Uniqueness, existence and concentration of positive ground state solutions for Kirchhoff type problems in [formula omitted

Abstract

In this paper, we first prove the uniqueness of positive ground state solution for the following Kirchhoff equation in R3 with constant coefficients{−(a+b∫R3|∇u|2)Δu+cu=d|u|p−1uinR3,u>0,u∈H1(R3), where a, b, c, d>0 are positive constants, 3<p<5. Then we use the uniqueness result to obtain the existence and concentration theorems of positive ground state solutions to the following Kirchhoff equation with competing potential functions−(ε2a+εb∫R3|∇u|2)Δu+V(x)u=K(x)|u|p−1uinR3, for a sufficiently small positive parameter ε.