Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 29, pp. 1-10.

Title: Homoclinic solutions for second-order non-autonomous
       Hamiltonian systems without global Ambrosetti-Rabinowitz
       conditions
       
Authors: Rong Yuan    (Beijing Normal Univ., Beijing, China)
         Ziheng Zhang (Beijing Normal Univ., Beijing, China)

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.