Borderline Variational Problems Involving Fractional Laplacians and Critical Singularities

We consider the problem of attainability of the best constant C > 0 in the following critical fractional Hardy-Sobolev inequality: For all where and γ ∈ ℝ. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on ℝ , where is the critical α-fractional Sobolev exponent, and , the latter being the best fractional Hardy constant on ℝ .