Radial and nonradial solutions of a strongly indefinite elliptic system on $\mathbb{R}^N$

This paper is concerned with the following system of elliptic equations {equation*} \{{array}{ll} -\Delta u+u= F_u(|x|,u,v), & \hbox{} -\Delta v+v=- F_v(|x|,u,v), & \hbox{} \,\,\,\,\,u,v\in H^1(\mathbb{R}^N). & \hbox{} {array}. {equation*} It is shown that if $F$ is odd in $(u,v)$ and satisfy some growth conditions, then $(\mathcal{S})$ has infinitely many both radial and nonradial solutions. The proof relies on the Principle of Symmetric Criticality and a generalized Fountain Theorem for strongly indefinite functionals.