Homoclinic solutions for a second-order Hamiltonian system with a positive semi-definite matrix

In this paper, we study homoclinic solutions for second-order Hamiltonian systems u¨-L(t)u+Wu(t,u)=0, where L(t)L(t) is allowed to be a positive semi-definite symmetric matrix for all t∈Rt∈R, and W∈C1(R×RN,R) is an indefinite potential satisfying asymptotically quadratic condition at infinity on u. We obtain some new results on the existence and multiplicity of homoclinic solutions for second-order systems. The proof is based on variational methods.