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Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent

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The authors are supported by NSFC grants 11371282 and 11571259.
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  • 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.

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


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