Infinitely many homoclinic orbits for Hamiltonian systems with indefinite sign subquadratic potentials

In this paper, we deal with the existence and multiplicity of homoclinic solutions of the second-order Hamiltonian system ü(t)−L(t)u(t)+∇W(t,u(t))=0, where L(t)L(t) and W(t,x)W(t,x) are neither autonomous nor periodic in tt. Under the assumption that W(t,x)W(t,x) is indefinite sign and subquadratic as |x|→+∞|x|→+∞ and L(t)L(t) is a N×NN×N real symmetric positive definite matrices for all t∈Rt∈R, we establish some existence criteria to guarantee that the above system has at least one or infinitely many homoclinic solutions by using the genus properties in critical theory.