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

1. Introduction

In this paper, we shall study the following semipositone problem involving the š‘-Laplacian: ī‚€||||āˆ’divāˆ‡š‘¢š‘āˆ’2ī‚āˆ‡š‘¢=šœ†š‘“(š‘¢)inĪ©,š‘¢=0onšœ•Ī©,(1.1) where Ī©āŠ‚ā„š‘ is a smooth bounded domain, šœ† is a positive parameter, and š‘“āˆ¶[0,+āˆž)ā†’ā„ is a continuous function satisfying the condition(F0)š‘“(0)=āˆ’š‘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 š‘“ was positive and monotone.

Under the case of š‘ā‰”2, 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 š‘-Laplacian. The main difficulty is in verifying (PS)š‘-condition because of the operator āˆ’Ī”š‘ is not self-adjoint linear.

We define the discontinuous nonlinearity š‘”(š‘ ) by ī‚»š‘”(š‘ )=0,ifš‘ ā‰¤0,š‘“(š‘ ),ifš‘ >0.(1.2) We shall consider the modified problem ī‚€||||āˆ’divāˆ‡š‘¢š‘āˆ’2ī‚āˆ‡š‘¢=šœ†š‘”(š‘¢)inĪ©,š‘¢=0onšœ•Ī©.(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)limš‘ ā†’+āˆž(š‘“(š‘ )/š‘ š‘āˆ’1)=0 (the sublinear case);(F2)š¹(š›æ)>0 for some š›æ>0, where āˆ«š¹(š‘¢)=š‘¢0š‘“(š‘ )š‘‘š‘ .Then, there exist 0<šœ†0ā‰¤šœ†āˆ— such that (1.3) has no nontrivial nonnegative solution for 0<šœ†<šœ†0, and has at least two nontrivial nonnegative solutions š‘¢šœ†, š‘£šœ† for all šœ†>šœ†āˆ—. Moreover, when Ī© is a ball šµš‘…=šµš‘…(0), these two solutions are non-increasing, radially symmetric and, if š‘ā‰„2, at least one of them is positive, hence a solution of (1.1).

Theorem 1.2. Assume (F0), (F2) and the following assumptions:(F3)limš‘ ā†’+āˆž(š‘“(š‘ )/š‘ š‘āˆ’1)=+āˆž, limš‘ ā†’+āˆž(š‘“(š‘ )/š‘ š‘āˆ—āˆ’2)=0 (the sublinear, subcritical case);(F4)šœƒš¹(š‘ )ā‰¤š‘“(š‘ )š‘ for all š‘ ā‰„š¾ and some šœƒ>š‘ (the AR-condition).Then, (1.3) has at least one nonnegative solution š‘£šœ† for all šœ†>0. IfĪ©=šµš‘… then š‘£šœ† is nonincreasing, radially symmetric and one of the two alternatives occurs. (i)There exists šœ†āˆ—>0 such that, for all 0<šœ†<šœ†āˆ—, š‘£šœ† is a positive solution of (1.1) having negative normal derivative on šœ•šµš‘…. (ii)For some sequence šœ‡š‘›ā†’0, problem (1.1) with šœ†ā‰”šœ‡š‘› has a positive solution š‘¤šœ‡š‘› with zero normal derivative on šœ•šµš‘….

2. Preliminaries

We start by recalling some basic results on variational methods for locally Lipschitz functionals. Let (š‘‹,ā€–ā‹…ā€–) be a real Banach space and š‘‹āˆ— is its topological dual. A function š‘“āˆ¶š‘‹ā†’ā„ is called locally Lipschitz if each point š‘¢āˆˆš‘‹ possesses a neighborhood Ī©š‘¢ such that |š‘“(š‘¢1)āˆ’š‘“(š‘¢2)|ā‰¤šæā€–š‘¢1āˆ’š‘¢2ā€– for all š‘¢1,š‘¢2āˆˆĪ©š‘¢, for a constant šæ>0 depending on Ī©š‘¢. The generalized directional derivative of š‘“ at the point š‘¢āˆˆš‘‹ in the direction š‘£āˆˆš‘‹ is š‘“0(š‘¢,š‘£)=limsupš‘¤ā†’š‘¢,š‘”ā†’0+1š‘”(š‘“(š‘¤+š‘”š‘£)āˆ’š‘“(š‘¤)).(2.1) The generalized gradient of š‘“ at š‘¢āˆˆš‘‹ is defined by ī€½š‘¢šœ•š‘“(š‘¢)=āˆ—āˆˆš‘‹āˆ—āˆ¶āŸØš‘¢āˆ—,šœ‘āŸ©ā‰¤š‘“0ī€¾,(š‘¢;šœ‘)āˆ€šœ‘āˆˆš‘‹(2.2) which is a nonempty, convex, and š‘¤āˆ—-compact subset of š‘‹, where āŸØā‹…,ā‹…āŸ© is the duality pairing between š‘‹āˆ— and š‘‹. We say that š‘¢āˆˆš‘‹ is a critical point of š‘“ if 0āˆˆšœ•š‘“(š‘¢). 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)ā€‰ā€‰Let š‘—āˆ¶š‘‹ā†’ā„ be a continuously differentiable function. Then šœ•š‘—(š‘¢)={š‘—ā€²(š‘¢)}, š‘—0(š‘¢;š‘§) coincides with āŸØš‘—ā€²(š‘¢),š‘§āŸ©š‘‹, and (š‘“+š‘—)0(š‘¢,š‘§)=š‘“0(š‘¢;š‘§)+āŸØš‘—ā€²(š‘¢),š‘§āŸ©š‘‹for all š‘¢,š‘§āˆˆš‘‹.
(2)ā€‰ā€‰šœ•(šœ†š‘“)(š‘¢)=šœ†šœ•(š‘¢) for all šœ†āˆˆā„.
(3)ā€‰ā€‰If š‘“ is a convex functional, then šœ•š‘“(š‘¢) coincides with the usual subdifferential of š‘“ in the sense of convex analysis.
(4)ā€‰ā€‰If š‘“ has a local minimum (or a local maximum) at š‘¢0āˆˆš‘‹, then 0āˆˆšœ•š‘“(š‘¢0).
(5)ā€‰ā€‰ā€–šœ‰ā€–š‘‹āˆ—ā‰¤šæā€‰ā€‰for all šœ‰āˆˆšœ•š‘“(š‘¢).
(6)ā€‰ā€‰š‘“0(š‘¢,ā„Ž)=max{āŸØšœ‰,ā„ŽāŸ©āˆ¶šœ‰āˆˆšœ•š‘“(š‘¢)} for all ā„Žāˆˆš‘‹.
(7)ā€‰ā€‰The function š‘š(š‘¢)=minš‘¤āˆˆšœ•š‘“(š‘¢)ā€–š‘¤ā€–š‘‹āˆ—,(2.3) exists and is lower semicontinuous; that is, limš‘¢ā†’š‘¢0infš‘š(š‘¢)ā‰„š‘š(š‘¢0).

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

Definition 2.2. One says šœ‘ that nonsmooth satisfies the (PS)š‘-condition if any sequence {š‘¢š‘›}āŠ‚š‘‹ such that šœ‘(š‘¢š‘›)ā†’š‘ and š‘š(š‘¢š‘›)ā†’0.
In what follows we write the (PS)š‘-condition as simply as the PS-condition if it holds for every level š‘āˆˆā„ for the Palais-Smale condition at level š‘.
We note that property (4) above says that a local minimum (or local maximum) of šœ‘ is a critical point of š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 š‘‹ is a reflexive Banach space, šœ‘āˆ¶š‘‹ā†’ā„ is a locally Lipschitz function which satisfies the nonsmooth (PS)š‘-condition, and for some š‘Ÿ>0 and š‘’1āˆˆš‘‹ with ā€–š‘’1ā€–>š‘Ÿ, max{šœ‘(0),šœ‘(š‘’1)}ā‰¤inf{šœ‘(š‘¢)āˆ¶ā€–š‘¢ā€–=š‘Ÿ}. Then šœ‘ has a nontrivial critical š‘¢āˆˆš‘‹ such that the critical value š‘=šœ‘(š‘¢) is characterized by the following minimax principle: š‘=infš›¾āˆˆĪ“max[]š‘”āˆˆ0,1šœ‘(š›¾(š‘”)),(2.4) where Ī“={š›¾āˆˆš¶([0,1],š‘‹)āˆ¶š›¾(0)=0,š›¾(1)=š‘’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 š‘‹=š‘Š01,š‘(Ī©) with the norm by āˆ«ā€–š‘¢ā€–=(Ī©|āˆ‡š‘¢|š‘š‘‘š‘„)1/š‘ and denote by š‘ and š‘š‘– 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 šœ‘šœ†(1š‘¢)=š‘ī€œĪ©||||ī€œāˆ‡š‘¢š‘‘š‘„āˆ’šœ†Ī©šŗ(š‘¢)š‘‘š‘„,(3.1) where āˆ«šŗ(š‘¢)=š‘¢0š‘”(š‘ )š‘‘š‘  and š‘”(š‘ ) were defined in (1.2). Clearly šŗāˆ¶ā„ā†’ā„ is a locally Lipschitz continuous function and satisfies šŗ(š‘ )=0 for š‘ ā‰¤0. In view of [3, Theorems 2.1 and 2.2], the above formula for šœ‘šœ†(š‘¢) defines a locally Lipschitz functional on š‘‹ whose critical points are solutions of the differential inclusion ī‚€||||āˆ’divāˆ‡š‘¢š‘āˆ’2ī‚āˆ‡š‘¢āˆˆšœ•šŗ(š‘¢)a.e.inĪ©.(3.2) In our present case, it follows that šœ•šŗ(š‘ )=š‘“(š‘ ) for š‘ >0, šœ•šŗ(š‘ )=0 for š‘ <0, and šœ•šŗ(š‘ )=[āˆ’š‘Ž,0] for š‘ =0.

We start with some preliminary lemmas.

Lemma 3.1. Assume (F0), (F1), and (F2). Then there exists šœ†0 such that problem (1.3) has no nontrivial solution 0ā‰¤š‘¢āˆˆš‘‹ for 0<šœ†<šœ†0.

Proof . If š‘¢ā‰„0 is a solution of problem (1.3), then, multiplying the equation by š‘¢ and integrating over Ī© yields ā€–š‘¢ā€–š‘=ī€œĪ©||||āˆ‡š‘¢š‘ī€œš‘‘š‘„=šœ†Ī©ī‚µī€œš‘”(š‘¢)š‘¢š‘‘š‘„=šœ†{š‘„āˆˆĪ©āˆ£š‘¢(š‘„)ā‰„š›æ0}ī€œš‘¢š‘”(š‘¢)š‘‘š‘„+{š‘„āˆˆĪ©āˆ£š‘¢(š‘„)ā‰„š›æ0}ī‚¶,š‘¢š‘”(š‘¢)š‘‘š‘„(3.3) hence ā€–š‘¢ā€–š‘ī€œā‰¤šœ†{š‘„āˆˆĪ©āˆ£š‘¢(š‘„)ā‰„š›æ0}š‘¢š‘”(š‘¢)š‘‘š‘„,(3.4) where we have chosen š›æ0>0 so that š‘”(š‘ )ā‰¤0 for 0ā‰¤š‘ ā‰¤š›æ0 (such a š›æ0 exists in view of (F0)). Now, since (F1) implies the existence of š‘>0 such that š‘ š‘”(š‘ )ā‰¤š‘(1+š‘ š‘),(3.5) for all š‘ >0, we obtain from (3.4) that ā€–š‘¢ā€–š‘ī€œā‰¤šœ†š‘{š‘„āˆˆĪ©āˆ£š‘¢(š‘„)ā‰„š›æ0}(1+š‘¢š‘īƒ©1)š‘‘š‘„ā‰¤šœ†š‘1+š›æš‘0īƒŖī€œ{š‘„āˆˆĪ©āˆ£š‘¢(š‘„)ā‰„š›æ0}š‘¢š‘š‘‘š‘„ā‰¤šœ†š‘1ī€œĪ©š‘¢š‘š‘‘š‘„,(3.6) so that ā€–š‘¢ā€–š‘ā‰¤šœ†š‘2ā€–š‘¢ā€–š‘,(3.7) where this last constant š‘2>0 is independent of both š‘¢ and šœ†. Therefore we must have 1šœ†ā‰„š‘2āˆ¶=šœ†0>0.(3.8)

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

Proof . Since (F1) or (F3) implies the existence of š‘3>0 such that šŗ(š‘ )ā‰¤š‘3ī€·1+|š‘ |š‘āˆ—ī€øāˆ€š‘ āˆˆā„,(3.9) recall also that šŗ(š‘ )=0 for š‘ ā‰¤0. Then, with š›æ0>0 as in the proof of Lemma 3.1 and noticing that šŗ(š‘ )ā‰¤0 for all āˆ’āˆž<š‘ ā‰¤š›æ0, we can write for an arbitrary š‘¢āˆˆš‘‹, šœ‘šœ†(1š‘¢)=š‘ā€–š‘¢ā€–š‘ī€œāˆ’šœ†Ī©ā‰„1šŗ(š‘¢)š‘‘š‘„š‘ā€–š‘¢ā€–š‘ī€œāˆ’šœ†{š‘„āˆˆĪ©āˆ£š‘¢(š‘„)ā‰„š›æ0}ā‰„1šŗ(š‘¢)š‘‘š‘„š‘ā€–š‘¢ā€–š‘āˆ’šœ†š‘3ī€œ{š‘„āˆˆĪ©āˆ£š‘¢(š‘„)ā‰„š›æ0}ī€·1+š‘¢š‘āˆ—ī€øā‰„1š‘‘š‘„š‘ā€–š‘¢ā€–š‘āˆ’šœ†š‘3īƒ©11+š›æš‘āˆ—0īƒŖī€œ{š‘„āˆˆĪ©āˆ£š‘¢(š‘„)ā‰„š›æ0}š‘¢š‘āˆ—š‘‘š‘„,(3.10) so that, using the Sobolev embedding theorem in the last inequality and with a constant š‘4>0 independent of š‘¢ and Ī©, we obtain šœ‘šœ†1(š‘¢)ā‰„š‘ā€–š‘¢ā€–š‘āˆ’šœ†š‘4ā€–š‘¢ā€–š‘āˆ—=1š‘ā€–š‘¢ā€–š‘ī‚€1āˆ’š‘šœ†š‘4ā€–š‘¢ā€–š‘āˆ—āˆ’š‘ī‚.(3.11) Therefore, for each 0<šœŒ<šœŒšœ†āˆ¶=1/(š‘šœ†š‘4)š‘āˆ—āˆ’š‘, it follows that šœ‘šœ†(š‘¢)ā‰„š›¼šœŒ>0 if ā€–š‘¢ā€–=šœŒ. This shows that š‘¢=0 is a strict local minimum of šœ‘šœ†.

Lemma 3.3. Under the same assumptions as in Lemma 3.2, let Ģ‚š‘¢āˆˆš‘‹ be a critical point of šœ‘šœ†. Then, Ģ‚š‘¢āˆˆš¶1,š›¼(Ī©) and Ģ‚š‘¢ is a nonnegative solution of problem (1.3), where 0<š›¼<1.

Proof . We shall follow some of the arguments in [3]. As mentioned earlier, if Ģ‚š‘¢ is a critical point of šœ‘šœ†, then it is shown in [3] that Ģ‚š‘¢ is a solution of the differential inclusion ī‚€||||āˆ’divāˆ‡Ģ‚š‘¢š‘āˆ’2ī‚āˆ‡Ģ‚š‘¢āˆˆšœ•šŗ(Ģ‚š‘¢)a.e.inĪ©.(3.12) Since š‘” is only discontinuous at š‘ =0, the above differential inclusion reduces to an equality, except possibly on the subset š“āŠ‚Ī© where Ģ‚š‘¢=0. Since š‘“ is subcritical, using the š¶1,š›¼ regularity results for quasilinear elliptic equations with š‘-growth condition (see, for example, [6]), we have Ģ‚š‘¢āˆˆš¶1,š›¼(Ī©ā§µš“). And, in this latter case, āˆ’div(|āˆ‡Ģ‚š‘¢|š‘āˆ’2āˆ‡Ģ‚š‘¢) takes on values in the bounded interval [āˆ’š‘Ž,0]. Therefore, by the Michael selection theorem (see Theoremā€‰ā€‰1.2.5 of [5]), we see that āˆ’Ī”š‘āˆ¶š‘¢ā†¦[āˆ’š‘Ž,0] admits a continuous selection. Using the š¶1,š›¼ regularity results for quasilinear elliptic equations with š‘-growth condition again, it follows that Ģ‚š‘¢āˆˆš¶1,š›¼(Ī©), 0<š›¼<1.
Next, using a Morrey-Stampacchia theorem [7, Theoremā€‰ā€‰3.2.2, page 69], we have that āˆ’div(|āˆ‡Ģ‚š‘¢|š‘āˆ’2āˆ‡Ģ‚š‘¢)=0 a.e. in š“. Therefore, since we defined š‘”(0)=0, it follows that ī‚€||||āˆ’divāˆ‡Ģ‚š‘¢š‘āˆ’2ī‚āˆ‡Ģ‚š‘¢=š‘”(Ģ‚š‘¢)a.e.inĪ©.(3.13) Replacing the inclusion (3.2) on Ģ‚š‘¢, we conclude that Ģ‚š‘¢āˆˆš¶1,š›¼(Ī©) is a solution of (1.3). Finally, recalling that š‘”(š‘ )=0 for š‘ ā‰¤0, it is clear that Ģ‚š‘¢ā‰„0. The proof of Lemma 3.3 is complete.

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

Proof. Let {š‘¢š‘›}š‘›ā‰„1āŠ†š‘‹ be a sequence such that |šœ‘šœ†(š‘¢š‘›)|ā‰¤š‘ for all š‘›ā‰„1 and š‘š(š‘¢š‘›)ā†’0 as š‘›ā†’āˆž. In the superlinear and subcritical case, from (F4), we have š‘ā‰„šœ‘šœ†ī€·š‘¢š‘›ī€ø=1š‘ī€œĪ©||āˆ‡š‘¢š‘›||š‘ī€œš‘‘š‘„āˆ’šœ†Ī©šŗī€·š‘¢š‘›ī€øā‰„ā€–ā€–š‘¢š‘‘š‘„š‘›ā€–ā€–š‘š‘āˆ’šœ†šœƒī€œĪ©ī«šœ‰ī€·š‘¢š‘›ī€ø,š‘¢š‘›ī¬ā‰„ī‚µ1š‘‘š‘„āˆ’š‘š‘āˆ’1šœƒī‚¶ā€–ā€–š‘¢š‘›ā€–ā€–š‘+ī€œĪ©1šœƒī€·ā€–ā€–š‘¢š‘›ā€–ā€–š‘ī«šœ‰ī€·š‘¢āˆ’šœ†š‘›ī€ø,š‘¢š‘›ā‰„ī‚µ1ī¬ī€øš‘‘š‘„āˆ’š‘š‘āˆ’1šœƒī‚¶ā€–ā€–š‘¢š‘›ā€–ā€–š‘āˆ’šœ†šœƒā€–šœ‰ā€–š‘‹āˆ—ā€–ā€–š‘¢š‘›ā€–ā€–āˆ’š‘,(3.14) where šœ‰(š‘¢š‘›)āˆˆšœ•šŗ(š‘¢š‘›). Hence {š‘¢š‘›}š‘›ā‰„1āŠ†š‘‹ is bounded.
Thus by passing to a subsequence if necessary, we may assume that š‘¢š‘›ā‡€š‘¢ in š‘‹ as š‘›ā†’+āˆž. We have ī«š½ī…žī€·š‘¢š‘›ī€ø,š‘¢š‘›ī¬āˆ’ī€œāˆ’š‘¢Ī©šœ‰ī€·š‘¢š‘›š‘¢ī€øī€·š‘›ī€øāˆ’š‘¢š‘‘š‘„ā‰¤šœ€š‘›ā€–ā€–š‘¢š‘›ā€–ā€–āˆ’š‘¢(3.15) with šœ€š‘›ā†“0, where šœ‰(š‘¢š‘›)āˆˆšœ•ĪØ(š‘¢š‘›) and āˆ«š½=(1/š‘)Ī©|āˆ‡š‘¢š‘›|š‘š‘‘š‘„. From (F3) and Chang [4] we know that šœ‰(š‘¢š‘›(š‘„))āˆˆšæ(š‘āˆ—āˆ’1)ā€²(Ī©) ((š‘āˆ—āˆ’1)ī…ž=(š‘āˆ—āˆ’1)/(š‘āˆ—āˆ’2)). Since š‘‹ is embedded compactly in šæš‘āˆ—āˆ’1(Ī©), we have that š‘¢š‘›ā†’š‘¢ as š‘›ā†’āˆž in šæš‘āˆ—āˆ’1(Ī©). So using the Hƶlder inequality, we have ī€œĪ©šœ‰š‘›(ī€·š‘¢š‘„)š‘›ī€øāˆ’š‘¢š‘‘š‘„āŸ¶0asš‘›āŸ¶+āˆž.(3.16)
Therefore, we obtain that limš‘›ā†’āˆžsupāŸØš½ā€²(š‘¢š‘›),š‘¢š‘›āˆ’š‘¢āŸ©ā‰¤0. But we know that š½ā€² is a mapping of type (š‘†+). Thus we have š‘¢š‘›āŸ¶š‘¢inš‘‹.(3.17) Using the similar method, we can more easily get the nonsmooth (PS)š‘-condition in the case of sublinear.

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

Lemma 3.6. Under assumptions (F0), (F1), and (F2), let Ī©=šµš‘…āŠ‚ā„š‘ with š‘ā‰„2, and let š‘¢āˆˆš¶1(šµš‘…) be a radially symmetric, nonincreasing function such that š‘¢ā‰„0 and š‘¢ is a minimizer of šœ‘šœ†(š‘¢) with šœ‘šœ†(š‘¢)=š‘š<0. Then, š‘¢ does not vanish in šµš‘…; that is, š‘¢(š‘„)>0 for all š‘„āˆˆšµš‘….

Proof. Since š‘” is discontinuous at zero, we note that the conclusion does not follow directly from uniqueness of solution for the Cauchy problem with data at š‘Ÿ=š‘… (in fact, writing š‘¢=š‘¢(š‘Ÿ), š‘Ÿ=|š‘„|, we may have š‘¢(š‘…)=š‘¢ā€²(š‘…)=0 and š‘¢ā‰¢0).
Now, since š‘¢ā‰¢0 by assumption, š‘…0āˆ¶=inf{š‘Ÿā‰¤š‘…āˆ¶š‘¢(š‘ )=0forš‘Ÿā‰¤š‘ ā‰¤š‘…} satisfies 0<š‘…0ā‰¤š‘…. If š‘…0=š‘… there is nothing to prove in view of the fact that š‘¢ is non-increasing. On the other hand, if š‘…0<š‘… then š‘¢ā€²(š‘…0)=0 and š‘¢(š‘Ÿ)>0 for š‘Ÿāˆˆ[0,š‘…0). It is not hard to prove that this contradicts that š‘¢ is a minimizer of šœ‘šœ†. Indeed, if š‘…0<š‘…then šœ‘šœ†(1š‘¢)=š‘ī€œšµš‘…0||||āˆ‡š‘¢š‘ī€œš‘‘š‘„āˆ’šœ†šµš‘…0šŗ(š‘¢)š‘‘š‘„=š‘š<0.(3.18) A simple calculation shows that the rescaled function š‘£(š‘Ÿ)=š‘¢(š‘…0š‘Ÿ/š‘…)āˆˆš‘Š01,š‘(šµš‘…)āˆ©š¶1(šµš‘…) satisfies šœ‘šœ†(š‘£)=š›æš‘āˆ’š‘īƒ¬1š‘ī€œšµš‘…0||||āˆ‡š‘¢š‘š‘‘š‘„āˆ’š›æāˆ’š‘šœ†ī€œšµš‘…0īƒ­šŗ(š‘¢)š‘‘š‘„,(3.19) where š›æāˆ¶=š‘…0/š‘… is less than 1. Therefore, since we are assuming 1<š‘ā‰¤š‘, we would reach the contradiction šœ‘šœ†(š‘£)<š‘š.

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

Proof of Theorem 1.1. We observe that the functional šœ‘šœ† is weakly lower semicontinuous on š‘‹. Moreover, the sublinearity assumption (F1) on š‘”(š‘¢) implies that šœ‘šœ† is coercive. Therefore, the infimum of šœ‘šœ† is attained at some š‘¢šœ†: infš‘¢āˆˆš‘‹šœ‘šœ†(š‘¢)=šœ‘šœ†ī€·š‘¢šœ†ī€ø.(3.20) And, in view of Lemma 3.3, š‘¢šœ†āˆˆš¶1,š›¼(Ī©) is a nonnegative solution of (1.3). We now claim that š‘¢šœ† is nonzero for all šœ†>0 large.

Claim 1. There exists Ī›>0 such that šœ‘šœ†(š‘¢šœ†)<0 for all šœ†ā‰„Ī›.
In order to prove the claim it suffices to exhibit an element š‘¤āˆˆš‘‹ such that šœ‘šœ†(š‘¤)<0 for all šœ†>0 large. This is quite standard considering that šŗ(š›æ)>0 by (F2). Indeed, letting Ī©šœ€āˆ¶={š‘„āˆˆĪ©āˆ¶dist(š‘„,šœ•Ī©)>šœ€} for šœ€>0 small, define š‘¤ so that š‘¤(š‘„)=š›æ for š‘„āˆˆĪ©šœ€ and 0ā‰¤š‘¤(š‘„)ā‰¤š›æ for š‘„āˆˆĪ©ā§µĪ©šœ€. Then šœ‘šœ†1(š‘¤)=š‘ā€–š‘¤ā€–š‘ī‚µī€œāˆ’šœ†Ī©šœ€ī€œšŗ(š‘¤)š‘‘š‘„+Ī©ā§µĪ©šœ€ī‚¶ā‰¤1šŗ(š‘¤)š‘‘š‘„š‘ā€–š‘¤ā€–š‘ī€·ī€·Ī©āˆ’šœ†šŗ(š›æ)measšœ€ī€øāˆ’š‘(1+š›æš‘ī€·)measĪ©ā§µĪ©šœ€,ī€øī€ø(3.21) where we note that the expression in the above parenthesis is positive if we choose šœ€>0 sufficiently small. Therefore, there exists Ī›>0 such that šœ‘šœ†(š‘¤)<0 for all šœ†ā‰„Ī›, which proves the claim.
On the other hand, when Ī©=šµš‘…, let š‘¢āˆ—šœ† denote the Schwarz symmetrization of š‘¢šœ†, namely, š‘¢āˆ—šœ† is the unique radially symmetric, nonincreasing, nonnegative function in š‘‹ which is equi-measurable with š‘¢šœ†. As is well known, ī€œšµš‘…šŗī€·š‘¢āˆ—šœ†ī€øī€œš‘‘š‘„=šµš‘…šŗī€·š‘¢šœ†ī€øš‘‘š‘„,(3.22) and ā€–š‘¢āˆ—šœ†ā€–ā‰¤ā€–š‘¢šœ†ā€– (the Faber-Krahn inequality; see [8]), so that šœ‘šœ†(š‘¢āˆ—šœ†)ā‰¤šœ‘šœ†(š‘¢šœ†). Therefore, we necessarily have šœ‘šœ†(š‘¢āˆ—šœ†)=šœ‘šœ†(š‘¢šœ†) and may assume that š‘¢šœ†=š‘¢āˆ—šœ†. Moreover, š‘¢šœ†>0 in Ī© by Lemma 3.6. Therefore, š‘¢šœ† is a positive solution of both problems (1.1) and (1.3).
Next, we recall that š‘¢=0 is a strict local minimum of šœ‘šœ† by Lemma 3.2. Therefore, since šœ‘šœ† satisfies the nonsmooth Palais-Smale condition by Lemma 3.4, we can use the minima š‘¢=0 and š‘¢=š‘¢šœ† of šœ‘šœ† to apply the nonsmooth mountain pass theorem and conclude that there exists a second nontrivial critical point š‘£šœ† with šœ‘šœ†(š‘£šœ†)>0. Again, š‘£šœ† is a nonnegative solution of problem (1.3) in view of Lemma 3.3. In addition, when Ī©=šµš‘…, arguments similar to above passage show that we may assume š‘£šœ†=š‘£āˆ—š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 š‘’šœ†āˆˆš‘‹ such that šœ‘šœ†(š‘’šœ†)ā‰¤0. On the other hand, Lemma 3.2 says that š‘¢=0 is a (strict) local minimum of šœ‘šœ† and Lemma 3.4 says that šœ‘šœ† satisfies nonsmooth (PS)š‘ for every š‘āˆˆā„. Therefore, an application of the nonsmooth mountain pass theorem stated in section 2 yields the existence of a critical point š‘£šœ† such that šœ‘šœ†ī€·š‘£šœ†ī€ø>0.(3.23) In particular, š‘£šœ†ā‰ 0, and it follows that š‘£š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 š‘£šœ†=š‘£āˆ—šœ† in the case of a ball Ī©=šµš‘….
Finally, still in the case of a ball Ī©=šµš‘…, we claim that there exists šœ†āˆ—>0 such that, for all 0<šœ†<šœ†āˆ—, š‘£šœ†=š‘£āˆ—šœ† is a positive solution of problem (1.3) (hence of problem (1.1)) having negative normal derivative on šœ•šµš‘…. Indeed, if that is not the case, then, for any given šœ†>0, we can find 0<šœ‡=šœ‡(šœ†)<šœ† such that the nonnegative solution š‘£šœ‡=š‘£āˆ—šœ‡ of problem (1.1) with šœ†=šœ‡ obtained a bove satisfies š‘£šœ‡ī€ŗ(š‘Ÿ)>0forš‘Ÿāˆˆ0,š‘…0ī€ø,š‘£ī…žšœ‡ī€·š‘…0ī€ø=0,š‘£šœ‡ī€ŗš‘…(š‘Ÿ)=0forš‘Ÿāˆˆ0ī€»,,š‘…(3.24) for some 0<š‘…0ā‰¤š‘…. Therefore, the rescaled function š‘¤šœ‡(š‘Ÿ)āˆ¶=š‘£šœ‡(š‘…0š‘Ÿ/š‘…) is a positive solution of (1.1) with šœ†=šœ‡0 (again in the ball šµš‘…), with šœ‡0āˆ¶=šœ‡š‘…š‘0/š‘…š‘ā‰¤šœ‡. This shows that we can always construct a decreasing sequence šœ‡š‘›>0 satisfying alternative (ii) of Theorem 1.2, in case alternative (i) does not hold.


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