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

Abstract

Let G \mathbb {G} be a nilpotent, stratified homogeneous group, and let X 1 X_{1} , …, X m X_{m} be left invariant vector fields generating the Lie algebra G \mathcal {G} associated to G \mathbb {G} . The main goal of this paper is to study the Yamabe type equations associated with the sub-Laplacian Δ G = ∑ k = 1 m X k 2 ( x ) \Delta _{\mathbb {G}} = \sum _{k=1}^m X_k^2(x) on G \mathbb {G} : (*) Δ G u + K ( x ) u p = 0. \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.