 Research
 Open Access
Fast homoclinic solutions for damped vibration problems with superquadratic potentials
 Xinhe Zhu^{1}Email author and
 Ziheng Zhang^{1}
 Received: 19 March 2018
 Accepted: 29 November 2018
 Published: 5 December 2018
Abstract
Keywords
 Homoclinic solutions
 Critical point
 Variational methods
 Mountain pass theorem
MSC
 34C37
 35A15
 35B38
1 Introduction
 (L):

\(L\in C(\mathbb{R}, \mathbb{R}^{n^{2}})\) is a symmetric and positive definite matrix for all \(t\in\mathbb{R}\), and there is a continuous function \(\beta:\mathbb{R}\rightarrow\mathbb{R}\) such that \(\beta(t)>0\) for all \(t\in\mathbb{R}\) and \((L(t)u,u)\geq\beta(t)u^{2}\) and \(\beta(t)\rightarrow+\infty\) as \(t\rightarrow+\infty\),
Compared with the literature available for \(W(t,u)\) being superquadratic as \(u\rightarrow+\infty\), the study of the existence of homoclinic solutions of (HS) under the assumption that \(W(t,u)\) is subquadratic at infinity is much more recent and the number of references is considerably smaller, see for instance [8, 20, 21, 28, 29], where some other types of coercive conditions on L are also utilized. In addition, the existence of homoclinic solutions under the condition that \(W(t,u)\) is asymptotically quadratic at infinity has also been investigated by many researchers, see for instance [9, 24, 30, 32].
Throughout this paper, we adopt the definition of [4] for fast homoclinic solution of (DS):
Definition 1.1
If (1.1) holds, a solution u of (DS) is called a fast homoclinic solution if \(u\in E\).
 (\(\mathrm{W}_{1}\)):

There is a constant \(\mu>2\) such that, for every \(t\in \mathbb{R}\) and \(u\in\mathbb{R}^{n}\backslash \{0 \}\),$$0< \mu W(t,u)\leq\bigl(W_{u}(t,u),u\bigr); $$
 (\(\mathrm{W}_{2}\)):

\(W_{u}(t,u)=o(u)\) as \(u\rightarrow0\) uniformly with respect to \(t\in\mathbb{R}\);
 (\(\mathrm{W}_{3}\)):

There exists \(\overline{W}\in C(\mathbb{R}^{n}, \mathbb {R})\) such that \(W_{u}(t,u)\leq\overline{W}(u)\) for every \(t\in\mathbb{R}\) and \(u\in\mathbb{R}^{n}\).
Theorem 1.2
 (\(\mathrm{W}_{4}\)):

\(W(t,u)=W(t,u)\) for all \(t\in\mathbb{R}\) and \(u\in\mathbb{R}^{n}\),
Remark 1.3
In (DS), if \(q(t)\equiv0\), then Theorem 1.2 (under the same hypothesis on L and \(W(t,u)\)) reaches the results in [15](see its Theorem 1 and Theorem 2). Therefore, we extend the results of [15] for (HS) to the more general situations (DS).
It is worth pointing out that an open problem was proposed in [31], explicitly, how to obtain the existence of fast homoclinic solutions of (1.3) for the case that W satisfies the (AR) condition using the mountain pass theorem. Here, Theorem 1.2 gives some partial answer to this open problem. For some recently related results, we refer the reader to [5, 27].
Remark 1.4
The remaining part of this paper is organized as follows. Some preliminary results are presented in Sect. 2. In Sect. 3, we are devoted to accomplishing the proof of our main result.
2 Preliminary results
Lemma 2.1
Suppose that (1.1) holds and L satisfies (L), then the embedding of E in \(L^{2}(e^{Q(t)})\) is compact.
Proof
Lemma 2.2
([4, Lemma 2.1])
Now we introduce more notations and some necessary definitions. Let \(\mathcal{B}\) be a real Banach space, \(I\in C^{1}(\mathcal{B}, \mathbb {R})\), which means that I is a continuously Fréchetdifferentiable functional defined on \(\mathcal{B}\). Recall that \(I\in C^{1}(\mathcal{B}, \mathbb{R})\) is said to satisfy the (PS) condition if any sequence \(\{u_{j} \}_{j\in\mathbb{N}}\subset\mathcal{B}\), for which \(\{I(u_{j}) \}_{j\in\mathbb{N}}\) is bounded and \(I'(u_{j})\rightarrow0\) as \(j\rightarrow+\infty\), possesses a convergent subsequence in \(\mathcal{B}\).
Moreover, let \(B_{\rho}\) be the open ball in \(\mathcal{B}\) with the radius ρ and centered at 0 and \(\partial B_{\rho}\) denote its boundary. To obtain the existence and multiplicity of fast homoclinic solutions of (DS), we appeal to the following wellknown mountain pass theorem, see [17].
Lemma 2.3
([17, Theorem 2.2 ])
 (A1)
there are constants ρ, \(\alpha>0\) such that \(I_{\partial B_{\rho}}\geq\alpha\), and
 (A2)
there is \(e\in\mathcal{B}\setminus\overline {B}_{\rho}\) such that \(I(e)\leq0\).
Lemma 2.4
([17, Theorem 9.12])
 (A3):

there are constants ρ, \(\alpha>0\) such that \(I_{\partial B_{\rho}\cap X}\geq\alpha\), and
 (A4):

for each finite dimensional subspace \(\tilde {E}\subset\mathcal{B}\), there is \(R=R(\tilde{E})\) such that \(I\leq 0\) on \(\tilde{E}\backslash B_{R(\tilde{E})}\),
3 Proof of Theorem 1.2
Lemma 3.1
Proof
Lemma 3.2
Suppose that (L), (\(\mathrm{W}_{2}\)), and (\(\mathrm{W}_{3}\)) are satisfied. If \(u_{j}\rightharpoonup u\) (weakly) in E, then there exists one subsequence still denoted by \(\{u_{j}\}_{j\in\mathbb{N}}\) such that \(W_{u}(t,u_{j})\rightarrow W_{u}(t,u)\) in \(L^{2}(e^{Q(t)})\).
Proof
Lemma 3.3
If (L), (\(\mathrm{W}_{1}\)), (\(\mathrm{W}_{2}\)), and (\(\mathrm{W}_{3}\)) hold, then I satisfies the (PS) condition.
Proof
Now we are in a position to give the proof of Theorem 1.2. We divide the proof into several steps.
Proof of Theorem 1.2
Step 1 It is clear that \(I(0)=0\) and \(I\in C^{1}(E,\mathbb{R})\) satisfies the (PS) condition by Lemmas 3.1 and 3.3.
Step 4 Now suppose that \(W(t,u)\) is even in u, i.e., (\(\mathrm{W}_{4}\)) holds, which implies that I is even. Furthermore, we have already known that \(I(0)=0\) and \(I\in C^{1}(E,\mathbb{R})\) satisfies the (PS) condition in Step 1.
Declarations
Acknowledgements
The authors would like to thank the referees for carefully reading the manuscript, giving valuable comments and suggestions to improve the results as well as the exposition of the paper.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Funding
The authors are supported financially by the National Natural Science Foundation of China (11771044).
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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