A note on a superlinear and periodic elliptic system in the whole space

Abstract

This paper is concerned with the following periodic Hamiltonianelliptic system$ -\Delta u+V(x)u=g(x,v)$ in $R^N,$$ -\Delta v+V(x)v=f(x,u)$ in $R^N,$ $ u(x)\to 0$ and $v(x)\to 0$ as $|x|\to\infty,$ where the potential $V$ is periodic and has a positive bound frombelow, $f(x,t)$ and $g(x,t)$ are periodic in $x$ and superlinear butsubcritical in $t$ at infinity. By using generalized Nehari manifoldmethod, existence of a positive ground state solution as wellas multiple solutions for odd $f$ and $g$ are obtained.