- About this Journal ·
- Abstracting and Indexing ·
- 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 2013 (2013), Article ID 769620, 6 pages

http://dx.doi.org/10.1155/2013/769620

## Infinitely Many Solutions of Superlinear Elliptic Equation

School of Mathematical Sciences, Qufu Normal University, Jining, Shandong 273165, China

Received 4 January 2013; Revised 27 April 2013; Accepted 28 April 2013

Academic Editor: Wenming Zou

Copyright © 2013 Anmin Mao and Yang Li. 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.

#### Abstract

Via the Fountain theorem, we obtain the existence of infinitely many solutions of the following superlinear elliptic boundary value problem: in on , where is a bounded domain with smooth boundary and is odd in and continuous. There is no assumption near zero on the behavior of the nonlinearity , and does not satisfy the Ambrosetti-Rabinowitz type technical condition near infinity.

#### 1. Introduction

Consider the following nonlinear problem: which has been receiving much attention during the last several decades. Here is a bounded smooth domain and is a continuous function on and odd in . We make the following assumptions on : there exist constants and such that , for all , and where ; there exists a constant such that

Note that Costa and Magalhães in [1] introduced a condition similar to , which also appeared in [2].

In this paper, we will study the existence of infinitely many nontrivial solutions of (1) via a variant of Fountain theorems established by Zou in [3]. Fountain theorems and their dual form were established by Bartsch in [4] and by Bartsch and Willem in [5], respectively. They are effective tools for studying the existence of infinitely many large or small energy solutions. It should be noted that the P.S. condition and its variants play an important role for these theorems and their applications.

We state our main result as follows.

Theorem 1. *Assume that hold and is odd in . Then problem (1) possesses infinitely many solutions.*

Problem (1) was studied widely under various conditions on ; see, for example, [6–10]. In 2007, Rabinowitz et al. [6] studied the problem where is a bounded smooth domain, and assumed, , , ,, , s.t. , for all and , and for small.

They got the existence of at least three nontrivial solutions. was given by Ambrosetti and Rabinowitz [11] to ensure that some compactness and the Mountain Pass setting hold.

However, there are many functions which are superlinear but do not necessarily need to satisfy . For example, It is easy to check that does not hold. On the other hand, in order to verify , it is usually an annoying task to compute the primitive function of and sometimes it is almost impossible, for example, More examples are presented in Remark 2.

We recall that implies a weaker condition In [12], Willem and Zou gave one weaker condition Note that is much weaker than the above conditions.

In [13], Schechter and Zou proved that under the hypotheses that problem (1) has a nontrivial weak solution.

Recently, Miyagaki and Souto in [2] proved that problem (1) has a nontrivial solution via the Mountain Pass theorem under the following conditions: and they adapted some monotonicity arguments used by Schechter and Zou [13]. This approach is interesting, but many powerful variational tools such as the Fountain theorem and Morse theory are not directly applicable. In addition, the monotonicity assumption on is weaker than the monotonicity assumption on .

As to the case in the current paper, we make some concluding remarks as follows.

*Remark 2. *To show that our assumptions and are weaker than , we give two examples:(1),(2),

which do not satisfy . Example can be found in [3]. So the case considered here cannot be covered by the cases mentioned in [6, 11].

*Remark 3. *Compared with papers [11, 12], we do not assume any superlinear conditions near zero. Compared with paper [2], we do not impose any kind of monotonic conditions. In addition, although we do not assume holds, we are able to check the boundedness of P.S. (or P.S.*) sequences. So, our result is different from those in the literature.

Our argument is variational and close to that in [2, 3, 13, 14]. The paper is arranged as follows. In Section 2 we formulate the variational setting and recall some critical point theorems required. We then in Section 3 complete the proof of Theorem 1.

#### 2. Variational Setting

In this section, we will first recall some related preliminaries and establish the variational setting for our problem. Throughout this paper, we work on the space equipped with the norm

Lemma 4. * embeds continuously into , for all , and compactly into , for all ; hence there exists such that
**
where .*

Define the Euler-Lagrange functional associated to problem (1), given by where . Note that () implies that

In view of (16) and Sobolev embedding theorem, and are well defined. Furthermore, we have the following.

Lemma 5 (see [15] or [16]). *Suppose that is satisfied. Then and is compact and hence . Moreover
**
for all , and critical points of on are solutions of (1).*

Lemma 6 (see [17]). * Assume that , and . Then for every , and the operator is continuous.*

Let be a Banach space equipped with the norm and , where for any . Set and . Consider the following functional defined by The following variant of the Fountain theorems was established in [3].

Theorem 7 (see [3, Theorem 2.1]). *Assume that the functional defined above satisfies the following: * * maps bounded sets to bounded sets uniformly for ; furthermore, for all ;* * for all ; moreover, or as ;* * there exists such that
* *Then
* *where and . Moreover, for a.e. , there exists a sequence such that
*

In order to apply the above theorem to prove our main results, we define the functionals , , and on our working space by for all and . Note that , where is the functional defined in (15).

From Lemma 5, we know that , for all . It is known that is a selfadjoint operator with a sequence of eigenvalues (counted with multiplicity) and the corresponding system of eigenfunctions forming an orthogonal basis in . Let , for all .

#### 3. Proof of Theorem 1

Lemma 8. *Assume that ()-() hold. Then there exists a positive integer and two sequences as such that
**
where and for all .*

*Proof**Step 1*. We first prove (25).

By (16) and (23), for all and , we have
where is the constant in (16). Let
Then
since is compactly embedded into . Combining (14), (27), and (28), we have
By (29), there exists a positive integer such that
since . Evidently,
Combining (30) and (31), direct computation shows
*Step *2. We then verify (26).

We claim that for any finite-dimensional subspace , there exists a constant such that
Here and in the sequel, always denotes the Lebesgue measure in .

If not, for any , there exists such that
Let , for all . Then , for all , and
Passing to a subsequence if necessary, we may assume , for some , since is of finite dimension. Evidently, . In view of Lemma 4 and the equivalence of any two norms on , we have
and .

By the definition of norm , there exists a constant such that
For any , let
Set . Then for large enough, by (36) and (38), we have
Consequently, for large enough, there holds
This is in contradiction to (37). Therefore (34) holds.

Consequently, for any , there exists a constant such that
where , for all , and . By (), for any , there exists a constant such that
Combining (23), (42), (43), and (), for any and , we have
with . Now for any , if we choose
then (44) implies
ending the proof.

*Proof of Theorem 1. *It follows from (16), (23), and Lemma 5 that maps bounded sets to bounded sets uniformly for . In view of the evenness of in , it holds that for all . Thus the condition () of Theorem 7 holds. Besides, as and since . Thus the condition () of Theorem 7 holds. And Lemma 8 shows that the condition () holds for all . Therefore, by Theorem 7, for any and a.e. , there exists a sequence such that
as , where
with and .

Furthermore, it follows from the proof of Lemma 8 that
where and by (32).*Claim 1.* possesses a strong convergent subsequence in , for and .

In fact, by the boundedness of the , passing to a subsequence, as , we may assume
By the Sobolev embedding theorem,
Lemma 6 implies that
Observe that
By (47), it is clear that
It follows from the Hölder inequality, (51), and (52) that
as . Thus by (53), (54), and (55), we have proved that
that is, in .

Thus, for each , we can choose such that the sequence obtained a convergent subsequence; passing again to a subsequence, we may assume
This together with (47) and (49) yields
*Claim *2. is bounded in for all .

For notational simplicity, we will set for all throughout this paragraph. If is unbounded in , we define . Since , without loss of generality we suppose that there is such that
Let . If , from it follows that
On the other hand, after a simple calculation, we have
We conclude that has zero measure and .

Moreover, from (49) and (58)
By (),
which contradicts (62). Hence is bounded.*Claim *3. possesses a convergent subsequence with the limit for all .

In fact, by Claim 2, without loss of generality, we have assume
By virtue of the Riesz Representation theorem, and can be viewed as and , respectively, where is the dual space of . Note that
that is
By Lemma 5, is also compact. Due to the compactness of and (64), the right-hand side of (66) converges strongly in and hence in .

Now for each , by (58), the limit is just a critical point of with . Since as in (49), we get infinitely many nontrivial critical points of . Therefore (1) possesses infinitely many nontrivial solutions by Lemma 5.

#### Acknowledgments

The authors would like to thank the referee for valuable comments and helpful suggestions. The first author would like to acknowledge the hospitality of Professor Y. Ding of the AMSS of the Chinese Academy of Sciences, where this paper was written during his visit. Anmin Mao was supported by NSFC (11101237) and ZR2012AM006.

#### References

- D. G. Costa and C. A. Magalhães, “Variational elliptic problems which are nonquadratic at infinity,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 23, no. 11, pp. 1401–1412, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - O. H. Miyagaki and M. A. S. Souto, “Superlinear problems without Ambrosetti and Rabinowitz growth condition,”
*Journal of Differential Equations*, vol. 245, no. 12, pp. 3628–3638, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Zou, “Variant fountain theorems and their applications,”
*Manuscripta Mathematica*, vol. 104, no. 3, pp. 343–358, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Bartsch, “Infinitely many solutions of a symmetric Dirichlet problem,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 20, no. 10, pp. 1205–1216, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Bartsch and M. Willem, “On an elliptic equation with concave and convex nonlinearities,”
*Proceedings of the American Mathematical Society*, vol. 123, no. 11, pp. 3555–3561, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. H. Rabinowitz, J. Su, and Z.-Q. Wang, “Multiple solutions of superlinear elliptic equations,”
*Rendiconti Lincei. Serie IX. Matematica e Applicazioni*, vol. 18, no. 1, pp. 97–108, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Jeanjean, “On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${R}^{N}$,”
*Proceedings of the Royal Society of Edinburgh A*, vol. 129, no. 4, pp. 787–809, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Ding and C. Lee, “Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms,”
*Journal of Differential Equations*, vol. 222, no. 1, pp. 137–163, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-S. Zhou, “Positive solution for a semilinear elliptic equation which is almost linear at infinity,”
*Zeitschrift für Angewandte Mathematik und Physik*, vol. 49, no. 6, pp. 896–906, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Schechter, “Superlinear elliptic boundary value problems,”
*Manuscripta Mathematica*, vol. 86, no. 3, pp. 253–265, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,”
*Journal of Functional Analysis*, vol. 14, pp. 349–381, 1973. View at Zentralblatt MATH · View at MathSciNet - M. Willem and W. Zou,
*On a Semilinear Dirichlet Problem and a Nonlinear Schrödinger Equation with Periodic Potential*, vol. 305 of*Séminaire de Mathématique*, Université Catholique de Louvain, 2000. - M. Schechter and W. Zou, “Superlinear problems,”
*Pacific Journal of Mathematics*, vol. 214, no. 1, pp. 145–160, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Y. Zhang and C. G. Liu, “Infinitely many periodic solutions for second order Hamiltonian systems,”
*Journal of Differential Equations*, vol. 251, no. 4-5, pp. 816–833, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Benci and P. H. Rabinowitz, “Critical point theorems for indefinite functionals,”
*Inventiones Mathematicae*, vol. 52, no. 3, pp. 241–273, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. H. Rabinowitz,
*Minimax Methods in Critical Point Theory with Applications to Differential Equations*, CBMS, 1986. View at MathSciNet - M. Willem,
*Minimax Theorems*, Birkhäuser, Boston, Mass, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet