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 (-Δ)^{\frac α2} u=λ|u|^{r-2}u+μ\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in }\quad Ω, \\ \\ u=0 &{\rm on }\quad \partialΩ, \end{array} \right. \end{equation} where $Ω$ is a smooth bounded domain in $\mathbb R^N$ with $0\in Ω$, $λ,μ>0,0<s\leqα$, $(-Δ)^{\frac α2}$ is the fractional Laplacian operator with $0<α<2$. We establish existence results of problem \eqref{eq:1} for subcritical, Sobolev critical and Hardy-Sobolev critical cases.

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