Homoclinic solutions for a class of second order Hamiltonian systems

Abstract

In this paper, we study homoclinic solutions for the nonperiodic second order Hamiltonian systems ü - L(t)u + Wu(t,u) = 0, ∀ t ∈ R, where L is unnecessarily coercive or uniformly positively definite, and W(t,u) is only locally defined near the origin with respect to u. Under some general conditions on L and W, we show that the above system has infinitely many homoclinic solutions near the origin. Some related results in the literature are extended and generalized.