Abstract and Applied Analysis

Volume 2013 (2013), Article ID 281949, 6 pages

http://dx.doi.org/10.1155/2013/281949

Research Article

The Existence of a Nontrivial Solution for a -Kirchhoff Type Elliptic Equation in

Zonghu Xiu1,2

¹Science and Information College, Qingdao Agricultural University, Qingdao 266109, China

²College of Science, Hohai University, Nanjing 210098, China

Received 11 March 2013; Revised 18 July 2013; Accepted 18 July 2013

Academic Editor: Dumitru Motreanu

Copyright © 2013 Zonghu Xiu. 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.

Using Mountain Pass lemma, under some appropriate assumptions, we establish the existence of one nontrivial solution for a class of p-Kirchhoff-type elliptic equations in .

1. Introduction and Main Result

In this paper, we consider the existence of solution for the following elliptic problem:

where , , and are positive functions, is continuous function, and further assumptions will be listed later. Problems like (1) originally came from the stationary problem of a model introduced by Kirchhoff [1]. Due to the existence of the integration over the whole space, problems like (1) are also called nonlocal problems.

In recent years, the Kirchhoff-type equations with -Laplacian operator has been considered by many authors; see [2–6]. When , the authors in [6] considered the following similar Kirchhoff-type elliptic problem on the bounded domain

The function in (2) is required to meet the condition of for some constants . The authors proved that problem (2) has at least one positive solution under some other additional conditions. Problem (2) was also considered in [3], where the function is odd about . For , we refer to [7–9]. The authors in [8] studied the following Kirchhoff type problem:

By the Fountain theorem, the author proved the existence of infinitely many solutions. Note that one of the assumptions made on the function in (3) is that for any . In the present paper, however, the function is not required to be odd about as that of [3, 8]. When and , problem (1) becomes the -Laplacian elliptic equations without nonlocal term, and this kind of problem is also studied by many authors. For these works, we refer to [10–14] and the references therein. In [15], Liu discussed the following elliptic problem:

The author proved that problem (4) has at least one ground state. The weight function is required to be bounded; more precisely, there exist constants such that

We point out that the weight function in problem (1) is permitted to tends to infinity.

In this paper, inspired by [4, 8, 15], we consider the existence of solution of problem (1). By the variational method, we will prove that problem (1) has at least one nontrivial weak solution. Since problem (1) is considered in the whole space , the loss of compactness of the Sobolev embedding renders the variational technique more delicate.

In this paper, we make the following hypothesis:

  , and as ,   there exists such that , where ,   there exist constants and such that .

Remark 1. Throughout this paper, we denote by the constant which may vary from line to line but remains independent of the relevant quantities. Note that there exist many functions , and such that satisfy the assumptions of Theorem 3, for example, , , .

Let be the usual Sobolev space with the norm of

Denote

Then, is a Sobolev space with the norm of

We give another space

endowed with the norm

It is not difficult to check that is a Banach space. The Euler functional of problem (1) is

where . Then, the assumptions – imply that , and for any , there holds

Particularly,

Definition 2. A function is said to be a weak solution of (1) if and only if (12) holds for any .

Our main result is listed later.

Theorem 3. Assume –. Then, problem (1) has at least one nontrivial weak solution in .

This paper is organized as follows. In Section 2, we introduce some definitions and prove several lemmas which will be used later. In Section 3, we give the proof of Theorem 3 by making use of the Mountain Pass lemma.

2. Preliminary Results

In this section, we give some important lemmas, which will be needed in the proof of our main result. Particularly, one result of compact embedding on unbounded domain will be proved.

Lemma 4. Assume and . Then, the embedding is compact.

Proof. We split the proof into two cases.

(). Let

and ; then shows that as . Furthermore, one gets that

which implies that

It follows from (16) that

Let be a bounded sequence of such that . Then, there exists such that . For any , it follows from (17) that there exists large enough such that

Note that the embedding is compact. Therefore, for these and , there exists large enough such that

for all . Thus, one can get from (18) and (19) that

which implies that strongly in .

  (). By the Hölder and Young inequalities, one can get from that

Since and as , then

Therefore, similar to the proof of  , the embedding is compact.

Now, we give the definitions of Palais-Smale (simply ) sequence and condition.

Definition 5. Let , , and be a Banach space. The sequence is said to be a sequence if there holds

A functional is said to satisfy the condition if any sequence in contains a convergent subsequence.

Next, we will prove that the functional satisfies the condition.

Lemma 6. Assume   –. Then, satisfies the condition on for any .

Proof. Let be an any sequence in . We divide the proof into two steps.

( is bounded in ). Note that ; then, it follows from (23) and that

which implies that is bounded in .

( converges strongly in ). Since is bounded in the separable , there exist and a subsequence of , still denoted by , such that in . Now, we want to prove that

In fact, it follows from that

Thus, in order to prove (25), we need only to prove that

Here, we divide the proof (27) into two cases:

(i)

By the Hölder inequality, we obtain that

It is easy to check that . Then, the compact embedding in Lemma 4 shows that

which gives (27).

(ii)

Let . Since and , there exist such that

The Hölder inequality shows that

Therefore, the embedding in Lemma 4 implies that

which also gives (27).

Note that and

Therefore, the previous cases (i) and (ii) imply that (25) holds for all .

Denote

Then, it follows from (23) and (25) that and

Since in , one obtains from the Hölder inequality that

Thus, it follows from (37) and (38) that

Since , , and , are positive functions, we get from (25) and (39) that

Therefore, we can deduce from (40) and the following standard inequalities in [16]:

that

Then, we complete the proof.

3. Existence of Solution

In this section, the proof of Theorem 3 is mainly based on the following Mountain Pass lemma [17] (also see [18]).

Lemma 7 (Mountain Pass lemma). Let be a Banach space and satisfies condition. Suppose that and

there are constant such that , there is an such that .

Then, possesses a critical value . Moreover, can be characterized as

where

In view of Lemma 6, the functional satisfies the condition. It is obvious that ; then, in order to apply Lemma 7, we need only to prove that satisfies the geometric conditions of the Mountain Pass lemma.

Lemma 8. Assume –. Then,

there exist such that with , there is an such that and .

Proof. It follows from , , and the embedding in Lemma 4 that

Since , there exists a sufficient small and such that with . On the other hand, for fixed and , it follows from that

Then, there is a large such that and . One may choose ; then, and . Thus, the proof of Lemma 8 is complete.

Proof of Theorem 3. It follows from Lemmas 7 and 8 that the functional has a critical point such that ; that is, problem (1) has at least one weak solution. On the other hand, ; then, problem (1) has at least one nontrivial weak solution.

Acknowledgment

The author is very grateful to the anonymous reviewer for valuable comments and suggestions.

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