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ISSN 1079-6762

 

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
Published electronically: August 28, 1997
MathSciNet review: 1465830
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Abstract: Let $\mathbb {G}$ be a nilpotent, stratified homogeneous group, and let $X_{1}$, $\dots ,X_{m}$ 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 $\bigtriangleup _{\mathbb {G}}=\sum _{k=1}^{m}X_{k}^{2}(x)$ on $\mathbb {G}$:

 \begin{equation}\label{0} \bigtriangleup _{\mathbb {G}}u+K(x)u^{p}=0. \end{equation}

Especially, we will establish the existence, nonexistence and asymptotic behavior 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.


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Additional Information

Guozhen Lu
Affiliation: Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435
Email: gzlu@math.wright.edu

Juncheng Wei
Affiliation: Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Email: wei@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S1079-6762-97-00029-2
PII: S 1079-6762(97)00029-2
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