Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials

In this paper we study the following nonperiodic second order Hamiltonian systems − u ¨ ( t ) + L ( t ) u ( t ) = ∇ R ( t , u ) , ∀ t ∈ R , where L ( t ) may not be uniformly positive definite for all t ∈ R . Under more general conditions on R u ( t , u ) , we prove that the above system has infinitely many homoclinic orbits provided R is even in u.