Concentration of Normalized Solutions for Mass Supercritical Kirchhoff Type Equations

In this paper, we study the existence and concentration behavior of normalized positive solutions to the following L 2 -supercritical Kirchhoff type equation - a ε 2 + b ε ∫ R 3 | ∇ v | 2 d x Δ v + λ v = K ( x ) | v | p - 2 v , x ∈ R 3 , ∫ R 3 | v | 2 d x = m ε 3 , v ∈ H 1 ( R 3 ) , where a , b , m > 0 , p ∈ 14 3 , 6 , ε is a small positive parameter, K is a positive continuous function possessing a local maximum point and λ ∈ R will arise as a Lagrange multiplier. We construct a family of positive normalized solutions v ε ∈ H 1 ( R 3 ) which concentrates around the local maximum of K as ε → 0 by using a combination of the variational approach and a penalization technique. Moreover, we also give the expression of the approximation solutions.