In this paper, we study the elliptic system \(\left\{\begin{array}{ll}-\Delta u +V(x) u=|v|^{p-2}v-\lambda_2|v|^{s_2-2}v, \\-\Delta v + V(x)v=|u|^{p-2}u-\lambda_1|u|^{s_1-2}u, \\ u,v \in H^1(\mathbb{R}^N)\end{array}\right.\) with strongly indefinite structure and sign-changing nonlinearity. We overcome the absence of the upper semi-continuity assumption which is crucial in traditional variational methods for strongly indefinite problems. By some new tools and techniques we proved the existence of infinitely many geometrically distinct solutions if parameters λ1, λ2 > 0 small enough. To the best of our knowledge, our result seems to be the first result about infinitely many solutions for Hamiltonian system involving sign-changing nonlinearity.
