Abstract
Keywords:
homoclinic solutions; variational methods; weighted space; pLaplacian systems1 Introduction
Consider homoclinic solutions of the following pLaplacian system:
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 secondorder Hamiltonian system:
If we take and , then (1.2) reduces to the following secondorder Hamiltonian system:
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 thataandWsatisfy the following conditions:
(A) Let, is a continuous, positive function on ℝ such that for all
(W1) and there exists a constantsuch that
(W2) asuniformly with respect to.
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 thataandWsatisfy (A), (W1)(W3) and the following condition:
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 1.1Suppose thataandWsatisfy the following conditions:
(A)′ Letand, is a continuous, positive function on ℝ such that for all
(W5) , , and there exists a constantsuch that
(W6) There is a constantsuch that
(W7) and there exists a constantsuch that
Then problem (1.1) has one nontrivial homoclinic solution.
Theorem 1.2Suppose thataandWsatisfy (A)′, (W6) and the following conditions:
(W7)′ and there exists a constantsuch that
Then problem (1.1) has one nontrivial homoclinic solution.
Theorem 1.3Suppose thataandWsatisfy (A)′ and (W4)(W7). Then problem (1.1) has an unbounded sequence of homoclinic solutions.
Theorem 1.4Suppose thataandWsatisfy (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 1.1Theorem 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
with the usual norms
Let
be the Sobolev space with the norm given by
If σ is a positive, continuous function on ℝ and , let
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
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
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]
Letandbe two sequences of measurable functions on a measurable setA, and let
If
and
then
The following lemma is an improvement result of [23] in which the author considered the case .
Lemma 2.2Ifasatisfies assumption (A)′, then
Moreover, there exists a Sobolev spaceZsuch that
where from (A)′, . Then (2.3) holds.
By (A)′, there exists a continuous positive function ρ such that as and
Since
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]
LetEbe a real Banach space andsatisfying (PS)condition. Supposeand:
(i) There exist constantssuch that.
(ii) There exists ansuch that.
ThenIpossesses a critical valuewhich can be characterized as
whereandis an open ball inEof radiusρcentered at 0.
Lemma 2.4[26]
LetEbe a real Banach space andwithIeven. Assume thatandIsatisfies (PS)condition, (i) of Lemma 2.3 and the following condition:
(iii) For each finite dimensional subspace, there issuch thatfor, is an open ball inEof radiusrcentered at 0.
ThenIpossesses an unbounded sequence of critical values.
Lemma 2.5Assume that (W6) and (W7) or (W7)′ hold. Then, for every,
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
It is well known [27] that there exists a constant such that
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
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
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
Hence, from (3.5), we have
where . Moreover, since is a positive continuous function on ℝ, and for almost every , we have
and
It follows from Lemma 2.1, (3.6) and the above inequalities that
This shows that
From (2.2), we have
It is easy to see that for any there exists a constant such that
Inequality (3.9) implies that
and
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
and
Hence, by (3.12) and (3.13), in E. This shows that φ satisfies (PS)condition.
Step 2. From (W5), there exists such that
By (3.14), we have
Let
Set and , it follows from (3.2) that
which shows that . From Lemma 2.5(i) and (3.16), we have
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
and for . For , from Lemma 2.5(i) and (3.19), we get
where . From (W7), (2.1), (3.18), (3.19), (3.20), we get for
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
where
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
By (3.23), we have
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
Assume that and is a base of such that
For any , there exists , such that
Let
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
where δ is given in (3.24). Let
Then by (3.25)(3.28), (3.30) and (3.31), we have
This shows that and there exists such that , which together with (3.29), implies that . Let and
Since for all and , and , it follows that . For any , from Lemma 2.5(i) and (3.2), we have
where , . Since , , it follows that there exists such that
where . Then, for with and , it follows from (3.27), (3.30), (3.31), (3.32) and (3.35) that
On the other hand, since for , then
Therefore, from (3.33), (3.36) and (3.37), we have
From (3.29) and (3.30), we have
By (2.1), (3.15), (3.34), (3.38), (3.39) and Lemma 2.5, we have for and
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
By (3.2) and (3.41), we get
From (W5), we can choose and such that
which implies that
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:
where , , , , and a satisfies (A)′. 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:
where , , , , and a satisfies (A)′. 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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
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).
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