Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems

In this paper we study the following nonperiodic second order Hamiltonian system−u¨(t)+L(t)u(t)=∇uR(t,u),∀t∈R, where L(t) may not be uniformly positive definite for all t∈R, and R(t,u) is asymptotically quadratic in u as |u|→∞. By applying a generalized linking theorem in Bartsch and Ding (2006) [8], we obtain the existence and multiplicity of homoclinic orbits for the above system.