The Brezis-Nirenberg problem for fractional systems with Hardy potentials

In this work, we study the existence of positive solutions to the following fractional elliptic systems with Hardy-type singular potentials and coupled by critical homogeneous nonlinearities {(-Delta)(s)u - mu(1)u/vertical bar x vertical bar(2)s = vertical bar u vertical bar(2s*-2)u + eta alpha/2(s)*vertical bar u vertical bar(alpha-2)|v vertical bar(beta)u + 1/2Q(u)(u, v) in Omega, (-Delta)(s)v - mu(2)v/vertical bar x vertical bar(2)s = vertical bar v vertical bar(2s*-2)v + eta beta/2(s)*vertical bar u vertical bar(alpha)vertical bar v vertical bar(beta-2)v + 1/2Q(v)(u, v) in Omega, u, v > 0 in Omega, u = v = 0 in R-N\Omega, where (- Delta)(s) denotes the fractional Laplace operator, Omega subset of R-N is a smooth bounded domain such that 0 is an element of Omega, mu(1), mu(2) is an element of [0, Lambda(N, s)) with Lambda(N, s) the sharp constant of the fractional Hardy inequality, and 2s* = 2N/N-2s is the fractional critical Sobolev exponent. In order to prove the main result, we study the related fractional Hardy-Sobolev type inequalities and then establish the existence of positive solutions to the systems through variational methods.