The Scientific World Journal

Volume 2013 (2013), Article ID 516093, 9 pages

http://dx.doi.org/10.1155/2013/516093

Research Article

Existence of Multiple Solutions for a -Kirchhoff Problem with Nonlinear Boundary Conditions

Zonghu Xiu1,2 and Caisheng Chen²

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

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

Received 12 April 2013; Accepted 7 July 2013

Academic Editors: H. Du and S. Momani

Copyright © 2013 Zonghu Xiu and Caisheng Chen. 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.

The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem , ,   = , on , where , . By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.

1. Introduction and Main Result

In this paper, we consider the existence of multiple solutions for the singular elliptic problem:

where , , , is an exterior domain of : that is, and , where is a bounded domain in with the smooth boundary , and . and are continuous functions, with the parameters .

Problem like (1) is usually called nonlocal problem because of the presence of the integral over the entire domain, and this implies that (1) is no longer a pointwise identity. In fact, such kind of problem can be traced back to the work of Kirchhoff. In [1], Kirchhoff investigated the model of the form

where , , , , and are all positive constants. This equation extends the classical d’Alembert’s wave equation by considering the effects of changes in the length of the strings during the vibrations. Problem (1) is related to the stationary analogue of problem (2). After Kirchhoff’s work, various models of Kirchhoff-type have been studied by many authors: we refer the readers to [2–9]. In [4], by the variational methods, Bensedik and Bouchekif considered the problem

where is a bounded domain in . One of the assumptions made on in (3) is that is continuous function on such that

The authors proved that problem (3) has a positive solution or has no solution when some other assumptions are fulfilled. In our paper, however, the weight functions and are permitted to change sign. Thus, the methods in [4] cannot be directly applied on (1).

In recent years, some other authors considered the Kirchhoff-type equations with -Laplacian [10–13]. In fact, motivated by [4, 5] and our previous work [14], we consider the existence of multiple solutions for problem (1) on the Nehari manifold by variational methods. We prove that problem (1) has at least two positive solutions. Since is an unbounded domain and the problem is singular, the loss of compactness of the Sobolev embedding renders variational technique more delicate.

In order to state our result, we introduce a weighted Sobolev space , which is the completion of the space with the norm of

For and in , we define the space as being the set of Lebesgue measurable functions , which satisfies

The following weighted Sobolev-Hardy inequality is due to Caffarelli et al. [15], which is called the Caffarelli-Kohn-Nirenberg inequality. There is a constant such that

where , , , and . Throughout this paper, we make the following assumptions:

  with , , , .

Now, we give the definition of weak solution for problem (1).

Definition 1. A function is said to be a weak solution of problem (1) if for any

The assumptions mean that all the integrals in (8) are well defined and convergent.

In view of , it follows from the compact trace embedding [16] that

for some constant , and .

Our main result is in the following.

Theorem 2. Assume ; there exists such that problem (1) has at least two positive solutions for all satisfying .

This paper is organized as follows. In Section 2, we give some properties of the Nehari manifold and set up the variational framework for problem (1). In Section 3, we consider the multiplicity results and prove Theorem 2.

2. Preliminary Results

It is clear that problem (1) has a variational structure. Let be the corresponding Euler functional of problem (1), which is defined by

where . Then, we see that the functional , and for , there holds

Particularly, we have

It is well known that the weak solution of problem (1) is the critical point of . Thus, to prove the existence of weak solutions for problem (1), it is sufficient to show that admits a sequence of critical points. Since is not bounded below on , it is useful to consider the functional on the Nehari manifold [17, 18]:

where denotes the usual duality. Then, it follows from (12) that if and only if

Then, we get from (10)–(14) that

We define

Then, (14) implies that

It is natural to split into three parts:

Now, we give some important properties of , , and .

Lemma 3. Let and . Then, is coercive and bounded below on .

Proof. For , we obtain from , the Hölder and Caffarelli-Kohn-Nirenberg inequalities that

where , . Thus, we get from (16) that

Then, one can obtain by the Young inequality and that is coercive and bounded below on .

Let

Then, we have the following result.

Lemma 4. Let . Then, for .

Proof. Suppose that there exists . If , then it follows from (19) and (21) that

which implies that

On the other hand, we can similarly get from (20) and (21) that

which yields that

Thus, inequalities (26) and (28) show that , which contradicts the hypothesis of .

For , we can similarly obtain from (19) and (21) that

Thus, (28) and (29) imply that , which is also a contradiction. Therefore, we complete the proof.

Lemma 5. If is a local minimizer of on and , then is a critical point of .

Proof. Let

Consider the following minimizing problem:

By the Lagrange multiplier principle, there exists such that

Since and , it follows from (33) that ; furthermore, .

Now, we write for and define

Denote

Then, the following results on and are established.

Lemma 6. Let , and ; then

(i),(ii)there exists constant such that .

Proof. (i) For , we have from (19) and (21) that

Thus, (36) and (16) give that

which implies that

(ii) Let ; one can deduce from (20) and (21) that

On the other hand, we obtain from (16), (26), and (39) that

Therefore, if , there exists such that .

For each with , we define

Then, , as , and gets its unique maximum at the critical point . Particularly, gets its unique maximum at

and we have the following results.

Lemma 7. Let , , and . For each with , one has the following:

(i)if , then there exists such that and

(ii)if , then there exists such that , and

Proof. (i) For , we define

Since , there exists unique such that

Thus, , which implies that . It follows from (20) that

that is, . By (47), we obtain that

which shows that increases for and decreases for . Therefore,

(ii) We firstly want to prove that . In fact,

Then, if , we have that

Since , there exists such that

and , . Similar to the proof of (i), we get that , . We can deduce from (50) that decreases for and increases for . Therefore,

Similarly,

Then, we complete the proof.

For each with , we define

Then, as and as . Furthermore, gets its unique maximum at some certain point . Particularly, gets its unique maximum at the critical point

then we have the following results.

Lemma 8. Let , , and . For each with one has the following:

(i)if , then there exists unique such that and

(ii)if , then there exists such that , , and

Proof. (i) Since , there exists unique such that

Note that (45) can be rewritten as

Thus, (61) shows that . By virtue of (14) and (46), we get that

which implies that . We have from (47) that

Then, decreases for and increases for ; that is,

(ii) We need also to prove that . In fact, we have that

for .

The rest of the proof is similar to that of (ii) in Lemma 7, and here we omit the proof.

Lemma 9. Assume . If in , then there exists a subsequence of , still denoted by , such that

Proof. By the assumption , we have , and then for any , there exists large enough such that

where , for .

The compact embedding theorem (Theorem 2.1 in [19]) implies that

Inequality (7) shows that is bounded in , which implies that in . Thus, we can obtain that

for some . On the other hand, we get from the Höder inequality and (68)–(70) that

for some constant and large .

Thus, (71) implies that

that is,

Since is compact and , we obtain by the trace embedding theorem in [20] that

This concludes the proof.

3. Existence of Solutions

In this part, we will give the proof of the existence of nonnegative and nontrivial solutions. Before this, we need to prove the following two important lemmas.

Lemma 10. Assume and . Then, the functional has a minimizer and

(i), (ii) is a positive weak solution of problem (1).

Proof. Since is bounded in , there exists a minimizing sequence of such that

We can get that is bounded in and weakly in since is coercive. It follows from Lemma 6 that

Furthermore, equality (16) and Lemma 9 imply that . In the following, we prove that in . Suppose otherwise that

Let

It is obvious that has the unique critical point and increases for and decreases for . Particularly, has the unique critical point at . Since , it follows from Lemma 8 that there exists such that and

By virtue of , we obtain that

Similarly, we set

In view of (77), we get that for large . Since and for , we obtain that and for . Then, by (77), (79), and Lemma 9, we get that

which is a contradiction. Hence, strongly in , and

Thus, is a minimum of on . Since and , we may assume by Lemma 5 that is a positive solution of problem (1).

Lemma 11. Assume and . Then, the functional has a minimizer and

(i), (ii) is a positive weak solution of problem (1).

Proof. Since is bounded on , there exists a minimizing sequence such that

Similar to the proof of Lemma 10, we may assume that weakly in . For , we deduce by Lemma 6 and (15) that ; furthermore, . We also want to prove that strongly in . In fact, if not, we have

By virtue of Lemma 7, there exists such that

and a simple transformation shows that

Therefore, Lemma 9 together with (84)–(87) gives that

which is a contradiction, and we complete the proof.

Proof of Theorem 2. We set . When , by Lemmas 10 and 11, we obtain that problem (1) has two nontrivial nonnegative solutions , . Lemma 4 and the assumptions of Theorem 2 imply that ; then and are distinct. Furthermore, since and , we can assume that the solutions and are positive. This completes the proof.

4. Conclusions

The object of this paper is to prove the existence of multiple solutions for the nonlinear Kirchhoff-type problem (1). By the variational methods, we discuss the problem on the Nehari manifold and give the sufficient conditions for the existence of solutions. We overcome the difficulty due to the loss of compactness on the unbounded domain. In the future work, we are interested to consider similar problems, but the term on the right will be replaced by abstract functions.

Acknowledgment

The authors would like to express their sincere gratitude to the anonymous reviewers for the valuable comments and suggestions.

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