In this paper, we study homoclinic solutions for the nonperiodic second order Hamiltonian systems u - L(t)u +W-u(t,u) =3D 0, for all t is an element of 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.
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