  We prove the existence, qualitative properties and asymptotic behavior of
positive solutions to the doubly critical problem $$ (-\Delta)^s
u=\vartheta\frac{u}{|x|^{2s}}+u^{2_s^*-1}, \quad u\in
\dot{H}^s(\mathbb{R}^N).$$ The technique that we use to prove the existence is
based on variational arguments. The qualitative properties are obtained by
using of the moving plane method, in a nonlocal setting, on the whole
$\mathbb{R}^N$ and by some comparison results.
  Moreover, in order to find the asymptotic behavior of solutions, we use a
representation result that allows to transform the original problem into a
different nonlocal problem in a weighted fractional space.
