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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 158, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/158\hfil Unique continuation]
{Unique continuation of positive solutions for doubly degenerate
quasilinear elliptic equations}

\author[G. Di Fazio, M. S. Fanciullo, P. Zamboni  \hfil EJDE-2017/158\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}

\dedicatory{Communicated by Giovanni Molica Bisci}

\thanks{Submitted May 17, 2017. Published June 29, 2017.}
\subjclass[2010]{35J70, 35B65}
\keywords{Grushin operator; strong $A_{\infty}$ weights; Stummel-Kato classes;
\hfill\break\indent unique continuation}

\begin{abstract}
 We consider quasilinear elliptic equations that are degenerate in two
 ways. One kind of degeneracy is due to the particular structure
 of the given vector fields. Another is a consequence of the weights
 that we impose to the quadratic form of the associated differential operator.
 Nonetheless we prove that positive solutions satisfy unique continuation property.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

This note is a further step in the study of qualitative properties of generalized
solutions of degenerate elliptic PDE's.
Indeed, in the last decades many papers have been devoted to investigations 
of qualitative and quantitative properties of solutions.
Among several kind of degeneracy, we focus our attention to the case where the
operator is associated to a non commuting system of vector fields of a particular 
kind.
Moreover, we also study the possibility of regularity for the weak 
solutions where minimal assumptions on the lower order coefficients are assumed
(see \cite{dan,dagani,dz,dz1,dz2}).
A different kind of degeneracy has been investigated by Fabes, Kenig and Serapioni
and Gutierrez in \cite{FKS} and \cite{g} respectively
(see also \cite{dfz1,z3}).  There the degeneracy was given by a Muckenhoupt weight
of the class $A_2$. However, degeneracy given by a $A_2$ weight is not the only
possibility. Studies about a different kind of degeneracy started in \cite{fgw1}
 and \cite{fgw2} (see also \cite{dfz2,dfz3,dfz4,dfz5,dz4,lu})
where the weight is a suitable power of a function that belongs to the class
of strong $A_\infty$ weights introduced by David and Semmes in \cite{ds}.

In this note we prove that the positive solutions of a quasilinear strongly
degenerate elliptic equation satisfy the unique continuation property.
To make the statement precise, we let $\Omega$ be a bounded domain in
$\mathbb{R}^N (N\ge 2)$, let $z=(x,y) \in \mathbb{R}^N$ and denote points 
in $\mathbb{R}^N$ in such
a way that $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^m$, $n+m=N$.

Let us consider a quasilinear elliptic equation of the  kind
\begin{equation} \label{eq:equazione-intro}
\operatorname{div}A(z,u,\nabla_\lambda u) + B(z,u,\nabla_\lambda u) = 0
\end{equation}
where $A$ and $B$ are measurable functions satisfying the following structure
conditions
\begin{gather*}
 |A(z,u,\xi)| \leq a \omega(z) [|\xi_x|^2+\lambda^2(x) |\xi_y|^2]^{\frac{p-1}{2}}
+ b |u|^{p-1}  \\
 |B(z,u,\xi)| \leq c [|\xi_x|^2+\lambda^2(x) |\xi_y|^2]^{\frac{p-1}{2}}
+ d |u|^{p-1}  \\
 \xi  \cdot  A(z,u,\xi) \geq \omega(z) [|\xi_x|^2+\lambda^2(x)
|\xi_y|^2]^{\frac{p}{2}} - d |u|^{p} \,.
\end{gather*}
Here $\omega = v^{1-\frac{p}{N}}$, $1<p<N$, $v$ is a strong $A_\infty$ weight,
$\nabla_\lambda u=(\nabla_xu(z)$, $\lambda(x)\nabla_yu(z))$ and $\lambda$ is
a function satisfying  (H1),  (H2),  (H3) (see Section 2).

Regarding equation \eqref{eq:equazione-intro} we will show that
if $u$ is a non negative solution of \eqref{eq:equazione-intro},
then $u$ cannot have zeros of infinite order unless it is identically
zero in $\Omega$.

We remark here that the equation we consider is doubly degenerate in
the following sense. It is a Grushin operator which exhibits further
degeneracy given by a strong $A_\infty$ weight.
Moreover, we assume the lower order coefficients to belong to suitable
Stummel - Kato classes (see Section 4). This kind of assumption is necessary
for regularity of solutions at least in some cases (see \cite{DF-MAN}).

We use the following two key ideas. The first is based on a method in \cite{cg}
to prove that a suitable power of any positive solution satisfies a
doubling inequality. The other one is the clever use of a Fefferman Poincare'
type inequality
\begin{equation}
\int_B |V(z)| |u(z)|^p \, \omega(z)\,dz
\leq C \int_B |\nabla_\lambda u(z)|^p \omega \,dz\,,
\end{equation}
proved in the Section 4 (see also \cite{dan,dz2002,dfz_CPAA})
that is needed in order to handle the lower order terms.
We then get our result because the solution has zeros of infinite order and,
satisfies a doubling inequality at the same time. This is possible if and only
if the solution is identically zero.

\section{Preliminaries}

To fix the notation we denote by $z=(x,y)$ points in $\mathbb{R}^N$, where $x$ and $y$
belong to $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively and $n+m=N$.
Let us denote by $\lambda=\lambda(x)$ a function such that verifies the
following assumptions
\begin{itemize}
\item[(H1)] $\lambda$ is a continuous nonnegative function vanishing only
at a finite number of points.

\item[(H2)] $\lambda^n$ is a strong $A_\infty$ weight (see
Definition \ref{pesostrong}).

\item[(H3)] $\lambda$ satisfies an infinite order reverse H\"older inequality,
i.e. for any $x_0\in \mathbb{R}^n$, $r>0$ we have
$$
\hbox{--}\hskip-9pt\int _{|x-x_0|<r}\lambda(x)dx \sim \max_{|x-x_0|<r}\lambda(x)\,.
$$
\end{itemize}
Let $u:\mathbb{R}^N\to \mathbb{R}$ be a almost everywhere differentiable function in $\mathbb{R}^N$.
We denote its $\lambda$-gradient by
$ \nabla_\lambda u(z)=(\nabla_xu(z),\lambda(x)\nabla_yu(z))$ and
$|\nabla_\lambda u|^2=|\nabla_xu|^2+\lambda^2(x)|\nabla_yu|^2$.

Then, we define the Carnot-Carath\'eodory metric (C-C metric) $\rho$ in $\mathbb{R}^N$
with respect to $\nabla_\lambda $ in the following way.

\begin{definition}[$\lambda$ sub unit curve] \rm
An absolutely continuous curve
$\gamma=(\gamma_1,\gamma_2,\dots,\gamma_N):[0,T]\to\mathbb{R}^N $
is said to be a $\lambda$-sub unitary curve if the following inequality holds
$$
\langle \gamma' (t),z\rangle^2
\le |x|^2+\lambda(\gamma_1(t),\dots,\gamma_n(t))|y|^2
$$
for any $z=(x,y)\in \mathbb{R}^N$, and almost any $t\in [0,T]$.
\end{definition}


\begin{definition}[$\lambda$ Carnot-Caratheodory distance] \rm
If $z_1$,$z_2$ are points in $\mathbb{R}^N$, let us consider the set of all
$\lambda$-sub unitary curves connecting $z_1$ and $z_2$.
We set
\begin{align*}
\rho(z_1,z_2)=\inf \bigl\{& T\ge 0: \exists \text{ a sub-unit curve
$\gamma:[0,T] \to \mathbb{R}^N$,} \\
&\text{such that } \gamma(0)=z_1 \text{ and} \; \gamma(T)=z_2 \bigr\}.
\end{align*}
If there are no such curves we set $\rho(z_1,z_2)=+\infty$.

We denote by $B(z_0,r)$ the $\lambda$ Carnot-Caratheodory metric ball 
centered at $z_0$ of radius $r$. We also write $B$ or $B_r$ if the center 
of the ball has no relevance.
\end{definition}

We recall the definition of the $A_q$ and the strong $A_\infty$ weights with 
respect to the $\lambda$ metric (see \cite{fgw1,fgw2}).

\begin{definition}[$A_q$ weights] \label{pesoA_q} \rm
Let $q>1$ and let $v$ be a nonnegative locally integrable function in $\mathbb{R}^N$.
 We say that $v$ is a weight of the Muckenhoupt class $A_q$ if
$$
\sup_B\Big( {\frac{1}{|B|}}\int_Bv(z)\,dz \Big)
\Big({\frac{1}{|B|}}\int_B [v(z)]^{\frac{-1}{q-1}} \,dz\Big)^{q-1} 
\equiv C_0<+\infty\,
$$
the supremum being taken over all Carnot-Carath\'eodory metric balls $B$ in $\mathbb{R}^N$. 
The number $C_0$ is called the $A_q$ constant of $v$.
\end{definition}

We recall the doubling property.

\begin{lemma}[Doubling property] \label{doubling1}
Let $v\in A_q$. There exists a positive constant $c$ such that for any 
C-C metric ball $B_r$,
$$ 
v(B_r)\le c \,v(B_{r/2}).
$$
\end{lemma}

For a proof of the above lemma, see \cite{c}.
Now we give the definition of strong $A_\infty$ weight with respect to
 the function $\lambda$.

\begin{definition}\label{pesostrong} \rm
Let $v$ be a $A_q$ weight for some $q> 1$. For any $z_1$, $z_2$ in $\mathbb{R}^N$ we set
$$
\delta(z_1,z_2)=\inf\Big(\int_{B}v(z)\lambda ^{\frac{m}{N-1}}(x)dz\Big)^{1/N}
$$
where the infimum is taken over the C-C balls $B$ such that $z_1,z_2\in B$.

For any curve $\gamma: [0,T]\to \mathbb{R}^N$, we define its $v$-length as
$$
l(\gamma)=\liminf_{|\sigma|\to 0}\sum_{i=0}^{p-1} \delta 
(\gamma(t_{i+1}),\gamma(t_i))
$$
where $\sigma=\{t_0,\dots, t_p\}$ is a partition of $[0,T]$.

Then we define a distance $d(z_1,z_2)$ as the infimum of the $v$-lengths 
of $\lambda$-sub unitary curves connecting $z_1$ and $z_2$.
If there exist positive constants $c_1$ and $c_2$ such that
$$
c_1\delta(z_1,z_2)\le d(z_1,z_2)\le c_2 \delta(z_1,z_2)
$$
we say that $v$ is a strong $A_\infty$ weight for the metric $\rho$.
\end{definition}

\begin{example} \rm
A strong $A_\infty$ weight is the function $v(z)=\rho(z,z_0)^\alpha$ with 
$\alpha\ge 0$ and $z_0\in \mathbb{R}^N$ (see \cite{fgw2}).
\end{example}

\begin{proposition}\label{5}
Let $0<\alpha<1$ and let $v$ be a strong $A_\infty$ weight. Then the 
measure $v^\alpha$ satisfies the doubling property.
\end{proposition}

\begin{proof}
 Since $v$ is a strong $A_\infty$ there exists $q$ such that $v\in A_q$. 
This implies that $v^\alpha\in A_{(q-1)\alpha+1}$ and, as a consequence, $v^\alpha$ 
satisfies the doubling property.
\end{proof}

\begin{lemma}\label{antidoubling}
Let $v$ be a strong $A_\infty$ weight. Then for every compact set $E$ there 
exist two positive constant $r_0$ and $\gamma$ such that
$$
v(B(z_0,r))\le \gamma r^N\,
$$
 for all $z_0\in E$ and for all $0<r<r_0$.
\end{lemma}

The above lemma is a consequence of \cite[Theorem 2.3]{f_box}.
We can define Lebesgue and Sobolev classes with respect to the strong 
$A_\infty$ weights.

\begin{definition} \rm
Let $v$ be a strong $A_\infty$ weight and $\omega = v^{1-p/N}$, 
$\Omega \subset \mathbb{R}^N$.
For any $u\in C^\infty_0(\Omega)$ we set
\begin{equation*}
\|u\|_{L^p_v(\Omega)} = \Big(\int_\Omega |u(z)|^p\,\omega(z)\,dz\Big)^{1/p}
 \quad 1\leq p<\infty\,.
\end{equation*}
We define $L^p_v(\Omega)$ to be the completion of $C^\infty_0(\Omega)$ with 
respect to the above norm.
In a similar way we define Sobolev classes.
Let $1<p<N$.
For any $u\in C^\infty(\Omega)$ we set
\begin{equation} \label{spazio-di-sobolev-pesato}
\|u\|_{H^{1,p}_{v}(\Omega)} = \Big(\int_\Omega |u(z)|^p\,\omega(z)\,dz\Big)^{1/p}
+\Big(\int_\Omega |\nabla_\lambda u(z)|^p\,\omega(z)\,dz\Big)^{1/p}\,.
\end{equation}
We define $H^{1,p}_{0,v}(\Omega)$ to be the completion of
$C^\infty_0(\Omega)$ with respect to the norm \eqref{spazio-di-sobolev-pesato} and
$H^{1,p}_{v}(\Omega)$ to be the completion of
$C^\infty(\Omega)$ with respect to the same norm.
\end{definition}

Now,  to recall the Sobolev embedding theorem and the representation 
formula proved in \cite[Theorem I and Corollary 3.2]{fgw2}, we need another
 assumption on strong $A_\infty$ weights.

A strong $A_\infty$ weight $v$ satisfies the local boundedness condition 
near the zeros of $\lambda$ if the following condition holds
\begin{itemize}
\item[(H4)] if $\lambda(x_1)=0$, then $v(x,y)$ is bounded as
$x\to x_1$ uniformly in $y$ for $y$ in any bounded set.
\end{itemize}

\begin{theorem}\label{sobolev}
Let $u_1$, $u_2\in L^1_{\rm loc}$ be positive functions such that $u_1$ is doubling, 
$1\le p<q<+\infty$.
Assume that there exist positive constants $c_1$ and $c_2$ such that, for all 
C-C balls $B_0=B(z_0,r_0)$ and $B_r=B(z,r)\subset c_1B_0$, we have
\begin{equation*}
\frac{r}{r_0}\Big( \frac{u_1(B_r)}{u_1(B_0)}\Big)^{1/q}
\le c_2\Big( \frac{u_2(B_r)}{u_2(B_0)}\Big)^{1/p}\,,
 \end{equation*}
where $u_1(B_0)=\int_{B_0}u_1(z)dz$.

If there exists a strong $A_\infty$ weight $w$ satisfying condition
 {\rm (H4)} such that $u_2 w^{-(N-1)/N}$ belongs to the class $A_p$ with
respect to the (doubling) measure $w^{(N-1)/N}dz$, then the following 
Sobolev-Poincar\'e inequality hold 
\begin{equation*}
\Big( \hbox{--}\hskip-9pt\int _{B(z_0,r)}|g-\mu|^qu_1(z)dz\Big)^{1/q}
\le Cr \Big( \hbox{--}\hskip-9pt\int _{B(z_0,r)}|\nabla_\lambda g|^pu_2(z)dz\Big)^{1/p}
\end{equation*}
for any Lipschitz continuous function $g$, where $\mu$ can be chosen to be 
the $u_1$-average of $g$ over $B(z_0,r)$.
\end{theorem}

If $v$ is a strong $A_\infty$ weight, by taking $u_1=u_2= v^{1-p/N}$ 
in Theorem \ref{sobolev} the result follows by observing that a positive 
power of a strong $A_\infty$ weight is doubling.

\begin{theorem}\label{sobolev1}
Let $1<p<N$ and let $v$ be a strong $A_\infty$.
If there exists a strong $A_\infty$ weight $w$ satisfying {\rm (H4)}
 such that $v^{1-p/N} w^{-(N-1)/N}$ belongs to $A_p$ with respect to the
 (doubling) measure $w^{(N-1)/N}dz$, then there exists a constant $q>p$ such that
\begin{equation}\label{nostrasobolev}
\Big( \hbox{--}\hskip-9pt\int _{B(z_0,r)}|g-\mu|^qv^{1-p/N}(z)dz\Big)^{1/q}
\le Cr \Big( \hbox{--}\hskip-9pt\int _{B(z_0,r)}|\nabla_\lambda g|^pv^{1-p/N}(z)dz\Big)^{1/p}
\end{equation}
for any Lipschitz continuous function $g$, where $\mu$ can be chosen to be the 
$v^{1-p/N}$-average of $g$ over $B(z_0,r)$.
\end{theorem}

\begin{remark} \rm
We stress that if we take the weights $v=w=\rho^\alpha(0,z)$ and the function 
$\lambda(x)=|x|^\sigma$, ($\alpha$, $\sigma>0$), the assumptions of 
Theorem \ref{sobolev1} are satisfied (see also \cite{fgw1}).
\end{remark}

\section{$BMO_\omega$, $A_2(\omega)$ and functions with zeros of infinite order }

In this section we denote by $v$ be a strong $A_\infty$ weight, $1<p<N$ and 
$\omega=v^{1-p/N}$.

\begin{definition}
Let $f$ be a locally integrable function in $\mathbb{R}^N$ with respect to $\omega$. 
We say that $f$ belongs to the class $BMO_\omega$ if
$$ \sup_{B}\frac{1}{\omega(B)}\int_{B}|f(\zeta)-f_B|\omega(\zeta)d\zeta <+\infty\,,
$$
where the supremum is taken over the balls $B$.
\end{definition}

\begin{definition}
A locally integrable function $u$ such that $\frac{1}{u}$ is also locally integrable 
$u$ belongs to the class $A_2(\omega)$ if
$$
\int_{B}u(\zeta)\omega(\zeta)d\zeta\int_{B}\frac{1}{u(\zeta)}\omega(\zeta)d\zeta\le C (\omega(B))^{2}
$$
for any C-C ball $B$.
\end{definition}

\begin{theorem}\label{doubling}
Let $u$ be a function in the class $A_2(\omega)$. Then
$$
\int_{B_{2r}}u\, \omega dz \le C\int_{B_r}u\,\omega dz
$$
for any C-C ball $B_r$.
\end{theorem}

The proof of the above theorem is similar to that of \cite[Theorem 4.5]{dz5}, 
and is omitted.

\begin{definition} \rm
Let $\Omega$ be a bounded domain of $\mathbb{R}^N$. Let $\mu$ be a locally integrable 
function such that $\frac{1}{\mu}$ is locally integrable and 
$u\in L^1_{\rm loc}(\Omega)$. We say that $u$ has a zero of infinite order 
at $z_0\in \Omega$ if
$$
\lim_{\sigma \to 0}\frac{\int_{B(z_0,\sigma)}u(z)\mu(z)dz}{(\mu(B(z_0,\sigma)))^k}=0\,,
\quad \forall k>0\,.
$$
\end{definition}

\begin{theorem}\label{zero}
Let $u\in L^1_{\rm loc}(\Omega)$, $u\ge 0$ a.e. in $\Omega$, $u\not\equiv 0$.
If there exist positive numbers $C$ and $\sigma_0$ such that
$$
\int_{B(z,2\sigma)}u\,\omega dz
\le C \int_{B(z, \sigma)}u \,\omega dz
$$
for any $0<\sigma<\sigma_0$ and any $z\in \Omega$
then $u$ cannot have zeros of infinite order in $\Omega$.
\end{theorem}

\begin{proof}
We argue by contradiction. Let $z_0\in \Omega$ be a zero of infinite order for $u$. 
Using Theorem \ref{doubling} $k$ times, H\"older inequality and Lemma \ref{antidoubling} $k$ 
times we obtain
\begin{equation}
\begin{aligned}
\int_{B_{r}}u\omega dz
&\le C^k\int_{B_{\frac{r}{2^k}}}u\omega dz\\
&= C^k \omega(B_{r/2^k})^\beta {1 /( \omega(B_{r/2^k})^\beta)}
 \int_{B_{r / 2^k}}u\omega\,dx  \\
&\le C^k\gamma^{\beta(1-p/N)} \left(\frac{r}{2^k} \right)^{\beta(N-p)}
|B_{r}|^{\frac{\beta p}{N}} {1 / \omega(B_{r / 2^k})^\beta}
 \int_{B_{r / 2^k}}u\omega\,dx.
\end{aligned}
\end{equation}
Now, if we choose $\beta= \log_{2^{N-p}} C$, we obtain
$$
\int_{B_{r}}u\omega dz \le \gamma^{\beta(1-p/N)} r^{\beta(N-p)}|B_{r}|
^{\beta p/N} {1/ \omega(B_{r/2^k})^\beta}
 \int_{B_{r/2^k}} u\omega\,dx. ,
$$
from which, as $k\to \infty$, we have that $u\omega\equiv 0$ in $B_r$ and,
since $\omega\not\equiv0$, $u\equiv 0$.
\end{proof}

\section{Stummel- Kato classes and embedding theorem}

In this section we give the definition of Stummel-Kato classes with respect 
to a strong $A_\infty$ weight and then prove our embedding result.
Our starting point is a representation formula 
(see \cite[Corollary 3.2]{fgw2}) proved by Franchi, Gutierrez and Wheeden.

\begin{lemma}[Representation formula]\label{formularappresentazione}
Let $\Omega$ be a bounded domain in $\mathbb{R}^N$, $v$ a strong $A_\infty$ weight 
satisfying {\rm (H4)} and $u$ a compactly supported smooth function in a metric
ball $B=B_R\subset\Omega$. Then there exists $c$ independent of $u$ such that
\begin{equation*}
|u(z)|\le c \int_{B}|\nabla_\lambda u(\xi)|v^{1-\frac{1}{N}}(\xi)k(z,\xi)d\xi
\end{equation*}
for almost all $z\in B$, where
$$ 
k(z,\xi)=\Big( \int_{B_{\rho(z,\xi)}(z)}v(\zeta)\lambda^{\frac{m}{N-1}}
(\chi)d\zeta\Big)^{\frac{1-N}{N}}\,.
$$
\end{lemma}

\begin{definition} \rm
Let $V$ be a locally integrable function in $\Omega$, $r>0$, and $p\in ]1,N[$. 
Let $v$ be a strong $A_{\infty}$ weight.
 We set $\omega(z)\equiv v^{1-\frac{p}{N}}(z)$ and
\[
\phi(V;r)\equiv
 \sup_{z\in \Omega} \Big( \int_{B(z,r)} k(z,\xi)v(\xi)
\Big(\int_{B(z,r)}|V(\zeta)|k(\zeta,\xi)\omega(\zeta)d \zeta\Big)^{\frac{1}{p-1}}
d\xi\Big)^{p-1}\,.
\]
We say that $V$ belongs to the class $\tilde S_v(\Omega)$ if $\phi(V;r)$ 
is bounded in a right neighborhood of the origin.
\end{definition}

\begin{remark} \rm
 If $v(z)=1$, $\lambda(x)=1$ and $p=2$ the previous definitions give back 
the classical Stummel-Kato class (see \cite{as}).
\end{remark}

By using the representation formula we now prove an embedding Theorem related 
to the Stummel-Kato classes.

\begin{theorem} \label{embedding-con-peso}
Let $v$ be a strong $A_{\infty}$ weight satisfying {\rm (H4)} and $1<p<N$. 
If $V$ belongs to the class $\tilde{S}_v(\Omega)$, then there exists a constant $C$
such that $\forall u \in C^\infty_0(\Omega)$
\begin{equation} \label{embedding}
\Big(\int_B |V(z)| |u(z)|^p  \omega(z)\,dz\Big)^{1/p}
\leq C \phi^{1/p}\left(V;2R\right)
\Big(\int_B |\nabla_\lambda u(z)|^p \omega \,dz\Big)^{1/p}\,,
\end{equation}
where $\omega(z)\equiv v^{1-\frac{p}{N}}(z)$ and $R$ is the radius of a metric ball
$B$, containing the support of $u$.
\end{theorem}

\begin{proof}
From the representation formula in Lemma \ref{formularappresentazione} it 
follows that
\begin{align*}
&\int_B |V(z)| |u(z)|^p  \omega(z)\,dz \\
&\leq C \int_B |V(z)| |u(z)|^{p-1}  
\Big(\int_B |\nabla_\lambda u(\xi)| k(z,\xi) v^{1-\frac{1}{N}}(\xi)\,d\xi\Big)
\omega(z)\,dz\\
&= C \int_B |\nabla_\lambda u(\xi)| v^{1-\frac{1}{N}}(\xi) 
\Big(\int_B |V(z)| |u(z)|^{p-1} k(z,\xi)\, \omega(z)\,dz \Big)\,d\xi\\
&\leq C \Big(\int_B |\nabla_\lambda u(\xi)|^p v^{1-\frac{p}{N}}(\xi) \,d\xi\Big)^{1/p} \\
&\quad\times\Big(\int_B \Big(\int_B |V(z)| |u(z)|^{p-1} k(z,\xi)
 \omega(z)\,dz\Big)^{p'}
v(\xi)\,d\xi\Big)^{1/p'}\,.
\end{align*}
By considering the last integral we can write
\begin{align*}
&\Big(\int_B |V(z)| |u(z)|^{p-1} k(z,\xi) \omega(z)\,dz\Big)^{p'}\\
&\le \Big(\int_B |V(z)| k(z,\xi) \omega(z)\,dz\Big)^{p'/p} 
\int_B |V(z)| |u(z)|^{p} k(z,\xi) \omega(z)\,dz
\end{align*}
and then, we obtain
\begin{align*}
&\int_B |V(z)| |u(z)|^p \omega(z)\, dz \\
&\leq
\Big(\int_B |\nabla_\lambda u(\xi)|^p v^{1-\frac{p}{N}}(\xi) \,d\xi\Big)^{1/p}
\Big(\int_B\Big(\int_B |V(\zeta)| k(\zeta,\xi) \omega(\zeta)\,d\zeta\Big)^{1/p-1}
\\
&\quad \times \int_B |V(z)|k(z,\xi) |u(z)|^p\,\omega(z)\,dz v(\xi)\,d\xi\Big)^{1/p'}
\end{align*}
so
\begin{align*}
&\int_B |V(z)| |u(z)|^p \omega(z)\, dz \\
&=\Big(\int_B |\nabla_\lambda u(\xi)|^p v^{1-\frac{p}{N}}(\xi) \,d\xi\Big)^{1/p}
\int_B |V(z)| |u(z)|^p \,\int_B k(z,\xi) \\
&\quad \times\Big(\int_B |V(\zeta)| k(\zeta,\xi) \omega(\zeta)\,d\zeta 
\Big)^{\frac{1}{p-1}} v(\xi)\,d\xi\,\omega(z)\,dz\\
&\leq \Big(\int_B |\nabla_\lambda u(\xi)|^p \omega(\xi) \,d\xi\Big)^{1/p}
\Big(\int_B \phi^{\frac{1}{p-1}}(V;R) |V(z)| |u(z)|^p\,\omega(z)\,dz  \Big)^{1/p'}\\
&= \phi^{1/p}(V;R)
\Big(\int_B |\nabla_\lambda u(\xi)|^p \omega(\xi) \,d\xi\Big)^{1/p}
\Big(\int_B |V(z)| |u(z)|^p\,\omega\,dz  \Big)^{1/p'}
\end{align*}
from which \eqref{embedding} easily follows.
\end{proof}

\section{Unique continuation property}

In this section we show how to apply the results obtained in the previous section 
in order to get unique continuation property for positive solutions of 
equation \eqref{eq:equazione-intro}.
Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ and $\lambda$ be a function 
satisfying {\rm (H1), (H2), (H3)}. Moreover, let $v$ be a strong $A_\infty$
 weight satisfying {\rm (H4)}.

We consider the quasilinear equation
\begin{equation} \label{eq:equazione-introduzione}
\operatorname{div}A(z,u,\nabla_\lambda u) + B(z,u,\nabla_\lambda u) = 0
\end{equation}
where $A$ and $B$ are given measurable functions that satisfy the following 
structure conditions
\begin{equation} \label{ipotesi-di-struttura}
\begin{gathered}
 |A(z,u,\xi)| \leq a \omega(z) [|\xi_x|^2+\lambda^2(x) |\xi_y|^2]^{\frac{p-1}{2}} 
+ b(z) |u|^{p-1}  \\
 |B(z,u,\xi)| \leq c(z) [|\xi_x|^2+\lambda^2(x) |\xi_y|^2]^{\frac{p-1}{2}}
 + d(z) |u|^{p-1}  \\
 \xi  \cdot \! A(z,u,\xi) \geq \omega(z) 
[|\xi_x|^2+\lambda^2(x) |\xi_y|^2]^{\frac{p}{2}} - d(z) |u|^{p} \,, 
\end{gathered}
\end{equation}
where $\omega = v^{1-\frac{p}{N}}$, $1<p<N$.

\begin{definition} \rm
A function $u\in H^{1,p}_{v}(\Omega)$ is a local weak solution of 
\eqref{eq:equazione-introduzione} in $\Omega$ if
\begin{equation} \label{def:soluzione_debole}
\int_{\Omega}A(z,u(z),\nabla_\lambda u(z))\,\nabla_\lambda\varphi(z)\,dz
+ \int_{\Omega}B(z,u(z),\nabla_\lambda u(z))\ \varphi(z)\,dz
= 0\,,
\end{equation}
for every $\varphi\in H^{1,p}_{0, v}(\Omega)$.
\end{definition}

\begin{theorem}
Let $u$ be a non negative weak solution of \eqref{eq:equazione-introduzione} 
that is not identically zero in $\Omega$.
Let us assume that the structure conditions \eqref{ipotesi-di-struttura} hold 
assuming that
\begin{equation} \label{ipotesi_stummel-harnack}
a \in \mathbb{R}\,, \quad
\Big(\frac{b}{\omega}\Big)^{p/p-1}\,,
\left(\frac{c}{\omega}\right)^p\,,  \frac{d}{\omega}\,,
  \in \tilde S_ v(\Omega)\,.
\end{equation}
Then $u$ has no zero of infinite order in $\Omega$.
\end{theorem}

\begin{proof}
Let $z_0\in \Omega$ and $B(z_0,r)$ be a ball such that $B(z_0,2r)\subset\Omega$. 
Let $B_h\subset B(z_0,r)$ and $\eta\in W^{1,p}_0(\Omega)$ be a non negative 
function supported in $B_{2h}$. Using $\varphi = \eta^p u^{1-p}$ as test function 
in \eqref{def:soluzione_debole} we obtain (see \cite{dfz4})
\begin{equation} \label{eq:betauguale}
\int_{\Omega}\eta^p|\nabla_\lambda \log u|^p\omega\,dz
\le C_1\Big\{\int_{\Omega}|\nabla_\lambda\eta|^p\omega\,dz
+ \int_{\Omega}V\eta^p\,dz\Big\}\,,
\end{equation}
where
$$ 
V=\frac{b^{\frac{p}{p-1}}}{\omega^{\frac{1}{p-1}}}+\frac{c^p}{\omega^{p-1}}+d\,.
$$
From \eqref{ipotesi_stummel-harnack} $\frac{V}{\omega}\in \tilde S_ v(\Omega)$ 
and by Theorem \ref{embedding-con-peso} we obtain
\begin{equation*}
\int_{\Omega}V\eta^p\,dz
\le C_2 \int_{\Omega}|\nabla_\lambda\eta|^p\omega\,dz
\end{equation*}
and then, from \eqref{eq:betauguale}, we have
\begin{equation*}
\int_{\Omega}\eta^p|\nabla_\lambda \log u|^p\omega\,dz
\le C_3(p,a,\phi_\frac{V}{\omega}, \operatorname{diam}\Omega)
\int_{\Omega}|\nabla_\lambda \eta|^p\omega\,dz\,.
\end{equation*}
 Choosing $\eta(z)$ so that $\eta(z)\equiv 1$ in $B_h$, $0\le\eta\le1$ in
$B_{2r}\setminus B_h$ and $|\nabla_\lambda \eta|\le 3/h$,
we obtain
\begin{equation*}
\int_{B_h}|\nabla_\lambda \log u|^p\omega\,dz
\le C_4(p,a,\phi_\frac{V}{\omega}, \operatorname{diam}\Omega)
{\frac{\omega(B_h) }{h^p} }\,.
\end{equation*}
Therefore, by Poincar\'e-Sobolev inequality \eqref{nostrasobolev} and 
John-Nirenberg lemma (see \cite{b}), we have $ \log {u} \in
BMO_\omega$.

Then there exists a constant $\delta>0$ such that $u^\delta \in A_2(\omega)$. 
By Theorem \ref{doubling}, $u^\delta$ has the doubling property. Then 
from Theorem \ref{zero} $u^{\delta }$ and $u$ have no zero of infinite order.
\end{proof}

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\end{document}