Homoclinic orbits of first order nonlinear Hamiltonian systems with asymptotically linear nonlinearities at infinity

Abstract

By using variational methods and critical point theory, in
particular, a generalized weak linking theorem, we study a first
order nonlinear Hamiltonian system with asymptotically linear
nonlinearity at infinity. We obtain the existence of ground state
homoclinic orbits for this nonlinear Hamiltonian system. In
particular, we obtain a {\it necessary and sufficient condition}
for the existence of ground state homoclinic orbits. To the best
of our knowledge, there is no published result focusing on
necessary and sufficient conditions of the existence of ground
state homoclinic orbits for this system.