Infinitely many homoclinic solutions for a second-order Hamiltonian system with locally defined potentials

In this paper, we study homoclinic solutions of the following second-order Hamiltonian system u¨(t)−L(t)u(t)+∇W(t,u(t))=0,where t∈R,u∈NR,t∈R,u∈RN,L:R→RN×NL:R→RN×N and W:R×NR→RW:R×RN→R. Applying a new symmetric Mountain Pass Theorem established by Kajikiya, we prove the existence of infinitely many homoclinic solutions for the above system in the case where L(t  ) is coercive but unnecessarily positive definite for all t∈R,t∈R, and W(t, x) is only locally defined near the origin with respect to x. Our results significantly generalize and improve related ones in the literature.