On ground state solutions of Nehari-Pohozaev type for the nonlinear Kirchhoff type problems with a general critical nonlinearity

Weihong Xie, Haibo Chen

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 some 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.

Keywords


Kirchhoff type problems; ground state solutions of Nehari--Pohozaev type; the least energy solutions; variational methods; critical Sobolev exponent

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