Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent

Peng Chen; Xiaochun Liu

doi:10.3934/cpaa.2018007

Article Contents

2018, Volume 17, Issue 1: 113-125. Doi: 10.3934/cpaa.2018007

This issue Previous Article Liouville results for fully nonlinear integral elliptic equations in exterior domains Next Article The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas

Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent

Peng Chen and
Xiaochun Liu^,

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

* Corresponding author
* Corresponding author

Received: January 2017

Revised: July 2017

Published: January 2018

The authors are supported by NSFC grants 11371282 and 11571259.

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Abstract

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.

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Mathematics Subject Classification: Primary:35B33, 35J60;Secondary:47J30.

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