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Let G be a nilpotent, stratied homogeneous group, and let X1, :::;Xm be left invariant vector elds generating the Lie algebra G associated to G. The main goal of this paper is to study the Yamabe type equations associated with the sub-Laplacian 4G = Pm k=1 X 2 k(x )o nG: 4G u + K(x)up =0 : (0.1) Especially, we will establish the existence, nonexistence and asymptotic behav- ior 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.

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