Infinitely many homoclinic solutions for perturbed second-order Hamiltonian systems with subquadratic potentials

Abstract

In this paper, we consider the following perturbed second-order Hamiltonian system $$ -\ddot{u}(t)+L(t)u=\nabla W(t,u(t))+\nabla G(t,u(t)), \qquad \forall \ t\in \mathbb{R}, $$ where $W(t,u)$ is subquadratic near origin with respect to $u$; the perturbation term $G(t,u)$ is only locally defined near the origin and may not be even in $u$. By using the variant Rabinowitz's perturbation method, we establish a new criterion for guaranteeing that this perturbed second-order Hamiltonian system has infinitely many homoclinic solutions under broken symmetry situations. Our result improves some related results in the literature.

Keywords

• broken symmetry
• hamiltonian system
• homoclinic solutions
• subquadratic potential
• rabinowitz's perturbation method