Fractional nonlinear Schr\"odinger equations with singular potential in $\mathbf R^n$

We are interested in nonlinear fractional Schr\"odinger equations with singular potential of form \begin{equation*} (-\Delta)^su=\frac{\lambda}{|x|^{\alpha}}u+|u|^{p-1}u,\quad \mathbf R^n\setminus\{0\}, \end{equation*} where $s\in (0,1)$, $\alpha>0$, $p\ge1$ and $\lambda\in \mathbf R$. Via Caffarelli-Silvestre extension method, we obtain existence, nonexistence, regularity and symmetry properties of solutions to this equation for various $\alpha$, $p$ and $\lambda$.