Fractional Hardy-Sobolev elliptic problems

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-\Delta)^{\frac \alpha 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in }\quad \Omega, \\ \\ u=0 &{\rm on }\quad \partial\Omega, \end{array} \right. \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$ with $0\in \Omega$, $\lambda,\mu>0,0<s\leq\alpha$, $(-\Delta)^{\frac \alpha 2}$ is the fractional Laplacian operator with $0<\alpha<2$. We establish existence results of problem \eqref{eq:1} for subcritical, Sobolev critical and Hardy-Sobolev critical cases.