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

\AtBeginDocument{{\noindent\small
Two nonlinear days in Urbino 2017,\newline
\emph{Electronic Journal of Differential Equations},
Conference 25 (2018), pp. 65--75.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document} \setcounter{page}{65}
\title[\hfilneg EJDE-2018/conf/25\hfil  Harnack inequality for degenerate]
{Harnack inequality for strongly degenerate elliptic operators with natural growth}

\author[G. Di Fazio, M. S. Fanciullo, P. Zamboni \hfil EJDE-2018/25\hfilneg]
{Giuseppe Di Fazio,  Maria Stella Fanciullo, Pietro Zamboni}

\address{Giuseppe Di Fazio  (corresponding author)\newline
Dipartimento di Matematica e Informatica,
Universit\`a di Catania,
Viale A. Doria 6, 95125, Catania, Italy}
\email{giuseppedifazio@unict.it}

\address{Maria Stella Fanciullo \newline
Dipartimento di Matematica e Informatica,
Universit\`a   di Catania,
Viale A. Doria 6, 95125, Catania, Italy}
\email{fanciullo@dmi.unict.it}

\address{Pietro Zamboni\newline
Dipartimento di Matematica e Informatica,
Universit\`a  di Catania,
Viale A. Doria 6, 95125, Catania, Italy}
\email{zamboni@dmi.unict.it}


\subjclass[2010]{35B45, 35B65}
\keywords{Harnack inequality, Stummel class}

\begin{abstract}
 We prove that  positive and bounded weak solutions of a strongly
 degenerate elliptic equation satisfy the Harnack inequality.
 The structure of the differential operator  includes a nonlinear
 term in the gradient with quadratic growth. Moreover, the lower order
 terms belong to some Stummel  classes defined in term of sum operators
 introduced in \cite{fpw2}. %[13]
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}
Recently the regularity of weak solutions of the  equation
\begin{equation} \label{vecchia-eq}
-X_j^*(a_{ij}X_iu +d_ju)+ b_i X_iu+cu = f - X_i^*h_i
\end{equation}
was studied in \cite{dfz_CPAA}. Here $X=(X_1,X_2,\dots, X_m)$ is a system of
first-order  locally Lipschitz vector fields in $\mathbb{R}^n$, and
the lower order terms  belong to suitable Stummel classes modeled on
 a special geometry introduced in \cite{fpw2}.
Namely, local boundedness and continuity of the weak solutions have been proved.
Later, in \cite{dfz_DIE_2018}, the results  in \cite{dfz_CPAA}
were generalized to a class of operators satisfying a weighted degeneracy
condition. There, the principal part of \eqref{vecchia-eq} is controlled by
 a $A_2$ Muckenhoupt weight, and the operators considered in 
\cite{dfz_CPAA, dfz_DIE_2018} have a controlled growth in the gradient.
 As it is well known, this implies that any weak solution is locally bounded.

 In this note we study an operator that is similar to those  in
\cite{dfz_CPAA,dfz_DIE_2018} but very different in the growth.
In fact, the operator considered here satisfies a quadratic growth with
respect to the gradient. It is worth to remark that adding such a term
destroy the local boundedness property of the weak solutions.
This phenomenon forces us to assume that the weak solutions are locally
bounded and then we can show regularity only for the bounded solutions.

To state our result, let us consider the  equation
\begin{equation} \label{eq-con-termine-quadratico}
-X_j^*(a_{ij}X_iu +d_ju)+ \frac{b_0}{\lambda}w |Xu|^2+ b_i X_iu+cu
= f - X_i^*h_i\,,
\end{equation}
where $X$ is as before and the coefficients $a_{ij}$ satisfy weighted
ellipticity condition with respect to  a Muckenhoupt $A_2$ weight. Assuming
the lower order terms   in appropriate weighted Stummel  classes
(see section \ref{par2} for definitions) we prove that the \textit{bounded}
positive solutions of equation \eqref{eq-con-termine-quadratico} satisfy a
Harnack inequality.
As a consequence, this will imply that the \textit{bounded} weak solutions
are continuous.

Some comments are now in order.
We  assume that $X$ satisfies  the following (1-2) weighted  Poincar\'e inequality
\begin{equation*}%\label{poincare}
\frac{1}{w(B)}\int_{B}|u-u_B|w\,dy
\le C_P  r\Big(\frac{1}{w(B)}\int_{B}|Xu|^2wd\,y\Big)^{1/2}
\quad
\forall u \in C^\infty\,.
\end{equation*}

As showed by Franchi Perez and Wheeden \cite{fpw2},
the above Poincar\'e inequality implies the following subrepresentation formula
\begin{equation*}
|u(x)-u_{B_0}|\le C   \sum_{j=0}^\infty r(B_j(x))
\Big(\frac{1}{w(B_j(x))}\int_{B_j(x)}|Xu|^2w(y)dy\Big)^{1/2}
\end{equation*}
where $\{B_j(x)\}_{j=1}^\infty$  is  special chain of balls related to a
fixed ball $B_0$.
The subrepresentation formula allowed us to prove
a Fefferman type inequality and to define suitable Stummel classes modeled
on  the geometry introduced in \cite{fpw2} (for more details see \cite{dfz5}).

The Fefferman inequality is a fundamental tool in our proof.
Indeed, following the classical pattern in Trudinger paper \cite{trudi}
(analogous results in different settings are shown in
\cite{dfz1,dfz3,27,28})
we are faced with products  between lower order terms and test functions.
Due to the low integrability of the lower order terms we use the Fefferman
inequality to complete the iteration process and obtain our results.

\section{Stummel type classes and  Fefferman  inequality} \label{par2}

Let $X=(X_1,X_2,\dots, X_m)$ be a system of locally Lipschitz vector fields
in $\mathbb{R}^n$ and  $d$  the associated  Carnot-Carath\'eodory distance.
We assume that $d$  is finite for any $x, y \in \mathbb{R}^n$ and denote by
$B=B_r=B(x,r)$ the Carnot-Carath\'eodory ball centered at $x$ of radius $r$.
Let us recall the definition of Muckenhoupt weight $A_p$.

\begin{definition}[$A_p$ Muckenhoupt weights] \rm
Let $w$ be a non negative and locally integrable function in $\mathbb{R}^n$
and $1<p<+\infty$.
We say that $w$ is an $A_p$ weight if
$$
[w]_p \equiv
\sup_B\Big( {\frac{1}{|B|}}\int_Bw(x)\,dx \Big)
\Big({\frac{1}{|B|}}\int_B [w(x)]^{\frac{-1}{p-1}} \,dx\Big)^{p-1} <+\infty
$$
where the supremum is taken over all metric balls $B$ in $\mathbb{R}^n$.
The number $[w]_p$ is called the $A_p$ constant of $w$.
\end{definition}

Throughout this article  we  assume the following.
\begin{itemize}
\item[(A1)] The distance $d$ is continuous with respect to the
Euclidean distance in $\mathbb{R}^n$.
\item[(A2)] There exists  a positive constant $C_D$ such that
$$
|B(x,2r)|\le C_D|B(x,r)|
\quad \forall x\in \mathbb{R}^n,  r>0 \,.
$$
\item[(A3)]
If $B_0$ is a given ball in $\mathbb{R}^n$ and $w\in A_2$,
there exists a positive constant $C_P$ such that
\begin{equation*}%\label{poincare}
\frac{1}{w(B)}\int_{B}|u-u_B|w\,dy
\le C_P  r\Big(\frac{1}{w(B)}\int_{B}|Xu|^2w\,dy\Big)^{1/2}
\end{equation*}
for all $B\subset B_0$ and all  $ u\in C^\infty(\overline B_0)$.
Here $u_B=\frac{1}{w(B)}\int_B u\, w\,dy$, $w (B)=\int_{B}w \,dy$
and $r$ is the radius of $B$.
\end{itemize}
The number $Q=\log_2C_D$ will be called homogeneous dimension of $\mathbb{R}^n$.

To state and prove our results we need to define the Sobolev classes
with respect to the measure $w\,dx$ where $w\in A_2$.

\begin{definition}[Sobolev spaces] \rm
Let $w\in A_2$  and $\Omega$ be a bounded domain in $\mathbb{R}^n$.
We say that $u$ belongs to $W^{1,2}(\Omega,w)$ if $u$,  $X_iu\in L^2(\Omega,w)$
for any $i=1,\dots m$.
Moreover, we denote by $W_0^{1,2}(\Omega,w)$ the closure of the smooth and
compactly supported functions in $W^{1,2}(\Omega,w)$
with respect to the norm
\begin{equation*}
\|u\|_{W^{1,2}(\Omega,w)}
=\|u\|_{L^2(\Omega,w)} +\sum_{i=1}^m\|X_iu\|_{L^2(\Omega,w)}\,.
\end{equation*}
and we say that $u$ belongs to $W^{1,2}_{\rm loc}(\Omega,w)$ if
$u\in W^{1,2}(\Omega',w)$ for any  $\Omega' \subset\subset \Omega$.
\end{definition}

We recall the useful embedding Theorem for Sobolev spaces
(see \cite{fhk,h,fssc}).


\begin{theorem}\label{sobolev}
Let $w\in A_2$  and $K$ be a compact subset of $\Omega$. Then there exist
$r_0>0$, $q_0>2$ and $C$ depending on $K$, $\Omega$ and $\{X_j\}$ such that
for any metric ball $B=B(x,r)$, $x\in K$, we have
\begin{equation*}
\Big(\frac{1}{w(B)}\int_B|u-u_B|^q w\,dy \Big)^{1/q}
\le C r\Big(\frac{1}{w(B)}\int_B |Xu|^2 w\,dy \Big)^{1/2}\,,
\quad \forall u\in C^\infty(\overline B)
\end{equation*}
provided  $0<r<r_0$ and $2<q<q_0$.
\end{theorem}

The following definition is useful for stating the subrepresentation
formula that we will use later (see \cite{fpw2}).

\begin{definition}\label{chain} \rm
Given $B_0=B(x_0,r)$ and $x\in B_0$,  let us  denote by $\{B_i\}=\{B_i(x)\}_{i=1}^\infty$ a chain of balls, of radius $r(B_i)$, such that
\begin{itemize}

\item[(H1)] $B_i\subset B_0$ for all $i\ge 0$

\item[(H2)] $r(B_i)\sim 2^{-i}r(B_0)$ for all $i\ge 0$

\item[(H3)] $\rho(B_i,x)\le C r(B_i)$ for all $i\ge 0$

\item[(H4)] for all $i\ge 0$, $B_i\cap B_{i+1}$ contains a ball $S_i$ with $r(S_i)\sim r(B_i)$.
\end{itemize}
\end{definition}


\begin{theorem}\label{representation}
 Given a weight $w\in A_2$ and a ball $B$ let $\{B_j(x)\}_{j=1}^\infty$
be a chain of balls as in Definition \ref{chain}.
Let $u\in  W^{1,2}(B_0,w)$ be such that for any ball $B\subset B_0$
\begin{equation}\label{ipo1}
\frac{1}{w (B)}\int_B|u-u_B|w\,dx\le C
s \Big(\frac{1}{w(B)}\int_{B}|Xu|^2 w\,dy\Big)^{1/2}\,
\end{equation}
where $s$ is the radius of $B$.
Then there exists $C'>0$ such that
\begin{equation*}
|u(x)-u_{B_0}|\le C'   \sum_{j=0}^\infty r(B_j(x))
\Big(\frac{1}{w(B_j(x))}\int_{B_j(x)}|Xu|^2w(y)dy\Big)^{1/2}
\end{equation*}
where $C'$ is a geometric constant which also depends on $C$.
\end{theorem}

Since we are interested to prove our result assuming low integrability
properties on the lower order term we introduce the Stummel and Morrey
classes adapted to our setting.

\begin{definition}[Stummel and Morrey classes] \rm
Let $w\in A_2$, $B_0$ be a ball and  $\{B_j(x)\}_{j=1}^\infty$ be a chain of
balls as in Definition \ref{chain}. We say that
$V\in L^1_{\rm loc}(\mathbb{R}^n,w)$ belongs to the class
$\tilde S({\mathbb{R}^n},w)$ if
$$
\eta_V(r)\equiv \sup_{x_0\in \mathbb{R}^n}
\sup_{y\in B_0}\int_{B_0}\sum_{j=0}^\infty
\frac{r^2(B_j(x))|V(x)|}{w(B_j(x))}\chi_{B_j(x)}(y)w(x)dx
$$
is finite for all $r>0$.
We say that $V$ belongs to  $S({\mathbb{R}^n},w)$ if, in addition, we have
$\lim_{r\to 0}\eta_V(r)=0$.
We say that $V \in S'(\mathbb{R}^n,w)$
if there exists $\delta>0$ such that
$$
\int_0^\delta \frac{\eta_V(t)}{t}dt<+\infty.
$$
We say that $V$ belongs to the Morrey space $M_\sigma(\mathbb{R}^n,w)$ if
there exist $C>0$ such that $\eta_V(r)\le C r^\sigma$.
\end{definition}

We close this section giving the proof of the weighted embedding result.
As we have already noted, it will allow us to get our main results.
The unweighted result and some corollaries has been proven in \cite{dfz5}
(see also \cite{cf}, \cite{dz2,dz4,l3,z1,zmanu1,z2}). Here we extend the
embedding to the weighted case.

\begin{theorem}\label{emb}
Let $w\in A_2$, $B_0$ be a ball and $V$  a function in $ \tilde S(\mathbb{R}^n,w)$.
Then, there exists a positive constant $C$ such that
\begin{equation*}
\int_{B_0}|V(x)|\,|u(x)-u_{B_0}|^2 w \,dx
\le  C \eta_V(r) \int_{B_0} |Xu(x)|^2 w\, dx
\end{equation*}
for any  $u\in C^\infty(B_0)$.
\end{theorem}

\begin{proof}
Let $u$ be a smooth function in $B_0$.
Theorem \ref{representation} yields  the following subrepresentation formula for $u$
\begin{equation}\label{representation2}
|u(x)-u_{B_0}|\le C \sum_{j=0}^\infty r(B_j(x))
\Big(\frac{1}{w(B_j(x))}\int_{B_j(x)}|Xu|^2w(y)dy\Big)^{1/2}
\end{equation}
for a.e. $x\in B_0$.
Now from \eqref{representation2} and H\"older inequality
\begin{align*}
&\int_{B_0}|V(x)||u(x)-u_{B_0}|^2w(x)dx \\
&\le \int_{B_0}|V(x)||u(x)-u_{B_0}|\sum_{j=0}^\infty  r(B_j(x))\\
&\quad\times \Big[ \frac{1}{w(B_j(x))}\int_{B_j(x)}|Xu(y)|^2 w(y)dy\Big]^{1/2}
 w(x)dx \\
&\le \Big[\int_{B_0}|V(x)||u(x)-u_{B_0}|^2w(x)dx  \Big]^{1/2} \\
&\quad\times \Big[ \int_{B_0}\sum_{j=0}^\infty |V(x)|\frac{r^2(B_j(x))}{w(B_j(x))}
 \int_{B_j(x)}|Xu(y)|^2w(y)dyw(x)dx\Big]^{1/2} \\
&\le \Big[\int_{B_0}|V(x)||u(x)-u_{B_0}|^2w(x)dx  \Big]^{1/2} \\
&\quad\times \Big[ \int_{B_0}\sum_{j=0}^\infty |V(x)|\frac{r^2(B_j(x))}{w(B_j(x))}
 \int_{B_0}|Xu(y)|^2\chi_{B_j(x)}(y) w(y)dyw(x)dx\Big]^{1/2}\\
&\le \Big[\int_{B_0}|V(x)||u(x)-u_{B_0}|^2w(x)dx  \Big]^{1/2} \\
&\quad\times \Big[ \int_{B_0}|Xu(y)|^2\int_{B_0}\sum_{j=0}^\infty |V(x)|
 \frac{r^2(B_j(x))}{w(B_j(x))}\chi_{B_j(x)}(y) w(x)dxw(y)dy\Big]^{1/2} \\
&\le \Big[\int_{B_0}|V(x)||u(x)-u_{B_0}|^2w(x)dx  \Big]^{1/2}\eta^{1/2}_V(r) \\
&\quad\times \Big[\int_{B_0}|Xu(y)|^2w(y)dy\Big]^{1/2}
\end{align*}
from which
\begin{equation*}
\int_{B_0}|V(x)||u(x)-u_{B_0}|^2w(x)dx\le C\eta_V(r)\int_{B_0}|Xu(x)|^2w(x)dx\,.
\end{equation*}
\end{proof}

From Theorem \ref{emb}  we obtain the following  corollaries.

\begin{corollary}\label{corollarioutilizzato}
Let  $V$ be a function in $ \tilde S(\mathbb{R}^n,w)$.
Then, there exists a positive constant $C$
such that
\begin{equation*}
\int_{\mathbb{R}^n}|V(x)|\,|u(x)|^2w\,dx \le  C \eta_V(r) \int_{\mathbb{R}^n}
|Xu(x)|^2w \, dx
\end{equation*}
for any compactly supported smooth function $u$ in $\mathbb{R}^n$.
\end{corollary}

\begin{corollary}\label{piccolograndestummel}
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $V$ in $S(\Omega,w)$.
Then, for any $\varepsilon >0$ there exists a positive function
$K(\varepsilon)\sim\frac{\varepsilon}{[\eta_V^{-1}(\varepsilon)]^{Q+2}}$
 (where $\eta_V^{-1}$ is the inverse function of $\eta_V$), such that
\begin{equation}\label{mezzopiccolo}
\int_{\Omega}|V(x)|\,|u(x)|^2w\,dx \le \, \varepsilon  \int_{\Omega}
|Xu(x)|^2 \, dx+K(\varepsilon)\int_{\Omega}|u(x)|^2\,dx
\end{equation}
for any compactly supported smooth function $u$ in $\Omega$.
\end{corollary}

\section{Harnack inequality for  strongly degenerate equations}

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Let $X=(X_1,X_2,\dots, X_m)$
be a system of locally Lipschitz vector fields in $\mathbb{R}^n$.
For $i=1,2,\dots,m$ we denote by $X^*_i$ the formal adjoint of the vector fields
$X_i$.
Let $\{a_{ij}(x)\}$ be a symmetric matrix of measurable functions in $\Omega$
 satisfying the  weighted ellipticity condition: There exists $\lambda >0$ such that
\begin{equation} \label{ellipcondition}
 \lambda^{-1}w(x)|\xi|^2\le a_{ij}(x)\xi_i\xi_j\le \lambda w(x)|\xi|^2\quad
\text{a.e. $x\in\Omega$ and } \forall \xi\in \mathbb{R}^m
\end{equation}
for some Muckenhoupt weight $w\in A_2$.

Let us consider the  strongly degenerate elliptic equation in divergence form
\begin{equation}\label{eqperharnack}
-X_j^*(a_{ij}X_iu +d_ju)+ \frac{b_0}{\lambda}w |Xu|^2+ b_i X_iu+cu = f - X_i^*h_i\,,
\end{equation}
where
\begin{equation}\label{ipotesicoefficienti}
b_0\in \mathbb{R}\setminus\{0\},\quad \big({\frac{b_i}{  w}}\big)^2, \quad
{\frac{c}{w}}, \quad \big({\frac{d_i}{ w}}\big)^2, \quad
{\frac{f}{w}}, \quad \big({\frac{h_i}{ w}}\big)^2
\in S'(\Omega,w)\,.
\end{equation}

To begin with we give the definition of weak super, sub solutions and solutions.

\begin{definition}[Weak supersolutions, subsolutions, solutions] \rm
Let $w\in A_2$ and $u\in W^{1,2}_{\rm loc}(\Omega,w)$.
We say that $u$ is a local weak supersolution (subsolution) of \eqref{eqperharnack}
if for any $ \varphi \in W^{1,2}_0(\Omega,w)$,  $\varphi\ge 0$
\begin{align*}
&\int_{\Omega} \Big[(a_{ij}X_{i}u+d_ju)X_{j}\varphi
 +\Big({\frac{b_0}{\lambda}}w|X u|^2 + b_i X_iu+cu \Big)\varphi \Big]dx \\
&\ge (\le) \int_{\Omega}(f\varphi +  h_iX_i\varphi)dx\,.
\end{align*}
We say that $u\in W^{1,2}_{\rm loc}(\Omega,w)$ is a local weak solution
 of \eqref{eqperharnack} if it is both a supersolution and a subsolution.
\end{definition}

Our first result is the weak Harnack inequality for supersolutions of
 \eqref{eqperharnack}. We follow the pattern drawn in \cite{trudi}.

\begin{theorem}\label{harnackinterno}
Let us assume conditions \eqref{ellipcondition} and \eqref{ipotesicoefficienti}
are satisfied, $w\in A_2$, and let $u$ be a weak nonnegative
supersolution of equation \eqref{eqperharnack} in a ball
$B_{3r}\subset\subset\Omega$.
Let $M>0$ be a constant such that $ u\le M$ in $B_{3r}$. Then, there
exists $C$ depending on $Q$, $M$, $\lambda$ and the $A_2$ constant of $w$, such that
\begin{align*}
&w^{-1} (B_{2r})\int_{B_{2r}}u\,w dx\\
&\le C \Big\{ {\rm min}_{B_r}u+r^\sigma \|\frac{f}{w}\|_{\sigma,B_{3r}}
+\Big(r^\sigma \sum_{i=1}^n \|\big({\frac{h_i}{w}}\big)^2\|_{\sigma,B_{3r}}\Big)^{1/2}
\Big\}.
\end{align*}
\end{theorem}

\begin{proof}
Let
\[
k=\|{\frac{f}{ w}}\|_{\sigma,B_{3r}}+
\Big(\sum_{i=1}^n \|\big({\frac{h_i} {w}}\big)^2\|_{\sigma,B_{3r}}\Big)^{1/2}
\]
and $v=u+k$.
For $\eta \in C^1_0(B_{3r})$, $\eta \ge 0$, we set
$\varphi(x)=\eta^2(x)v^\beta(x) e^{- |b_0|v(x)}$, $\beta < 0$, as a test
function in \eqref{eqperharnack}.
Since $u$ is a supersolution in $B_{3r}$ of \eqref{eqperharnack} we have
\begin{align*}
&\int_{B_{3r}}\Big[2\eta(a_{ij}X_iu+d_ju-h_j)X_j\eta v^{\beta}e^{- |b_0|v}\\
&+(-|\beta|v^{\beta-1}-|b_0|v^\beta)\eta^2e^{- |b_0|v}(a_{ij}X_i u+d_ju-h_j)X_j v\\
&+{\frac{b_0}{\lambda}}w |Xu|^2\eta^2v^\beta e^{- |b_0|v}
 + (b_i X_i u+cu -f )\eta^2 v^\beta e^{- |b_0|v} \Big]dx \ge 0
\end{align*}
and
\begin{align*}
&\int_{B_{3r}}\eta^2e^{- |b_0|v}(b_0v^\beta+|\beta|v^{\beta-1})|Xv|^2w dx\\
&\le \int_{B_{3r}}\eta^2e^{- |b_0|v}(|b_0|v^\beta+|\beta|v^{\beta-1})|Xv|^2w dx \\
&\le  \lambda\int_{B_{3r}}\eta^2e^{-|b_0|v}(|b_0|v^\beta+|\beta|v^{\beta-1})a_{ij}
 X_ivX_jv\,dx \\
&\le \lambda \int_{B_{3r}}\eta^2 e^{- |b_0|v}(|\beta|v^{\beta-1}
 +|b_0|v^\beta)(h_j-d_ju)X_jv\,dx \\
&\quad+2\lambda \int_{B_{3r}}\eta(a_{ij}X_iv+d_ju-h_j)X_j\eta v^\beta e^{-|b_0|v}dx \\
&\quad + \int_{B_{3r}}b_0w |Xv|^2\eta^2 v^\beta e^{- |b_0|v} dx \\
&\quad +\lambda \int_{B_{3r}}(b_iX_iv+cu -f)\eta^2 v^\beta e^{- |b_0|v} dx\,.
\end{align*}
From this inequality it follows that
\begin{align*}
&\int_{B_{3r}}\eta^2e^{- |b_0|v}|\beta|v^{\beta-1}|Xv|^2w dx \\
&\le \lambda \int_{B_{3r}}\eta^2 e^{- |b_0|v}(|\beta|v^{\beta-1}+|b_0|v^\beta)
(h_j-d_ju)X_jv\,dx \\
&+ 2\lambda \int_{B_{3r}}\eta(a_{ij}X_i v+d_ju-h_j)X_j\eta v^\beta e^{- |b_0|v}dx \\
&+\lambda \int_{B_{3r}}(b_iX_iv+c u -f)\eta^2 v^\beta e^{- |b_0|v} dx\,.
\end{align*}
Since $v$ is bounded, we may drop the exponential to obtain
\begin{align*}
&\int_{B_{3r}}\eta^2|\beta|v^{\beta-1}|Xv|^2 w dx \\
&\le C(M,b_0)\Big[2\lambda \int_{B_{3r}}\eta a_{ij}X_ivX_j\eta v^\beta dx
 +\lambda |\beta|\int_{B_{3r}} |d_j||X_jv|v^\beta\eta^2dx \\
&\quad + 2\lambda \int_{B_{3r}}|d_j|v^{\beta+1}X_j\eta\eta dx
 +2\lambda \int_{B_{3r}}|h_j|v^{\beta}X_j\eta\eta dx
 + \lambda \int_{B_{3r}} |b_i||X_iv\eta ^2 v^\beta \\
&\quad +\lambda  \int_{B_{3r}}|c|\eta^2 v^{\beta+1}dx
 +\lambda   \int_{B_{3r}} |f|\eta^2v^\beta dx \\
&\quad +\lambda |\beta| \int_{B_{3r}}h_jX_jvv^{\beta-1}\eta^2dx
 +\lambda  \int_{B_3r}|d_j||v_{x_{i}}|\eta^2v^\beta dx\Big]\,.
\end{align*}
Now, set
$$
V= \sum_{i=1}^n{\frac{|b_i|^2} { w}}+|c|+\sum_{j=1}^n{\frac{|d_j|^2} { w}}+
k^{-1}|f|+k^{-2}\sum_{i=1}^n{\frac{|h_i|^2 }{ w}}\,.
$$
Using Young's inequality yields
\begin{equation}\label{4}
\begin{aligned}
&\int_{B_{3r}}\eta^2v^{\beta-1}|Xv|^2 w dx  \\
&\le  C(M, b_0, \lambda)\Big[\frac{|\beta|+1}{\beta^2}
 \int_{B_{3r}}v^{\beta+1}|X\eta|^2 w dx
 + \big(\frac{|\beta|+1}{\beta}\big)^2
 \int_{B_{3r}}V\eta^2 v^{\beta+1} dx\Big] \\
&\le  C(M, b_0, \lambda) \big(\frac{|\beta|+1}{\beta}\big)^2
 \Big[\int_{B_{3r}}v^{\beta+1}|X\eta|^2 w dx+ \int_{B_{3r}}
 V\eta^2 v^{\beta+1} dx\Big] .
\end{aligned}
\end{equation}

Now we set
\begin{equation*}
\mathcal{U}(x) =
\begin{cases}
v^{\frac{\beta+1}{2}}(x) & \text{if }  \beta\neq -1\\
\log v(x) & \text{if }  \beta=-1
\end{cases}
\end{equation*}
and by \eqref{4} we have
\begin{equation} \label{eq:betadiverso}
\begin{aligned}
&\int_{B_{3r}}\eta^2|X \mathcal{U}|^2 w\,dx \\
&\le C (\beta+1)^2\Big(\frac{|\beta|+1}{\beta}\Big)^2
\Big\{\int_{B_{3r}}|X \eta|^2\mathcal{U}^2 w\,dx
+\int_{B_{3r}}V\eta^2 \mathcal{U}^2\,dx\Big\}\,, \quad \beta \neq -1
\end{aligned}
\end{equation}
while
\begin{equation} \label{eq:betauguale}
\int_{B_{3r}}\eta^2|X \mathcal{U}|^2w\,dx
\le C\Big\{\int_{B_{3r}}|X\eta|^2w\,dx
+\int_{B_{3r}}V\eta^2\,dx\Big\}
\end{equation}
if  $\beta=-1$.

Let us start with the case $\beta = -1$. By Corollary \ref{corollarioutilizzato}
we have
\begin{equation*}
\int_{B_{3r}}\eta^2|X \mathcal{U}|^2w\,dx
\le C\Big( \int_{B_{3r}}|X \eta|^2w\,dx+ \int_{B_{3r}}\eta^2w dx\Big)\,.
\end{equation*}

Let $B_h$ be a ball contained in $B_{2r}$. Choosing
$\eta(x)$ so that $\eta(x)=1$ in $B_h$, $0\le\eta\le1$ in
$B_{3r}\setminus B_h$ and $|X\eta|\le \frac{3}{h}$,
we obtain
\begin{equation*}
\|X \mathcal{U}\|_{L^2(B_h,w)}
\le C\frac{w(B_h)^{1/2}}{ h}\,.
\end{equation*}
By  Theorem \ref{sobolev}  and John-Nirenberg lemma
(see \cite{b}) we have $\mathcal{U}(x) = \log v(x) \in BMO$.
Then there exist two positive constants $p_0$ and $C$, such that
\begin{equation} \label{eq:pesoexp}
\Big(-\hspace{-0.38cm}\int_{B_{2r}}e^{p_0 \mathcal{U}}w\,dx\Big)
^{1/p_0} \Big(-\hspace{-0.38cm}\int_{B_{2r}} e^{-{p_0 \mathcal{U}}} w\,dx\Big) ^{1/ p_0} \le C\,.
\end{equation}

Let us consider the family of seminorms
\begin{equation*}
\Phi(p,h)=\Big(\int_{B_h}|v|^pw\,dx\Big)^{1/p}\,,
\quad p \neq 0\,.
\end{equation*}
By \eqref{eq:pesoexp} we have
\begin{equation*}
\frac{1}{w(B_{2r})^{1/p_0}}\Phi(p_0,2r)
\le Cw(B_{2r})^{1/p_0} \Phi(-p_0,2r)\,.
\end{equation*}
Now we consider $\beta \neq 1$ (see inequality \eqref{eq:betadiverso}).
By Corollary  \ref{piccolograndestummel} we obtain
\begin{equation}
\begin{aligned}
&\int_{B_{3r}}|X\mathcal{U}|^2\eta^2w\,dx\\
&\le C \Big\{\big[ \big( \frac{\beta+1}{2}\big)^2+1\big]
\big(1+{\frac{1}{ |\beta|}}\big)^2
\int_{B_{3r}}|X \eta|^2\mathcal{U}^2w\,dx \\
&\quad + \Big[{\frac{1}{ \phi^{-1}\Big(\frac{V}{w};\big(\frac{\beta+1}{2}\big)^{-2}
\big(1+{\frac{1}{|\beta|}}\big)^{-2}\Big) }}\Big]^{Q+2}
\int_{B_{3r}}\eta^2\mathcal{U}^2w\,dx\Big\}\,.
\end{aligned}
\end{equation}
By  Theorem \ref{sobolev} we have
\begin{equation}
\begin{aligned}
&\Big(\int_{B_{3r}}|\eta \mathcal{U}|^{\tau p}w\,dx\Big)^{1/\tau} \\
&\le c w(B_{3r})^{\frac{1}{\tau}-1}
\Big\{ \big[\big(  \frac{\beta+1}{2} \big)^2 + 2\big]
\big(1+{\frac{1}{ |\beta|}}\big)^2  \int_{B_{3r}}|X \eta|^2\mathcal{U}^2w\,dx \\
&\quad + \Big[{\frac{1}{ \phi^{-1}
\Big(\frac{V}{w};\big(\frac{\beta+1}{2}\big)^{-2}\big(1+{\frac{1}{
|\beta|}}\big)^{-2}\Big)} }\Big]^{Q+2}
\int_{B_{3r}}\eta^2\mathcal{U}^2w\,dx\Big\}
\end{aligned}
\end{equation}
where $c$ is a positive constant independent of $w$.

Now we choose the function $\eta$. For $r_1$ and $r_2$ such that
$r\le r_1 <r_2 \le 2r$ we choose $\eta$ such that $\eta(x) =1$ in
$B_{r_1}$, $0\le \eta(x) \le 1$ in $B_{r_2}$, $\eta(x)=0$ outside
$B_{r_2}$, $|X \eta| \le {\frac{c}{r_2-r_1}}$ for some fixed constant $c$.
Then we have
\begin{align*}
&\Big(\int_{B_{r_1}}\mathcal{U}^{2\tau}w\,dx\Big)^{1/\tau}\\
&\le c w(B_{3r})^{\frac{1}{\tau}-1}
\frac{1}{(r_2 -r_1)^2}\Big[\big(\frac{\beta+1}{2}\big)^2 + 2\Big] \\
&\quad\times
\big(1+{\frac{1}{ |\beta|}}\big)^2 \Big[{\frac{1}{  \phi^{-1}
\Big(\frac{V}{w};\big(\frac{\beta+1}{2}\big)^{-2}\big(1+{\frac{1}{
|\beta|}}\big)^{-2}\Big)}}\Big]^{Q+2} \int_{B_{r_2}}\mathcal{U}^2w\,dx\,.
\end{align*}
Setting $\gamma =  \beta + 1$ and recalling that
$\mathcal{U}(x)=v^{\frac{\beta+1}{2}}(x)$, we obtain
\begin{equation}\label{eq:iterazione_negativa}
\begin{aligned}
\Phi (\tau \gamma,r_1)
&\ge c^{1/\gamma} w(B_{3r})^{\frac{1}{\gamma}(\frac{1}{\tau}-1)}
\Big[\big(\frac{\beta+1}{2}\big)^2 + 2\Big]^{1/\gamma} \\
&\quad\times \Big[{\frac{1}{  \phi^{-1}
\Big(\frac{V}{w};\big(\frac{\beta+1}{2}\big)^{-2}\Big)}}\Big]^{\frac{Q+2}{\gamma}}
{\frac{1}{(r_2-r_1)^{2/\gamma}}} \Phi (\gamma,r_2)\,,
\end{aligned}
\end{equation}
for negative $\gamma$.

We are going to iterate the inequality just obtained.
Setting $\gamma_i = \tau^i p_0$ and $r_i = r + {\frac{r}{ 2^i}}$, $i=1,2,\dots$
iteration of \eqref{eq:iterazione_negativa} and use of
\cite[Lemma 3.4]{ragzam} yield
\begin{equation*}
\Phi(-\infty,r)\ge c\big(\phi_\frac{V}{w},\operatorname{diam}\Omega\big)
w(B_{3r})^{1/p_0} \Phi(-p_0,2r)\,.
\end{equation*}
Therefore, by H\"older inequality,
\begin{equation*}
\Phi(p_0',2r)
\le \Phi(p_0,2r)w(B_{3r})^{{\frac{1} {p_0'}}-{\frac{1}{ p_0}}}\,,
\quad p'_0\le p_0
\end{equation*}
so we obtain
\begin{equation*}
w^{-1}(B_{2r})\Phi(1,2r)\le c\Phi(-\infty,r)
\end{equation*}
 and the result follows.
\end{proof}

The following weak Harnack inequality for subsolutions can be obtained
in a similar way.

\begin{theorem}\label{harnacksottosoluzioni}
Let $u$ be a weak nonnegative subsolution of \eqref{eqperharnack} in
$B_{3r}\subset\subset\Omega$. Assume \eqref{ellipcondition} and
\eqref{ipotesicoefficienti}. Let $M>0$ be a constant such that
$ u\le M$ in $B_{3r}$. Then there exists $C$ depending on $Q$, $M$, $\lambda$
and the $A_2$ constant of $w$, such that
\[
\max_{B_r}u
\le C \Big\{ w^{-1} (B_{2r})\int_{B_{2r}}u\,w dx+r^\sigma
\|\frac{f}{w}\|_{\sigma,B_{3r}}
\Big(r^\sigma \sum_{i=1}^n
\|\big({\frac{h_i} { w}}\big)^2\|_{\sigma,B_{3r}}\Big)^{1/2}\Big\}.
\]
\end{theorem}

Now, from our previous results, we obtain the Harnack inequality for solutions.

\begin{theorem}\label{harnacksoluzioni}
Let us assume conditions \eqref{ellipcondition} and \eqref{ipotesicoefficienti} are
 satisfied, $w\in A_2$, and let $u$ be a weak nonnegative supersolution of
 \eqref{eqperharnack} in a ball $B_{3r}\subset\subset\Omega$.
Let $M>0$ be a constant such that $ u\le M$ in $B_{3r}$.	
Then, there exists $C$ depending on $Q$, $M$, $\lambda$ and the $A_2$ constant
of $w$ such that
\begin{equation*}
\max_{B_r}u \le  C \Big\{\min_{B_r}u +r^\sigma \|\frac{f}{w}\|_{\sigma,B_{3r}}
+\Big(r^\sigma \sum_{i=1}^n
\|\big({\frac{h_i} {w}}\big)^2\|_{\sigma,B_{3r}}\Big)^{1/2}\Big\}.
\end{equation*}
\end{theorem}

As a consequence of Harnack inequality we can show that the weak solutions
of \eqref{eqperharnack} are continuous with respect to the Carnot-Carath\'eodory
metric.

\begin{theorem}
Let us assume conditions \eqref{ellipcondition} and \eqref{ipotesicoefficienti} are
satisfied, $w\in A_2$. Let $u$ be a weak  solution of  \eqref{eqperharnack} in
$\Omega$ and let $\sup_\Omega |u|=L<+\infty$.
Then $u$ is continuous in $\Omega$.
\end{theorem}

The next result is a natural consequence of the previous one if we assume
the lower order terms to belong to the Morrey classes $M_\sigma$.

\begin{theorem}
Let us assume condition \eqref{ellipcondition} is  satisfied, 
$w\in A_2$. Let $u$ be a weak  solution of \eqref{eqperharnack}
in $\Omega$, let $\sup_\Omega |u|=L<+\infty$ and moreover
$$
\big({\frac{b_i}{  w}}\big)^2,\quad
{\frac{c}{w}}, \quad \big({\frac{d_i}{ w}}\big)^2, \quad
{\frac{f}{w}}, \quad \big({\frac{h_i}{ w}}\big)^2
\in M_\sigma(\Omega,w)\,.
$$
Then $u$ is locally H\"older continuous in $\Omega$.
\end{theorem}


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\end{document}