Journal of Applied Mathematics

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.