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 $X_{1}$, …, $X_{m}$ 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 $\Delta _{\mathbb {G}} = \sum _{k=1}^m X_k^2(x)$ on $\mathbb {G}$: \begin{equation}\tag {*} \Delta _{\mathbb {G}} u+K(x)u^{p}=0. \end{equation} Especially, we will establish the existence, nonexistence and asymptotic behavior of positive solutions to ($*$). 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.