Abstract and Applied Analysis

Abstract and Applied Analysis / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 834918 | 7 pages | https://doi.org/10.1155/2013/834918

Existence and Multiplicity of Nonnegative Solutions for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

Academic Editor: Julio Rossi
Received21 May 2013
Revised27 Oct 2013
Accepted27 Oct 2013
Published24 Nov 2013

Abstract

Existence and multiplicity results are established for quasilinear elliptic problems with nonlinear boundary conditions in an exterior domain. The proofs combine variational methods with a fibering map, due to the competition between the different growths of the nonlinearity and nonlinear boundary term.

1. Introduction

Consider the following quasilinear elliptic problem: where is a smooth exterior domain in , , and    is the unit vector of the outward normal on the boundary .

Equations of the type (1) arise in many and diverse contexts like differential geometry [1], nonlinear elasticity [2], non-Newtonian fluid mechanics [3], glaciology [4], and mathematical biology [5]. As a result, questions concerning the solvability of problem (1) have received great attention; see [610].

For with , , by using the fibering method, Kandilakis and Lyberopoulos [6] studied the existence of nonnegative solutions for problem (1) in unbounded domains with a noncompact boundary. When with , Lyberopoulos [7] studied the existence versus absence of nontrivial weak solutions for problem (1). Similar consideration can be found in Kandilakis and Magiropoulos [8]. In [9], Filippucci et al. established existence and nonexistence results for problem (1) via variational methods combined with the geometrical feature, where . Recently, Chen et al. [10] considered the existence and multiple of solutions for problem (1) by the variational principle and the mountain pass lemma.

Motivated by these findings, we consider the following quasilinear elliptic problem: where is a smooth exterior domain in and   is the unit vector of the outward normal on the boundary . Since , problem (2) is essentially different from problem (1). Using the Nehari manifold and fibering map, Wu [11] considered problem (2) for ; Afrouzi and Rasouli [12] considered problem (2) for .

Throughout this paper, we make the following assumptions.(), , and , where and   .() The function and .() The function and .() The function satisfies and with .() The function ,   in and with .

The purpose of this paper is to find existence and multiplicity of nonnegative solutions to problem (2). Our proofs are based on the variational method. The main difficulty is the lack of compactness of the Sobolev embeddings in unbounded domains. To overcome this difficulty, we impose the integrality conditions ()-() on and to establish compact Sobolev embedding theorems (see Lemmas 3 and 4).

The rest of the paper is organized as follows. In Section 2, we set up the variational framework of the problem and give some preliminaries. Section 3 is devoted to the existence results for problem (2). The multiplicity of nonnegative solutions for problem (2) is considered in the last section.

2. Variational Framework and Some Preliminaries

In this section, we set up the variational framework and give some preliminaries.

Define the weighted Sobolev space as the completion of under the norm which is equivalent to the standard one under assumptions ()-(). Moreover, denote by and the weighted Lebesgue spaces equipped with the norm: respectively. The definition of the weak solution of problem (2) reads as follows.

Definition 1. One says is a weak solution of problem (2) if holds for all .

The energy functional corresponding to problem (2) is where , . It is well known that the weak solutions of (2) are the critical points of the energy functional . If is a critical point of , then necessarily belongs to the Nehari manifold: where For all , we have , where

Moreover

The variational framework that we adopt is based on the so-called one-dimensional fibering method proposed by Pohozaev [13]. The central idea of this strategy consists in embedding the original variational problem into the “wider” space and then investigating the conditional solvability of the new problem in under an appropriately imposed constraint. To this end, we define the extended functional by setting for any If is a critical point of , then necessarily ; that is, In particular, if , then (12) is equivalent to where

Now, suppose that solves (13) for all ; then . Furthermore, if exists and is unique for all , then (13) generates a bijection between and . Moreover, the following proposition holds.

Lemma 2 (see [13]). If is a conditional critical point of , under the constraint , then is a critical point of , where and is a nonnegative solution of (13).

In view of Lemma 2, the problem of finding solutions of (2) will be reduced to that of locating the critical point of under the constraint .

The following compact embedding theorems play an important role in the proof of our main results.

Lemma 3. Assume ()–(). Then the embedding is compact.

Proof. Let . Since , it follows that . Let be the standard Banach space endowed with the norm . By assumptions ()-(), . Similar to the proof of [10, Lemma 2] (see also the proof of [14, Theorem 7.9]), we can prove that is compact and so is . Let be the best trace embedding constant; that is, By Hölder's inequality, we have This shows that the embedding is continuous.
Assume is a bounded sequence in . Then by the compact embedding , there exist and a subsequence of (not relabelled) such that strongly in .
By Hölder's inequality again, we infer This completes the proof.

Lemma 4. Assume ()-() and (). Then the embedding is compact.

Proof. Let . Since , it follows that . Hence the embedding is compact (see [15, 16]). Let be the best trace embedding constant; that is, By Hölder's inequality, we have This shows that the embedding is continuous.
Assume is a bounded sequence in . Then by the compact embedding , there exist and a subsequence of (not relabelled) such that strongly in .
By Hölder's inequality again, we infer This completes the proof.

We also need the following mountain pass lemma (see [17, 18]).

Lemma 5. Let be a real Banach space and with . Suppose() there are such that for ;() there is , such that .
Define Then is finite and possess a sequence at level . Furthermore, if satisfies the (PS) condition, then is a critical value of .

To get multiplicity results, we need the following fountain theorem due to Bartsch [19] and a critical point theorem in [20, 21].

Let be a reflexive and separable Banach space. It is well known that there exist and () such that(1), where for and for ,(2) and .

For convenience, we write

Lemma 6 (fountain theorem [19]). Assume is an even functional that satisfies the condition. If for every there exist such that() as ,(), then J has a sequence of critical points with .

Lemma 7 (see [20, 21]). Let , where is a Banach space. Assume that satisfies the condition and is even and bounded from below, and . If for any there exists a -dimensional subspace and such that then has a sequence of critical values satisfying as .

3. Existence of Nonnegative Solutions

In this section, the existence results are established for problem (2). The proofs combine variational methods with a fibering map. Since , we may suppose that the solution to problem (2) is nonnegative throughout this paper.

Theorem 8. Let ()–() hold with either or . Then problem (2) admits a nonnegative nontrivial weak solution which is also a ground state.

Proof. Suppose . Rewriting (13) as we immediately see that for every (where is defined by (9)) there exists a unique satisfying (25). Moreover, it can be easily checked that
Consider now the variational problem Let be a minimizing sequence in with . Then there exists such that in . By Lemmas 3 and 4, we have and .
We first assert that . Suppose the contrary; then . In view of (25),
Letting , it follows that . Thus which contradicts .
Next, we prove . If not, then . So, there exists such that . From (25), we have This and (26) yield
On the other hand, it follows from (29) that is bounded and so there exists a subsequence (not relabelled) such that . Thus by (25), we have
Hence . Notice that is strictly decreasing for all ; we have which is a contradiction. So, and is a critical point of . By Lemma 2, is a nontrivial solution of problem (2). Since is a unique solution of (25), then (25) generates a bijection between and and so the obtained solution is actually a ground state.
The case can be treated in a similar way.

Remark 9. Afrouzi and Rasouli [12] consider the following problem: where is a bounded domain in and . The functions and are continuous functions which change sign in . Using the Nehari manifold and fibering map, they proved that problem (36) has at least two nontrivial nonnegative solutions if is sufficiently small. In fact, by slight modification, we can prove that the result they established is still true if is a smooth exterior domain or the parameters satisfy . But for , we can prove that problem (36) has at least one nontrivial nonnegative solution for is sufficiently small via the method used in [12]. Notice that our result (Theorem 8) does not need to be small.

Theorem 10. Let ()–() hold with . Then problem (2) has a nonnegative nontrivial weak solution.

Proof. From Lemma 3, we have Thus So is coercive by .
By Lemmas 3 and 4, it is easy to verify that is weakly lower semicontinuous. So has a minimum point in and is a weak solution of (2).
In the following, we prove . Let . Then Thus for is sufficiently small. Notice that ; we obtain Thus the minimum point of is nontrivial.

Theorem 11. Let ()–() hold with . Then problem (2) has a nonnegative nontrivial weak solution in .

Proof. Let . Then Thus for small and for large.
Let satisfy in and in . Then This shows that is bounded in . Up to a subsequence, we obtain in . Thus It follows from Lemmas 3 and 4 that
Hence Then (45) give that . Notice that ; we have where
Using the standard inequality in given by we have from (46) that in . Thus satisfies (PS) condition. Then the assertion of this theorem follows from Lemma 5.

Next, we seek for a solution in with . In this case, we find it necessary to strengthen our hypothesis by assuming that the function is positive. That is, () will be replaced by() the function and with .

Theorem 12. Let ()–() and () hold with . Suppose also where . Then problem (2) has a nonnegative nontrivial weak solution in .

Proof. Define Fix with . In view of assumption (), we have . For all , there exist and with such that . Moreover, . Thus as . Hence is coercive in . By Lemmas 3 and 4, it is easy to verify that is weakly lower semicontinuous. So has a minimum point in and is a weak solution of (2).
In the following, we prove that . Notice that as , as , and ; we infer that attain its maximum at , where
If , then (13) has exactly two solutions and with . Let . We have from (11) and (12) that
Let . It follows that which ensures . Thus . This implies that the weak solution of (2) is nontrivial.

Remark 13. Condition (49) may be viewed as grading the “strength” of interaction induced by and . Hence, qualitatively speaking, one may rephrase Theorem 12 as saying that if , then problem (2) admits a nontrivial weak solution provided that “prevails” over .

4. Multiplicity of Nonnegative Solutions

In this section, we establish multiplicity results for the cases and by Lemmas 6 and 7, respectively. To this purpose, the assumption () will be replaced by the following:() the function satisfies in and with .

Theorem 14. Let ()–(), (), and () hold with . Then problem (2) has a sequence of solutions in E with as .

Proof. We will prove this theorem by fountain theorem. The proof is divided into three steps.
(1) Let and be defined by (23) and . Then it follows that (see [22]). Therefore, we have Choosing , we obtain that if ,, then Thus in Lemma 6 is proved.
(2) Since in the finite dimensional space all norms are equivalent, there exist such that hold for all . Thus by (19), Therefore in Lemma 6 is satisfied for every large enough.
(3) Let be a sequence of . Then we have Therefore is bounded in . Similar to the proof of Theorem 11, we can verify that satisfies (PS) condition.
Obviously, is an even functional and . Thus the assertion of Theorem 14 follows from Lemma 6.

Theorem 15. Let ()–(), (), and () hold with . Then problem (2) has a sequence of solutions in E with and as .

Proof. Since and ,   is coercive and bounded from below. As before, we can verify that (PS) condition holds.
Let be defined by (23) and . Since is a finite dimensional space and , we can choose small enough such that We obtain a sequence of solutions by Lemma 7.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (2013B09914).

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Copyright © 2013 Jincheng Huang. 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.

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