Homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign nonlinearities

In this paper, we obtain the multiplicity of homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign potentials. The concentration-compactness principle is applied to show the compactness. As a byprod-uct, we obtain the uniqueness of the positive ground state solution for a class of au-tonomous Hamiltonian systems and the best constant for Sobolev inequality which are of independent interests.