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January  2018, 17(1): 113-125. doi: 10.3934/cpaa.2018007

## Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent

 School of mathematics and statistics, Wuhan University, Wuhan, 430072, China

* Corresponding author

Received  January 2017 Revised  July 2017 Published  September 2017

Fund Project: The authors are supported by NSFC grants 11371282 and 11571259.

This paper is concerned with the existence and multiplicity of solutions to the following Kirchhoff type elliptic equations with critical nonlinearity:
 $\begin{cases} -(a+b \int_{Ω}|\nabla u|^{2}dx)Δ u=f(x, u)+μ|u|^{4}u &\; \; \mbox{in }Ω, \\ u=0 &\; \; \mbox{on }\partial Ω, \end{cases}$
where $Ω\subset\mathbb{R}^3$ is a bounded smooth domain, $μ$ is a positive parameter and $f:Ω×\mathbb{R}\to \mathbb{R}$ is a Carathéodory function satisfying some further conditions. Our approach is based on concentration-compactness principle and symmetry mountain pass theorem.
Citation: Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007
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