- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 492025, 36 pages
Dirichlet Problems with an Indefinite and Unbounded Potential and Concave-Convex Nonlinearities
1Faculty of Mathematics and Computer Science, Jagiellonian University, ulica Łojasiewicza 6, 30-348 Kraków, Poland
2Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece
Received 5 March 2012; Accepted 18 April 2012
Academic Editor: Yuriy Rogovchenko
Copyright © 2012 Leszek Gasiński and Nikolaos S. Papageorgiou. 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.
We consider a parametric semilinear Dirichlet problem with an unbounded and indefinite potential. In the reaction we have the competing effects of a sublinear (concave) term and of a superlinear (convex) term. Using variational methods coupled with suitable truncation techniques, we prove two multiplicity theorems for small values of the parameter. Both theorems produce five nontrivial smooth solutions, and in the second theorem we provide precise sign information for all the solutions.
Let be a bounded domain with a -boundary . In this paper we study the following parametric nonlinear Dirichlet problem: Here with is a potential function which may change sign (indefinite potential). Also is a parameter, and are Carathéodory functions (i.e., for all , functions and are measurable and for almost all , functions and are continuous). We assume that for almost all , the function is strictly sublinear near , while the function is superlinear near . So, problem () exhibits competing nonlinearities of the concave-convex type. This situation was first studied with and the right hand side nonlinearity being with by Ambrosetti et al. . In , the authors focus on positive solutions and proved certain bifurcation-type phenomena as varies. Further results in this direction can be found in the works of I'lyasov , Li et al. , Lubyshev , and Rădulescu and Repovš . In all the aforementioned works . We should also mention the recent work of Motreanu et al. , where the authors consider equations driven by the -Laplacian, with concave term of the form (where ) and a perturbation exhibiting an asymmetric behaviour at and at (-superlinear near and -sublinear near ). They prove multiplicity results producing four nontrivial solutions with sign information.
In this work, we prove two multiplicity results for problem () when the parameter is small. In both results we produce five nontrivial smooth solutions, and in the second we provide precise sign information for all the solutions. For the superlinear (“convex”) nonlinearity , we do not employ the usual in such cases Ambrosetti-Rabinowitz condition. Instead, we use a more general condition, which incorporates in our framework also superlinear perturbations with “slow” growth near , which do not satisfy the Ambrosetti-Rabinowitz condition. We should point out that none of the works mentioned earlier provide sign information for all the solutions (in particular, none of them produced a nodal (sign changing) solution) and all use the Ambrosetti-Rabinowitz condition to express the superlinearity of the “convex” contribution in the reaction.
Our approach is variational based on the critical point theory which is combined with suitable truncation techniques. In the next section we recall the main mathematical tools we will use in the analysis of problem (). We also introduce the hypotheses on the terms and .
2. Mathematical Background and Hypotheses
Let be a Banach space and let be its topological dual. By we denote the duality brackets for the pair . Let . We say that satisfies the Cerami condition, if the following is true.
Every sequence , such that is bounded and
admits a strongly convergent subsequence.
This compactness-type condition is in general weaker than the usual Palais-Smale condition. However, the Cerami condition suffices to prove a deformation theorem and from it we derive the minimax theory for certain critical values of (see Gasiński and Papageorgiou  and Motreanu and Rădulescu ). In particular, we can have the following theorem, known in the literature as the mountain pass theorem.
Theorem 2.1. If is a Banach space, satisfies the Cerami condition, , : where Then and are a critical value of .
In the analysis of problem (), in addition to the Sobolev space , we will also use the Banach space This is an ordered Banach space with positive cone: This cone has a nonempty interior given by where is the outward unit normal on .
In the proof of the second multiplicity theorem and in order to produce a nodal (sign changing) solution, we will also use critical groups. So, let us recall their definition. Let and let . We introduce the following sets: Also, if is a topological pair with and is an integer, by we denote the th relative singular homology group for the pair with integer coefficients. The critical groups of at an isolated critical point of with are defined by where is a neighbourhood of , such that . The excision property of singular homology implies that this definition is independent of the particular choice of the neighbourhood .
Using the spectral theorem for compact self-adjoint operators, we can show that the differential operator has a sequence of distinct eigenvalues , such that The first eigenvalue is simple and admits the following variational characterization: where Moreover, the corresponding eigenfunction does not change sign, and in fact we can take for all (see Gasiński and Papageorgiou ). Using (2.11) and this property of the principal eigenfunction, we can have the following lemma (see Gasiński and Papageorgiou [9, Lemma 2.1]).
Lemma 2.2. If , for almost all , , then there exists , such that
Also, from Gasiński and Papageorgiou  we know that there exist , such that with , the norm of the Sobolev space .
Let: is a Carathéodory function, such that for almost all .(i)For every , there exists a function , such that (ii)We have (iii)There exist constants , and , such that (iv)For every , we can find , such that for almost all
Remark 2.3. Hypothesis (ii) implies that for almost all , the function is strictly sublinear near . Hence is the “concave” component in the reaction of () (the terminology “concave” and “convex” nonlinearities is due to Ambrosetti ). Note that hypothesis (iii) implies that for almost all , the function has a similar growth near 0 that is, we have a concave term near zero. Hypothesis (iv) is weaker than assuming the monotonicity of for almost all .: is a Carathéodory function, such that for almost all .(i)There exist , and , such that (ii)We have (iii)There exist functions , such that for almost all , and uniformly for almost all .(iv)For every , we can find , such that for almost all , we have
Remark 2.4. Hypothesis (ii) implies that for almost all , the function is superquadratic near . Evidently, this is satisfied if the function is superlinear near , that is, when So, is the “convex” component of the reaction which “competes” with the “concave” component .
Note that in , we did not include the Ambrosetti-Rabinowitz condition to characterize the superlinearity of . We recall that the Ambrosetti-Rabinowitz condition says that there exist and , such that Integrating (2.25) and using (2.26), we obtain the weaker condition Therefore, for almost all , the function is superquadratic with at least -growth near . Hence the Ambrosetti-Rabinowitz condition excludes superlinear perturbations with “slower” growth near . For this reason, here we employ a weaker condition. So, let
We employ the following hypothesis. For every , there exists a function , such that
Remark 2.5. Hypothesis is a generalized version of a condition first introduced by Li and Yang , where the reader can find other possible extensions of the Ambrosetti-Rabinowitz condition and comparisons between them. Hypothesis is a quasimonotonicity condition on , and it is satisfied if there exists , such that for almost all , the function is increasing on and is decreasing on (see Li and Yang ).
Example 2.6. The following pairs of functions satisfy hypotheses , , and (for the sake of simplicity we drop the -dependence):
with , , .
Note that and do not satisfy the Ambrosetti-Rabinowitz condition (see (2.25)-(2.26)).
For every , we set (by virtue of the Poincaré inequality). We mention that the notation will be also used to denote the -norm. It will always be clear from the context which norm is used. For , let . Then for , we set . We have , , and . For a given measurable function (e.g., a Carathéodory function), we set Finally, let be the operator, defined by For the properties of the operator we refer to Gasiński and Papageorgiou [11, Proposition 3.1, page 852].
3. Solutions of Constant Sign
In this section, for small, we generate four nontrivial smooth solutions of constant sign (two positive and two negative). To this end, we introduce the following modifications of the nonlinearities and : with as in (2.14). These modifications are Carathéodory functions. We set and consider the -functionals , defined by
Proposition 3.1. If hypotheses , , and hold and , then the functionals satisfy the Cerami condition.
Proof. We do the proof for , the proof for being similar.
So, let be a sequence, such that for some and From (3.5), we have with . In (3.6) we choose . Then so (see (2.14)), and hence From (3.4) and (3.9), we have for some . Also, if in (3.6) we choose , then Adding (3.10) and (3.11), we obtain for some (see (2.29) for the definition of ).
Claim 1. The sequence is bounded.
Arguing by contradiction, suppose that the claim is not true. Then by passing to a subsequence if necessary, we may assume that Let Then And so, passing to a subsequence if necessary, we may assume that If , then (recall that ). So, by virtue of hypotheses (ii) and (ii), for almost all , we have Then Fatou's lemma implies that But from (3.4) and (3.9), we have for some , so for some (since the sequence is bounded in ). Comparing (3.20) and (3.22), we reach a contradiction.
So, we have . We fix and set Evidently (see (3.17)). Hence by Krasnoselskii's theorem (see Gasiński and Papageorgiou [12, Proposition 1.4.14, page 87] and hypotheses (i) and (i)), we have Since , we can find , such that Let be such that By virtue of (3.26), we have Note that This fact together with (3.25) and (3.28) implies that for some . Since is arbitrary, we infer that Note that for some (see (3.4) and (3.9)).
Hence (3.31) implies that there exists , such that And so from the choice of , we have so thus and hence Hypothesis implies that so (see (3.37)), and thus (see (3.12)).
Comparing (3.31) and (3.40), we reach a contradiction. This proves the claim.
By virtue of the claim and (3.9), we have that the sequence is bounded. So, we may assume that In (3.6) we choose , pass to the limit as , and use (3.42). Then so (see Gasiński and Papageorgiou [11, Proposition 3.1, page 852]), and thus (by the Kadec-Klee property of Hilbert spaces). This proves that satisfies the Cerami condition.
Similarly we show that also satisfies the Cerami condition.
Our aim is to apply Theorem 2.1 (the mountain pass theorem) to two functionals and . We have checked that both functionals satisfy the Cerami condition. So, it remains to show that they satisfy the mountain pass geometry as it is described in Theorem 2.1.
The next proposition is a crucial step in satisfying the mountain pass geometry for the two functionals and .
Proposition 3.2. If hypotheses and hold, then there exist , such that for every , we can find , such that
Proof. Hypotheses (i) and (ii) imply that for a given , we can find , such that
Similarly hypotheses (i) and (iii) imply that for a given , we can find , such that
Then, for , we have
for some (see (3.48), (3.50), (2.11), (2.14), and Lemma 2.2).
Choosing , we have for some . Let Since , we have Also is continuous in . Therefore, we can find , such that so and thus Evidently Hence we can find , such that so (see (3.52)).
Similarly, for , we can find , such that for all there exists , such that
Proposition 3.3. If hypotheses and hold, and with , then
Proof. By virtue of hypotheses (i) and (ii), for a given , we can find , such that
Similarly, hypotheses (i) and (ii) imply that for any given , we can find , such that
Then, we have
for some (see (3.63), (3.64) and recall that , and ).
Since is arbitrary, choosing from (3.65), we infer that
Now we are ready to produce the first two nontrivial smooth solutions of constant sign. In what follows, we set
Proof. Propositions 3.1, 3.2, and 3.3 permit the application of the mountain pass theorem (see Theorem 2.1) for the functional , and so we obtain , such that
From (3.70), we see that . From (3.71), we have On (3.72) we act with and obtain so (see (2.14)); hence , .
Therefore (3.72) becomes so From the regularity theory for Dirichlet problems (see Struwe [13, pp. 217–219]), we have that . Moreover, invoking the weak Harnack inequality of Pucci and Serrin [14, page 154], we have that for all .
Similarly working with , this time we obtain a nontrivial smooth negative solution with for all .
We can improve the conclusion of this proposition by strengthening the condition on the potential : with and .
Remark 3.5. So, the potential function is bounded from above but in general can be unbounded from below.
Proof. From Proposition 3.4, we already have two solutions:
(see hypothesis (iii)). Let and let be as postulated by hypothesis (iv). Then from (3.79), we have
and thus (see Vázquez  and Pucci and Serrin [14, page 120]).
Similarly for the negative solution .
To continue and produce additional nontrivial smooth solutions of constant sign, we need to keep hypotheses .
Proof. From Proposition 3.6, we already have two solutions:
We introduce the following truncation perturbation of the reaction of the problem ():
This is a Carathéodory function. We set
and consider the -functional , defined by
Claim 2. We have , where
Let . Then On (3.88) we act with . Then (see (3.84)), so thus hence (see (2.14)), and so finally . This proves Claim 2.Claim 3. We may assume that is a local minimizer of .
Let , and consider problem . As we did in the proof of Proposition 3.4, via the mountain pass theorem, we obtain a nontrivial smooth positive solution , and by virtue of the strong maximum principle, we have (see the proof of Proposition 3.7). Then (see (iii) and recall that ).
We consider the following truncation perturbation of the reaction of problem (): This is a Carathéodory function. We set and consider the -functional , defined by From (3.94), it is clear that is coercive. Also, using the Sobolev embedding theorem, we check that is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find , such that By virtue of hypothesis (iii), we can find and , such that Let and let be small, such that (recall that ). Then, we have (see (3.94), hypothesis (iii), and (3.98)). Since , by choosing even smaller if necessary, we have so (see (3.97)); hence .
From (3.97), we have so Acting on (3.104) with , we obtain that , . Also, acting on (3.104) with , we have (see (3.94) and (3.93)), so thus hence (see (2.14)) and so finally So, we have proved that Hence (3.104) becomes (see (3.94)), so thus (as in the proof of Proposition 3.6) and it is a solution of ().
If , then this is the desired second nontrivial positive smooth solution of ().
So, we may assume that and that there is no other solution of () in the order interval We introduce the following truncation of : This is a Carathéodory function. We set and consider the -functional , defined by From (3.94) and (3.114), it follows that is coercive. Also, it is sequentially weakly lower semicontinuous. Hence, we can find , such that so and thus As before, acting on (3.119) with and with , we show that . Then, from (3.94) and (3.114), it follows that so