Singular critical elliptic problems with fractional Laplacian

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.

Keywords

• Fractional Laplacian
• singular critical problem
• non-contractible domain