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

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 $Lin C(mathbb{R},mathbb{R}^{n^2})$ is a symmetric and positive definite matrix for all $tin mathbb{R}$. The function $Win C^{1}(mathbb{R}imesmathbb{R}^{n},mathbb{R})$ 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.

Keywords

• Homoclinic solutions
• critical point
• variational methods
• mountain pass theorem