Nonradial symmetric bound states for a system of coupled Schrödinger equations

We consider bound state solutions of the coupled elliptic system \Delta u - u + u^3 +\beta v^2 u=0 \ \text{in} \ \R^N, \Delta v - v + v^3 +\beta u^2 v=0 \ \text{in} \ \R^N, u >0, v >0, u, v \in {\mathbb H}^1 (\R^N), where N=2,3 . It is known ([13]) that when \beta <0 , there are no ground states, i.e., no least energy solutions. We show that, for certain finite subgroups of O(N) acting on {\mathbb H}^1 (\R^N) , least energy solutions can be found within the associated subspaces of symmetric functions. For \beta\le -1 these solutions are nonradial. From this we deduce, for every \beta \le -1 , the existence of infinitely many nonradial bound states of the system.