Abstract and Applied Analysis
Volume 2014 (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

Jincheng Huang1,2

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, [1–7].

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 [3–6].

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

Thus (H₂)–(H₄) are satisfied.

By applying Theorem 10, we obtain the following.

Theorem 12. If (i) or (ii) is satisfied, then there exists such that, for , system (93) has at least two nonnegative solutions in and in such that .

Proof. Notice that (25)-(26) and (51)-(52) are satisfied provided is sufficiently small; we obtain all assumptions of Theorem 10 which are satisfied. Thus, by Theorem 10, there exist at least two solutions and . Moreover, since and , we may assume . In the proof of Propositions 8 and 9, we have proved that and , respectively; thus .

(II) We consider the multiplicity of nonnegative solutions for the following -Laplacian system with Dirichlet boundary conditions:

where , is a bounded domain with smooth boundary, are parameters in , and the weight functions satisfy the following condition:

(A₂) , , and .

We consider problem (103) in the framework of the Sobolev space .

A pair of functions is said to be a weak solution of problem (103) if

hold for all .

If , let ,  ; then, for all , we define potential operators given by

Thus we have

It is clear that the corresponding energy functional of (103) is defined by

By standard Sobolev embedding theorems, we have

where

Thus (H₁)–(H₄) are satisfied.

If , let ,  , and define potential operators given by

Similarly, we have

where

Thus (H₁)–(H₄) are satisfied.

Arguing as before, we obtain the following.

Theorem 13. If or , then there exists such that, for , system (103) has at least two nonnegative solutions in and in such that .

(III) Finally, we consider the multiplicity of nonnegative solutions for the following -Laplacian system with boundary conditions containing both convex and concave nonlinearities:

where , is a bounded domain with smooth boundary, are parameters in , and the weight functions satisfy the following condition:

(A₃) , , and .

We consider problem (113) in the framework of the Sobolev space .

If , then we let ,  , and define potential operators given by

for all . By standard Sobolev embedding theorems, we have

where

Thus (H₁)–(H₄) are satisfied.

If , then we let ,  , and define potential operators given by

for all . By standard Sobolev embedding theorems, we have

where

Thus (H₁)–(H₄) are satisfied.

Arguing as before, we obtain the following.

If or , then there exists such that, for , system (113) has at least two nonnegative solutions in and in such that .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

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

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