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The Scientific World Journal
Volume 2013 (2013), Article ID 516093, 9 pages
Existence of Multiple Solutions for a -Kirchhoff Problem with Nonlinear Boundary Conditions
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.
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 , 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 , 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  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 , 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. , 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  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 ) 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  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.
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.
The authors would like to express their sincere gratitude to the anonymous reviewers for the valuable comments and suggestions.
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