On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups

Let $\mathbb {G}$ be a nilpotent, stratified homogeneous group, and let $X_{1}$, $\dots ,X_{m}$ be left invariant vector fields generating the Lie algebra $\mathcal {G}$ associated to $\mathbb {G}$. The main goal of this paper is to study the Yamabe type equations associated with the sub-Laplacian $\bigtriangleup _{\mathbb {G}}=\sum _{k=1}^{m}X_{k}^{2}(x)$ on $\mathbb {G}$:

 

Especially, we will establish the existence, nonexistence and asymptotic behavior of positive solutions to (0.1). Our results include the Yamabe type problem on the Heisenberg group as a special case, which is of particular importance and interest and also appears to be new even in this case.