Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 128, pp. 1-9.
Title: Homoclinic solutions for a class of second order
non-autonomous systems
Authors: Rong Yuan (Beijing Normal Univ., Beijing, China)
Ziheng Zhang (Beijing Normal Univ., Beijing, China)
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 07, 2009.
Math Subject Classifications: 34C37, 35A15, 37J45.
Key Words: Homoclinic solutions; critical point; variational methods;
mountain pass theorem.