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

• Main Authors: Ziheng Zhang, Rong Yuan
• Format: Article
• Language: English
• Published: Texas State University 2009-10-01
• Series: Electronic Journal of Differential Equations
• Subjects: Homoclinic solutions; critical point; variational methods; mountain pass theorem
• Online Access: http://ejde.math.txstate.edu/Volumes/2009/128/abstr.html

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 $tin mathbb{R}$, $Win C^{1}(mathbb{R}imesmathbb{R}^{n},mathbb{R})$. 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.