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# Existence of homoclinic orbits for a class of p-Laplacian systems in a weighted Sobolev space

Boundary Value Problems20132013:137

DOI: 10.1186/1687-2770-2013-137

Accepted: 7 May 2013

Published: 24 May 2013

## Abstract

By applying the mountain pass theorem and symmetric mountain pass theorem in critical point theory, the existence of at least one or infinitely many homoclinic solutions is obtained for the following p-Laplacian system:

where , , , , and are not periodic in t.

MSC:34C37, 35A15, 37J45, 47J30.

### Keywords

homoclinic solutions variational methods weighted space p-Laplacian systems

## 1 Introduction

Consider homoclinic solutions of the following p-Laplacian system:
(1.1)

where , , , , , . As usual, we say that a solution u of (1.1) is a nontrivial homoclinic (to 0) if such that , as .

When , (1.1) reduces to the following second-order Hamiltonian system:
(1.2)
If we take and , then (1.2) reduces to the following second-order Hamiltonian system:
(1.3)

The existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been recognized by Poincaré [1]. Up to the year of 1990, a few of isolated results can be found, and the only method for dealing with such a problem was the small perturbation technique of Melnikov.

Recently, the existence and multiplicity of homoclinic solutions and periodic solutions for Hamiltonian systems have been extensively studied by critical point theory. For example, see [220] and references therein. However, few results [21, 22] have been obtained in the literature for system (1.2). In [22], by introducing a suitable Sobolev space, Salvatore established the following existence results for system (1.2) when .

Theorem A [22]

Assume that a and W satisfy the following conditions:
1. (A)
Let , is a continuous, positive function on such that for all

(W1) and there exists a constant such that

(W2) as uniformly with respect to .

(W3) There exists such that

Then problem (1.2) has one nontrivial homoclinic solution.

When is an even function in x, Salvatore [22] obtained the following existence theorem of an unbounded sequence of homoclinic orbits for problem (1.2) by the symmetric mountain pass theorem.

Theorem B [22]

Assume that a and W satisfy (A), (W1)-(W3) and the following condition:

(W4) , .

Then problem (1.2) has an unbounded sequence of homoclinic solutions.

In [21], Chen and Tang improved Theorem A and Theorem B by relaxing conditions (W1) and (W2) and removing condition (W3). Motivated mainly by the ideas of [18, 2123], we will consider homoclinic solutions of (1.1) by the mountain pass theorem and symmetric mountain pass theorem. Precisely, we obtain the following main results.

Theorem 1.1 Suppose that a and W satisfy the following conditions:

(A)′ Let and , is a continuous, positive function on such that for all
(W5) , , and there exists a constant such that

uniformly in .

(W6) There is a constant such that
(W7) and there exists a constant such that

Then problem (1.1) has one nontrivial homoclinic solution.

Theorem 1.2 Suppose that a and W satisfy (A)′, (W6) and the following conditions:

(W5)′ , , and

uniformly in .

(W7)′ and there exists a constant such that

Then problem (1.1) has one nontrivial homoclinic solution.

Theorem 1.3 Suppose that a and W satisfy (A)′ and (W4)-(W7). Then problem (1.1) has an unbounded sequence of homoclinic solutions.

Theorem 1.4 Suppose that a and W satisfy (A)′, (W4), (W5)′, (W6), (W7)′. Then problem (1.1) has an unbounded sequence of homoclinic solutions.

Remark 1.1 When , condition (A)′ reduces to condition (A). Obviously, Theorem 1.1-Theorem 1.4 generalize and improve Theorem A, Theorem B and the corresponding results in [21]. It is easy to see that our results hold true even if . To the best of our knowledge, similar results for problem (1.1) cannot be seen in the literature; from this point, our results are new.

Remark 1.2 If and , then problem (1.1) reduces to problem (1.3). As pointed out in [23], Theorem A can be proved by replacing (A) with the more general assumption: as .

The rest of this paper is organized as follows. In Section 2, some preliminaries are presented and we establish an embedding result. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.

## 2 Preliminaries

We set, for any real number ,
with the usual norms
Let
be the Sobolev space with the norm given by
If σ is a positive, continuous function on and , let
equipped with the norm
is a reflexive Banach space. When , we set
with the norm given by
Set , where a is the function given in condition (A)′. Then E with its standard norm is a reflexive Banach space. The functional φ corresponding to (1.1) on E is given by
(2.1)
Clearly, it follows from (W5) or (W5)′ that . By Theorem 2.1 of [24], we can deduce that the map
is continuous from in the dual space , where . As the embeddings for all are continuous, if (A)′ and (W5) or (W5)′ hold, then and one can easily check that
(2.2)

Furthermore, the critical points of φ in E are classical solutions of (1.1) with .

To prove our results, we need the following generalization of the Lebesgue dominated convergence theorem.

Lemma 2.1 [25]

Let and be two sequences of measurable functions on a measurable set A, and let
If
and
then

The following lemma is an improvement result of [23] in which the author considered the case .

Lemma 2.2 If a satisfies assumption (A)′, then
(2.3)
Moreover, there exists a Sobolev space Z such that
(2.4)
(2.5)
Proof Let , , we have

where from (A)′, . Then (2.3) holds.

By (A)′, there exists a continuous positive function ρ such that as and
Since

(2.4) holds by taking .

Finally, as is the weighted Sobolev space , it follows from [24] that (2.5) holds. □

The following two lemmas are the mountain pass theorem and symmetric mountain pass theorem, which are useful in the proofs of our theorems.

Lemma 2.3 [26]

Let E be a real Banach space and satisfying (PS)-condition. Suppose and:
1. (i)

There exist constants such that .

2. (ii)

There exists an such that .

Then I possesses a critical value which can be characterized as

where and is an open ball in E of radius ρ centered at 0.

Lemma 2.4 [26]

Let E be a real Banach space and with I even. Assume that and I satisfies (PS)-condition, (i) of Lemma  2.3 and the following condition:
1. (iii)

For each finite dimensional subspace , there is such that for , is an open ball in E of radius r centered at 0.

Then I possesses an unbounded sequence of critical values.

Lemma 2.5 Assume that (W6) and (W7) or (W7)′ hold. Then, for every ,
1. (i)

is nondecreasing on ;

2. (ii)

is nonincreasing on .

The proof of Lemma 2.5 is routine and we omit it.

## 3 Proofs of theorems

Proof of Theorem 1.1 Step 1. The functional φ satisfies (PS)-condition. Let satisfying be bounded and as . Hence, there exists a constant such that
(3.1)
It is well known [27] that there exists a constant such that
(3.2)
From (2.1), (2.2), (3.1), (W6) and (W7), we have
It follows from Lemma 2.2, , and the above inequalities that there exists a constant such that
(3.3)
Now we prove that in E. Passing to a subsequence if necessary, it can be assumed that in E. From Lemma 2.2, we have in . From (3.2) and (3.3), we have
(3.4)
Inequality (3.4) implies that for all . By (W5), we know that
which implies that for any given constant , there exists a constant related to C such that
Hence, there exists a constant such that
(3.5)
Hence, from (3.5), we have
(3.6)
where . Moreover, since is a positive continuous function on , and for almost every , we have
and
It follows from Lemma 2.1, (3.6) and the above inequalities that
This shows that
(3.7)
From (2.2), we have
(3.8)
It is easy to see that for any there exists a constant such that
(3.9)
Inequality (3.9) implies that
(3.10)
and
(3.11)
where and are positive constants. Since as , in E and the embeddings for all are continuous, it follows from Lemma 2.2, (3.7), (3.8), (3.10) and (3.11) that
(3.12)
and
(3.13)

Hence, by (3.12) and (3.13), in E. This shows that φ satisfies (PS)-condition.

Step 2. From (W5), there exists such that
(3.14)
By (3.14), we have
(3.15)
Let
(3.16)
Set and , it follows from (3.2) that
which shows that . From Lemma 2.5(i) and (3.16), we have
(3.17)
It follows from (W7), (3.15), (3.17) that

Therefore, we can choose a constant depending on ρ such that for any with .

Step 3. From Lemma 2.5(ii) and (3.2), we have for any
(3.18)
where , . Take such that
(3.19)
and for . For , from Lemma 2.5(i) and (3.19), we get
(3.20)
where . From (W7), (2.1), (3.18), (3.19), (3.20), we get for
(3.21)
Since and , it follows from (3.21) that there exists such that and . Let , then , and . By Lemma 2.3, φ has a critical value given by
(3.22)
where
Hence, there exists such that

The function is a desired solution of problem (1.1). Since , is a nontrivial homoclinic solution. The proof is complete. □

Proof of Theorem 1.2 In the proof of Theorem 1.1, the condition in (W7) is only used in the proofs of (3.3) and Step 2. Therefore, we only need to prove that (3.3) and Step 2 still hold if we use (W5)′ and (W7)′ instead of (W5) and (W7). We first prove that (3.3) holds. From (W6), (W7)′, (2.1), (2.2) and (3.1), we have
which implies that there exists a constant such that (3.3) holds. Next, we prove Step 2 still holds. From (W5)′, there exists such that
(3.23)
By (3.23), we have
(3.24)
Let and , it follows from (3.2) that
which shows that . It follows from (2.1) and (3.24) that

Therefore, we can choose a constant depending on ρ such that for any with . The proof of Theorem 1.2 is complete. □

Proof of Theorem 1.3 Condition (W4) shows that φ is even. In view of the proof of Theorem 1.1, we know that and satisfies (PS)-condition and assumptions (i) of Lemma 2.3. Now, we prove that (iii) of Lemma 2.4. Let be a finite dimensional subspace of E. Since all norms of a finite dimensional space are equivalent, there exists such that
(3.25)
Assume that and is a base of such that
(3.26)
For any , there exists , such that
(3.27)
Let
(3.28)
It is easy to see that is a norm of . Hence, there exists a constant such that . Since , by Lemma 2.2, we can choose such that
(3.29)
where δ is given in (3.24). Let
(3.30)
Hence, for , let such that
(3.31)
Then by (3.25)-(3.28), (3.30) and (3.31), we have
(3.32)
This shows that and there exists such that , which together with (3.29), implies that . Let and
(3.33)
Since for all and , and , it follows that . For any , from Lemma 2.5(i) and (3.2), we have
(3.34)
where , . Since , , it follows that there exists such that
(3.35)
where . Then, for with and , it follows from (3.27), (3.30), (3.31), (3.32) and (3.35) that
(3.36)
On the other hand, since for , then
(3.37)
Therefore, from (3.33), (3.36) and (3.37), we have
(3.38)
From (3.29) and (3.30), we have
(3.39)
By (2.1), (3.15), (3.34), (3.38), (3.39) and Lemma 2.5, we have for and
(3.40)
Since , we deduce that there exists such that
It follows that
which shows that (iii) of Lemma 2.4 holds. By Lemma 2.4, φ possesses an unbounded sequence of critical values with , where is such that for  . If is bounded, then there exists such that
(3.41)
By (3.2) and (3.41), we get
(3.42)
From (W5), we can choose and such that
which implies that
(3.43)
Hence, by (2.1) and (3.43), we have
which, together with (3.41), implies that

This contradicts the fact that is unbounded, and so is unbounded. The proof is complete. □

Proof of Theorem 1.4 In view of the proofs of Theorem 1.2 and Theorem 1.3, the conclusion of Theorem 1.4 holds. The proof is complete. □

## 4 Examples

Example 4.1 Consider the following system:
(4.1)
where , , , , and a satisfies (A)′. Let
where , , , . Let

Then it is easy to check that all the conditions of Theorem 1.3 are satisfied with and . Hence, problem (4.1) has an unbounded sequence of homoclinic solutions.

Example 4.2 Consider the following system:
(4.2)
where , , , , and a satisfies (A)′. Let
where , , . Let

Then it is easy to check that all the conditions of Theorem 1.4 are satisfied with and . Hence, by Theorem 1.4, problem (4.2) has an unbounded sequence of homoclinic solutions.

## Declarations

### Acknowledgements

XS and QZ are supported by the Scientific Research Foundation of Guangxi Education Office (No. 201203YB093), Guangxi Natural Science Foundation (Nos. 2013GXNSFBA019004 and 2012GXNSFBA053013) and the Scientific Research Foundation of Guilin University of Technology. QMZ is supported by the NNSF of China (No. 11201138) and the Scientific Research Fund of Hunan Provincial Education Department (No. 12B034).

## Authors’ Affiliations

(1)
College of Science, Guilin University of Technology
(2)
College of Science, Hunan University of Technology

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