Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials

We study the existence of infinitely many homoclinic orbits for some second-order Hamiltonian systems: ü−L(t)u(t)+∇F(t,u(t))=0,∀t∈R, by the variant fountain Theorem, where F(t,u)F(t,u) satisfies the super-quadratic condition F(t,u)/|u|2→∞F(t,u)/|u|2→∞ as |u|→∞|u|→∞ uniformly in t, and need not satisfy the global Ambrosetti–Rabinowitz condition.