# Nonradial symmetric bound states for a system of coupled Schrödinger equations

### Juncheng Wei

University of British Columbia, Vancouver, Canada### Tobias Weth

Universität Giessen, Germany

## Abstract

We consider bound state solutions of the coupled elliptic system

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

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

$u >0, v >0, u, v \in {\mathbb H}^1 (\R^N),$

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 ${\mathbb H}^1 (\R^N)$, 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.

## Cite this article

Juncheng Wei, Tobias Weth, Nonradial symmetric bound states for a system of coupled Schrödinger equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 3, pp. 279–294

DOI 10.4171/RLM/495