Positive Solutions of a Kind of Equations Related to the Laplacian and -Laplacian

Abstract

Positive solutions of a kind of equations related to the Laplacian and -Laplacian on a bounded domain in with are studied by using variational method. A sufficient condition of the existence of positive solutions is characterized by the eigenvalues of linear and another nonlinear eigenvalue problems.

1. Introduction

In this paper, we study the equation where , is a smooth bounded domain in for , and satisfies the following conditions: (f₁) for any and for any ;(f₂)for , the limits  exist uniformly for .

The asymptotic behaviors of near zero and infinity lead us to define where and are the usual Sobolev spaces defined as the completion of with respect to the norms and , respectively. Then it is well known that is the first eigenvalue of the nonlinear eigenvalue problem Moreover is a simple eigenvalue of (4), the associated eigenfunction can be chosen as positive in , and any eigenfunction corresponding to an eigenvalue larger than must change sign. The reader is referred to [1–3] for details.

By a solution of (1), we mean that solves (1) in the weak sense; that is, satisfies Moreover, by a positive solution of (1), we mean that is a weak solution of (1), and for .

Our main result is the following theorem.

Theorem 1. Suppose that satisfies and with , . Then (1) has a positive solution.

Assume that . Equation (1) can be viewed as combination of the following equations: In the last decade or so, there was an extensive effort in studying the existence of solutions of (6); see [4–8].

Before concluding this section, we recall a theorem from [9], which will be used to prove our main theorem in this paper.

Theorem 2. Let be a Banach space and an interval. Consider the family of functionals on , with , nonnegative and either or as . For any , we set If for every the set is nonempty and then for almost every there exists a sequence such that (i) is bounded;(ii) as ;(iii) in the dual as .

Throughout this paper, we denote by the Sobolev space with the norm , by the duality space of , by   the weak convergence in , and by the duality pairing between and . The letters will denote various positive constants whose exact values are not essential to the analysis of the problem. Let , a.e. and if or if .

2. Proof of Theorem 1

In this section, we always assume and hold with and . Hence, there exist and such that where . In the following, we utilize Theorem 2 to complete the proof of Theorem 1. In the setting of Theorem 2 we have with , and It is easy to verify that Firstly, we show that satisfies the conditions of Theorem 2 by proving several lemmas.

Lemma 3. for any .

Proof. Let be a -eigenfunction. For , we have by (10) that where . Noting that , we can choose large enough so that , where is independent of . The proof is completed.

Lemma 4. There exists a constant such that for any .

Proof. For any , it follows from (11) that By Sobolev’s embedding theorem, we conclude that there exist and such that for and Fix and . By the definition of , we have that . Hence, there exists such that . So The proof is completed.

Lemma 5. For any , if is bounded and in as , then admits a convergent subsequence.

Proof. Given , assume that is bounded, in as . By extracting a subsequence, we may suppose that there exists such that as It follows from and that there exist such that Hence, by Hölder’s inequality and Sobolev’s embedding theorem, we have Similarly, we have Noting that and the inequality deduced from an inequality in Appendix of [3], it follows from (20) and (21) that where we have used the fact that Hence in . The proof is completed.

Lemma 6. There exists a sequence with as and such that , .

Proof. We only need to show that for almost every there exists such that and . By Theorem 2, for almost each , there exists a bounded sequence such that By Lemma 5, we may assume that in as . Then the continuity of and implies that and . The proof is completed.

Define , . Then we have the following.

Lemma 7. Supposing and hold, then

Proof. By , , for every , there is a constant such that For , letting , by Hölder’s inequality and Sobolev’s embedding theorem, we have where is independent of . The proof is completed.

Proof of Theorem 1. By Lemma 6, there exists a sequence with as and such that By Lemma 4 and (30), we have and . Hence . In the following, we first claim that is bounded in . Suppose by contradiction that . Let . Hence, we have, for , Since is bounded in , we may assume that in and a.e. on as . Letting in (31) and , we get It follows from [5, Theorem 10] that in as . Passing to limit in (31), we obtain by Lemma 7 that From (33) and the fact that , we know that , which contradicts the assumption . Since , we can show that In fact, for any , it follows from (19), Hölder’s inequality, and Sobolev’s embedding theorem that Furthermore, (30) implies that Hence, in as . By Lemma 5, has a convergent subsequence. Without loss of generality, we may assume that as . According to Lemma 4, (30) and we have The standard process shows that is a positive solution to (1). The proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is partially supported by National Natural Science Foundation of China (Grant nos. 11071149, 11301313) and Science Council of Shanxi Province (2012011004-2, 2013021001-4, and 2014021009-1).