Even homoclinic orbits for super quadratic Hamiltonian systems

We study the existence of even homoclinic orbits for the second‐order Hamiltonian system ü + V u ( t, u )=0. Let V ( t, u )=− K ( t, u )+ W ( t, u )∈ C 1 (ℝ × ℝ n , ℝ), where K is less quadratic and W is super quadratic in u at infinity. Since the system we considered is neither autonomous nor periodic, the ( PS ) condition is difficult to check when we use the Mountain Pass theorem. Therefore, we approximate the homoclinic orbits by virtue of the solutions of a sequence of nil‐boundary‐value problems. Copyright © 2010 John Wiley & Sons, Ltd.