Existence of Homoclinic Orbits for Hamiltonian Systems with Superquadratic Potentials

Abstract

This paper concerns solutions for the Hamiltonian system: $\dot{z}=𝒥H_z\left(t,z\right)$.Here $H\left(t,z\right)=\left(1/2\right)z\cdot Lz+W\left(t,z\right)$, $L$ is a $2N\times 2N$ symmetric matrix, and $W\in C^1\left(ℝ\times {ℝ}^{2N},ℝ\right)$. We consider the case that $0\in {\sigma }_c\left(-\left(𝒥\left(d/dt\right)+L\right)\right)$ and $W$ satisfies some superquadratic condition different from the type of Ambrosetti-Rabinowitz. We study this problem by virtue of some weak linking theorem recently developed and prove the existence of homoclinic orbits.