Multiple solutions for superlinear elliptic systems of Hamiltonian type

Abstract

This paper is concerned with the following periodic Hamiltonianelliptic system$\-\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)$ in $\mathbb{R}^N,$$\-\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)$ in $\mathbb{R}^N,$$\varphi(x)\to 0$ and $\psi(x)\to0$ as $|x|\to\infty.$Assuming the potential $V$ is periodic and $0$ lies in a gap of$\sigma(-\Delta+V)$, $G(x,\eta)$ is periodic in $x$ andsuperquadratic in $\eta=(\varphi,\psi)$, existence and multiplicityof solutions are obtained via variational approach.