Local super-quadratic conditions on homoclinic solutions for a second-order Hamiltonian system

In this paper, we study the homoclinic solutions of the following second-order Hamiltonian system ü−L(t)u+∇W(t,u)=0,where t∈R,u∈RN, L:R→RN×N and W:R×RN→R. Applying the Mountain Pass Theorem, we prove the existence of nontrivial homoclinic solutions under new weaker conditions. In particular, we use a local super-quadratic condition lim|x|→∞W(t,x)|x|2=∞, uniformly in t∈(a,b) for some −∞