Research

# Existence of nontrivial solutions for p-Kirchhoff type equations

Chunhan Liu*, Jianguo Wang and Qingling Gao

Author Affiliations

Department of Mathematics, Qilu Normal University, Jinan, 250013, P.R. China

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Boundary Value Problems 2013, 2013:279  doi:10.1186/1687-2770-2013-279

 Received: 13 July 2013 Accepted: 29 November 2013 Published: 30 December 2013

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, the linking theorem and the mountain pass theorem are used to show the existence of nontrivial solutions for the p-Kirchhoff equations without assuming Ambrosetti-Rabinowitz type growth conditions, nontrivial solutions are obtained.

MSC: 35J60, 35J25.

##### Keywords:
linking theorem; mountain pass theorem; nontrivial solutions

### 1 Introduction

In this paper, we consider the nonlocal elliptic problem of the p-Kirchhoff type given by

(1)

where is a bounded domain, and is the p-Laplacian with .

Recently, the equation

(2)

began to attract the attention of several researchers only after Lion [1] had proposed an abstract framework for this problem. Perera and Zhang [2] obtained a nontrivial solution of (2) by using the Yang index and critical group. They revisited (2) via invariant sets of decent flow and obtained the existence of a positive solution, a negative, and a sign-changing solutions in [3].

The study of Kirchhoff-type equations has been extended to the following case involving the p-Laplacian:

for details see [4-6]. One of the authors has done some related work on this field. Liu [7] gave infinite solutions to the following equation via the fountain theorem and the dual fountain theorem:

However, to the best of our knowledge, there have been few papers dealing with equation (1) using the linking theorem and the mountain pass theorem. This paper will make some contribution to this research field.

It is well known (see [8]) that the eigenvalue problem

has the first eigenvalue , which is simple, and has an associated eigenfunction . It is also known that is an isolated point of , the spectrum of , which contains at least an eigenvalue sequence and .

Let

be a Banach space with the norm for . be the one-dimensional eigenspace associated with , where . Let , where , there exists such that

When , we can take , the second eigenvalue of −△ in .

In this paper, the weak solutions of (1) are the critical points of the energy functional

where , . Obviously, and for all ,

In this paper we use the following notation: denotes the Lebesgue space with the norm ; denotes the Lebesgue measure of the set ; is the dual pairing of the space and ; → (resp. ⇀) denotes strong (resp. weak) convergence. denote positive constants (possibly different).

Definition 1[9]

Let , we say that Φ satisfies the Cerami condition at the level if any sequence , along with

possesses a convergent subsequence; Φ satisfies the (C) condition if Φ satisfies for all .

Definition 2[9]

A subset A of E is link (with respect to Φ ) to B of E if , for every , there is such that .

Letbe a Banach space with. Let, and letbe such that. Define

Letbe such that

If Φ satisfies the (PS) condition with

thencis a critical value of Φ.

Remark 1 If Φ satisfies the (C) condition, then Theorem 1 still holds.

Theorem 2[10] (Mountain pass theorem)

LetXbe a real Banach space, and letsatisfy the (C) condition. Suppose, for some, and, ,

Then Φ has a critical valuecharacterized by

where

### 2 Main results

In this section, we give our main theorem. Near the origin, we make the following assumptions.

Suppose that is a continuous function satisfying the following conditions:

() there exists a constant such that for all ;

() there exists a constant such that for all and , .

Caratheodory function f satisfies:

() For some , there exists a constant such that

() There exist and such that

() uniformly in .

uniformly in .

() uniformly in .

uniformly in .

() , .

The main results of this paper are the following.

Theorem 3Assume that (), () and ()-() hold, then problem (1) has at least one nontrivial weak solution in.

Theorem 4Assume that (), () and (), (), , hold, then problem (1) has at least one nontrivial weak solution in.

### 3 Proofs of theorems

First, we give several lemmas.

Lemma 1[7]

Under assumptions () and (), any bounded sequencesuch thatinashas a convergent subsequence.

Lemma 2Under assumptions () and (), the functionalsatisfies the (C) condition.

Proof Let , for every ,

(3)

We claim that is bounded in . For this purpose, we can suppose that . By (), there exists such that

(4)

For large n, set , (3) and (4) imply that there exists such that

This is a contradiction. Then is bounded in . By Lemma 1, we see that has a convergent subsequence in . □

Proof of Theorem 3 We obtain from assumptions (), () and () that for some small, there exists such that

Taking , using the inequality and the Sobolev inequality , we have

Then there exists such that .

Let , and , .

For every , if , and , then . By (), we know that

Since , there exists such that

By () and (), there exists such that , . Let , then we have

Therefore, , we have

Hence, for large enough, we have .

Then there exists such that

By Lemmas 1 and 2, Φ satisfies the (C) condition. Then the conclusion follows from Theorem 1 and Remark 1. □

Remark 2 (i) There exists such that , , , which implies , ,

Then

Hence () is much weaker than Ambrosetti-Rabinowitz type growth conditions.

(ii) If , as uniformly in , then

Example 1 Set

where , . Then it is easy to verify that satisfies ()-() with . When , we can use odd expansion to .

Example 2 Set

where

, , . Then it is easy to verify that satisfies ()-() with . When , we can use odd expansion to .

Proof of Theorem 4 We obtain from assumptions (), (), that for some small, there exists such that

Taking , using the inequality and the Sobolev inequality , we have

Then there exists such that .

By () and , there exist , , , and such that ,

Let , then we have

For , we have

Therefore, for , let , , we have

Hence there exists , such that .

Then

Summing up Lemma 1 and Lemma 2, satisfies all the conditions of Theorem 2, then the conclusion follows from Theorem 2. □

Remark 3 The result of Theorem 1.1 in [11] corresponds to our results for the case and replaces (). It is easy to see that () is much weaker than , hence the results of Theorems 3 and 4 extend the results of [11].

Example 3 Set

where p is odd, ,

where p is even, .

Then similar to [12], it is easy to verify that satisfies (), (), , with .

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

CL and JW obtained the results in a joint research. All the authors read and approved the final manuscript.

### Acknowledgements

This work was supported by the National Nature Science Foundation of China (10971179) and the Project of Shandong Province Higher Educational Science and Technology Program (J09LA55, J12LI53).

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