Borderline variational problems involving fractional Laplacians and critical singularities

We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} \mu_{\gamma,s}(\R^n):= \inf\limits_{u \in H^{\frac{\alpha}{2}} (\R^n)\setminus {0}} \frac{ \int_{\R^n} |({-}{ \Delta})^{\frac{\alpha}{4}}u|^2 dx - \gamma \int_{\R^n} \frac{|u|^2}{|x|^{\alpha}}dx }{(\int_{\R^n} \frac{|u|^{2_{\alpha}^(s)}}{|x|^{s}}dx)^\frac{2}{2_{\alpha}^(s)}}, \end{equation*} where $0\le s<\alpha <2$, $n>\alpha$, $2_{\alpha }^{\ast }(s):=\frac{2(n-s)}{n-\alpha },$ and $\gamma \in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on ${\mathbb{R}}^n$, \begin{equation*} \left{\begin{array}{lll} ({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}&= |u|^{2_{\alpha}^-2} u + {\frac{|u|^{2_{\alpha}^(s)-2}u}{|x|^s}} & $\text{in }$ {\R^n}\ \hfill u&>0 & $\text{in }$ \R^n, \end{array}\right. \end{equation*} where $2_{\alpha }^{\ast }:=\frac{2n}{n-\alpha }$ is the critical $\alpha$-fractional Sobolev exponent, and $\gamma < \gamma_H:=2^\alpha \frac{\Gamma^2(\frac{n+\alpha}{4})}{\Gamma^2(\frac{n-\alpha}{4})}$, the latter being the best fractional Hardy constant on ${\mathbb{R}}^n$.