Homoclinic orbits for the second-order Hamiltonian systems

We deal with the following second-order Hamiltonian systems ü−L(t)u+∇W(t,u)=0, where L∈C(R,RN2) is a symmetric and positive define matrix for all t∈R, W∈C1(R×RN,R) and ∇W(t,u) is the gradient of W with respect to u. Under the superquadratic condition, we obtain the existence of ground state homoclinic orbits by means of the generalized Nehari manifold developed by Szulkin and Weth. Under the subquadratic condition, we employ variational techniques and the concentration-compactness principle to establish new criteria guaranteeing the multiplicity of classical homoclinic orbits. Recent results in literature are generalized and significantly improved.