Nonradial symmetric bound states for a system of coupled Schrödinger equations

Abstract

We consider bound state solutions of the coupled elliptic system

\[ \Delta u - u + u^3 +\beta v^2 u=0 \ \mbox{in} \ \R^N, \] \[ \Delta v - v + v^3 +\beta u^2 v=0 \ \mbox{in} \ \R^N, \]

where $N=2,3$. It is known (\cite{lw1}) 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 ${\mathbb{H}}^1({\mathbb{R}}^N)$, 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.