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:
MSC: 34C37, 35A15, 37J45, 47J30.
Keywords:
fast homoclinic solutions; variational methods; critical point1 Introduction
Consider fast homoclinic solutions of the following problem:
where
When
When
If we take
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
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
Then X is a Hilbert space with the norm given by
It is obvious that
with the embedding being continuous. Here
If σ is a positive, continuous function on ℝ and
is a reflexive Banach space.
Set
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
is continuous from
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
(W1)
uniformly in
(W2) There is a constant
(W3)
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:
(W1)′
uniformly in
(W3)′
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:
(W4)
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
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
Lemma 2.1For any
and
where
The following lemma is an improvement result of [16].
Lemma 2.2Ifasatisfies assumption (A), then
Moreover, there exists a Hilbert spaceZsuch that
Proof Let
where from (A) and (1.2),
By (A), there exists a continuous positive function ρ such that
Since
(2.5) holds by taking
Finally, as
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 and
(i) There exist constants
(ii) There exists an
ThenIpossesses a critical value
where
Lemma 2.4[44]
LetEbe a real Banach space and
(iii) For each finite dimensional subspace
ThenIpossesses an unbounded sequence of critical values.
Lemma 2.5Assume that (W2) and (W3) or (W3)′ hold. Then for every
(i)
(ii)
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
From (1.6), (1.7), (3.1), (W2), and (W3), we have
It follows from Lemma 2.2,
Now we prove that
It follows from (2.2), (3.2), and (3.3) that
Similarly, by (2.3), (3.2), and (3.3), we have
Since
Since
For any given number
From (3.8), we have
Moreover, since
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
tends to 0 as
From (1.7), we have
where
and
Hence,
Step 2. From (W1), there exists
By (3.15) and
Let
Set
By (W3), (3.16), and (3.18), we have
Therefore, we can choose a constant
Step 3. From Lemma 2.5(ii) and (2.1), we have for any
where
and
where
Since
where
Hence, there exists
The function
Proof of Theorem 1.2 In the proof of Theorem 1.1, the condition
which implies that there exists a constant
By (3.24) and
Let
Therefore, we can choose a constant
Proof of Theorem 1.3 Condition (W4) shows that φ is even. In view of the proof of Theorem 1.1, we know that
Assume that
For any
Let
It is easy to see that
where δ is given in (3.25). Let
Hence, for
Then by (3.26)(3.29), (3.31), and (3.32), we have
This shows that
Since
where
Then for
On the other hand, since
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
Since
It follows that
which shows that (iii) of Lemma 2.4 holds. By Lemma 2.4, φ possesses an unbounded sequence
In a similar fashion to the proof of (3.4) and (3.5), for the given δ in (3.16), there exists
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
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
where
Then it is easy to check that all the conditions of Theorem 1.3 are satisfied with
Example 4.2 Consider the following system:
where
where
Then it is easy to check that all the conditions of Theorem 1.4 are satisfied with
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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