Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations

We study the uniqueness of positive solutions of the followingcoupled nonlinear Schrödinger equations:{eqnarray*}Δ u_1-λ_1 u_1+μ_1u_1^3+β u_1u_2^2=0 in R^N,\\Δ u_2-λ_2u_2+μ_2u_2^3+β u_1^2u_2=0 in^N, _1>0, u_2>0, u_1, u_2 ∈ H^1 (R^N),{eqnarray*}where N≤3, λ₁,λ₂,μ₁,μ₂ are positive constants and β≥ 0 is a coupling constant. We provefirst the uniqueness of positive solution for sufficiently smallβ > 0. Secondly, assuming that λ₁=λ₂, we show that u₁=u₂√β-μ₁/√β-μ₂ whenβ > max\μ₁,μ₂\ and thus obtain the uniqueness ofpositive solution using the corresponding result of scalar equation. Finally, for $N=1$ and λ₁=λ₂,we prove the uniqueness of positive solution when 0≤ β∉ [min\μ₁,μ₂\,max\μ₁,μ₂\]and thus give a complete classification of positive solutions.