Electron. J. Diff. Equ., Vol. 2016 (2016), No. 97, pp. 1-11.

Multiple solutions for critical elliptic problems with fractional Laplacian

Guowei Lin, Xiongjun Zheng

Abstract:
This article is devoted to the study of the nonlocal fractional equation involving critical nonlinearities
$$\displaylines{
 (-\Delta)^{\alpha/2} u=\lambda u+|u|^{2^{\ast}_{\alpha}-2}u \quad 
 \text{in } \Omega,\cr
 u=0 \quad \text{on } \partial \Omega,
 }$$
where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $N \geq 2\alpha$, $\alpha\in(0,2)$, $\lambda\in(0,\lambda_{1})$ and $2^*_{\alpha}=\frac{2N}{N-\alpha}$ is critical exponent. We show the existence of at least $\hbox{cat}_{\Omega}(\Omega) $ nontrivial solutions for this problem.

Submitted January 11, 2016. Published April 14, 2016.
Math Subject Classifications: 35J60, 35J61, 35J70, 35J99.
Key Words: Fractional Laplacian, critical exponent, multiple solutions, category.

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Guowei Lin
Department of Mathematics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: lgw2008@sina.cn
Xiongjun Zheng
Department of Mathematics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: xjzh1985@126.com

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