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)L(t) may not be uniformly positive definite for all t∈Rt∈R. Under more general conditions on Ru(t,u)Ru(t,u), we prove that the above system has infinitely many homoclinic orbits provided R is even in u.