On ground state solutions for the nonlinear Kirchhoff type problems with a general critical nonlinearity

Abstract

In this paper, we are concerned with the following Kirchhoff type problem with critical growth: \begin{equation*} -\bigg(a+b\int_{\mathbb R^3}|\nabla u|^2dx\bigg)\Delta u+V(x)u=f(u)+|u|^4u, \quad u\in H^1(\mathbb R^3), \end{equation*} where $a,b > 0$ are constants. Under certain assumptions on $V$ and $f$, we prove that the above problem has a ground state solution of Nehari-Pohozaev type and a least energy solution via variational methods. Furthermore, we also show that the mountain pass value gives the least energy level for the above problem. Our results improve and extend some recent ones in the literature.