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.

Show me the PDF file (241 KB), TEX file for this article.

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

Return to the EJDE web page