Electron. J. Diff. Equ., Vol. 2010(2010), No. 29, pp. 1-10.

Homoclinic solutions for second-order non-autonomous Hamiltonian systems without global Ambrosetti-Rabinowitz conditions

Rong Yuan, Ziheng Zhang

Abstract:
This article studies the existence of homoclinic solutions for the second-order non-autonomous Hamiltonian system
$$ \ddot q-L(t)q+W_{q}(t,q)=0, $$
where $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$. The function $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$ is not assumed to satisfy the global Ambrosetti-Rabinowitz condition. Assuming reasonable conditions on $L$ and $W$, we prove the existence of at least one nontrivial homoclinic solution, and for $W(t,q)$ even in $q$, we prove the existence of infinitely many homoclinic solutions.

Submitted January 14, 2010. Published February 25, 2010.
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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