A note on uniqueness of the ground state for nonlinear Schrödinger equations

Abstract We are concerned with the following nonlinear Schrodinger equation − e 2 Δ u + V ( x ) u = | u | p − 2 u , u ∈ H 1 ( R N ) , where N ≥ 3 , 2 p 2 N N − 2 . For e small enough and a class of V ( x ) , we show the uniqueness of the positive ground state under certain assumptions on asymptotic behavior of V ( x ) and its first derivatives. Here our results are suitable for a kind of V ( x ) which has different increasing rates at different directions.