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In this paper, we study the following nonlinear fractional Laplacian system with critical Sobolev‐Hardy exponent false(−normalΔfalse)su−γufalse|xfalse|2s=λffalse(xfalse)false|ufalse|q−2ufalse|xfalse|α+2ηη+θhfalse(xfalse)false|ufalse|η−2ufalse|vfalse|θfalse|xfalse|β1emin0.3em0.3em0.3emnormalΩ,false(−normalΔfalse)sv−γvfalse|xfalse|2s=μgfalse(xfalse)false|vfalse|q−2vfalse|xfalse|α+2θη+θhfalse(xfalse)false|ufalse|ηfalse|vfalse|θ−2vfalse|xfalse|β1emin0.3em0.3em0.3emnormalΩ,u=v=01emin0.3em0.3em0.3emdouble-struckRN∖normalΩ, where 0∈normalΩ is a smooth bounded domain in double-struckRN, 01 satisfy η+θ=2s*false(βfalse), 2s*false(βfalse)=2false(N−βfalse)N−2s is the critical Sobolev‐Hardy exponent, λ, μ>0 are parameters, f, g and h are nonnegative functions on normalΩ. Using the variational methods and analytic techniques, we prove that the critical fractional Laplacian system admits at least two positive solutions when the pair of parameters false(λ,μfalse) belongs to a suitable subset of double-struckR+2.

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where - Δ G is a sub-Laplacian on Carnot group G, μ ∈ [ 0 , μ G ) , d is the Δ G -natural gauge, ψ is the weight function defined as ψ : = | ∇ G d | . By analytic technics and variational methods, the extremals of the corresponding best Sobolev constant are found, the existence of positive solution to the system is established. Moreover, by the Moser iteration method, some asymptotic properties of its nontrivial solution at the singular point are verified.