‹ possesses a neighborhood ฮฉ๐‘ข such that |๐‘“(๐‘ข1)โˆ’๐‘“(๐‘ข2)|โ‰ค๐ฟโ€–๐‘ข1โˆ’๐‘ข2โ€– for all ๐‘ข1,๐‘ข2โˆˆฮฉ๐‘ข, for a constant ๐ฟ>0 depending on ฮฉ๐‘ข. The generalized directional derivative of ๐‘“ at the point ๐‘ขโˆˆ๐‘‹ in the direction ๐‘ฃโˆˆ๐‘‹ is ๐‘“0(๐‘ข,๐‘ฃ)=limsup๐‘คโ†’๐‘ข,๐‘กโ†’0+1๐‘ก(๐‘“(๐‘ค+๐‘ก๐‘ฃ)โˆ’๐‘“(๐‘ค)).(2.1) The generalized gradient of ๐‘“ at ๐‘ขโˆˆ๐‘‹ is defined by ๎€ฝ๐‘ข๐œ•๐‘“(๐‘ข)=โˆ—โˆˆ๐‘‹โˆ—โˆถโŸจ๐‘ขโˆ—,๐œ‘โŸฉโ‰ค๐‘“0๎€พ,(๐‘ข;๐œ‘)โˆ€๐œ‘โˆˆ๐‘‹(2.2) which is a nonempty, convex, and ๐‘คโˆ—-compact subset of ๐‘‹, where โŸจโ‹…,โ‹…โŸฉ is the duality pairing between ๐‘‹โˆ— and ๐‘‹. We say that ๐‘ขโˆˆ๐‘‹ is a critical point of ๐‘“ if 0โˆˆ๐œ•๐‘“(๐‘ข). For further details, we refer the reader to Chang [4].

We list some fundamental properties of the generalized directional derivative and gradient that will be possibly used throughout the paper.

Proposition 2.1 (see [4, 5]). (1)โ€‰โ€‰Let ๐‘—โˆถ๐‘‹โ†’โ„ be a continuously differentiable function. Then ๐œ•๐‘—(๐‘ข)={๐‘—โ€ฒ(๐‘ข)}, ๐‘—0(๐‘ข;๐‘ง) coincides with โŸจ๐‘—โ€ฒ(๐‘ข),๐‘งโŸฉ๐‘‹, and (๐‘“+๐‘—)0(๐‘ข,๐‘ง)=๐‘“0(๐‘ข;๐‘ง)+โŸจ๐‘—โ€ฒ(๐‘ข),๐‘งโŸฉ๐‘‹for all ๐‘ข,๐‘งโˆˆ๐‘‹.
(2)โ€‰โ€‰๐œ•(๐œ†๐‘“)(๐‘ข)=๐œ†๐œ•(๐‘ข) for all ๐œ†โˆˆโ„.
(3)โ€‰โ€‰If ๐‘“ is a convex functional, then ๐œ•๐‘“(๐‘ข) coincides with the usual subdifferential of ๐‘“ in the sense of convex analysis.
(4)โ€‰โ€‰If ๐‘“ has a local minimum (or a local maximum) at ๐‘ข0โˆˆ๐‘‹, then 0โˆˆ๐œ•๐‘“(๐‘ข0).
(5)โ€‰โ€‰โ€–๐œ‰โ€–๐‘‹โˆ—โ‰ค๐ฟโ€‰โ€‰for all ๐œ‰โˆˆ๐œ•๐‘“(๐‘ข).
(6)โ€‰โ€‰๐‘“0(๐‘ข,โ„Ž)=max{โŸจ๐œ‰,โ„ŽโŸฉโˆถ๐œ‰โˆˆ๐œ•๐‘“(๐‘ข)} for all โ„Žโˆˆ๐‘‹.
(7)โ€‰โ€‰The function ๐‘š(๐‘ข)=min๐‘คโˆˆ๐œ•๐‘“(๐‘ข)โ€–๐‘คโ€–๐‘‹โˆ—,(2.3) exists and is lower semicontinuous; that is, lim๐‘ขโ†’๐‘ข0inf๐‘š(๐‘ข)โ‰ฅ๐‘š(๐‘ข0).

In the following we need the nonsmooth version of the Palais-Smale condition.

Definition 2.2. One says ๐œ‘ that nonsmooth satisfies the (PS)๐‘-condition if any sequence {๐‘ข๐‘›}โŠ‚๐‘‹ such that ๐œ‘(๐‘ข๐‘›)โ†’๐‘ and ๐‘š(๐‘ข๐‘›)โ†’0.
In what follows we write the (PS)๐‘-condition as simply as the PS-condition if it holds for every level ๐‘โˆˆโ„ for the Palais-Smale condition at level ๐‘.
We note that property (4) above says that a local minimum (or local maximum) of ๐œ‘ is a critical point of ๐œ‘.Finally, we point out that many of the results of the classical critical point theory have been extended by Chang [4] to this setting of locally Lipschitz functionals. For example, one has the following celebrated theorem.

Theorem 2.3 (nonsmooth mountain pass theorem; see [4, 5]). If ๐‘‹ is a reflexive Banach space, ๐œ‘โˆถ๐‘‹โ†’โ„ is a locally Lipschitz function which satisfies the nonsmooth (PS)๐‘-condition, and for some ๐‘Ÿ>0 and ๐‘’1โˆˆ๐‘‹ with โ€–๐‘’1โ€–>๐‘Ÿ, max{๐œ‘(0),๐œ‘(๐‘’1)}โ‰คinf{๐œ‘(๐‘ข)โˆถโ€–๐‘ขโ€–=๐‘Ÿ}. Then ๐œ‘ has a nontrivial critical ๐‘ขโˆˆ๐‘‹ such that the critical value ๐‘=๐œ‘(๐‘ข) is characterized by the following minimax principle: ๐‘=inf๐›พโˆˆฮ“max[]๐‘กโˆˆ0,1๐œ‘(๐›พ(๐‘ก)),(2.4) where ฮ“={๐›พโˆˆ๐ถ([0,1],๐‘‹)โˆถ๐›พ(0)=0,๐›พ(1)=๐‘’1}.

We would like to point out that we can obtain the same results for problem (1.3) through approximation of the discontinuous nonlinearity by a sequence of continuous functions. Variational methods were then applied to the corresponding sequence of problems and limits were taken. For the rest of this paper, we write ๐‘‹=๐‘Š01,๐‘(ฮฉ) with the norm by โˆซโ€–๐‘ขโ€–=(ฮฉ|โˆ‡๐‘ข|๐‘๐‘‘๐‘ฅ)1/๐‘ and denote by ๐‘ and ๐‘๐‘– the generic positive constants for simplicity.

3. Proof of the Main Results

Now, having listed some basic results on critical point theory for the Lipschitz functionals, let us consider the functional ๐œ‘๐œ†(1๐‘ข)=๐‘๎€œฮฉ||||๎€œโˆ‡๐‘ข๐‘‘๐‘ฅโˆ’๐œ†ฮฉ๐บ(๐‘ข)๐‘‘๐‘ฅ,(3.1) where โˆซ๐บ(๐‘ข)=๐‘ข0๐‘”(๐‘ )๐‘‘๐‘  and ๐‘”(๐‘ ) were defined in (1.2). Clearly ๐บโˆถโ„โ†’โ„ is a locally Lipschitz continuous function and satisfies ๐บ(๐‘ )=0 for ๐‘ โ‰ค0. In view of [3, Theorems 2.1 and 2.2], the above formula for ๐œ‘๐œ†(๐‘ข) defines a locally Lipschitz functional on ๐‘‹ whose critical points are solutions of the differential inclusion ๎‚€||||โˆ’divโˆ‡๐‘ข๐‘โˆ’2๎‚โˆ‡๐‘ขโˆˆ๐œ•๐บ(๐‘ข)a.e.inฮฉ.(3.2) In our present case, it follows that ๐œ•๐บ(๐‘ )=๐‘“(๐‘ ) for ๐‘ >0, ๐œ•๐บ(๐‘ )=0 for ๐‘ <0, and ๐œ•๐บ(๐‘ )=[โˆ’๐‘Ž,0] for ๐‘ =0.

We start with some preliminary lemmas.

Lemma 3.1. Assume (F0), (F1), and (F2). Then there exists ๐œ†0 such that problem (1.3) has no nontrivial solution 0โ‰ค๐‘ขโˆˆ๐‘‹ for 0<๐œ†<๐œ†0.

Proof . If ๐‘ขโ‰ฅ0 is a solution of problem (1.3), then, multiplying the equation by ๐‘ข and integrating over ฮฉ yields โ€–๐‘ขโ€–๐‘=๎€œฮฉ||||โˆ‡๐‘ข๐‘๎€œ๐‘‘๐‘ฅ=๐œ†ฮฉ๎‚ต๎€œ๐‘”(๐‘ข)๐‘ข๐‘‘๐‘ฅ=๐œ†{๐‘ฅโˆˆฮฉโˆฃ๐‘ข(๐‘ฅ)โ‰ฅ๐›ฟ0}๎€œ๐‘ข๐‘”(๐‘ข)๐‘‘๐‘ฅ+{๐‘ฅโˆˆฮฉโˆฃ๐‘ข(๐‘ฅ)โ‰ฅ๐›ฟ0}๎‚ถ,๐‘ข๐‘”(๐‘ข)๐‘‘๐‘ฅ(3.3) hence โ€–๐‘ขโ€–๐‘๎€œโ‰ค๐œ†{๐‘ฅโˆˆฮฉโˆฃ๐‘ข(๐‘ฅ)โ‰ฅ๐›ฟ0}๐‘ข๐‘”(๐‘ข)๐‘‘๐‘ฅ,(3.4) where we have chosen ๐›ฟ0>0 so that ๐‘”(๐‘ )โ‰ค0 for 0โ‰ค๐‘ โ‰ค๐›ฟ0 (such a ๐›ฟ0 exists in view of (F0)). Now, since (F1) implies the existence of ๐‘>0 such that ๐‘ ๐‘”(๐‘ )โ‰ค๐‘(1+๐‘ ๐‘),(3.5) for all ๐‘ >0, we obtain from (3.4) that โ€–๐‘ขโ€–๐‘๎€œโ‰ค๐œ†๐‘{๐‘ฅโˆˆฮฉโˆฃ๐‘ข(๐‘ฅ)โ‰ฅ๐›ฟ0}(1+๐‘ข๐‘๎ƒฉ1)๐‘‘๐‘ฅโ‰ค๐œ†๐‘1+๐›ฟ๐‘0๎ƒช๎€œ{๐‘ฅโˆˆฮฉโˆฃ๐‘ข(๐‘ฅ)โ‰ฅ๐›ฟ0}๐‘ข๐‘๐‘‘๐‘ฅโ‰ค๐œ†๐‘1๎€œฮฉ๐‘ข๐‘๐‘‘๐‘ฅ,(3.6) so that โ€–๐‘ขโ€–๐‘โ‰ค๐œ†๐‘2โ€–๐‘ขโ€–๐‘,(3.7) where this last constant ๐‘2>0 is independent of both ๐‘ข and ๐œ†. Therefore we must have 1๐œ†โ‰ฅ๐‘2โˆถ=๐œ†0>0.(3.8)

Lemma 3.2. Assume (F0) and either (F1) or (F3). Then ๐‘ข=0 is a strict local minimum of the functional ๐œ‘๐œ†.

Proof . Since (F1) or (F3) implies the existence of ๐‘3>0 such that ๐บ(๐‘ )โ‰ค๐‘3๎€ท1+|๐‘ |๐‘โˆ—๎€ธโˆ€๐‘ โˆˆโ„,(3.9) recall also that ๐บ(๐‘ )=0 for ๐‘ โ‰ค0. Then, with ๐›ฟ0>0 as in the proof of Lemma 3.1 and noticing that ๐บ(๐‘ )โ‰ค0 for all โˆ’โˆž<๐‘ โ‰ค๐›ฟ0, we can write for an arbitrary ๐‘ขโˆˆ๐‘‹, ๐œ‘๐œ†(1๐‘ข)=๐‘โ€–๐‘ขโ€–๐‘๎€œโˆ’๐œ†ฮฉโ‰ฅ1๐บ(๐‘ข)๐‘‘๐‘ฅ๐‘โ€–๐‘ขโ€–๐‘๎€œโˆ’๐œ†{๐‘ฅโˆˆฮฉโˆฃ๐‘ข(๐‘ฅ)โ‰ฅ๐›ฟ0}โ‰ฅ1๐บ(๐‘ข)๐‘‘๐‘ฅ๐‘โ€–๐‘ขโ€–๐‘โˆ’๐œ†๐‘3๎€œ{๐‘ฅโˆˆฮฉโˆฃ๐‘ข(๐‘ฅ)โ‰ฅ๐›ฟ0}๎€ท1+๐‘ข๐‘โˆ—๎€ธโ‰ฅ1๐‘‘๐‘ฅ๐‘โ€–๐‘ขโ€–๐‘โˆ’๐œ†๐‘3๎ƒฉ11+๐›ฟ๐‘โˆ—0๎ƒช๎€œ{๐‘ฅโˆˆฮฉโˆฃ๐‘ข(๐‘ฅ)โ‰ฅ๐›ฟ0}๐‘ข๐‘โˆ—๐‘‘๐‘ฅ,(3.10) so that, using the Sobolev embedding theorem in the last inequality and with a constant ๐‘4>0 independent of ๐‘ข and ฮฉ, we obtain ๐œ‘๐œ†1(๐‘ข)โ‰ฅ๐‘โ€–๐‘ขโ€–๐‘โˆ’๐œ†๐‘4โ€–๐‘ขโ€–๐‘โˆ—=1๐‘โ€–๐‘ขโ€–๐‘๎‚€1โˆ’๐‘๐œ†๐‘4โ€–๐‘ขโ€–๐‘โˆ—โˆ’๐‘๎‚.(3.11) Therefore, for each 0<๐œŒ<๐œŒ๐œ†โˆถ=1/(๐‘๐œ†๐‘4)๐‘โˆ—โˆ’๐‘, it follows that ๐œ‘๐œ†(๐‘ข)โ‰ฅ๐›ผ๐œŒ>0 if โ€–๐‘ขโ€–=๐œŒ. This shows that ๐‘ข=0 is a strict local minimum of ๐œ‘๐œ†.

Lemma 3.3. Under the same assumptions as in Lemma 3.2, let ฬ‚๐‘ขโˆˆ๐‘‹ be a critical point of ๐œ‘๐œ†. Then, ฬ‚๐‘ขโˆˆ๐ถ1,๐›ผ(ฮฉ) and ฬ‚๐‘ข is a nonnegative solution of problem (1.3), where 0<๐›ผ<1.

Proof . We shall follow some of the arguments in [3]. As mentioned earlier, if ฬ‚๐‘ข is a critical point of ๐œ‘๐œ†, then it is shown in [3] that ฬ‚๐‘ข is a solution of the differential inclusion ๎‚€||||โˆ’divโˆ‡ฬ‚๐‘ข๐‘โˆ’2๎‚โˆ‡ฬ‚๐‘ขโˆˆ๐œ•๐บ(ฬ‚๐‘ข)a.e.inฮฉ.(3.12) Since ๐‘” is only discontinuous at ๐‘ =0, the above differential inclusion reduces to an equality, except possibly on the subset ๐ดโŠ‚ฮฉ where ฬ‚๐‘ข=0. Since ๐‘“ is subcritical, using the ๐ถ1,๐›ผ regularity results for quasilinear elliptic equations with ๐‘-growth condition (see, for example, [6]), we have ฬ‚๐‘ขโˆˆ๐ถ1,๐›ผ(ฮฉโงต๐ด). And, in this latter case, โˆ’div(|โˆ‡ฬ‚๐‘ข|๐‘โˆ’2โˆ‡ฬ‚๐‘ข) takes on values in the bounded interval [โˆ’๐‘Ž,0]. Therefore, by the Michael selection theorem (see Theoremโ€‰โ€‰1.2.5 of [5]), we see that โˆ’ฮ”๐‘โˆถ๐‘ขโ†ฆ[โˆ’๐‘Ž,0] admits a continuous selection. Using the ๐ถ1,๐›ผ regularity results for quasilinear elliptic equations with ๐‘-growth condition again, it follows that ฬ‚๐‘ขโˆˆ๐ถ1,๐›ผ(ฮฉ), 0<๐›ผ<1.
Next, using a Morrey-Stampacchia theorem [7, Theoremโ€‰โ€‰3.2.2, page 69], we have that โˆ’div(|โˆ‡ฬ‚๐‘ข|๐‘โˆ’2โˆ‡ฬ‚๐‘ข)=0 a.e. in ๐ด. Therefore, since we defined ๐‘”(0)=0, it follows that ๎‚€||||โˆ’divโˆ‡ฬ‚๐‘ข๐‘โˆ’2๎‚โˆ‡ฬ‚๐‘ข=๐‘”(ฬ‚๐‘ข)a.e.inฮฉ.(3.13) Replacing the inclusion (3.2) on ฬ‚๐‘ข, we conclude that ฬ‚๐‘ขโˆˆ๐ถ1,๐›ผ(ฮฉ) is a solution of (1.3). Finally, recalling that ๐‘”(๐‘ )=0 for ๐‘ โ‰ค0, it is clear that ฬ‚๐‘ขโ‰ฅ0. The proof of Lemma 3.3 is complete.

Lemma 3.4. Assume either (F1) or (F3), (F4). Then ๐œ‘๐œ† satisfies the nonsmooth (๐‘ƒ๐‘†)๐‘-condition at every ๐‘โˆˆโ„.

Proof. Let {๐‘ข๐‘›}๐‘›โ‰ฅ1โІ๐‘‹ be a sequence such that |๐œ‘๐œ†(๐‘ข๐‘›)|โ‰ค๐‘ for all ๐‘›โ‰ฅ1 and ๐‘š(๐‘ข๐‘›)โ†’0 as ๐‘›โ†’โˆž. In the superlinear and subcritical case, from (F4), we have ๐‘โ‰ฅ๐œ‘๐œ†๎€ท๐‘ข๐‘›๎€ธ=1๐‘๎€œฮฉ||โˆ‡๐‘ข๐‘›||๐‘๎€œ๐‘‘๐‘ฅโˆ’๐œ†ฮฉ๐บ๎€ท๐‘ข๐‘›๎€ธโ‰ฅโ€–โ€–๐‘ข๐‘‘๐‘ฅ๐‘›โ€–โ€–๐‘๐‘โˆ’๐œ†๐œƒ๎€œฮฉ๎ซ๐œ‰๎€ท๐‘ข๐‘›๎€ธ,๐‘ข๐‘›๎ฌโ‰ฅ๎‚ต1๐‘‘๐‘ฅโˆ’๐‘๐‘โˆ’1๐œƒ๎‚ถโ€–โ€–๐‘ข๐‘›โ€–โ€–๐‘+๎€œฮฉ1๐œƒ๎€ทโ€–โ€–๐‘ข๐‘›โ€–โ€–๐‘๎ซ๐œ‰๎€ท๐‘ขโˆ’๐œ†๐‘›๎€ธ,๐‘ข๐‘›โ‰ฅ๎‚ต1๎ฌ๎€ธ๐‘‘๐‘ฅโˆ’๐‘๐‘โˆ’1๐œƒ๎‚ถโ€–โ€–๐‘ข๐‘›โ€–โ€–๐‘โˆ’๐œ†๐œƒโ€–๐œ‰โ€–๐‘‹โˆ—โ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘,(3.14) where ๐œ‰(๐‘ข๐‘›)โˆˆ๐œ•๐บ(๐‘ข๐‘›). Hence {๐‘ข๐‘›}๐‘›โ‰ฅ1โІ๐‘‹ is bounded.
Thus by passing to a subsequence if necessary, we may assume that ๐‘ข๐‘›โ‡€๐‘ข in ๐‘‹ as ๐‘›โ†’+โˆž. We have ๎ซ๐ฝ๎…ž๎€ท๐‘ข๐‘›๎€ธ,๐‘ข๐‘›๎ฌโˆ’๎€œโˆ’๐‘ขฮฉ๐œ‰๎€ท๐‘ข๐‘›๐‘ข๎€ธ๎€ท๐‘›๎€ธโˆ’๐‘ข๐‘‘๐‘ฅโ‰ค๐œ€๐‘›โ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ข(3.15) with ๐œ€๐‘›โ†“0, where ๐œ‰(๐‘ข๐‘›)โˆˆ๐œ•ฮจ(๐‘ข๐‘›) and โˆซ๐ฝ=(1/๐‘)ฮฉ|โˆ‡๐‘ข๐‘›|๐‘๐‘‘๐‘ฅ. From (F3) and Chang [4] we know that ๐œ‰(๐‘ข๐‘›(๐‘ฅ))โˆˆ๐ฟ(๐‘โˆ—โˆ’1)โ€ฒ(ฮฉ) ((๐‘โˆ—โˆ’1)๎…ž=(๐‘โˆ—โˆ’1)/(๐‘โˆ—โˆ’2)). Since ๐‘‹ is embedded compactly in ๐ฟ๐‘โˆ—โˆ’1(ฮฉ), we have that ๐‘ข๐‘›โ†’๐‘ข as ๐‘›โ†’โˆž in ๐ฟ๐‘โˆ—โˆ’1(ฮฉ). So using the Hรถlder inequality, we have ๎€œฮฉ๐œ‰๐‘›(๎€ท๐‘ข๐‘ฅ)๐‘›๎€ธโˆ’๐‘ข๐‘‘๐‘ฅโŸถ0as๐‘›โŸถ+โˆž.(3.16)
Therefore, we obtain that lim๐‘›โ†’โˆžsupโŸจ๐ฝโ€ฒ(๐‘ข๐‘›),๐‘ข๐‘›โˆ’๐‘ขโŸฉโ‰ค0. But we know that ๐ฝโ€ฒ is a mapping of type (๐‘†+). Thus we have ๐‘ข๐‘›โŸถ๐‘ขin๐‘‹.(3.17) Using the similar method, we can more easily get the nonsmooth (PS)๐‘-condition in the case of sublinear.

Remark 3.5. Note that we cannot directly apply the results of [4] because the operator โˆ’ฮ”๐‘ is not self-adjoint linear.

Lemma 3.6. Under assumptions (F0), (F1), and (F2), let ฮฉ=๐ต๐‘…โŠ‚โ„๐‘ with ๐‘โ‰ฅ2, and let ๐‘ขโˆˆ๐ถ1(๐ต๐‘…) be a radially symmetric, nonincreasing function such that ๐‘ขโ‰ฅ0 and ๐‘ข is a minimizer of ๐œ‘๐œ†(๐‘ข) with ๐œ‘๐œ†(๐‘ข)=๐‘š<0. Then, ๐‘ข does not vanish in ๐ต๐‘…; that is, ๐‘ข(๐‘ฅ)>0 for all ๐‘ฅโˆˆ๐ต๐‘….

Proof. Since ๐‘” is discontinuous at zero, we note that the conclusion does not follow directly from uniqueness of solution for the Cauchy problem with data at ๐‘Ÿ=๐‘… (in fact, writing ๐‘ข=๐‘ข(๐‘Ÿ), ๐‘Ÿ=|๐‘ฅ|, we may have ๐‘ข(๐‘…)=๐‘ขโ€ฒ(๐‘…)=0 and ๐‘ขโ‰ข0).
Now, since ๐‘ขโ‰ข0 by assumption, ๐‘…0โˆถ=inf{๐‘Ÿโ‰ค๐‘…โˆถ๐‘ข(๐‘ )=0for๐‘Ÿโ‰ค๐‘ โ‰ค๐‘…} satisfies 0<๐‘…0โ‰ค๐‘…. If ๐‘…0=๐‘… there is nothing to prove in view of the fact that ๐‘ข is non-increasing. On the other hand, if ๐‘…0<๐‘… then ๐‘ขโ€ฒ(๐‘…0)=0 and ๐‘ข(๐‘Ÿ)>0 for ๐‘Ÿโˆˆ[0,๐‘…0). It is not hard to prove that this contradicts that ๐‘ข is a minimizer of ๐œ‘๐œ†. Indeed, if ๐‘…0<๐‘…then ๐œ‘๐œ†(1๐‘ข)=๐‘๎€œ๐ต๐‘…0||||โˆ‡๐‘ข๐‘๎€œ๐‘‘๐‘ฅโˆ’๐œ†๐ต๐‘…0๐บ(๐‘ข)๐‘‘๐‘ฅ=๐‘š<0.(3.18) A simple calculation shows that the rescaled function ๐‘ฃ(๐‘Ÿ)=๐‘ข(๐‘…0๐‘Ÿ/๐‘…)โˆˆ๐‘Š01,๐‘(๐ต๐‘…)โˆฉ๐ถ1(๐ต๐‘…) satisfies ๐œ‘๐œ†(๐‘ฃ)=๐›ฟ๐‘โˆ’๐‘๎ƒฌ1๐‘๎€œ๐ต๐‘…0||||โˆ‡๐‘ข๐‘๐‘‘๐‘ฅโˆ’๐›ฟโˆ’๐‘๐œ†๎€œ๐ต๐‘…0๎ƒญ๐บ(๐‘ข)๐‘‘๐‘ฅ,(3.19) where ๐›ฟโˆถ=๐‘…0/๐‘… is less than 1. Therefore, since we are assuming 1<๐‘โ‰ค๐‘, we would reach the contradiction ๐œ‘๐œ†(๐‘ฃ)<๐‘š.

Remark 3.7. Note that the condition (F2) is necessary because of which guarantee ๐บ can have positive values.

Proof of Theorem 1.1. We observe that the functional ๐œ‘๐œ† is weakly lower semicontinuous on ๐‘‹. Moreover, the sublinearity assumption (F1) on ๐‘”(๐‘ข) implies that ๐œ‘๐œ† is coercive. Therefore, the infimum of ๐œ‘๐œ† is attained at some ๐‘ข๐œ†: inf๐‘ขโˆˆ๐‘‹๐œ‘๐œ†(๐‘ข)=๐œ‘๐œ†๎€ท๐‘ข๐œ†๎€ธ.(3.20) And, in view of Lemma 3.3, ๐‘ข๐œ†โˆˆ๐ถ1,๐›ผ(ฮฉ) is a nonnegative solution of (1.3). We now claim that ๐‘ข๐œ† is nonzero for all ๐œ†>0 large.

Claim 1. There exists ฮ›>0 such that ๐œ‘๐œ†(๐‘ข๐œ†)<0 for all ๐œ†โ‰ฅฮ›.
In order to prove the claim it suffices to exhibit an element ๐‘คโˆˆ๐‘‹ such that ๐œ‘๐œ†(๐‘ค)<0 for all ๐œ†>0 large. This is quite standard considering that ๐บ(๐›ฟ)>0 by (F2). Indeed, letting ฮฉ๐œ€โˆถ={๐‘ฅโˆˆฮฉโˆถdist(๐‘ฅ,๐œ•ฮฉ)>๐œ€} for ๐œ€>0 small, define ๐‘ค so that ๐‘ค(๐‘ฅ)=๐›ฟ for ๐‘ฅโˆˆฮฉ๐œ€ and 0โ‰ค๐‘ค(๐‘ฅ)โ‰ค๐›ฟ for ๐‘ฅโˆˆฮฉโงตฮฉ๐œ€. Then ๐œ‘๐œ†1(๐‘ค)=๐‘โ€–๐‘คโ€–๐‘๎‚ต๎€œโˆ’๐œ†ฮฉ๐œ€๎€œ๐บ(๐‘ค)๐‘‘๐‘ฅ+ฮฉโงตฮฉ๐œ€๎‚ถโ‰ค1๐บ(๐‘ค)๐‘‘๐‘ฅ๐‘โ€–๐‘คโ€–๐‘๎€ท๎€ทฮฉโˆ’๐œ†๐บ(๐›ฟ)meas๐œ€๎€ธโˆ’๐‘(1+๐›ฟ๐‘๎€ท)measฮฉโงตฮฉ๐œ€,๎€ธ๎€ธ(3.21) where we note that the expression in the above parenthesis is positive if we choose ๐œ€>0 sufficiently small. Therefore, there exists ฮ›>0 such that ๐œ‘๐œ†(๐‘ค)<0 for all ๐œ†โ‰ฅฮ›, which proves the claim.
On the other hand, when ฮฉ=๐ต๐‘…, let ๐‘ขโˆ—๐œ† denote the Schwarz symmetrization of ๐‘ข๐œ†, namely, ๐‘ขโˆ—๐œ† is the unique radially symmetric, nonincreasing, nonnegative function in ๐‘‹ which is equi-measurable with ๐‘ข๐œ†. As is well known, ๎€œ๐ต๐‘…๐บ๎€ท๐‘ขโˆ—๐œ†๎€ธ๎€œ๐‘‘๐‘ฅ=๐ต๐‘…๐บ๎€ท๐‘ข๐œ†๎€ธ๐‘‘๐‘ฅ,(3.22) and โ€–๐‘ขโˆ—๐œ†โ€–โ‰คโ€–๐‘ข๐œ†โ€– (the Faber-Krahn inequality; see [8]), so that ๐œ‘๐œ†(๐‘ขโˆ—๐œ†)โ‰ค๐œ‘๐œ†(๐‘ข๐œ†). Therefore, we necessarily have ๐œ‘๐œ†(๐‘ขโˆ—๐œ†)=๐œ‘๐œ†(๐‘ข๐œ†) and may assume that ๐‘ข๐œ†=๐‘ขโˆ—๐œ†. Moreover, ๐‘ข๐œ†>0 in ฮฉ by Lemma 3.6. Therefore, ๐‘ข๐œ† is a positive solution of both problems (1.1) and (1.3).
Next, we recall that ๐‘ข=0 is a strict local minimum of ๐œ‘๐œ† by Lemma 3.2. Therefore, since ๐œ‘๐œ† satisfies the nonsmooth Palais-Smale condition by Lemma 3.4, we can use the minima ๐‘ข=0 and ๐‘ข=๐‘ข๐œ† of ๐œ‘๐œ† to apply the nonsmooth mountain pass theorem and conclude that there exists a second nontrivial critical point ๐‘ฃ๐œ† with ๐œ‘๐œ†(๐‘ฃ๐œ†)>0. Again, ๐‘ฃ๐œ† is a nonnegative solution of problem (1.3) in view of Lemma 3.3. In addition, when ฮฉ=๐ต๐‘…, arguments similar to above passage show that we may assume ๐‘ฃ๐œ†=๐‘ฃโˆ—๐œ†. The proof of Theorem 1.1 is complete.

Proof of Theorem 1.2. As is well-known, the AR condition (F4) readily implies the existence of an element ๐‘’๐œ†โˆˆ๐‘‹ such that ๐œ‘๐œ†(๐‘’๐œ†)โ‰ค0. On the other hand, Lemma 3.2 says that ๐‘ข=0 is a (strict) local minimum of ๐œ‘๐œ† and Lemma 3.4 says that ๐œ‘๐œ† satisfies nonsmooth (PS)๐‘ for every ๐‘โˆˆโ„. Therefore, an application of the nonsmooth mountain pass theorem stated in section 2 yields the existence of a critical point ๐‘ฃ๐œ† such that ๐œ‘๐œ†๎€ท๐‘ฃ๐œ†๎€ธ>0.(3.23) In particular, ๐‘ฃ๐œ†โ‰ 0, and it follows that ๐‘ฃ๐œ† is a nonnegative solution of problem (1.3) by Lemma 3.3. As in the proof of Theorem 1.1, we may assume that ๐‘ฃ๐œ†=๐‘ฃโˆ—๐œ† in the case of a ball ฮฉ=๐ต๐‘….
Finally, still in the case of a ball ฮฉ=๐ต๐‘…, we claim that there exists ๐œ†โˆ—>0 such that, for all 0<๐œ†<๐œ†โˆ—, ๐‘ฃ๐œ†=๐‘ฃโˆ—๐œ† is a positive solution of problem (1.3) (hence of problem (1.1)) having negative normal derivative on ๐œ•๐ต๐‘…. Indeed, if that is not the case, then, for any given ๐œ†>0, we can find 0<๐œ‡=๐œ‡(๐œ†)<๐œ† such that the nonnegative solution ๐‘ฃ๐œ‡=๐‘ฃโˆ—๐œ‡ of problem (1.1) with ๐œ†=๐œ‡ obtained a bove satisfies ๐‘ฃ๐œ‡๎€บ(๐‘Ÿ)>0for๐‘Ÿโˆˆ0,๐‘…0๎€ธ,๐‘ฃ๎…ž๐œ‡๎€ท๐‘…0๎€ธ=0,๐‘ฃ๐œ‡๎€บ๐‘…(๐‘Ÿ)=0for๐‘Ÿโˆˆ0๎€ป,,๐‘…(3.24) for some 0<๐‘…0โ‰ค๐‘…. Therefore, the rescaled function ๐‘ค๐œ‡(๐‘Ÿ)โˆถ=๐‘ฃ๐œ‡(๐‘…0๐‘Ÿ/๐‘…) is a positive solution of (1.1) with ๐œ†=๐œ‡0 (again in the ball ๐ต๐‘…), with ๐œ‡0โˆถ=๐œ‡๐‘…๐‘0/๐‘…๐‘โ‰ค๐œ‡. This shows that we can always construct a decreasing sequence ๐œ‡๐‘›>0 satisfying alternative (ii) of Theorem 1.2, in case alternative (i) does not hold.

Acknowledgments

The author is very grateful to an anonymous referee for his or her valuable suggestions. Research supported by NSFC (no. 11061030, no. 10971087).

References

  1. A. Castro and R. Shivaji, โ€œNonnegative solutions for a class of nonpositone problems,โ€ Proceedings of the Royal Society of Edinburgh A, vol. 108, no. 3-4, pp. 291โ€“302, 1988. View at: Publisher Site | Google Scholar
  2. H. B. Keller and D. S. Cohen, โ€œSome positone problems suggested by nonlinear heat generation,โ€ Journal of Mathematics and Mechanics, vol. 16, pp. 1361โ€“1376, 1967. View at: Google Scholar | Zentralblatt MATH
  3. D. G. Costa, H. Tehrani, and J. Yang, โ€œOn a variational approach to existence and multiplicity results for semipositone problems,โ€ Electronic Journal of Differential Equations, vol. 11, pp. 1โ€“10, 2006. View at: Google Scholar | Zentralblatt MATH
  4. K. C. Chang, โ€œVariational methods for nondifferentiable functionals and their applications to partial differential equations,โ€ Journal of Mathematical Analysis and Applications, vol. 80, no. 1, pp. 102โ€“129, 1981. View at: Publisher Site | Google Scholar
  5. L. Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, vol. 8, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005.
  6. G. M. Lieberman, โ€œBoundary regularity for solutions of degenerate elliptic equations,โ€ Nonlinear Analysis, vol. 12, no. 11, pp. 1203โ€“1219, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Springer, New York, NY, USA, 1966.
  8. W. G. Faris, โ€œWeak Lebesgue spaces and quantum mechanical binding,โ€ Duke Mathematical Journal, vol. 43, no. 2, pp. 365โ€“373, 1976. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2012 Guowei Dai and Chunfeng Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views568
Downloads414
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.