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\begin{document}
\title[ON POSITIVE ENTIRE SOLUTIONS TO THE YAMABE-TYPE PROBLEM]
{On positive entire solutions to the Yamabe-type problem 
on the Heisenberg and  stratified groups}
%\markboth{GUOZHEN LU AND JUNCHENG WEI}{ON POSITIVE ENTIRE SOLUTIONS TO 
%THE YAMABE-TYPE PROBLEM}

\author{Guozhen Lu}
\address{Department of Mathematics and Statistics, 
Wright State University, Dayton, OH 45435}
\email{gzlu@@math.wright.edu}
\thanks{The work of the first author was supported in part by the 
National Science Foundation 
Grant \#DMS96-22996.}

\author{Juncheng Wei}
\address{Department of Mathematics, Chinese University of Hong Kong, 
Shatin, N.T., Hong Kong}
\email{wei@@math.cuhk.edu.hk}
\thanks{The work of the second author was supported in part by an 
Earmarked Grant from RGC of Hong Kong.}

\subjclass{Primary 35H05; Secondary 35J70}

\keywords{Heisenberg group, stratified group, Yamabe problem, 
a priori estimates, asymptotic behavior, positive entire solutions}

\commby{Thomas Wolff}

\date{June 12, 1997}

%\issueinfo{3}{1}{August}{1997}

\copyrightinfo{1997}{American Mathematical Society}

%\begin{center}
%{\large \bf On   positive entire 
%solutions to the Yamabe-type problem on the Heisenberg and  stratified 
%groups}
%\vspace{0.2cm}
%Guozhen Lu\footnote{Supported in part by the National Science 
%Foundation Grant \#DMS96-22996.}
%\\Juncheng Wei\footnote{Supported in part by an Earmarked Grant from 
%RGC of Hong Kong. }
%\end{center}
%\vspace{0.2cm}
%{\bf Abstract:} 
%{\footnotesize 

\begin{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 $ %{\cal 
\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.
\end{abstract}

\maketitle



%1991 Mathematics Subject Classification: Primary 35H05; Secondary 35J70.

%Keywords and phrases: Heisenberg group, Stratified group, Yamabe 
%problem, 
%A priori estimates, Asymptotic behaviors, Positive entire solutions.}



\section{Introduction}

In a series of papers \cite{JM1},
\cite{JM2},
\cite{JM3} by D. Jerison and J. M. Lee, the Yamabe problem 
on CR manifolds was first studied, in conjunction with finding the
best constant and extremal for the Sobolev inequality on the Heisenberg 
group. In particular, they studied the problem of conformally changing 
the contact
form to one with constant Webster curvature in the compact setting.
The Webster curvature is a scalar invariant of pseudo-Hermitian 
manifolds, which is defined
independently by B. Webster \cite{W} and N. Tanaka \cite{T}. 

 

In this paper, we investigate the existence, nonexistence and asymptotic
behavior of positive entire solutions to the following Yamabe-type 
problem
associated with the sub-Laplacian $\G=\sub $ on the stratified group 
$\bg$:
\addtocounter{theorem}{1}
\be\label{10}
\G u+K(x)u^{\frac{Q+2}{Q-2}}=0,
\end{equation}
where $X_1,\dots, X_m$ are left-invariant vector fields
on $\bg$, and $Q$ is the homogeneous dimension of $\bg$ (see 
definition
below), and $K(x)$ is locally H\"{o}lder continuous on $\bg$ with respect 
to the Carnot metric as defined in 
\cite{FS}. The results we obtain in this paper, of course, include
the Heisenberg group $\H$ as a special case. These results are new 
even on $\H$.

Equation (\ref{10}) on the Heisenberg group $\H$ arises naturally 
in the study of
the Cauchy-Riemann manifold $(\H, \theta)$
with contact form $\theta$, where $\H$ is a homogeneous group of real 
dimension $2n+1$ with the underlying manifold 
%$$
\[\H=\c^n\times \r %$$
\]
and whose group structure is given by
\addtocounter{theorem}{1}
\be\label{mul}
(z,t)\cdot (z',t')=(z+z', t+t'+2Im(z\cdot z')),
\end{equation}
for any two points $(z,t)$ and $(z',t')$ in $\H$.

The Lie algebra of $\H$ is generated by the left invariant vector 
fields
%$$
\[X_i=\frac{\partial }{\partial x_i}+2y_i\frac{\partial}{\partial t},\qquad
Y_i=\frac{\partial }{\partial y_i}-2x_i\frac{\partial}{\partial t} %$$
\]
for $i=1, \dots,n$.
This is a prominent example of stratified Lie group of step two 
\cite{FS}.
The Kohn sub-Laplacian on $\H$ is defined
as 
%$$
\[\HL=\subh, %$$
\]
which has received extensive attention for the past several decades.

The generators $\{X_i\}_{i=1}^n$ and $\{Y_i\}_{i=1}^n$ induce a
canonical structure with a contact form $\theta$, which
is determined up to
a transformation 
\addtocounter{theorem}{1}
\be\label{11}
\tilde{\theta}=u^{\frac{4}{Q-2}}\theta
\end{equation}
for some positive $u$. Each choice of $\theta$ corresponds to a
pseudo-Hermitian structure on $\H$ and 
we denote 
%$$
\[\theta_0=2\sum_{i=1}^{n}(x_jdy_j-y_jdx_j). %$$
\]
Given any contact form $\theta$ on $\H$, it gives rise to a Webster 
scalar curvature  $R_\theta$. By the
 tranformation (\ref{11}), the relationship
between $R_\theta$ and $R_{\tilde{\theta}}$ is described as
\addtocounter{theorem}{1}
\be\label{12}
\frac{2Q}{Q-2}\bigtriangleup_{\theta}u+
R_{\theta}u=R_{\tilde{\theta}}u^{\frac{Q+2}{Q-2}},
\end{equation}
where $\bigtriangleup_{\theta}$ is the Laplacian on the pseudo-Hermitian
manifold $(\H,\theta)$.

When we take $\theta=\theta_0$, $R_\theta$ becomes zero, and
the equation (\ref{12}) reduces to
\addtocounter{theorem}{1}
\be\label{13}
\frac{2Q}{Q-2}\bigtriangleup_{\theta}u=R_{\tilde{\theta}}u^{\frac{Q+
2}{Q-2}},
\end{equation}
under the transformation $\tilde{\theta}=u^{\frac{4}{Q-2}}\theta_0$.

Thus, equation (\ref{13}) is the CR analogue of the Yamabe problem in 
the Euclidean setting. 

The problem of  finding a contact form $\tilde{\theta}$ on $\H$
with prescribed Webster scalar curvature  $R_{\tilde{\theta}}$  is 
equivalent to finding a 
$C^{\infty}$ positive function $u$ on $\H$ such that 
$\tilde{\theta}=u^{\frac{4}{Q-2}}\theta_0$ 
and $u(x)$ satisfies the equation (\ref{13}).


To state our theorems on the stratified group $\bg$, we need to 
recall briefly some notions about groups.
Let $ %{\cal 
\mathcal{G}$ be a finite-dimensional, stratified, nilpotent Lie 
algebra.
Assume that
%$$
\[
%{\cal 
\mathcal{G}=\bigoplus_{i=1}^{s}V_{i}\;,
 %$$
\]
%\noindent 
and $[V_{i},V_{j}]\subset V_{i+j}$ for $i+j\le s$,
$[V_{i},V_{j}]=0 $ for $i+j > s$. Let $X_{1},\dots,X_{m}$ be a
basis for $V_{1}$ and suppose that $X_{1},\dots,X_{m}$
generate $ %\cal 
\mathcal{G}$ as a Lie algebra. Thus we can choose a basis
$\{X_{ij}\}$, for $1\le j\le s$, $1\le i\le m_{j}$ for $V_{j}$,
consisting of vectors of the form $X_{\alpha}$ for some multi-indices
$\alpha$ of length $j$. In particular, $X_{i1}=X_{i},\ i=1,\dots,m$
and $m=m_{1}$. 

Let $\bg$ be the simply connected Lie group associated to $ %{\cal 
\mathcal{G}$. 
Since the
exponential mapping is a global diffeomorphism from $ %{\cal 
\mathcal{G}$ to 
$\bg$, for each
$g\in \bg$, there is $x=(x_{ij})\in %{\bb 
\mathbb{R}^{N}$, $1\le i\le m_{j},\ 1\le
j\le s,\ N=\sum_{j=1}^{s}m_{j}$, such that
%$$
\[
g=\exp(\sum x_{ij}X_{ij})\;.
 %$$
\]
Thus we define a homogeneous norm function $| \cdot |$ on $\bg$ by
%$$
\[
|g|=(\sum\vert x_{ij}\vert^{2s!/j})^{1/2s!}\;.
 %$$
\]
On the Heisenberg group $\H$, for each $(z,t)\in \H$, we
have
%$$
\[|(z,t)|=\rho(z,t)=\left(|z|^4+t^2\right)^{\frac{1}{4}}. %$$
\]

Let $\delta_{t}$ be a dilation on $\bg$ defined by
%$$
\[
\delta_{t}x=(t^{j}x_{ij})_{1\le i\le m_{j},1\le j\le s}
 %$$
\]
%\noindent 
for each $t > 0$. It is easy to see that $\delta_{t}$ is
an automorphism of $\bg$ for each $t > 0$. Lebesgue measure $dy$ is
the bi-invariant Haar measure of $\bg$ and the Jacobian of $\delta_{t}$,
$J\delta_{t}$, is equal to $t^{Q}$, where $Q=\sum_{j=1}^{s}jm_{j}$ is
called the {\it homogeneous dimension} of $\bg$, which is usually greater
than $\dim \bg=N$. On the Heisenberg group $\H$, $Q=2n+2$.


We mention in passing that some
(existence and nonexistence) results, different 
from ours,  on the Heisenberg group $\H$ were obtained by  Garofalo and 
Lanconelli 
\cite{GL} and Citti \cite{C} on certain bounded or unbounded domains.
We should also mention
the earlier works by D. Jerison on solvability of the Dirichlet problem 
for a sub-Laplacian on the Heisenberg group
\cite{J1-2}. In the recent
announcement\footnote{We have not been able to obtain a copy of the 
full version of the paper 
containing the results announced in \cite{BRS}.}  of Brandolini, Rigoli, 
and Setti 
\cite{BRS}, they also considered
the existence and nonexistence of positive entire solutions on the 
Heisenberg group and obtained some interesting results on 
$\H$. They  appear to have
used  the method of comparison
with radial equations in conjunction with the sub-super solution 
method. To be more precise,
in \cite{BRS} the authors consider, for nonnegative functions $a(x)$ and 
$b(x)$,
\addtocounter{theorem}{1}
\be\label{brs}
\HL u+a(x)u-b(x)|u|^{\sigma-1}u=0, \ x=(z,t).
\end{equation}
In the polar coordinates on $\H$, the sub-Laplacian
can be written as
\addtocounter{theorem}{1}
\be\label{polar}
\HL u=\frac{|z|^2}{\rho(z,t)^2}\left(u''(\rho)+
\frac{Q-1}{\rho}u'(\rho)\right).
\end{equation}
Thus, the study of (\ref{brs}) will resemble 
 the elliptic case (see the first general work due to W.-M. Ni \cite{N} 
and many other subsequent works 
\cite{CN}, \cite{LN}, etc.). 
However, the presence
of the density factor $\Psi(x)=\Psi(z,t)=\frac{|z|^2}{\rho(z,t)^2}$ on 
$\H$
in (\ref{polar}) makes the situation here more delicate, and one will need  
to  assume
that $a(x)$ and $b(x)$ satisfy
%$$
\[a_1(|x|)\Psi(x)\le a(x)\le a_2(|x|) \Psi(x), \qquad b_1(|x|)\Psi(x)\le 
b(x)\le b_2(|x|)\Psi(x), %$$
\]
where $ x=(z,t)\in \H$ and $|x|=\rho(z,t)$ (see \cite{BRS}).
Existence and nonexistence results of this type for positive entire 
solutions, obtained by using the comparison
with the radial equations and also the method of sub-super solutions (as was
done, e.g., in \cite{N} by Ni), were also independently derived on $\H$ 
by Lu in
 \cite{L}
for the 
equation
%$$
\[\HL u+K(x)u^p=0, \ \textrm{on } \H. %$$
\]

Our methods in this paper use mainly an a priori estimate of the 
solutions,
 and thus
avoid using the comparison with radial equations. As a result, we do
not  need
to assume any condition on $K(x)$ involved with the degenerate factor 
$\Psi(x)$.
Moreover, our method does not need the explicit structure of the 
Heisenberg group $\H$ and thus allows us to study the Yamabe type 
equation on a more general
stratified, nilpotent group $\bg$:
%$$
\[\G u+K(x)u^p=0. %$$
\] 

The theorems obtained in this paper consist of several  parts.
In Section 2, we obtain some existence results. 
In Theorems \ref{21} and \ref{22} there, we assume that 
$K(x)$ decays
at infinity not more slowly than $|x|^{l}$ for some $l<-2$. 
These two theorems 
give existence results for all $11
\end{equation}
and 
%$$
\[u\in C_{loc}^2(G), \qquad \lim_{|x|\to \infty}u(x)=0. %$$
\]
We have assumed here again that
$K$ is a nonnegative, nontrivial and locally H\"{o}lder continuous function
in $\bg$, which satisfies 
%$$
\[|K(x)|\le C|x|^l \ \textrm{at } \infty %$$
\]
for some $-2\le l<0$. 


\begin{theorem}\label{23}
If we assume that $\frac{Q+2+2l}{Q-2}0$, with
$\liminf_{|x|\to\infty}u(x)=0$. Suppose that 
%$$
\[|K(x)|\le C|x|^{l} %$$
\]
at infinity 
for some $l<-2$. Then for any positive $\epsilon$ we have 
%%$$u(x)\le \cases{C|x|^{2-Q}\ at\ \infty, & if\ $p>\frac{Q+l}{Q-2}$,\cr
%C_\epsilon |x|^{\frac{(1-\epsilon)(l+2)}{1-p}}\ at\ \infty, &if \ 
%$p\le\frac{Q+l}{Q-2}$.\cr} %$$
%$$
\[u(x)\le \left\{
\begin{array}{ll}
C|x|^{2-Q}\ at\ \infty & %\hbox{\rm 
\text{if } p>\frac{Q+l}{Q-2},\\
C_\epsilon |x|^{\frac{(1-\epsilon)(l+2)}{1-p}}\ at\ \infty & 
%%\hbox{\rm i
\text{if } p\le\frac{Q+l}{Q-2},
\end{array}
\right. %$$
\] 
where $C_\epsilon$ only depends on $\epsilon$.
\end{theorem}



\begin{theorem}\label{25}
Let $u$ be a bounded positive solution of \eqref{0} in $\bg$, and $p>0$.
 Suppose that 
%$$
\[|K(x)|\le C|x|^{l} %$$
\]
at infinity 
for some $l<-2$. 
Then $u_\infty=\lim_{|x|\to\infty}u(x)$  exists. If we assume
further that $K$ never changes sign in $\bg$ and $|K(x)|\simeq |x|^l$ 
at infinity, then  we have
%$$
\[|u(x)-u_{\infty}|\simeq 
\left\{
\begin{array}{ll}
C|x|^{2-Q} & %\hbox{\rm 
\text{if } l<-Q,\\
C|x|^{2-Q}\log |x| & %\hbox{\rm 
\text{if } l=-Q,\\
C|x|^{2+Q} & %\hbox{\rm 
\text{if } -Q0$.
\end{theorem}

\begin{theorem}\label{26}
Let $u$ be a bounded positive solution of \eqref{0} in $\bg$ for which  
\linebreak
$\liminf_{|x|\to\infty}u(x)=0$. Let $\alpha=(\alpha_1,\dots, 
\alpha_k)$ be any multi-index. Suppose that $K(x)$ is smooth enough, 
say
$K(x)\in C^{|\alpha|}(\bg)$ and 
%$$
\[|K(x)|\le C|x|^{l} %$$
\]
at infinity 
for some $l<-2$, and  $p>\max\{0, \frac{Q+l}{Q-2}\}$. 
Then 
%$$
\[|X^{\alpha}u(x)|\le C(1+|x|)^{|\alpha|-Q}, \ for\ all\ x\ in \ \bg, %$$
\]
where $X^\alpha=X_{i_1}^{\alpha_1}\cdot\cdot\cdot X_{i_k}^{\alpha_k}$,
$1\le i_j\le m$, $|\alpha |=\alpha_1+\alpha_2+\cdot\cdot\cdot+\alpha_k$,
and $C$ is a positive constant.
\end{theorem}

\section{Nonexistence results on the Heisenberg group}

All the results stated in the previous sections apply as well to the 
Heisenberg group.
However, by taking advantage of the special structure of $\H$, we can 
do more precise analysis for (\ref{10}).
We now state a nonexistence theorem on the Heisenberg group $\H$. We 
denote 
%$$
\[XK=\sum_{i=1}^n\left(x_i\frac{\p K}{\p x_i}+y_i\frac{\p K}{\p y_i}
\right)+2t\frac{\p K}{\p t}. %$$
\]

\begin{theorem}\label{31}
Suppose that $|K(x)|\le C|x|^{l}$ at infinity for some $l<-2$, $K\in 
C^1(\H)$, and the function $L(x)=\left[Q-(Q-2)(p+1)/2\right]K(x)+XK(x)$
never changes sign in $H^n$. Then equation \eqref{10} does not possess
any positive bounded solution $u$ satisfying $\liminf_{|x|\to 
\infty}u=0$.
\end{theorem}


To state the next two theorems, we  need to introduce some notation on the 
Heisenberg group $\H$.
Let 
%$$
\[B_r(x)=B(x,r)=\{y\in \H: \rho(x,y)=|x\cdot y^{-1}|0$,
and 
%$$
\[\partial B_r(x)=\{y\in \H: \rho(x,y)=r\}. %$$
\]
Given any nonzero function $K$ we define
%$$
\[\overline{K}(r)=\left(\frac{1}{|\partial B_r(0)|}\int_{\partial B_r(0)}
\frac{\Psi(z,t)}{K(z,t)^{\frac{Q-2}{4}}}\frac{d\sigma}{|%
\bigtriangledown \rho|}\right)^{-\frac{4}{Q-2}}, %$$
\]
where $d\sigma$ is the surface measure on $\partial B_r(0)$, and 
$|\bigtriangledown \rho|$ is the classical gradient of the distance 
function
$\rho(z,t)$, $\Psi(z,t)=\frac{|z|^2}{\rho(z,t)^2}$.

If $K(x)=K(z,t)=K(|(z,t)|)=K(\rho(z,t))$, then an easy calculation shows
that 
\be\label{kk}
\overline{K}(\rho)=C_QK(\rho),
\end{equation}
with $C_Q$ an absolute constant. 

\begin{theorem}\label{32}
If $K(x)\ge 0$ in $\H$, and $\overline{K}(\rho)\ge C\rho^p$ at infinity 
for some
constants $C>0$ and $p>2$, then the equation 
%$$
\[\HL u+K(x)\Psi u^{\frac{Q+2}{Q-2}}=0, \ on \ \H %$$
\]
does not possess any positive entire solutions in $\H$.
\end{theorem}

\begin{theorem}\label{33}
If $K(x)\ge 0$ in $\H$, and $\overline{K}(\rho)\ge \frac{C}{\rho^k}$ at 
infinity for some
constants $C>0$ and $k<2$, then the equation 
%$$
\[\HL u=K(x)\Psi u^{\frac{Q+2}{Q-2}}, \ on \ \H %$$
\]
does not possess any positive entire solutions in $\H$.
\end{theorem}

%\noindent{\bf Remarks:}
\begin{remarks}
By the observation (\ref{kk}), the $\overline{K}(\rho)$ can be
replaced by $K(\rho)$ for radial $K$ in the statements of the above two 
theorems.  We can extend 
as well Theorems \ref{32} and \ref{33}
to the case where the critical exponent $\frac{Q+2}{Q-2}$ is replaced by
any $p>1$, with the appropriate growth or decay conditions on $K$.
\end{remarks}




\begin{thebibliography}{MMW}
\bibitem[BRS]{BRS} L. Brandolini, M. Rigoli, and A. G. Setti, On the 
existence
of positive solutions of Yamabe-type equations on the Heisenberg group, 
Electronic research announcements of the AMS, Dec. 1996, 2 (1996), 101-107.
\bibitem[C]{C} G. Citti, Semilinear Dirichlet problem involving critical
exponenent for the Kohn Laplacian, Annali di Mate. Pura  Appl. 169 (1995), 
375-392. \MR{96j:35071}
\bibitem[CN]{CN} K. S. Cheng and W.-M. Ni, On the structure of the 
conformal
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\end{thebibliography}

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


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




\end{document}



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