New existence and multiplicity results of homoclinic orbits for a class of second order Hamiltonian systems

In this paper, we study the nonperiodic second order Hamiltonian systemsu¨(t)-λL(t)u(t)+∇W(t,u(t))=0,∀t∈R,where λ⩾1λ⩾1 is a parameter, the matrix L(t)L(t) is not necessarily positive definite for all t∈Rt∈R nor coercive. Replacing the Ambrosetti–Rabinowitz condition by general superquadratic assumptions, we establish the existence and multiplicity results for the above system when λ>1λ>1 large. We also consider the situation where W is a combination of subquadratic and superquadratic terms, and obtain infinitely many homoclinic solutions.