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 (Formula presented.).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