Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems

Abstract

We study the following nonperiodic Hamiltonian system $\dot{z}=\mathcal{J}{H}_z(t,z)$, where $H\in C^1(\mathbb{R}\{\backslash times\}{\mathbb{R}}^{2N},\mathbb{R})$ is the form $H\left(t,z\right)=\left(1/2\right)B\left(t\right)z\cdot z+R(t,z)$. We introduce a new assumption on $B\left(t\right)$ and prove that the corresponding Hamiltonian operator has only point spectrum. Moreover, by applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of homoclinic orbits for asymptotically quadratic nonlinearity as well as the existence of infinitely many homoclinic orbits for superquadratic nonlinearity.