LAUSR

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 (−Δ)su−μ1u|x|2s=|u|2s∗−2u+ηα2s∗|u|α−2|v|βu+12Qu(u,v)inΩ,(−Δ)sv−μ2v|x|2s=|v|2s∗−2v+ηβ2s∗|u|α|v|β−2v+12Qv(u,v)inΩ,u,v>0inΩ,u=v=0inℝN\Ω, where (− Δ)s denotes the fractional Laplace operator, Ω⊂ℝN is a smooth bounded domain such that 0 ∈ Ω,  μ1, μ2 ∈ [0, ΛN, s) with ΛN, s the sharp constant of the fractional Hardy inequality, and 2s∗=2NN−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.