We study the second order Hamiltonian system , where and denotes the gradient with respect to of a smooth potential , -periodic in time, having an unstable equilibrium point for all . Without loss of generality we can take and . Thus, is a trivial solution. We look for homoclinic orbits to 0, namely non-zero solutions of the problem
The potential has the form , where and satisfy some technical assumptions. We prove that the problem (P) admits infinitely many solutions.