Existence of homoclinic solution for the second order Hamiltonian systems ✩

An existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order Hamiltonian systems ¨u(t )− L(t)u(t)+∇W(t,u(t)) = 0, ∀t ∈ R, by the minimax methods in the critical point theory, specially, the generalized mountain pass theorem, where L(t) is unnecessary uniformly positively definite for all t ∈ R, and W(t,x) satisfies the superquadratic condition W(t,x)/|x|2 →+∞ as |x| →∞ uniformly in t , and need not satisfy the global Ambrosetti– Rabinowitz condition.  2003 Elsevier Inc. All rights reserved.