Critical growth problems with singular nonlinearities on Carnot groups

We provide regularity, existence and non existence results for the semilinear subelliptic problem with critical growth −ΔGu=ψα|u|2∗(α)−2ud(ξ)α+λu in ΩΩ, u=0u=0 on ∂Ω∂Ω, where ΔGΔG is a sublaplacian on a Carnot group GG, 0<α<20<α<2, 2∗(α)=2(Q−α)/(Q−2)2∗(α)=2(Q−α)/(Q−2), ΩΩ is a bounded domain of GG, dd is the natural gauge associated with the fundamental solution of −ΔG−ΔG on GG and ψ:=|∇Gd|ψ:=|∇Gd|, ∇G∇G being the subelliptic gradient associated to ΔGΔG, λλ is a real parameter.