Multiple solutions for a superlinear and periodic elliptic system on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^N}$$\end{document}

This paper is concerned with the following periodic Hamiltonian elliptic system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left \{\begin{array}{l}-\Delta u+V(x)u=g(x,v)\, {\rm in }\,\mathbb{R}^N,\\-\Delta v+V(x)v=f(x,u)\, {\rm in }\, \mathbb{R}^N,\\ u(x)\to 0\, {\rm and}\,v(x)\to0\, {\rm as }\,|x|\to\infty,\end{array}\right.$$\end{document}where the potential V is periodic and 0 lies in a gap of the spectrum of −Δ + V, f(x, t) and g(x, t) depend periodically on x and are superlinear but subcritical in t at infinity. By establishing a variational setting, existence of a ground state solution and multiple solution for odd f and g are obtained.