Multiple Solutions for Kirchhoff Equations under the Partially Sublinear Case
We prove the infinitely many solutions to a class of sublinear Kirchhoff type equations by using an extension of Clark’s theorem established by Zhaoli Liu and Zhi-Qiang Wang.
1. Introduction and Main Results
In this paper we study the existence and multiplicity of solutions for the following Kirchhoff type equations:where , are positive constants.
When is a smooth bounded domain in , the problemhas been studied in several papers. Perera and Zhang  considered the case where is asymptotically linear at 0 and asymptotically 4-linear at infinity. They obtained a nontrivial solution of the problems by using the Yang index and critical group. Then, in  they considered the cases where is 4-sublinear, 4-superlinear, and asymptotically 4-linear at infinity. By various assumptions on near 0, they obtained multiple and sign changing solutions. Cheng and Wu  and Ma and Rivera  studied the existence of positive solutions of (2) and He and Zou  obtained the existence of infinitely many positive solutions of (2), respectively; Mao and Luan  obtained the existence of signed and sign-changing solutions for problem (2) with asymptotically 4-linear bounded nonlinearity via variational methods and invariant sets of descent flow; Sun and Tang  studied the existence and multiplicity results of nontrivial solutions for problem (2) with the weaker monotony and 4-superlinear nonlinearity. For (2), Sun and Liu  considered the cases where the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity, and the nonlinearity is asymptotically linear near zero but 4-superlinear at infinity. By computing the relevant critical groups, they obtained nontrivial solutions via Morse theory.
Comparing with (1) and (2), is in place of the bounded domain . This makes the study of problem (1) more difficult and interesting. Wu  considered a class of Schrödinger Kirchhoff type problem in and a sequence of high energy solutions are obtained by using a symmetric Mountain Pass Theorem. In , Alves and Figueiredo study a periodic Kirchhoff equation in ; they get the nontrivial solution when the nonlinearity is in subcritical case and critical case. Liu and He  obtained multiplicity of high energy solutions for superlinear Kirchhoff equations in . Li et al. in  proved the existence of a positive solution to a Kirchhoff type problem on by using variational methods and cutoff functional technique.
In , Jin and Wu consider the following problem: where constants , , or 3, and . By using the Fountain Theorem, they obtained the following theorem.
Theorem A (see ). Assume that the following conditions hold.
If the following assumptions are satisfied,() as uniformly for any ,()there are constants and such that where ()there exists such that() () for each and for each , where is the group of orthogonal transformations on ,() for any ,then problem (3) has a sequence of radial solutions.
Recently, Liu and Wang  obtained an extension of Clark’s theorem as follows.
Theorem B (see ). Let be a Banach space, . Assume is even and satisfies the (PS) condition, bounded from below, and . If, for any , there exists a -dimensional subspace of and such that , where , then at least one of the following conclusions holds.(i)There exists a sequence of critical points satisfying for all and as .(ii)There exists such that for any there exists a critical point such that and .
Theorem A obtained the existence of infinitely many solutions under the case that is sublinear at infinity in . It is worth noticing that there are few papers concerning the sublinear case up to now. Motivated by the above fact, in this paper our aim is to study the existence of infinitely many solutions for (1) when satisfies sublinear condition in at infinity. Our tool is extension of Clark’s theorem established in . Now, we state our main result.
Theorem 1. Assume that satisfies () and the following conditions:()There exist , , such that and .()Consider uniformly in some ball , where .() is a positive continuous function such that .
Then (1) possesses infinitely many solutions such that as .
Remark 2. Throughout the paper we denote by various positive constants which may vary from line to line and are not essential to the problem.
The paper is organized as follows: in Section 2, some preliminary results are presented. Section 3 is devoted to the proof of Theorem 1.
In this section, we will give some notations that will be used throughout this paper.
Let be the completion of with respect to the inner product and norm Moreover, we denote the completion of with respect to the norm by . To avoid lack of compactness, we need to consider the set of radial functions as follows: Here we note that the continuous embedding is compact for any .
Define a functional by Then we have from () that is well defined on and is of , and It is standard to verify that the weak solutions of (1) correspond to the critical points of functional .
3. Proofs of the Main Result
Proof of Theorem 1. Choose such that is odd in , for and , and for and . In order to obtain solutions of (1) we consider Moreover, (13) is variational and its solutions are the critical points of the functional defined in by From (), it is easy to check that is well defined on and , and Note that is even, and . For , Hence, it follows from (14) that We now use the same ideas to prove the (PS) condition. Let be a sequence in so that is bounded and . We will prove that contains a convergent subsequence. By (17), we claim that is bounded. Assume without loss of generality that converges to weakly in . Observe that Hence, we have It is clear that and as . In the following, we will estimate , by using (), for any , which implies Therefore, converges strongly in and the (PS) condition holds for . By () and (), for any , there exists such that if and then , and it follows from (14) that This implies, for any , if is a -dimensional subspace of and is sufficiently small then , where . Now we apply Theorem B to obtain infinitely many solutions for (13) such that Finally we show that as . Let be a solution of (13) and . Let and set . Multiplying both sides of (13) with implies By using the iterating method in , we can get the following estimate: where is a number in and is independent of and . By (23) and Sobolev Imbedding Theorem , we derive that as . Therefore, are the solutions of (1) as is sufficiently large. The proof is completed.
Conflict of Interests
The authors declare no conflict of interests regarding the publication of this paper.
The authors would like to express their sincere gratitude to one anonymous referee for his/her constructive comments for improving the quality of this paper. This work was supported by the National Natural Science Foundation of China (Grant nos. 11071149, 11271299, and 11301313), Natural Science Foundation of Shanxi Province (2012011004-2, 2013021001-4, and 2014021009-1), and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (no. 2015101).
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