## Topological Methods in Nonlinear Analysis

### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 53, Number 2 (2019), 519-545.

Dates
First available in Project Euclid: 10 May 2019

https://projecteuclid.org/euclid.tmna/1557453831

Digital Object Identifier
doi:10.12775/TMNA.2019.010

Mathematical Reviews number (MathSciNet)
MR3983984

Zentralblatt MATH identifier
07130709

#### Citation

Xie, Weihong; Chen, Haibo. On ground state solutions for the nonlinear Kirchhoff type problems with a general critical nonlinearity. Topol. Methods Nonlinear Anal. 53 (2019), no. 2, 519--545. doi:10.12775/TMNA.2019.010. https://projecteuclid.org/euclid.tmna/1557453831

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