Multiple solutions for a superlinear and periodic elliptic system on $${mathbb{R}^N}$$

This paper is concerned with the following periodic Hamiltonian elliptic system

$$left {begin{array}{l}-Delta u+V(x)u=g(x,v), {rm in },mathbb{R}^N,-Delta v+V(x)v=f(x,u), {rm in }, mathbb{R}^N, u(x)to 0, {rm and},v(x)to0, {rm as },|x|toinfty,end{array}right.$$

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.