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*} μ_{γ,s}(\R^n):= \inf\limits_{u \in H^{\fracα{2}} (\R^n)\setminus \{0\}} \frac{ \int_{\R^n} |({-}{ Δ})^{\fracα{4}}u|^2 dx - γ\int_{\R^n} \frac{|u|^2}{|x|^α}dx }{(\int_{\R^n} \frac{|u|^{2_α^*(s)}}{|x|^{s}}dx)^\frac{2}{2_α^*(s)}}, \end{equation*} where $0\leq s<α<2$, $n>α$, ${2_α^*(s)}:=\frac{2(n-s)}{n-α},$ and $γ\in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on $\R^n$, \begin{equation*} \left\{\begin{array}{lll} ({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α}&= |u|^{2_α^*-2} u + {\frac{|u|^{2_α^*(s)-2}u}{|x|^s}} & \text{in } {\R^n}\\ \hfill u&>0 & \text{in } \R^n, \end{array}\right. \end{equation*} where $2_α^*:=\frac{2 n}{n-α}$ is the critical $α$-fractional Sobolev exponent, and $γ< γ_H:=2^α\frac{Γ^2(\frac{n+α}{4})}{Γ^2(\frac{n-α}{4})}$, the latter being the best fractional Hardy constant on $\R^n$.

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