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

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.