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\documentclass{era-l} \pagespan{101}{107} \PII{S 1079-6762(96)00014-5} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \renewcommand\a{\alpha} \renewcommand\b{\beta} \newcommand\s{\sigma} \newcommand\f{\phi} \newcommand\g{\gamma} \newcommand\p{\psi} \newcommand\e{\epsilon} \renewcommand\d{\delta} \newcommand\D{\Delta} \renewcommand\o{\omega} \renewcommand\O{\Omega} \newcommand\ty{\infty} \newcommand\ddfrac[2]{{\displaystyle {\frac{#1}{#2}}}} \newcommand\dint{\displaystyle \int } \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \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