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

We prove existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$(−Delta)_s u = artheta rac{u}{|x|^{2s}} + u^{2^*−1}, qquad u in dot{H}^s (mathbb{R}^N ), N > 2s, 0 < s < 1.$$ The technique that we use to prove the existence is based on variational arguments. The qualitative properties are obtained by using the moving plane method, in a nonlocal setting, on the whole RN and 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.