\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 130, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/130\hfil Uniqueness of positive solution]
{Uniqueness of a positive solution for quasilinear elliptic equations
in Heisenberg group}

\author[K. Bal \hfil EJDE-2016/130\hfilneg]
{Kaushik Bal}

\address{Kaushik Bal \newline
Department of Mathematics and Statistics,
Indian Institute of Technology,
Kanpur, Uttar Pradesh-280016 India}
\email{kaushik@iitk.ac.in}

\thanks{Submitted January 18, 20016. Published June 6, 2016.}
\subjclass[2010]{35R03, 35J92, 35J70}
\keywords{Quasilinear elliptic equation; Picone's identity; Heisenberg group}

\begin{abstract}
 In this note we address the question of  uniqueness of the Brezis-Oswald
 problem for the p-Laplacian operator  in Heisenberg Group. The non-availability
 of $C^{1,\alpha}$  regularity for all $1<p<\infty$  is the problem to extend
 the proof of Diaz-Saa \cite{DiSa} in the Heisenberg Group case. We overcome
 the problem by proving directly a generalized version of Diaz-Sa\'a inequality
 in the Heisenberg Group.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

The well-known paper of Br\'ezis and Oswald \cite{BrOs} provides
the necessary and sufficient condition for the existence and uniqueness
of positive solutions for the equation
\begin{gather*}
 -\Delta u=f(x,u)\quad \text{in } \Omega,\\
 u=0\quad \text{on }\partial\Omega
\end{gather*}
where $\Omega$ is a bounded open domain in $\mathbb{R}^n$ and under some
hypothesis on $f$.
Almost immediately the result was extended by Di\'az and Sa\'a \cite{DiSa}
to the p-Laplacian case by introducing a new inequality which came to be
later known as the D\'iaz-Sa\'a inequality.

The purpose of this paper is to extend the result of \cite{DiSa} in the context
of Heisenberg Group, which we will denote by $\mathcal{H}^n$.
Consider the problem:
\begin{equation} \label{QH}
\begin{gathered}
 -\Delta_{H,p} u=f(x,u)\quad \text{in }\Omega \\
u\geq 0,\quad u\not\equiv 0\quad \text{in }\Omega\\
 u=0\quad \text{on }\partial\Omega
\end{gathered}
\end{equation}
where $\Omega$ is an open bounded domain of $\mathcal{H}^n$ and $1<p<\infty$.

We consider $f:\Omega\times[0,\infty)\to (0,\infty)$ satisfying the following
 hypothesis:
\begin{itemize}
 \item[(I)] The function $r\mapsto f(x,r)$ is continuous on
$[0,\infty)$ for a.e. $x\in\Omega$ and for every $r\geq0$, the function
 $x\to f(x,r)$ is in $L^{\infty}(\Omega)$.

 \item[(II)] The function $r\mapsto \frac{f(x,r)}{r^{p-1}}$ is strictly
decreasing on $(0,\infty)$ for a.e $x\in\Omega$.

 \item[(III)] There exists $C>0$ such that
 $f(x,r)\leq C(r^{p-1}+1)$ for a.e. $x\in\Omega$ and for all $r$.
\end{itemize}

By weak solution of \eqref{QH} we mean
$u\in D^{1,p}_0(\Omega)\cap L^{\infty}(\Omega)$ such that
\begin{equation}
\int_{\Omega} |\nabla_H  u|^{p-2}\langle \nabla_H u(x), \phi(x)\rangle dx
=\int_{\Omega}f(x,u)\phi(x) dx
\end{equation}
holds for all $\phi\in C^{\infty}_c(\Omega)$.
In this note we aim to establish the  uniqueness of the weak solution to \eqref{QH}.

Before we start with our results let us briefly recall some basic notions
regarding the Heisenberg Group $\mathcal{H}^n$ along
with some literature which is available on the study of elliptic equation
on Heisenberg group.

The Heisenberg group $\mathcal{H}^n=(\mathcal{R}^{2n+1},\cdot)$ is nilpotent Lie
group endowed with the group structure:
\begin{equation}
 (x,y,t)\cdot(x',y',t')=(x+x',y+y',t+t'+2(\langle y,x'\rangle-\langle x,y'\rangle))
\end{equation}
where $x,y,x',y'\in \mathcal{R}^n,\;t,t'\in\mathcal{R}$ and $\langle,\rangle$
denotes the standard inner product in $\mathcal{R}^n$.

The left invariant vector field generating the Lie algebra is given by
\begin{equation}
\mathcal{T}=\frac{\partial}{\partial t},\quad
\mathcal{X}_i=\frac{\partial}{\partial x_i}+2y_i\frac{\partial}{\partial t},\quad
\mathcal{Y}_i=\frac{\partial}{\partial y_i}-2x_i\frac{\partial}{\partial t},
\quad i=1,2,\dots,n.
\end{equation}
and satisfy the  relationship
\begin{equation}
 [\mathcal{X}_i,\mathcal{Y}_i]=-4\delta_{ij}T,\quad
 [\mathcal{X}_i,\mathcal{X}_j]=[\mathcal{Y}_i,\mathcal{Y}_j]=
 [\mathcal{X}_i,\mathcal{T}]=[\mathcal{Y}_i,\mathcal{T}]=0.
\end{equation}

The generalized gradient is given by
$\nabla_H=(\mathcal{X}_1,\mathcal{X}_2,\dots,\mathcal{X}_n,
\mathcal{Y}_1,\mathcal{Y}_2,
\dots,\mathcal{Y}_n)$.
Hence the sub-Laplacian $\Delta_H$ and the p-sub-Laplacian $\Delta_{H,p}$
are denoted by
\begin{gather*}
 \Delta_H=\sum\limits_{i=1}^{n}\mathcal{X}_i^2+\mathcal{Y}_i^2
=\nabla_H\cdot\nabla_H,\\
 \Delta_{H,p}=\nabla_H\cdot(|\nabla_H |^{p-2}\nabla_H ),\quad p>1\,.
\end{gather*}
We also denote the space $D^{1,p}(\Omega)$ and $D^{1,p}_0(\Omega)$ as
$\{u:\Omega\to \mathcal{R};u,|\nabla_H u|\in L^p(\Omega)\}$
and the closure of $C^{\infty}_0(\Omega)$ with respect to the norm
$\|u\|_{D^{1,p}_0(\Omega)}=(\int_{\Omega}|\nabla_H u|^p \,dx\,dy\,dt)^{1/p}$
respectively.
Some results on the Laplacian and the p-Laplacian has been generalized to
the Heisenberg Group with various degree of success.
Consider the  problem
\begin{equation}\label{NIU}
\begin{gathered}
-\Delta_{H,p} u=f(u)\quad \text{in }\Omega, \\
 u=0\quad \text{on }\partial\Omega
\end{gathered}
\end{equation}

Some of the very first results obtained regarding the above problem for $p=2$
is by Garofalo-Lanconelli \cite{GL}, where existence and nonexistence results
 were derived using integral identities of Rellich-Pohozaev type.
In Birindelli et al \cite{BDC}, Liouville theorems for semilinear equations
are proved. One can also find monotonicity and symmetry results in Birindelli
and Prajapat \cite{BP}.
As for the p-sub-Laplacian case, Niu et al \cite{NZW} considered the question
of non-uniqueness of the \eqref{NIU} using the Picone Identity and the Pohozaev
Identities for the p-sub-Laplacian on Heisenberg Group. Results on p-sub-Laplacian
involving singular indefinite weight can be found in Dou \cite{DOU} and
Tyagi \cite{TY} and the reference therein.
For more details about Heisenberg Group the reader may consult \cite{CDPT}.

One of the problems when dealing with p-sub-Laplacian is the non-availability
of the $C^{1,\alpha}$ regularity for $1<p<\infty$, although
it has been proved in Marchi \cite{Marchi} to exist for $p$ near $2$.
It is worth mentioning that the methods of D\'iaz-S\'aa \cite{DiSa}
can not be directly applied here due to the non-availability
of $C^{1,\alpha}$ regularity in the Heisenberg Group. In this work we bypass
that problem by using a generalized
version of D\'iaz-S\'aa Inequality in Heisenberg Group.

\section{Preliminary Results}

We start this section with the generalized Picone's Identity for $p$-sub-Laplacian
in Heisenberg Group, which is extension of the main
result in Euclidean space obtained in \cite{KB}.
In what follows we assume $g:(0,\infty)\to(0,\infty)$ is a locally Lipchitz
function that satisfies the differential inequality
 \begin{equation}\label{PC}
  g'(x)\geq (p-1)[g(x)]^{\frac{p-2}{p-1}}\quad \text{a.e. in }(0,\infty)
 \end{equation}

 \begin{remark} \label{rmk2.1} \rm
Example of functions satisfying \eqref{PC} are $g(x)=x^{p-1}$
(where the equality holds) and $e^{(p-1)x}$.
\end{remark}

In what follows we  use $\nabla$ to denote $\nabla_H$ and $\Delta_p$
to denote $\Delta_{H,p}$.

\begin{theorem}[Generalized Picone identity] \label{PICN}
 Let $1<p\leq Q$ and $\Omega$ be any domain in $\mathcal{H}^n$.
Let $u$ and $v$ be differentiable functions with $v>0$ a.e in $\Omega$.
Also assume $g$ satisfies \eqref{PC}.
Define
 \begin{gather*}
 L(u,v)=|\nabla u|^p-p\frac{|u|^{p-2}u}{g(v)}\nabla u\cdot\nabla v|\nabla v|^{p-2}
 +\frac{g'(v)|u|^p}{[g(v)]^2}|\nabla v|^p\;\;\text{a.e}\;\text{in}\;\Omega.\\
 R(u,v)=|\nabla u|^p-\nabla(\frac{|u|^p}{g(v)})|\nabla v|^{p-2}\nabla v
\quad \text{a.e. in }\Omega.
 \end{gather*}
Then $L(u,v)=R(u,v)\geq 0$. Moreover $L(u,v)=0$ a.e. in $\Omega$
if and only if $\nabla (\frac{u}{v})=0$ a.e. in $\Omega$.
\end{theorem}

\begin{remark} \label{rmk2.3} \rm
Note that there is no restriction on the sign of $u$, as one can find in
\cite[Proposition 3]{Ch}. When $g(x)=x^{p-1}$ and $u\geq 0$,
we obtain the Picone identity \cite[Lemma 2.1]{DOU}.
\end{remark}

\begin{proof}[Proof of Theorem \ref{PICN}]
Expanding $\nabla(\frac{|u|^p}{g(v)})$ we have
\begin{align*}
\nabla(\frac{|u|^p}{g(v)})
&=\frac{pg(v)|u|^{p-2}u\nabla u-g'(v)|u|^p\nabla v}{[g(v)]^2}\\
&=p\frac{|u|^{p-2}u\nabla u}{g(v)}-\frac{g'(v)|u|^p\nabla v}{[g(v)]^2}.
\end{align*}
Plugging it in $R(u,v)$ we have $R(u,v)=L(u,v)$.

To show positivity of $L(u,v)$ we proceed as follows,
\begin{equation}\label{mod}
 \frac{|u|^{p-2}u}{g(v)}\nabla u.\nabla v|\nabla v|^{p-2}
\leq \frac{|u|^{p-1}}{g(v)}|\nabla v|^{p-1}|\nabla u|.
\end{equation}
By Young's inequality we have
\begin{equation}\label{young}
 p\frac{|u|^{p-1}}{g(v)}|\nabla v|^{p-1}|\nabla u|
\leq |\nabla u|^p+(p-1)\frac{|u|^p|\nabla v|^p}{[g(v)]^{\frac{p}{p-1}}}\,.
\end{equation}
Using \eqref{mod} and \eqref{young} we have
\[
 L(u,v)\geq -(p-1)\frac{|u|^p|\nabla v|^p}{[g(v)]^{\frac{p}{p-1}}}
+\frac{g'(v)|u|^p}{[g(v)]^2}|\nabla v|^p
\]
Now since $g$ satisfies \eqref{PC} i.e.,
$g'(x)\geq (p-1)[g(x)]^{\frac{p-2}{p-1}}$ we have $L(u,v)\geq0$.

Equality holds when the following occurs simultaneously:
\begin{gather}
g'(x)= (p-1)[g(x)]^{\frac{p-2}{p-1}},\label{PEC}\\
\frac{|u|^{p-2}u}{g(v)}\nabla u.\nabla v|\nabla v|^{p-2}
= \frac{|u|^{p-1}}{g(v)}|\nabla v|^{p-1}|\nabla u|, \\
\label{hi}
|\nabla u|=\frac{|u\nabla v|}{{g(v)}^{\frac{1}{p-1}}}
\end{gather}
Set
\begin{equation}
 \mathcal{X}=\{x\in\Omega:\frac{|u\nabla v|}{{g(v)}^{\frac{1}{p-1}}}=0\}
\end{equation}
By  \eqref{hi} we have
\begin{equation}\label{ko}
 \frac{|u\nabla v|}{{g(v)}^{\frac{1}{p-1}}}
=|\nabla u|=0\quad\text{a.e. on } \mathcal{X}.
\end{equation}
From \eqref{ko} and \eqref{PEC}, for $g(x)=x^{p-1}$  we have
\begin{equation}\label{pii}
 \frac{u}{v}\nabla v=\nabla u=0\quad \text{a.e. on } \mathcal{X}.
\end{equation}
On $\mathcal{X}^c$, let
\begin{equation}
 w=\frac{|\nabla u|[g(v)]^{\frac{1}{p-1}}}{|u\nabla v|}
\end{equation}
Hence from the fact that $L(u,v)=0$ a.e. in $\Omega$ we have
\begin{equation}\label{odd}
 w^p-pw+p-1=0
\end{equation}
which holds if and only if $w=1$.

Again taking into account \eqref{PEC}, for $g(x)=x^{p-1}$ we have
\begin{equation}\label{pi}
 \nabla u\cdot\big(\nabla u-\nabla v\frac{u}{v})=0\quad
\text{a.e. in }\mathcal{X}^c
\end{equation}
Combining \eqref{pii} and \eqref{pi} we can easily conclude that $L(u,v)=0$
if and only if $\nabla\big(\frac{u}{v})=0$ a.e. in $\Omega$.
\end{proof}

With the generalized Picone's identity in our hands we can now proceed to prove
the Picone's inequality which is the vital ingredient for the proof of
 D\'iaz-Sa\'a Inequality. We will present a non-linear version of the
inequality and will closely follow the proof of Abdellaoui-Peral \cite{AbPe}.

\begin{theorem}[Generalized Picone inequality] \label{PIn}
Let $1<p\leq Q$ and $\Omega$ be a bounded domain in $\mathcal{H}^n$.
If $u,v\in D^{1,p}_0(\Omega)$ such that $-\Delta_p v=\mu$ where $\mu$
is a positive, bounded and measurable function satisfying the hypothesis (III) with
$v|_{\partial\Omega}=0$, $v(\not\equiv 0)\geq0$ and $g$ satisfies \eqref{PC}.
Then we have
\begin{equation}
\int_{\Omega} |\nabla u|^p\geq \int_{\Omega}\Big(\frac{|u|^p}{g(v)}\Big)
(-\Delta_p v).
\end{equation}
\end{theorem}

\begin{remark} \rm
 When $g(u)=u^{p-1}$, we get Picone's inequality in Heisenberg Group in
 Dou \cite{DOU}.
\end{remark}

Before we prove our theorem we need the following lemma.


\begin{lemma}\label{p1}
 Let $p>1$ and $\Omega$ be any domain in $\mathcal{H}^n$ and let
 $v\in D^{1,p}(\Omega)$ be such that $v\geq\delta>0$. Then for all
 $u\in C_c^{\infty}(\Omega)$ we have
\begin{equation}
 \int_{\Omega}|\nabla u|^p
\geq \int_{\Omega}\Big(\frac{|u|^p}{g(v)}\Big)(-\Delta_p v).
\end{equation}
\end{lemma}

\begin{proof}
 Since $v\in D^{1,p}(\Omega)$, we can choose $v_n\in C^1(\Omega)$ such that 
the following holds:
 \begin{equation}
  v_n>\frac{\delta}{2} \text{ in }\Omega,\quad 
  v_n\to v\text{ in } D^{1,p}(\Omega), \quad 
  v_n\to v\text{ a.e. in }\Omega.
 \end{equation}
Employing Theorem \ref{PICN} with $v_n$ and $u$ we have
 \begin{equation}
  \int_{\Omega}R(u,v_n)\geq 0\quad \text{since $R(u,v_n)\geq 0$ a.e. in
$\Omega$ and for all $n\in\mathbf{N}$}.
 \end{equation}
 i.e.,
\[
 \int_{\Omega}|\nabla u|^p\geq \int_{\Omega}\nabla\Big(\frac{|u|^p}{g(v_n)}\Big)
 |\nabla v_n|^{p-2}\nabla v_n
 =\int_{\Omega}\frac{|u|^p}{g(v_n)}(-\Delta_p v_n).
\]

Note that since $-\Delta_p$ is a continuous function from $D^{1,p}(\Omega)$ 
to $D^{-1,p'}(\Omega)$ for $p'=\frac{p}{p-1}$,
we have $-\Delta_p v_n\to -\Delta_p v$ in $D^{1,p}(\Omega)$ and for $g$ 
locally Lipchitz continuous in $(0,\infty)$
we have $g(v_n)\to g(v)$ a.e.
Hence using Lebesgue dominated convergence theorem and the fact that $g$ 
is increasing on $(0,\infty)$ we have
\begin{equation}
 \int_{\Omega}|\nabla u|^p\geq \int_{\Omega}\frac{|u|^p}{g(v)}(-\Delta_p v)
\end{equation}
for any $u\in C^{\infty}_c(\Omega)$.
\end{proof}

Before we proceed with the proof of Theorem \ref{PIn}, we state the 
Strong Maximum Principle from \cite{DOU} which was proved using the Harnack
Inequality of \cite{CaDaGa}.

\begin{lemma}[Strong maximum principle] \label{smp}
 Let $p>1$ and $\Omega\subset \mathcal{H}^n$ be a bounded domain and 
$u\in D^{1,p}_0(\Omega)$ be nonnegative solution of the equation
 \begin{equation}
  -\Delta_p u=h(x,u)\quad \text{in }\Omega;\;u|_{\partial\Omega}=0
 \end{equation}
where $h:\Omega\times\mathbb{R}\to\mathbb{R}$ is a measurable function such 
that $|h(x,u)|\leq C(u^{p-1}+1)$.
 Then $u\equiv0$ or $u>0$ in $\Omega$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref {PIn}]
 Using the Strong Maximum Principle we have $v>0$ in $\Omega$. 
Denote, $v_n(x)=v(x)+\frac{1}{n},\;n\in\mathbb{N}$.
 Thus we have the following:
 \begin{itemize}
  \item $\Delta_p v_n=\Delta_p v$.
  \item $v_n\to v$ a.e in $\Omega$ and in $D^{1,p}(\Omega)$.
  \item $g(v_n)\to g(v)$ a.e in $\Omega$.
 \end{itemize}
Hence using Lemma \ref{p1}  for $u\in C^{\infty}_c(\Omega)$, we have
\begin{equation}
\int_{\Omega}|\nabla u|^p\geq \int_{\Omega}\frac{|u|^p}{g(v)}(-\Delta_p v)
\end{equation}
Now to conclude our theorem for $u\in D_0^{1,p}(\Omega)$, we use 
$u_n\in C_c^{\infty}(\Omega)$
such that $u_n\to u$ in $D^{1,p}_0(\Omega)$. 
Choosing $u_n$ and $v_n$ in Lemma \ref{p1}, we have
\begin{equation}
 \int_{\Omega}|\nabla u_n|^p
\geq \int_{\Omega}\Big(\frac{|u_n|^p}{g(v_n)}\Big)(-\Delta_p v_n)\,.
\end{equation}
Now using the fact that $g$ satisfies \eqref{PC}  by Fatou's lemma, we have
\begin{equation}
 \int_{\Omega}|\nabla u|^p\geq \int_{\Omega}\Big(\frac{|u|^p}{g(v)}\Big)(-\Delta_p v)
\end{equation}
which completes our proof.
\end{proof}

We  conclude this section with the D\'iaz-Sa\'a Inequality in Heisenberg Group. 
For this part we use $g(u)=u^{p-1}$

\begin{theorem}[D\'iaz-Sa\'a inequality]\label{diazsaa}
Let $1<p\leq Q$ and $\Omega$ be a bounded domain in $\mathcal{H}^n$. 
If $u_i\in D^{1,p}_0(\Omega)$ s.t $-\Delta_p u_i=\mu_i$, where 
$\mu_{i}$ is a positive, bounded and measurable function satisfying the hypothesis (III) 
with $u_i|_{\partial\Omega}=0$ and $u_i(\not\equiv 0)\geq0$
a.e. in $\Omega$ for $i=1,2$. Then we have
\begin{equation}
\int_{\Omega} \Big(-\frac{\Delta_p u_1}{u_1^{p-1}}+\frac{\Delta_p u_2}{u_2^{p-1}}
\Big)(u_1^p-u_2^p)\geq0.
\end{equation}
\end{theorem}

Note that above theorem is not true for a general $g$ satisfying \eqref{PC}.

\begin{proof}[Prof of Theorem \ref{diazsaa}]
Choosing $u_i$ for $i=1,2$ satisfying the hypothesis of Theorem \ref{diazsaa} 
and then plugging the couple $(u_1,u_2)$ into Theorem \ref{PIn} we obtain
\begin{equation}
  \int_{\Omega} |\nabla u_1|^p\geq \int_{\Omega}
\Big(-\frac{\Delta_p u_2}{u_2^{p-1}}\Big)u_1^p
 \end{equation}
Using integration by parts on right-hand side, we have
\begin{equation}\label{df}
 \int_{\Omega}\Big(-\frac{\Delta_p u_1}{u_1^{p-1}}
+\frac{\Delta_p u_2}{u_2^{p-1}}\Big)u_1^p\geq0.
\end{equation}
Now interchanging the couple $(u_1,u_2)$ with $(u_2,u_1)$ in Theorem \ref{PIn} we 
obtain
\begin{equation}
  \int_{\Omega} |\nabla u_2|^p
\geq \int_{\Omega}\Big(-\frac{\Delta_p u_1}{u_1^{p-1}}\Big)u_2^p
 \end{equation}
Again using integration by parts on the right-hand side, we have
\begin{equation}\label{dfg}
 \int_{\Omega}\Big(-\frac{\Delta_p u_2}{u_2^{p-1}}
+\frac{\Delta_p u_1}{u_1^{p-1}}\Big)u_2^p\geq0.
\end{equation}
Adding \eqref{df} and \eqref{dfg} we have
\begin{equation}
 \int_{\Omega}\Big(-\frac{\Delta_p u_1}{u_1^{p-1}}
+\frac{\Delta_p u_2}{u_2^{p-1}}\Big)(u_1^p-u_2^p)\geq0.
\end{equation}
Hence the proof is complete.
\end{proof}

\section{Main results}
In this section  we  state and prove our main result.

\begin{theorem}[Uniqueness of a solution] \label{uni}
There exists at most one positive weak solution to \eqref{PC} in 
$D^{1,p}_0(\Omega)\cap L^{\infty}(\Omega)$ for $1<p\leq Q$.
\end{theorem}

\begin{proof}
 Let $u$ and $v$ be two non-negative solutions of \eqref{QH}. 
Then using Lemma \ref{smp} we have, $u,v>0$ in $\Omega$.
 Moreover since $f(x,u)$ is positive and satisfy hypothesis (I) and (III), 
we have for $u\neq v$ by Theorem \ref{diazsaa},
\begin{equation}
0\leq\int_{\Omega}\Big(-\frac{\Delta_p u}{u^{p-1}}
+\frac{\Delta_p v}{v^{p-1}}\Big)(u^p-v^p)=\int_{\Omega}\Big(\frac{f(x,u)}{u^{p-1}}-
\frac{f(x,v)}{v^{p-1}}\Big)(u^p-v^p)<0.
\end{equation}
Hence we arrive at a contradiction.
\end{proof}

We conclude this article with a few comments:
\begin{itemize}
 \item Because of the lack of regularity  we are forced to put the positivity 
condition on $f$, which was not present in the assumptions of Diaz-Saa \cite{DiSa}.
It will be interesting to know if one can conclude the same results for uniqueness 
without the positivity condition on $f$.

\item The restriction on $p$ is due to the fact that 
 Lemma \ref{smp} (Strong Maximum Principle) is valid for $1<p\leq Q$.

\item The statements proved in Theorems \ref{PICN} and  \ref{PIn} were valid 
for a wide range of functions satisfying \eqref{PC}. This results are
 new in the contexts of Heisenberg group and one can actually obtain 
Nonexistence results and Comparison Principles for p-sub-Laplacian
similar to those in \cite{KB} and the reference therein.
\end{itemize}

\subsection*{Acknowledgements}
This work has been carried under the project INSPIRE Faculty Award [IFA-13 MA-29].

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