Abstract
By applying the mountain pass theorem and the symmetric mountain pass theorem in critical point theory, the existence and multiplicity of fast homoclinic solutions are obtained for the following secondorder nonautonomous problem: , where , , , , are not periodic in t and is a continuous function and with .
MSC: 34C37, 35A15, 37J45, 47J30.
Keywords:
fast homoclinic solutions; variational methods; critical point1 Introduction
Consider fast homoclinic solutions of the following problem:
where , , , , are not periodic in t, and is a continuous function and with
When , problem (1.1) reduces to the following special secondorder Hamiltonian system:
When , problem (1.1) reduces to the following secondorder damped vibration problem:
If we take and , then problem (1.1) reduces to the following secondorder Hamiltonian system:
The existence of homoclinic orbits plays an important role in the study of the behavior of dynamical systems. If a system has transversely intersected homoclinic orbits, then it must be chaotic. If it has smoothly connected homoclinic orbits, then it cannot stand the perturbation, and its perturbed system probably produces chaotic phenomena. The first work about homoclinic orbits was done by Poincaré [1].
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 [230] and references therein. In [6,16,17], the authors considered homoclinic solutions for the special Hamiltonian system (1.3) in weighted Sobolev space. Later, Shi et al.[31] obtained some results for system (1.3) with a pLaplacian, which improved and generalized the results in [6,16,17]. However, there is little research as regards the existence of homoclinic solutions for damped vibration problems (1.4) when . In 2008, Wu and Zhou [32] obtained some results for damped vibration problems (1.4) with some boundary value conditions by variational methods. Zhang and Yuan [33,34] studied the existence of homoclinic solutions for (1.4) when is a constant. Later, Chen et al.[35] investigated fast homoclinic solutions for (1.4) and obtained some new results under more relaxed assumptions on , which resolved some open problems in [33]. Zhang [36] obtained infinitely many solutions for a class of general secondorder damped vibration systems by using the variational methods. Zhang [37] investigated subharmonic solutions for a class of secondorder impulsive systems with damped term by using the mountain pass theorem.
Motivated by [21,23,3234,3842], we will establish some new results for (1.1) in weighted Sobolev space. In order to introduce the concept of fast homoclinic solutions for problem (1.1), we first state some properties of the weighted Sobolev space E on which the certain variational functional associated with (1.1) is defined and the fast homoclinic solutions are the critical points of the certain functional.
Let
where is defined in (1.2) and for , let
Then X is a Hilbert space with the norm given by
It is obvious that
with the embedding being continuous. Here () denotes the Banach spaces of functions on ℝ with values in under the norm
If σ is a positive, continuous function on ℝ and , let
is a reflexive Banach space.
Set , where a is the function given in condition (A). Then E with its standard norm is a reflexive Banach space. Similar to [33,35], we have the following definition of fast homoclinic solutions.
Definition 1.1 If (1.2) holds, a solution of (1.1) is called a fast homoclinic solution if .
The functional φ corresponding to (1.1) on E is given by
Clearly, it follows from (W1) or (W1)′ that . By Theorem 2.1 of [43], we can deduce that the map
is continuous from in the dual space , where . As the embeddings for all are continuous, if (A) and (W1) or (W1)′ hold, then and one can easily check that
Furthermore, the critical points of φ in E are classical solutions of (1.1) with .
Now, we state our main results.
Theorem 1.1Suppose thata, q, andWsatisfy (1.2) and the following conditions:
(A) Let, is a continuous, positive function on ℝ such that for all
(W1) , , and there exists a constantsuch that
(W2) There is a constantsuch that
(W3) and there exists a constantsuch that
Then problem (1.1) has at least one nontrivial fast homoclinic solution.
Theorem 1.2Suppose thata, q, andWsatisfy (1.2), (A), (W2), and the following conditions:
(W3)′ and there exists a constantsuch that
Then problem (1.1) has at least one nontrivial fast homoclinic solution.
Theorem 1.3Suppose thata, q, andWsatisfy (1.2), (A), (W1)(W3), and the following assumption:
Then problem (1.1) has an unbounded sequence of fast homoclinic solutions.
Theorem 1.4Suppose thata, q, andWsatisfy (1.2), (A), (W1)′, (W2), (W3)′, and (W4). Then problem (1.1) has an unbounded sequence of fast homoclinic solutions.
Remark 1.1 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. As pointed out in [17], condition (A) can be replaced by more general assumption: as .
The rest of this paper is organized as follows: in Section 2, some preliminaries are presented. In Section 3, we give the proofs of our results. In Section 4, some examples are given to illustrate our results.
2 Preliminaries
Let E and be given in Section 1, by a similar argument in [41], we have the following important lemma.
and
The following lemma is an improvement result of [16].
Lemma 2.2Ifasatisfies assumption (A), then
Moreover, there exists a Hilbert spaceZsuch that
where from (A) and (1.2), . Then (2.4) 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 [43] that (2.6) holds. □
The following two lemmas are the mountain pass theorem and the symmetric mountain pass theorem, which are useful in the proofs of our theorems.
Lemma 2.3[44]
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[44]
LetEbe a real Banach space andwithIeven. Assume thatandIsatisfies (PS)condition, assumption (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 (W2) and (W3) or (W3)′ 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 the (PS)condition. Let satisfying is bounded and as . Then there exists a constant such that
From (1.6), (1.7), (3.1), (W2), and (W3), 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. Since as , we can choose such that
It follows from (2.2), (3.2), and (3.3) that
Similarly, by (2.3), (3.2), and (3.3), we have
Since in E, it is easy to verify that converges to pointwise for all . Hence, it follows from (3.4) and (3.5) that
Since on , the operator defined by is a linear continuous map. So in . The Sobolev theorem implies that uniformly on J, so there is such that
For any given number , by (W1), we can choose such that
From (3.8), we have
Moreover, since is a positive continuous function on ℝ and converges to pointwise for all , it follows from (3.9) that
and
From Lebesgue’s dominated convergence theorem, (3.4), (3.5), (3.6), (3.9), and the above inequalities, we have
From Lemma 2.2, we have in . Hence, by (3.10),
tends to 0 as , which together with (3.7) shows that
From (1.7), we have
where is a positive constant. It follows from (3.11) and (3.12) that
and
Hence, in E by (3.13) and (3.14). This shows that φ satisfies (PS)condition.
Step 2. From (W1), there exists such that
Let
Set and , it follows from Lemma 2.1 that for . From Lemma 2.5(i) and (3.17), we have
By (W3), (3.16), and (3.18), we have
Therefore, we can choose a constant depending on ρ such that for any with .
Step 3. From Lemma 2.5(ii) and (2.1), we have for any
and for . For , from Lemma 2.5(i) and (3.20), we get
where . From (W3), (1.6), (3.19), (3.20), and (3.21), we get for
Since and , it follows from (3.22) that there exists such that and . Set , then , , and . It is easy to see that . By Lemma 2.3, φ has a critical value given by
where
The function is the desired solution of problem (1.1). Since , is a nontrivial fast homoclinic solution. The proof is complete. □
Proof of Theorem 1.2 In the proof of Theorem 1.1, the condition in (W3) is only used in the proofs of (3.2) and Step 2. Therefore, we only need to prove that (3.2) and Step 2 still hold if we use (W1)′ and (W3)′ instead of (W1) and (W3). We first prove that (3.2) holds. From (W2), (W3)′, (1.6), (1.7), and (3.1), we have
which implies that there exists a constant such that (3.2) holds. Next, we prove that Step 2 still holds. From (W1)′, there exists such that
Let , it follows from Lemma 2.1 that . It follows from (1.6) and (3.25) 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 assumption (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 basis 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 2.1, we can choose such that
where δ is given in (3.25). Let
Then by (3.26)(3.29), (3.31), and (3.32), we have
This shows that and there exists such that , which together with (3.30), implies that . Let and
Since for all and , and , it follows that . For any , from Lemma 2.1 and Lemma 2.5(i), we have
where , . Since , , it follows that there exists such that
Then for with and , it follows from (3.28), (3.31), (3.32), (3.33), and (3.36) that
On the other hand, since for , then
Therefore, from (3.34), (3.37), and (3.38), we have
By (3.30) and (3.31), we have
By (1.6), (3.16), (3.35), (3.39), (3.40), 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
In a similar fashion to the proof of (3.4) and (3.5), for the given δ in (3.16), there exists such that
Hence, by (1.6), (2.1), (3.16), (3.42), and (3.43), we have
which, together with (3.42), 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 fast 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 fast homoclinic solutions.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
Qiongfen Zhang was supported by the NNSF of China (No. 11301108), Guangxi Natural Science Foundation (No. 2013GXNSFBA019004) and the Scientific Research Foundation of Guangxi Education Office (No. 201203YB093). QiMing Zhang was supported by the NNSF of China (No. 11201138). Xianhua Tang was supported by the NNSF of China (No. 11171351).
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