Uniqueness results for semilinear elliptic systems on R n

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form− Δ u=f ( | x | , u , v )inR n ,− Δ v=f ( | x | , v , u )inR n .As an application we consider the nonlinear Schrödinger system− Δ u + u=u 2 q − 1 + b u q − 1v qinR n ,− Δ v + v=v 2 q − 1 + b v q − 1u qinR n ,for b > 0 and exponents q which satisfy 1 < q < ∞ in case n ∈ { 1 , 2 } and 1 < q < n n − 2in case n ≥ 3 . Generalizing the results of Wei and Yao for q = 2 we find new sufficient conditions and necessary conditions on b , q , n such that precisely one positive solution exists. Our results dealing with the special case n = 1 are optimal. Finally, an application to a multi‐component nonlinear Schrödinger system is given.