Spike-layer solutions to singularly perturbed semilinear systems of coupled Schrödinger equations

Let Ω be a bounded domain in RN(N⩽3), we are concerned with the interaction and the configuration of spikes in a double condensate by analyzing the least energy solutions of the following two couple Schrödinger equations in Ω(Sε){−ε2Δu+u=μ1u3+βuv2,−ε2Δv+v=μ2v3+βu2v,u>0,v>0, where μ1,μ2 are positive constants. We prove that under Neumann or Dirichlet boundary conditions, for any ε>0, when −∞<βmax{μ1,μ2}, system (Sε) has a least energy solution (uε,vε) and when min{μ1,μ2}<β0 or for β<0, thus our results are an extension of the results in Lin and Wei (2005) [10].