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These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

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\begin{document}

\title[Positive solutions of Yamabe-type equations]{On the existence of
   positive solutions of Yamabe-type equations on the Heisenberg group}

\author{L. Brandolini} \address{Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy}
\email{brandolini@vmimat.mat.unimi.it}

\author{ M. Rigoli}
\address{Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy}
\email{rigoli@vmimat.mat.unimi.it}

\author{A. G. Setti}
\address{Dipartimento di Matematica, Via Saldini 50, 20133 Milano, Italy}
\email{setti@vmimat.mat.unimi.it}

\subjclass{Primary 35H05; Secondary 35J70}

\commby{Richard Schoen}
\date{March 8, 1996}
\keywords{Heisenberg group, hypoelliptic equations, CR-Yamabe problem}

\begin{abstract}
We study nonexistence, existence and uniqueness of positive
solutions of the equation $\Delta _{H^n}u+a(x)u-b(x)u^\sigma =0$ with $\sigma >1$ on the Heisenberg group $H^n$. Our results hold, with essentially
no changes, also for the Euclidean version of the above equation. Even in
this case they appear to be new.
\end{abstract}
\maketitle
\section*{Introduction}
Let $H^n$ be the Heisenberg group of real dimension $2n+1,$ i.e. the
nilpotent Lie group which as a manifold is the product
$$
H^n={\mathbb C}^n\times {\mathbb R}
$$
and whose group structure is given by
$$
(z,t)\circ (z^{\prime },t^{\prime })=\left( z+z^{\prime },t+t^{\prime }+2
\Im (z,z^{\prime })\right) ,
$$
$$
 (z,t),\,(z^{\prime },t^{\prime })\in
H^n,
$$
where $(\,,\,)$ denotes the usual Hermitian product on ${\mathbb C}^n.$

A (real) basis for the Lie algebra of left-invariant vector fields on $H^n$
is given by
$$
X_j=2\Re\frac \partial {\partial z_j}+2\Im z_j\frac
\partial {\partial t},\quad Y_j=2\Im\frac \partial {\partial z_j}-2
\Re z_j\frac \partial {\partial t},\quad \frac \partial {\partial t},
$$
for $j=1,2,\dots ,n.$ The above basis satisfies Heisenberg's canonical
commutation relations for position and momentum
$$
\left[ X_j,Y_k\right] =-4\delta _{j\,k}\frac \partial {\partial
t},
$$
all other commutators being $0.$ It follows that the vector fields $X_j,$ $Y_k$ satisfy H\"ormander's condition, and the real part of the Kohn-Spencer
Laplacian, defined by
\begin{equation}
\label{pre.3}\Delta _{H^n}=\sum_{j=1}^n\left( X_j^2+Y_j^2\right) ,
\end{equation}
is hypoelliptic by H\"ormander's theorem (\cite{H}).

In $H^n$ one has a natural origin $0=(0,0)$ and a distinguished distance
function from $0$ defined by
$$
\rho(x)=\rho(z,t)=\left( |z|^4+t^2\right) ^{1/4},
$$
which is homogeneous of degree one with respect to the Heisenberg dilations $(z,t)\to (\delta z,\delta ^2t).$ The distance between two points $x,$ $x^{\prime }\in H^n$ is then given by $d(x,x^{\prime })=\rho(x^{-1}x^{\prime }).$

We also define the density function with respect to $0$ by
$$
\psi (x)=\psi (z,t)=\frac{|z|^2}{\rho(z,t)^2},\quad \mbox{for }x\neq 0,
$$
and note that $0\leq \psi (x)\leq 1.$ If $u$ is a ``radial
function'', that is, $u(z,t)=f\left( \rho(z,t)\right) $ for $f\,:\,[0,+\infty
)\to {\mathbb R}$ of class $C^2$, then
$$
\Delta _{H^n}u=\psi \left\{ f^{\prime \prime }(\rho)+\frac{2n+1}\rho f^{\prime }(\rho)\right\} .\nonumber
$$

In this paper we consider the equation
\begin{equation}
\label{maineq}\Delta _{H^n}u\,+\,a(x)u\,-\,b(x)|u|^{\sigma -1}u\,=\,0,
\end{equation}
with $\sigma >1$ constant, and determine conditions on the coefficients $a(x),$ $b(x)$ in order to guarantee the existence (resp., nonexistence) of
positive solutions on $H^n$.

Our problem is motivated by the following geometric fact. The vector fields $Z_j=X_j+iY_j$ span a subbundle $T_{1,0}$ of the complexified tangent
bundle of $H^n,$ and give rise to its canonical CR structure with contact
form $\theta $, which is determined modulo the transformation
\begin{equation}
\label{conftrans}\tilde \theta \,=\,u^{2/n}\theta
\end{equation}
for $01$, and let $a,b\in C^0(H^n)$ satisfy
\begin{equation}
\label{0.1}\left\{
\begin{array}{l}
a(x)\leq \psi (x)a_2\left( \rho(x)\right),  \\
\\
b(x)\geq \psi (x)b_1\left( \rho(x)\right)
\end{array}
\right. \mbox{on }\,H^n,
\end{equation}
with $a_2,$ $b_1\in C^0([0,+\infty )).$ Assume that for some constant $A\leq
n,$
$$
a_2(t)\leq \frac{A^2}{t^2},
$$
that $b_1(t)\geq 0$ on $[0,+\infty ),$ and that for some integer $k,$
$$
\left\{
\begin{array}{lc}
\displaystyle{\liminf_{t\to +\infty }b_1(t)\ddfrac{(\log t)^{\sigma +1}\log
(\log t)\cdots \log ^{(k)}(t)}{t^{n(\sigma -1)-2}}>0} & \mbox{if }A=n, \\  &
\\
\displaystyle{\liminf_{t\to +\infty }b_1(t)\ddfrac{(\log t)\log (\log
t)\cdots \log ^{(k)}(t)}{t^{(n-\sqrt{n^2-A^2})(\sigma -1)-2}}>0} &
\mbox{if }A0,$ and that there exist an integer $k
$ and a constant $C>0$ such that
$$
\left\{
\begin{array}{l}
b_1(t)\geq 0\quad
\mbox{on}\,\,[0,+\infty ), \\  \\
\displaystyle{\liminf_{t\to +\infty }t^2\log t\log (\log t)\cdots \log
^{(k)}(t)\,b_1(t)\geq C>0.}
\end{array}
\right.
$$
Then \eqref{0.4} has no positive solution on $H^n$.
\end{theorem}


\section{Existence results}

\begin{theorem}
\label{tC}Let $a$, $b\in C^\infty (H^n)$, $\mu \le 2$, $1<\sigma \le \frac{n+2}n$,
$$
A_\mu >\left\{
\begin{array}{ccc}
0 & \text{if} & \mu <2, \\
\ddfrac{2n}{\sigma -1} & \text{if} & \mu =2,
\end{array}
\right. ,
$$
and let $\gamma \in {\mathbb R}$. Assume that $a$ and $b$ satisfy 
$$
\psi (x)a_1(\rho(x))\le a(x)\le \psi (x)a_2(\rho(x))\;\;\text{on }H^n
$$
and
$$
\psi (x)b_1(\rho(x))\le b(x)\le \psi (x)b_2(\rho(x))\;\;\text{on }H^n,
$$
respectively, for suitable $a_1,a_2,b_1,b_2\in C^0([0,+\infty ))$ with
$$
a_1(t)=A_\mu t^{-\mu }\qquad \text{for }t\gg 1,
$$
$$
\begin{array}{ll}
\text{i)} & b_1(t)\ge 0
\text{ on }[0,+\infty )\text{ and }b_1(t)>0\text{ in }[t_0,+\infty ), \\  &
\\
\text{ii)} & b_2(t)\le c_1t^{-\frac 2n\gamma -\mu }\text{ for }t\gg 1,
\end{array}
$$
with $c_1>0$ and $t_0$ such that $a_2(t)\le \frac{n^2}{t^2}$ on $(0,t_0]$.
Then there exists a positive solution $u$ of \eqref{0.4} on $H^n$ satisfying
the further requirement
$$
u(x)\ge c_2\rho(x)^{\frac{2\gamma }{n(\sigma -1)}}
$$
for some constant $c_2>0$ and $\rho(x)$ sufficiently large.
\end{theorem}

\begin{definition}
We say that a solution $U$ of equation \eqref{0.4} on $H^n$ is \emph{maximal}
if for any other solution $u$ on $H^n$ we have$$
u(x)\le U(x)\text{\ \ for }x\in H^n.
$$
\end{definition}

\begin{theorem}
\label{E}Let $a,b\in C^0(H^n)$. Suppose there exist $a_2,b_1,b_2\in
C^0([0,+\infty ))$ satisfying
$$
a_2(t)\le \frac{n^2}{t^2}
$$
and
$$
\begin{array}{ll}
\text{i)} & b_1(t)\ge 0\;
\text{on }[0,+\infty )\text{ and }b_1(t)>0\text{ for }t\gg 1, \\  &  \\
\text{ii)} & tb_2(t)\in L^1(+\infty ),
\end{array}
$$
such that
$$
0\le a(x)\le \psi (x)a_2(\rho(x))\;\;\text{on }H^n
$$
and
$$
\psi (x)b_1(\rho(x))\le b(x)\le \psi (x)b_2(\rho(x))\;\;\text{on }H^n.
$$
Then, there exists a unique, positive, maximal solution $U$ of \eqref{0.4} on $H^n$ satisfying
$$
\lim _{\rho(x)\rightarrow +\infty }U(x)=+\infty .
$$

Furthermore, if for some constant $c>0$
$$
b_1(t)\ge ct^{-k}\text{, for }k>2\text{ and }t\gg 1,
$$
then$$
U(x)\le C\rho(x)^{\frac{k-2}{\sigma -1}}
$$
for some constant $C>0$ and $\rho(x)\gg 1$.

Similarly, if for some constant $c>0$$$
b_2(t)\le ct^{-h}\text{, for }h>2\text{ and }t\gg 1,
$$
then$$
U(x)\ge C\rho(x)^{\frac{h-2}{\sigma -1}}
$$
for some constant $C>0$ and $\rho(x)\gg 1$.

In particular, in the case where$$
C_1\psi (x)\left[ 1+\rho(x)\right] ^{-k}\le b(x)\le C_2\psi (x)\left[
1+\rho(x)\right] ^{-k}
$$
for some constants $C_1,C_2>0$, we find that$$
U(x)\asymp \rho(x)^{\frac{k-2}{\sigma -1}}\;
\text{as }\rho(x)\rightarrow +\infty .
$$
\end{theorem}

\begin{theorem}
Let $a,b\in C^0(H^n).$ Suppose there exist $a_2,b_1\in C^0([0,+\infty ))$
satisfying
$$
a_2(t)\le \frac{n^2}{t^2},
$$
$$
b_1(t)\ge 0\;\text{on }[0,+\infty )\text{ and }b_1(t)>0\text{ for }t\gg 1,
$$
such that
$$
0\le a(x)\le \psi (x)a_2(\rho(x))\;\;\text{on }H^n
$$
and
$$
\psi (x)b_1(\rho(x))\le b(x)\;\;\text{on }H^n.
$$
If there exists a positive subsolution $u$ of \eqref{0.4}, then there
exists a unique, positive, maximal solution $U$ of \eqref{0.4} on $H^n$
satisfying
$$
\lim _{\rho(x)\rightarrow +\infty }U(x)=+\infty .
$$
\end{theorem}

\section{A uniqueness result}

\begin{theorem}
\label{tU} Let $a(x),$ $b(x)\in C^0(H^n)$ satisfy $b(x)\geq 0$ on $H^n,$ and
$$
a(x)\leq \psi (x)a_2\left( \rho(x)\right) \quad \text{on }H^n,
$$
where $a_2\in C^0([0,+\infty )$ satisfies
\begin{equation}
\label{U.2}a_2(t)\leq \frac{A^2}{t^2}\qquad \text{on }(0,+\infty ),
\end{equation}
with $A\le n$. Let $u,v\in C^2(H^n)$ be positive solutions of equation \eqref
{0.4}. If
\begin{equation}
\label{U.3}(u-v)(x)=\left\{
\begin{array}{ll}
o\left( \rho(x)^{-n}\log (\rho(x))\right) & \text{for }A=n, \\
\\
o\left( \rho(x)^{-n+\sqrt{n^2-A^2}}\right) & \text{for }A