Electron. J. Diff. Equ., Vol. 2016 (2016), No. 153, pp. 1-20.

Multiple solutions for Brezis-Nirenberg problems with fractional Laplacian

Hui Guo

In this article, we prove the multiplicity of nontrivial solutions to the critical problem with fractional Laplacian
 (-\Delta)^{\alpha/2}  u=|u|^{2^*_{\alpha}-2}u+\lambda u\quad\text{in } \Omega,\cr
 u=0\quad \text{on } \partial \Omega,
where $0<\alpha<2,\; N>(1+\sqrt{2})\alpha,\; \Omega\subset \mathbb{R}^N$ is a smooth bounded domain. More precisely, for any $\lambda>0$, this problem has at least $[(N+1)/2]$ pairs of nontrivial weak solutions.

Submitted April 9, 2016. Published June 20, 2016.
Math Subject Classifications: 35J20, 35J60, 35J67, 35R11.
Key Words: Fractional Laplacian; multiple solutions; Brezis-Nirenberg problem.

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Hui Guo
College of Mathematics and Econometrics
Hunan University
Changsha 410082, China
email: huiguo_math@163.com

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