Multiple homoclinic orbits for a class of Hamiltonian systems

Abstract.

In this paper, we obtain the existence of at least two nontrivial homoclinic orbits for a class of second order autonomous Hamiltonian systems. This multiplicity result is obtained by a new variational method based on the relative category: to overcome the lack of compactness of the problem, we first solve perturbed nonautonomous problems and study the limit of the solutions as the nonautonomous perturbation goes to 0. This method allows to get rid of some assumptions on the potential used in the work of Ambrosetti and Coti-Zelati.