Multiple Solutions for Asymptotically Linear Elliptic Systems

Abstract

This paper is concerned with the following periodic Hamiltonian elliptic system $$\left\{ {\begin{array}{lll} { - \Delta u + V(x)u = g(x,v)} & {{\text{in}}} & {\mathbb{R}^N }, { - \Delta u + V(x)v = f(x,v)} & {{\text{in}}} & {\mathbb{R}^N }, {u(x) \to 0\,\,\text{and}\,\,v(x) \to 0} & {{\text{as}}} & {|x| \to \infty }, \end{array} } \right.$$ where the potential V is periodic and has a positive bound from below, f(x, t) and g(x, t) are periodic in x, asymptotically linear in t as \(|t| \rightarrow \infty\). By using critical point theory of strongly indefinite functionals, existence of a positive ground state solution as well as infinitely many geometrically distinct solutions for odd f and g are obtained.