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

### Guozhen Lu and Juncheng Wei

**Abstract.**
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.

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