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Abstract and Applied Analysis
Volume 2014, Article ID 947139, 11 pages
http://dx.doi.org/10.1155/2014/947139
Research Article

Multiplicity of Solutions to a Potential Operator Equation and Its Applications

1College of Sciences, Hohai University, Nanjing 210098, China
2Department of Mathematics and Physics, Hohai University, Changzhou Campus, Changzhou 213022, China

Received 12 July 2014; Accepted 27 August 2014; Published 22 October 2014

Academic Editor: M. Victoria Otero-Espinar

Copyright © 2014 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.

Abstract

We consider the multiplicity of solutions for operator equation involving homogeneous potential operators. With the help of Nehari manifold and fibering maps, we prove that such equation has at least two nontrivial solutions. Furthermore, we apply this result to prove the existence of two nonnegative solutions for three types of quasilinear elliptic systems involving (p, q)-Laplacian operator and concave-convex nonlinearities.

1. Introduction

Let be a bounded domain and . We define the Sobolev space equipped with the norm Then we denote and, for , It is well known that is a reflexive Banach space. The dual space of is denoted by .

In this paper, we consider the multiplicity results for nonzero solutions of the operator equation where is a -Laplacian operator defined by and are homogeneous operators of degrees and . It is well known that the functionals corresponding to , and are given by

Throughout the paper, we will assume the following.(H1). (H2)There exist such that (H3) are strongly continuous.(H4)There exist positive numbers and such that

There are several studies for existence and multiplicity of solutions of the quasilinear elliptic system where is a smooth bounded domain of . See, for example, [17].

In [5], Brown and Wu considered the semilinear elliptic system where is a smooth bounded domain of ,   . They found that the above problem has at least two positive solutions if the pair is below a certain subset of .

Recently, Afrouzi and Rasouli [3] considered the semilinear -Laplacian system where ,  . They proved that the above problem has at least two solutions if . Similar considerations can be found in [1, 4, 6].

We note that few researchers had used variational method with the help of Nehari manifold to consider semilinear elliptic system involving -Laplacian operator to the best of our knowledge. In this paper, we will use variational method with the help of Nehari manifold and fibering maps (see [8]) to prove the existence of at least two nonzero solutions of problem (3) and then apply this result to three types of -Laplacian systems. These applications extended some of the results in [36].

This paper is organized as follows. In Section 2, we discuss some of the properties of the Nehari manifold for (3). In Section 3, we prove that (3) has at least two nontrivial solutions. Applications to -Laplacian systems with nonlinearities involving both concave and convex terms are given in the last section.

2. The Properties of the Nehari Manifold

We say is a weak solution of problem (3) if holds for all .

Problem (3) has a variational structure. Let be defined by where , and are defined by (4). Clearly, the critical points of are the weak solutions of problem (3).

As the energy functional is not bounded below on , it is useful to consider the functional on the Nehari manifold: Thus if and only if

Moreover, we have the following result.

Lemma 1. The energy functional is coercive and bounded below on .

Proof. If , then Thus, by (6), Without loss of generality, we suppose ; then where and are constants independent of . Thus is coercive and bounded below on .

Define Then, for , we have Now, we split into three parts , , and defined by

We now derive some basic properties of , and .

Lemma 2. Suppose that is a local minimizer for J on and . Then in .

Proof. Consider the optimization problem By the theory of Lagrange multiplier principle, there exists such that Thus
Since , we obtain . However, the fact implies that . Thus and so . This completes the proof.

Lemma 3. One has provided where

Proof. Let . By (18) and (6),
If , then it follows from (28)-(29) that which contradicts (23).
Similarly, is impossible.
In view of (29), we have either or is satisfied. In the following, two cases are considered.
Case 1. Suppose that (31) holds. Then By (23), we obtain and so The last inequality is equivalent to Hence, by (28), we infer that which is equivalent to
On the other hand, we obtain from (29) that where Obviously, attains its minimum at and attains its maximum at . Furthermore, is increasing on ,   is decreasing on , and In view of (24)-(25), we have and and so which contradicts (39). Thus .
Case 2. Suppose that (32) holds. Then, by (24), which is equivalent to Thus (28) and (44) imply Hence In view of (23) and (26), we obtain , and so , which contradicts (39). Thus .

In order to get a better understanding of the Nehari manifold, we consider the function defined by for , where is defined in (4). Obviously, ,  , and If , then there is unique such that . Furthermore, for and for . By direct computation, we can deduce that Moreover if and . Then the following lemma holds.

Lemma 4. For each with , one has provided

Proof . The proof is divided into the following four cases.
Case 1 (). In this case, Thus, by (6), we infer that In view of (52), we have and so .
Case 2 (). Then and so Thus follows from (51).
Case 3 (). It follows from (49), (50), and (6) that Since , we have Thus Hence provided which is equivalent to (52).
Case 4 (). In this case, (58) still holds. Thus, in view of (6), we infer Hence follows from (51).

By Lemma 4, we have the following.

Lemma 5. Suppose that (51)-(52) hold. Then, for each with , one has the following.(i)If , then there is unique such that is decreasing on and increasing on . Moreover and (ii)If , then there are and with such that is decreasing on , increasing on , and decreasing on . Moreover ,  , and

Proof. Fix with . By (17) and (19), we infer that (i)Suppose that . Then has unique solution and . Hence , , and so . Moreover, since we obtain for and for . Thus is decreasing on and increasing on and .(ii)Suppose that . Since , the equation has exactly two solutions with such that . Thus there are exactly two multiples of lying in ; namely, and . Since we have is decreasing on , increasing on , and decreasing on and (64) holds.

Similarly, we define where is defined in (4). Clearly ,  , and If , then there is unique such that . Furthermore, for and for .

Using arguments similar to those in the proof of Lemmas 4 and 5, we have the following.

Lemma 6. For each with , one has provided (51)-(52) hold.

Lemma 7. Suppose that (51)-(52) hold. Then, for each with , one has the following.(i)If , then there is unique such that is increasing on and decreasing on . Moreover and (ii)If , then there are and with such that is decreasing on , increasing on , and decreasing on . Moreover ,  , and

3. Existence of Nonzero Solutions

In this section, we will give simple proofs of the existence of two nonzero weak solutions, one in and one in .

Proposition 8. Suppose that (25)-(26) and (51)-(52) hold. Then there exists a minimizer of on and .

Proof. Since is bounded below on and so on , there exists a minimizing sequence such that
Since is coercive, is bounded in . Thus we may assume, without loss of generality, that weakly in . By (H3),
Since , we can infer from (18) that Then This implies Notice that and, letting , we obtain . Hence .
Now we prove that strongly in . Suppose otherwise; then
Let be the unique solution of and let be the unique solution of , where is defined by (47). Since , we have from (19) that and so . Thus .
Since , it follows from Lemma 5 that there is unique such that
From (79), we have This implies for sufficiently large.
Since is increasing on and so on , is also increasing on .
In view of and for sufficiently large, we deduce that and so a contradiction. Hence strongly in . This implies as . Thus is a minimizer for on .

Proposition 9. Suppose that (25)-(26) and (51)-(52) hold. Then there exists a minimizer of in and .

Proof. Similarly as the proof of Proposition 8, there exists a minimizing sequence and such that
Since , we can infer from (18) and (6) that Then either or is satisfied. That is to say, either or is satisfied, where and are defined in Lemma 3. Hence, for all , we obtain from (85) that Letting , we get that Hence and so .
Now we prove that strongly in . Suppose on the contrary; then
Let be the unique solution of and let be the unique solution of , where is defined by (69).
By Lemma 7, there is unique such that . In view of (90), we infer that This implies for sufficiently large.
Since ,  , and it is clear from Lemmas 5 and 7 that is increasing on ; that is, .
Thus which is a contradiction. Hence strongly in . This implies as . Thus is a minimizer for on .

By Propositions 8 and 9, we can prove our main result read as follows.

Theorem 10. Problem (3) has at least two solutions and such that if (25), (26), (51), and (52) are satisfied.

Proof. By Propositions 8 and 9, there exist and such that , , and . By Lemma 2, are critical points of on and hence are weak solutions of (3).

Remark 11. Inequalities (25), (26), (51), and (52) can be fulfilled provided that or is sufficiently small.

4. Applications

In this section, we give some applications of Theorem 10.

(I) We consider the multiplicity of nonnegative solutions for the following -Laplacian system with nonlinear boundary conditions: where , is a bounded domain with smooth boundary, are parameters in , and the weight functions and satisfy the following condition:

(A1) , , and .

We consider problem (93) in the framework of the Sobolev space . A pair of functions is said to be a weak solution of problem (93) if hold for all , where is the measure on the boundary. In the following, two cases are considered.

Case 1 ( ( if , if )). Let ,  . Then (H1) is satisfied.

For all , we define potential operators given by Then we have By assumption (A1), we have (H2) is satisfied.

It is clear that the corresponding energy functional of (93) is defined by

Let and be the best Sobolev constant for the embedding of for and the best Sobolev trace constant for the embedding of for , respectively, where if and if . Then we have where ,  , and Thus (H3)–(H4) are satisfied.

Case 2 (). Let . Then (H1) is satisfied.

For all , we define potential operators given by Similarly as before, we have where