Uniqueness of positive solutions to some Schrödinger systems

Abstract This paper is devoted to the study of the uniqueness of positive solutions of the following coupled Schrodinger equations: − Δ u + λ u = μ 1 u 3 + β v 2 u − γ v in Ω , − Δ v + λ v = μ 2 v 3 + β u 2 v − γ u in Ω , u > 0 , v > 0 in Ω , u = v = 0 on ∂ Ω ( or u , v ∈ H 1 ( R N ) as Ω = R N ) , where N ≤ 3 , Ω ⊂ R N is a smooth domain, λ , μ 1 , μ 2 are positive constants, β and γ are nonlinear and linear coupling constants respectively. We prove some new results on the uniqueness of positive solutions in several cases, improving and strengthening earlier works in the literature.