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

Abstract

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 −∞<β<min{μ1,μ2} or β>max{μ1,μ2}, system (Sε) has a least energy solution (uε,vε) and when min{μ1,μ2}<β<max{μ1,μ2}, system (Sε) has no solution. Suppose Pε,Qε are the local maximum points of uε,vε respectively. Then under Neumann boundary conditions, as ε small enough, both of Pε,Qε locate on the boundary of Ω. Furthermore, when β⩾0, |Pε−Qε|ε→0 as ε→0 and for N=2 and N=3, Pε,Qε converge to the same point on the boundary which is the maximum point of mean curvature of the boundary. However, when β<0, |Pε−Qε|ε→∞ as ε→0 and suppose Pε→P and Qε→Q, then for N=2 and N=3, P,Q must be the maximum points of the mean curvature on the boundary and P,Q might be a same point if the mean curvature of the boundary has only one maximum point. Under Dirichlet boundary conditions, we can prove that as long as the least energy solution (Sε) exists, the same asymptotic behavior of the least energy solution (uε,vε) holds as described in Lin and Wei (2005) [10] for β>0 or for β<0, thus our results are an extension of the results in Lin and Wei (2005) [10].