Journal of Applied Mathematics

Volume 2013 (2013), Article ID 914210, 8 pages

http://dx.doi.org/10.1155/2013/914210

Research Article

Existence Results for a -Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition

Libo Wang1,2 and Minghe Pei²

¹Institute of Mathematics, Jilin University, Chang'chun 130012, China

²Department of Mathematics, Beihua University, Ji'lin 132013, China

Received 25 November 2012; Accepted 28 April 2013

Academic Editor: Jaime Munoz Rivera

Copyright © 2013 Libo Wang and Minghe Pei. 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.

We consider the existence and multiplicity of solutions for the -Kirchhoff-type equations without Ambrosetti-Rabinowitz condition. Using the Mountain Pass Lemma, the Fountain Theorem, and its dual, the existence of solutions and infinitely many solutions were obtained, respectively.

1. Introduction

The Kirchhoff equation

was introduced by Kirchhoff [1] in the study of oscillations of stretched strings and plates, where and are constants. The stationary analogue of the Kirchhoff equation, that is, (1), is as follows:

After the excellent work of Lions [2], problem (2) has received more attention; see [3–10] and references therein.

The -Laplace operator arises from various phenomena, for instance, the image restoration [11], the electro-rheological fluids [12], and the thermoconvective flows of non-Newtonian fluids [13, 14]. The study of the -Laplace operator is based on the theory of the generalized Lebesgue space and the Sobolev space , which we called variable exponent Lebesgue and Sobolev space. We refer the reader to [15–19] for an overview on the variable exponent Sobo-lev space, and to [20–29] for the study of the -Laplacian-type equations.

Recently, there has been an increasing interest in studying the Kirchhoff equation involving the -Laplace operator. Autuori et al. [30, 31] have dealt with the nonstationary Kirchhoff-type equation involving the -Laplacian of the form

In [32–35], applying variational method and Ambrosetti-Rabinowitz (AR) condition, Guowei Dai has studied the existence and multiplicity of solutions for the -Kirchhoff-type equations with Dirichlet or Neumann boundary condition. In [36], by using mapping theory and the Mountain Pass Lemma, Fan has discussed the nonlocal -Laplacian Dirichlet problem with the nonvariational form

and the -Kirchhoff-type equation with the variational form

under (AR) condition, where are two functionals defined on , and .

Motivated by the above works, the purpose of this paper is to study the -Kirchhoff-type equation

without (AR) condition, where is a smooth bounded domain in , are two positive constants, , for some , and

By taking the famous Mountain Pass Lemma, the Fountain Theorem, and its dual, we obtain the existence of solutions and infinitely many solutions for the -Kirchhoff-type equation (6) under no (AR) condition.

2. Preliminary

We recall in this section some definitions and properties of variable exponent Lebesgue-Sobolev space. The variable exponent Lebesgue space is defined by

with the norm

The variable exponent Sobolev space is defined by

with the norm

Denote by the closure of in . is an equivalent norm on . In this paper we use the notation for . Define

We refer the reader to [36–38] for the elementary properties of the space .

Proposition 1 (see [38]). Set . For any , then the following are given:

(1)  if  ; (2);(3)  if  ; (4)  if    ; (5);(6).

3. Positive Energy Solution

In this section we discuss the existence of weak solutions of (6). For simplicity we write .

First, we state the assumptions on as follows.

  Let be a continuous function, and there exist positive constants such that

where and for all .  Let be a continuous function, and there exist positive constants such that

where and for all ; for all .  Let , uniformly for , where .  There exists such that for and , where

  Let , uniformly on .  There exists , such that for , .  Let for and .  Let , uniformly on , where satisfies for .

Remark 2. Condition was first introduced by Jeanjean [39] for the case . Let , then

It is easy to see that the function does not satisfy (AR) condition, but it satisfies – and .

Define , where

Then .

Proposition 3 (see [38]). Assume that hold, then the functional is sequentially weakly lower semicontinuous, is sequentially weakly continuous, and is sequentially weakly lower semicontinuous.

Proposition 4 (see [37]). Assume that hold, and let be a local minimizer (resp., a strictly local minimizer) of in the topology. Then is a local minimizer (resp., a strictly local minimizer) of in the topology.

Definition 5. We say that is a weak solution of (6), if

for any .

Definition 6. Let be a Banach space and . Given . we say that satisfies the Cerami condition (we denote by the condition), if

(i)any bounded sequence such that and has a convergent subsequence; (ii)there exist constants such that

If satisfies condition for every , then we say that satisfies condition.

Remark 7. Although (PS) condition is stronger than condition, the Deformation Theorem is still valid under condition; we see that the Mountain Pass Lemma, the Fountain Theorem, and its dual are true under condition.

Lemma 8. Assume that conditions – hold. Then satisfies condition.

Proof. From [36, Proposition 3.1], satisfies (i) of condition.

Now we check that satisfies (ii) of condition. Arguing by contradiction, we may assume that, for some ,

Then we have

Let , then up to a subsequence we may assume that

If , inspired by [13, 14], then we define

For any , let . Since in and

by the continuity of , in , thus,

Then for large enough, and

That is, . From and , we know that and

Therefore, from , we have

This contradicts (21).

If , from (20), when ,

Then from we have

For , . By we have

Note that the Lebesgue measure of is positive; using the Fatou Lemma, we have

This contradicts (30).

The technique used in this lemma was first introduced by [39, 40].

Theorem 9. Assume that conditions – and (or ) hold. Then (6) has a nontrivial solution with positive energy.

Proof. From Lemma 8, satisfies condition. Let us show that the functional has a Mountain-Pass-type geometry.

Note that . By , there exists ,  and  for  any with ,

This shows that 0 is a strictly local minimizer of in the topology, and hence 0 is a strictly local minimizer of in the topology. By [37, Theorem 1.1], 0 is a strictly local minimizer of in the topology. Thus there exists such that for every with .

We claim that . To prove this claim, arguing by contradiction, assume that there exists a sequence with such that as . We may assume that in . Since is sequentially weakly lower semicontinuous, we have that , and hence . Since is sequentially weakly continuous, then we have that , and hence . It follows from this that in which contradicts with .

Let with in and . By and , for we have

We set . Then for large, we obtain

Hence by the famous Mountain Pass Lemma, problem (6) has a nontrivial weak solution with positive energy.

4. Infinitely Many Solutions

Since is a reflexive and separable Banach space, then there exists and such that

For convenience, we write , , .

Lemma 10 (see [21]). If ,   for any , denote

Then .

Proposition 11 (Fountain Theorem). Assume that is an even functional. If, for any , there exists such that

 ,    as  ,   satisfies    condition for every , then    has an unbounded sequence of critical values.

Proposition 12 (Dual Fountain Theorem). Assume that is an even functional. If, for any , there exists such that

 ,  ,   as  ,   satisfies    condition for every  ,  then    has a sequence of negative critical values converging to .

Theorem 13. Assume that the conditions , – hold. Then (6) has infinitely many solutions such that as .

Proof. By conditions , , and , for any , there exists such that

For , when ,

Then for some large enough,

On the other hand, by and , there exists such that

Let . From Lemma 10,   as . For , when and small enough,

If we choose as , then, for with ,

which implies that as .

Theorem 14. Assume that conditions , , , , and hold. Then (6) has infinitely many solutions such that as .

Proof. By conditions , , and , for any , there exists such that

For , when is large enough,

Then for some large enough,

On the other hand, by , there exists such that

Let , then as . For , when and small enough,

If we choose as , then, for with ,

which implies that as .

Furthermore, if with , then

which implies that as .

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (11126339, 11201008).

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