Electron. J. Diff. Equ., Vol. 2015 (2015), No. 297, pp. 1-12.

Singular critical elliptic problems with fractional Laplacian

Xueqiao Wang, Jianfu Yang

Abstract:
In this article, we consider the existence of solutions of the critical problem with a Hardy term for fractional Laplacian
$$\displaylines{ (-\Delta)^s u -\mu \frac u{|x|^{2s}}= u^{2^*_s-1} \quad \text{in }\Omega,\cr u>0 \quad \text{in }\Omega, \cr u=0 \quad \text{on }\partial \Omega, }$$
where $\Omega\subset \mathbb{R}^N$ is a smooth bounded domain and $0\in\Omega$, $\mu$ is a positive parameter, $N>2s$ and $s\in(0,1)$, $2^*_{s} =\frac{2N}{N-2s}$ is the critical exponent. $(-\Delta)^s$ stands for the spectral fractional Laplacian. Assuming that $\Omega$ is non-contractible, we show that there exists $\mu_0>0$ such that $0<\mu<\mu_0$, there exists a solution. We also discuss a similar problem for the restricted fractional Laplacian.

Submitted September 6, 2015. Published December 3, 2015.
Math Subject Classifications: 35J20, 35J25, 35J61.
Key Words: Fractional Laplacian; singular critical problem; non-contractible domain.

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  Xueqiao Wang
Department of Mathematics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: wangxueqiao1989@126.com
Jianfu Yang
Department of Mathematics
Jiangxi Normal University
Nanchang, Jiangxi 330022, China
email: jfyang_2000@yahoo.com

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