Homoclinic solutions for a class of second order Hamiltonian systems with locally defined potentials

In this paper, we study the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems ü−L(t)u+Wu(t,u)=0, ∀t∈R, where L(t)L(t) is unnecessarily positive definite for all t∈Rt∈R, and W(t,u)W(t,u) is only locally defined near the origin with respect to uu and not assumed to be periodic with respect to tt.