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.