Critical problems with Hardy potential on Stratified Lie groups

We prove existence and nonexistence results for positive solutions to the subelliptic Brezis-Nirenberg type problem with Hardy potential −ΔGu−μψ2/d2u=u2∗−1+λuinΩ,u=0on∂Ω, extending to the Stratified setting well-known Euclidean results by Jannelli [J. Diff. Equ. 156, 1999]. Here, ΔG is a sub-Laplacian on an arbitrary Carnot group G, Ω is a bounded domain of G, 0∈Ω, d is the ΔG-gauge, ψ:=|ΔGd|, where ΔG is the horizontal gradient associated to ΔG, 0≤μ<¯μ, where ¯μ=(Q−22)2 is the best Hardy constant on G and λ∈R. The main difficulty in this abstract framework is the lack of knowledge of the ground state solutions to the limit problem −ΔGu−μψ2/d2u=u2∗−1onG, whose explicit expression is not known, except for the case when μ=0 and G is a group of Iwasawa-type. So, a preliminary refined analysis of qualitative properties of solutions to the above problem on the whole space is required, which has independent interest. In particular, Lorentz regularity and a priori decay estimates are obtained.