JournalsrlmVol. 18 , No. 3DOI 10.4171/rlm/495

Nonradial symmetric bound states for a system of coupled Schrödinger equations

  • Juncheng Wei

    University of British Columbia, Vancouver, Canada
  • Tobias Weth

    Universität Giessen, Germany
Nonradial symmetric bound states for a system of coupled Schrödinger equations cover

Abstract

We consider bound state solutions of the coupled elliptic system

Δuu+u3+βv2u=0 \mboxin RN,\Delta u - u + u^3 +\beta v^2 u=0 \ \mbox{in} \ \R^N,
Δvv+v3+βu2v=0 \mboxin RN,\Delta v - v + v^3 +\beta u^2 v=0 \ \mbox{in} \ \R^N,
u>0,v>0,u,vH1(RN),u >0, v >0, u, v \in {\mathbb H}^1 (\R^N),

where N=2,3N=2,3. It is known (\cite{lw1}) that when β<0\beta <0, there are no ground states, i.e., no least energy solutions. We show that, for certain finite subgroups of O(N)O(N) acting on H1(RN){\mathbb H}^1 (\R^N), least energy solutions can be found within the associated subspaces of symmetric functions. For β1\beta\le -1 these solutions are nonradial. From this we deduce, for every β1\beta \le -1, the existence of infinitely many nonradial bound states of the system.