LAUSR

Abstract In this paper, we study the following Kirchhoff-type equations with critical growth − ( e 2 a + e b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = P ( x ) f ( u ) + Q ( x ) | u | 4 u , x ∈ R 3 , where e > 0 , a > 0 , b > 0 and f is a continuous superlinear but subcritical nonlinearity. Under suitable assumptions on the potentials V ( x ) , P ( x ) and Q ( x ) , we obtain the existence and concentration of positive solutions and prove that the semiclassical solutions w e with maximum points x e concentrating at a special set S p characterized by V ( x ) , P ( x ) and Q ( x ) . Furthermore, for any sequence x e → x 0 ∈ S p , v e ( x ) : = w e ( e x + x e ) converges in H 1 ( R 3 ) to a ground state solution v of − ( a + b ∫ R 3 | ∇ v | 2 d x ) Δ v + V ( x 0 ) v = P ( x 0 ) f ( v ) + Q ( x 0 ) | v | 4 v , x ∈ R 3 .