Ground states for Kirchhoff‐type equations with critical or supercritical growth

In this paper, we study the following Kirchhoff‐type equation with critical or supercritical growth− ( a + b ∫ R 3| ∇ u | 2 d x ) △ u + V ( x ) u = f ( x , u ) + λ | u | p − 2 u , x ∈ R 3 ,where a >0, b >0, λ >0, p ≥6 and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x , we prove that the equation has a ground state solution for small λ >0 by Nehari method. Moreover, we regard b as a parameter and obtain a convergence property of the ground state solution as b ↘0. Our main contribution is related to the fact that we are able to deal with the case p >6.