Multiple Homoclinics for a Class of Singular Hamiltonian Systems

We consider a Hamiltonian system u¨q=V u.s0 where the potential V: RN _ SªR has a unique strict global maximum at a point pgRN and a singular set S;RN _ p4 such that RN _ S is open, path-connected and the fundamental group Gsp1 RN _ S. is nontrivial. Under some compactness conditions on V at infinity and around the singular set S we study the existence of homoclinic orbits to p which link with S. When V and G satisfy suitable geometrical conditions, we can prove the existence of multiple homoclinics, each one belonging to a different homotopy class of G.