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.