On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups
Authors:
Guozhen Lu and Juncheng Wei
Journal:
Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 83-89
MSC (1991):
Primary 35H05; Secondary 35J70
DOI:
https://doi.org/10.1090/S1079-6762-97-00029-2
Published electronically:
August 28, 1997
MathSciNet review:
1465830
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Abstract: 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.
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Additional Information
Guozhen Lu
Affiliation:
Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435
MR Author ID:
322112
Email:
gzlu@math.wright.edu
Juncheng Wei
Affiliation:
Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong
MR Author ID:
339847
ORCID:
0000-0001-5262-477X
Email:
wei@math.cuhk.edu.hk
Keywords:
Heisenberg group,
stratified group,
Yamabe problem,
a priori estimates,
asymptotic behavior,
positive entire solutions
Received by editor(s):
June 12, 1997
Published electronically:
August 28, 1997
Additional Notes:
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.
Communicated by:
Thomas Wolff
Article copyright:
© Copyright 1997
American Mathematical Society