Electron. J. Diff. Equ., Vol. 2009(2009), No. 128, pp. 1-9.

Homoclinic solutions for a class of second order non-autonomous systems

Rong Yuan, Ziheng Zhang

Abstract:
This article concerns the existence of homoclinic solutions for the second order non-autonomous system
$$ \ddot q+A \dot q-L(t)q+W_{q}(t,q)=0, $$
where $A$ is a skew-symmetric constant matrix, $L(t)$ is a symmetric positive definite matrix depending continuously on $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$. We assume that $W(t,q)$ satisfies the global Ambrosetti-Rabinowitz condition, that the norm of A is sufficiently small and that $L$ and $W$ satisfy additional hypotheses. We prove the existence of at least one nontrivial homoclinic solution, and the existence of infinitely many homoclinic solutions if $W(t,q)$ is even in $q$. Recent results in the literature are generalized and improved.

Submitted October 14, 2008. Published October 7, 2009.
Math Subject Classifications: 34C37, 35A15, 37J45.
Key Words: Homoclinic solutions; critical point; variational methods; mountain pass theorem.

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Rong Yuan
School of Mathematical Sciences, Beijing Normal University
Laboratory of Mathematics and Complex Systems, Ministry of Education
Beijing 100875, China
email: ryuan@bnu.edu.cn
Ziheng Zhang
School of Mathematical Sciences, Beijing Normal University
Laboratory of Mathematics and Complex Systems, Ministry of Education
Beijing 100875, China
email: zhzh@mail.bnu.edu.cn

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