Homoclinic orbits for first-order Hamiltonian system with local super-quadratic growth condition

Abstract

In this paper we study the first-order Hamiltonian system Moreover precisely, assuming that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition, we obtain new existence results of ground state homoclinic orbits and infinitely many geometrically distinct homoclinic orbits by using a variational method. An interesting problem is that the nonlinearity may be super-quadratic on some domains and asymptotically quadratic on other domains under local super-quadratic condition. Since we are without more global information on the nonlinearity, we apply a perturbation approach and some special techniques in the proofs.