Local uniqueness of multi-peak solutions to a class of Schrödinger equations with competing potential

Abstract

In this paper, we consider the nonlinear Schrödinger equations. Let A(x)≔[V(x)]pp−2−N2[K(x)]−2p−2. Under some conditions on A, we show the local uniqueness of positive multi-peak solutions concentrating near k(k ≥ 2) distinct non-degenerate critical points of A by using the local Pohozaev identity. We generalize Cao–Li–Luo’s results to the competing potential cases and show how these two potentials impact the uniqueness of concentrated solutions.