Fractional Hardy equations with critical and supercritical exponents.

We study the existence, nonexistence and qualitative properties of the solutions to the problem (P) (- Δ) s u - θ u | x | 2 s = u p - u q in R N u > 0 in R N u ∈ H ˙ s (R N) ∩ L q + 1 (R N) , where s ∈ (0 , 1) , N > 2 s , q > p ≥ (N + 2 s) / (N - 2 s) , θ ∈ (0 , Λ N , s) and Λ N , s is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions, we mean both the radial symmetry, that is obtained by using the moving plane method in a nonlocal setting on the whole R N , and a suitable upper bound behavior of the solutions. To this last end, we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space. [ABSTRACT FROM AUTHOR]