Fractional Hardy equations with critical and supercritical exponents

Abstract

We study the existence, nonexistence and qualitative properties of the solutions to the problem where (p) {(-Delta)(s)u - theta u/vertical bar x vertical bar(2s) = u(p) - u(q) in R-N u > 0 in R-N u is an element of H-s (R-N) boolean AND Lq+1 (R-N), where s is an element of (0, 1), N > 2s, q > p >= (N + 2s)/(N - 2s), theta is an element of (0, Lambda(N,s)) and Lambda(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.