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We prove the existence, qualitative properties and asymptotic behavior ofpositive solutions to the doubly critical problem $$ (-\Delta)^su=\vartheta\frac{u}{|x|^{2s}}+u^{2_s^*-1}, \quad u\in\dot{H}^s(\mathbb{R}^N).$$ The technique that we use to prove the existence isbased on variational arguments. The qualitative properties are obtained byusing 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 arepresentation result that allows to transform the original problem into adifferent nonlocal problem in a weighted fractional space.

Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential