Uniqueness of Single Peak Solutions for a Kirchhoff Equation

Abstract

We deal with the following singular perturbation Kirchhoff equation: −ϵ2a+ϵb∫R3|∇u|2dyΔu+Q(y)u=|u|p−1u,u∈H1(R3), where constants a,b,ϵ>0 and 1<p<5. In this paper, we prove the uniqueness of the concentrated solutions under some suitable assumptions on asymptotic behaviors of Q(y) and its first derivatives by using a type of Pohozaev identity for a small enough ϵ. To some extent, our result exhibits a new phenomenon for a kind of Q(x) which allows for different orders in different directions.