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Let $\bg$ 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 $\bg$. The main goal of this paper is to study the Yamabe type equations associated with the sub-Laplacian $\G=\sub $ on $\bg$: \addtocounter{theorem}{1} \be\label{0} \G u+K(x)u^{p}=0. \end{equation} Especially, we will establish the existence, nonexistence and asymptotic behavior of positive solutions to (\ref{0}). 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.

Copyright 1997 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 03 (1997), pp. 83-89
• Publisher Identifier: S 1079-6762(97)00029-2
• 1991 Mathematics Subject Classification. Primary 35H05; Secondary 35J70
• Key words and phrases. Heisenberg group, stratified group, Yamabe problem, a priori estimates, asymptotic behavior, positive entire solutions
• Received by the editors June 12, 1997
• Posted on August 28, 1997
• Communicated by Thomas Wolff
• Comments (When Available)

Guozhen Lu

Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435

E-mail address: gzlu@math.wright.edu

Juncheng Wei

Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong

E-mail address: wei@math.cuhk.edu.hk

The work of the first author was supported in part by the National Science Foundation Grant #DMS96-22996.
The work of the second author was supported in part by an Earmarked Grant from RGC of Hong Kong.