A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system

We study the set of solutions of the nonlinear elliptic systemequation(P){−Δu+λ1u=μ1u3+βv2uin Ω,−Δv+λ2v=μ2v3+βu2vin Ω,u,v>0in Ω,u=v=0on ∂Ω, in a smooth bounded domain Ω⊂RNΩ⊂RN, N⩽3N⩽3, with coupling parameter β∈Rβ∈R. This system arises in the study of Bose–Einstein double condensates. We show that the value β=−μ1μ2 is critical for the existence of a priori bounds for solutions of (P). More precisely, we show that for β>−μ1μ2, solutions of (P) are a priori bounded. In contrast, when λ1=λ2λ1=λ2, μ1=μ2μ1=μ2, (P) admits an unbounded sequence of solutions if β⩽−μ1μ2.