Existence of homoclinic orbits for second order Hamiltonian systems without (AR) condition

The existence of homoclinic orbits is obtained for a class of the second order Hamiltonian systems ü(t)−L(t)u(t)+∇W(t,u(t))=0,∀t∈R, by the local linking lemma, where W(t,x)W(t,x) is superquadratic and need not be nonnegative globally.