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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 07, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/07\hfil Unique continuatio]
{Unique continuation for solutions of $p(x)$-Laplacian equations}

\author[J. Cuadro,  G. L\'opez \hfil EJDE-2012/07\hfilneg]
{Johnny Cuadro,  Gabriel L\'opez}  % in alphabetical order

\address{Johnny Cuadro M. \newline
Universidad Aut\'onoma Metropolitana,
M\'exico D. F., M\'exico}
\email{jcuadrom@yahoo.com}

\address{Gabriel L\'opez G. \newline
Universidad Aut\'onoma Metropolitana,
M\'exico D. F., M\'exico}
\email{gabl@xanum.uam.mx}

\thanks{Submitted September 8, 2011. Published January 12, 2012.}
\subjclass[2000]{35D05, 35J60, 58E05}
\keywords{$p(x)$-Laplace operator; unique continuation}

\begin{abstract}
 We study the unique continuation  property for solutions
 to the quasilinear elliptic equation
 $$
 \operatorname{div}(|\nabla u|^{p(x)-2}\nabla  u)
 +V(x)|u|^{p(x)-2}u=0\quad \text{in }\Omega,
 $$
 where $\Omega$  is a smooth bounded domain in $\mathbb{R}^N$ and
 $1<p(x)<N $ for $x$ in $\Omega$.
\end{abstract}

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

\newcommand{\slashint}{\rlap{--}\!\! \int}

\section{Introduction and preliminary results}

In the recent years increasing attention has been paid to the study
of differential and partial differential equations involving
variable exponent conditions. The interest in studying such problems
was stimulated by their applications in elastic mechanics, fluid
dynamics and calculus of variations. For  information on
modelling physical phenomena by equations involving $p(x)$-growth
condition we refer to \cite{ac,ru,wi}. The understanding of
such physical models has been facilitated by the development of
variable Lebesgue and Sobolev spaces, $L^{p(x)}$ and $W^{1,p(x)}$,
where $p(x)$ is a real-valued function. Variable exponent
Lebesgue spaces appeared for the first time in literature as
early as 1931 in an article by  Orlicz \cite{or}. The spaces $L^{p(x)}$ are special
cases of Orlicz spaces $L^{\varphi}$ originated by Nakano \cite{na}
and developed by Musielak and Orlicz \cite{mu,mu1}, where $f\in
L^{\varphi}$ if and only if $\int \varphi (x,| f(x)|)dx<\infty$ for
a suitable $\varphi$. Variable exponent Lebesque spaces on the real
line have been independently developed by Russian researchers. In
that context we refer to the studies of Tsenov \cite{ts},
Sharapudinov \cite{sh} and Zhikov \cite{zh1,zh2}.

This article is motivated by the phenomena that can be modelled with the
equation
\begin{equation}\label{e1}
 \begin{gathered}
  -\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) \quad
  \text{in } \Omega \\
   u=0 \quad\text{on } \partial\Omega,
 \end{gathered}
\end{equation}
where $\Omega\subset \mathbb{R}^N$ $(N\geq 3)$ is a bounded domain
with smooth boundary and $1<p(x)$, $p(x)\in C(\overline{\Omega})$.
Our goal to show strong unique continuation  nontrivial for weak solutions for \eqref{e1}
in the generalized Sobolev space $W^{1,p(x)}(\Omega)$ for some
particular nonlinearities of the type $f(x,u)$. Problems of type
\eqref{e1} have been intensively studied in the past decades. We
refer to \cite{al,fa1,fa2,mii,mi0,mi1,mi2,ra1,ra2,zh}, for
some interesting results. We point out the presence in \eqref{e1} of
the $p(x)$-Laplace operator. This is a natural extension of the
$p$-Laplace operator, with $p$ a positive constant. However, such
generalizations are not trivial since the $p(x)$-Laplace operator
possesses a more complicated structure than $p$-Laplace operator,
for example it is inhomogeneous.

We recall some definitions and  properties of the variable exponent
Lebesgue-Sobolev spaces $L^{p(\cdot)}(\Omega)$ and
$W_0^{1,p(\cdot)}(\Omega)$, where $\Omega$ is a bounded domain in
$\mathbb{R}^N$. Roughly speaking, anisotropic Lebesgue and Sobolev
spaces are functional spaces of Lebesgue's and Sobolev's type in
which different space directions have different roles.

Set $C_+(\overline\Omega)=\{h\in
C(\overline\Omega):\min_{x\in\overline\Omega}h(x)>1\}$. For any
$h\in C_+(\overline\Omega)$ we define
$$
h^+=\sup_{x\in\Omega}h(x)\quad\text{and}\quad
h^-=\inf_{x\in\Omega}h(x).
$$
For $p\in C_+(\overline\Omega)$, we introduce {\it the variable
exponent Lebesgue space}
\begin{align*}
L^{p(\cdot)}(\Omega)=\big\{&u: u \text{ is a
 measurable real-valued function}\\
&\text{such that }\int_\Omega |u(x) |^{p(x)}\,dx<\infty\big\},
\end{align*}
endowed with the so-called {\it Luxemburg norm}
$$
|u|_{p(\cdot)}=\inf\big\{\mu>0;\;\int_\Omega
|\frac{u(x)}{\mu}|^{p(x)}\,dx\leq 1\big\},
$$
which is a separable and reflexive Banach space. For basic
properties of the variable exponent Lebesgue spaces we refer to
\cite{ko}. If $0 <|\Omega|<\infty$ and $p_1$, $p_2$ are variable
exponents in $C_+(\overline\Omega)$ such that $p_1 \leq p_2$  in
$\Omega$, then the  embedding $L^{p_2(\cdot)}(\Omega)\hookrightarrow
L^{p_1(\cdot)}(\Omega)$ is continuous, \cite[Theorem~2.8]{ko}.

Let $L^{p'(\cdot)}(\Omega)$ be the conjugate space of
$L^{p(\cdot)}(\Omega)$, obtained by conjugating the exponent
pointwise that is,  $1/p(x)+1/p'(x)=1$, \cite[Corollary~2.7]{ko}.
For any $u\in L^{p(\cdot)}(\Omega)$ and $v\in L^{p'(\cdot)}(\Omega)$
the following H\"older type inequality
\begin{equation}\label{Hol}
\big|\int_\Omega uv\,dx\big|\leq\Big(\frac{1}{p^-}+
\frac{1}{{p'}^-}\Big)|u|_{p(\cdot)}|v|_{p'(\cdot)}
\end{equation}
is valid.

An important role in manipulating the generalized Lebesgue-Sobolev
spaces is played by the {\it $p(\cdot)$-modular} of the
$L^{p(\cdot)}(\Omega)$ space, which is the mapping
 $\rho_{p(\cdot)}:L^{p(\cdot)}(\Omega)\to\mathbb{R}$ defined by
$$
\rho_{p(\cdot)}(u)=\int_\Omega|u|^{p(x)}\,dx.
$$
If $(u_n)$, $u\in L^{p(\cdot)}(\Omega)$ then the following relations
hold
\begin{gather}\label{L40}
|u|_{p(\cdot)}<1\;(=1;\,>1)\;\Leftrightarrow\;\rho_{p(\cdot)}(u)
<1\;(=1;\,>1)
\\ \label{L4}
|u|_{p(\cdot)}>1 \;\Rightarrow\;
|u|_{p(\cdot)}^{p^-}\leq\rho_{p(\cdot)}(u) \leq|u|_{p(\cdot)}^{p^+}
\\ \label{L5}
|u|_{p(\cdot)}<1 \;\Rightarrow\; |u|_{p(\cdot)}^{p^+}\leq
\rho_{p(\cdot)}(u)\leq|u|_{p(\cdot)}^{p^-}
\\ \label{L6}
|u_n-u|_{p(\cdot)}\to 0\;\Leftrightarrow\;\rho_{p(\cdot)} (u_n-u)\to
0,
\end{gather}
since $p^+<\infty$. For a proof of these facts see \cite{ko}. Spaces
with $p^{+}=\infty$ have been studied by Edmunds, Lang and Nekvinda
\cite{ed}.

Next, we define $W_0^{1,p(x)}(\Omega)$ as the closure of
$C_0^{\infty}(\Omega)$ under the norm
\[
\| u\|_{p(x)}=|\nabla u|_{p(x)}.
\]
The space $(W_0^{1,p(x)}(\Omega),\| \cdot \|_{p(x)})$ is a separable
and reflexive Banach space. We note that if $q\in
C_+(\overline{\Omega})$ and $q(x)<p^*(x)$ for all $x\in
\overline{\Omega}$ then the embedding
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$ is compact
and continuous, where $p^*(x)=Np(x)/(N-p(x))$ if $p(x)<N$ or
$p^*(x)=+\infty$ if $p(x)\geq N$ \cite[Theorem 3.9 and 3.3]{ko} (see
also \cite [Theorem 1.3 and 1.1]{fa}).

The bounded variable exponent $p$ is said to be Log-H\"older continuous
if there is a constant $C>0$ such that
\[
  |p(x)-p(y)|\leq  \frac{C}{-\log (|x-y|)}
\]
for all  $x,y \in \mathbb{R}^{N}$, such that $|x-y|\le 1/2$.
A bounded exponent $p$ is Log-H\"older continuous in $\Omega$ if and
only if there exists a con\-stant $C>0$ such that
\[
|B|^{p^{-}_{B}-p^{+}_{B}}\le C
\]
for every ball $B\subset\Omega$  \cite[Lemma 4.1.6, page 101]{dil}.
As a result of the condition  Log-H\"older continuous we have
\begin{gather}\label{*l}
r^{-(p^{+}_{B}-p^{-}_{B})}\le C,\\
\label{*2}
 C^{-1}r^{-p(y)}\le r^{p(x)}\le Cr^{-p(y)}
\end{gather}
 for all $x,y\in\ B:=B(x_0,r)\subset\Omega$ and the constant $C$
 depends only on the constant Log-H\"older continuous.
 Under the Log-H\"older condition smooth function are dense
 in variable exponent
 Sobolev space \cite[Proposition 11.2.3, page 346]{dil}.

Concerning to the Unique Continuation in his paper
on  Schr\"odinger semigroup \cite {sim}, B.Simon formulated
the following conjecture:

\begin{quote}
 Let $\Omega$ be a bounded subset $\mathbb{R}^{N}$ and $V$ a
function defined in  $\Omega$ whose extension with values outside
$\Omega$ belong to the Stummel-Kato $\mathrm{S}(\mathbb{R}^{N})$.
Then the  Schr\"odinger operator $H:= -\Delta + V$ has the unique
continuation property.
\end{quote}

That is, $u\in H^{1}(\Omega)$ is a solutions of equations $Hu=0$
which vanishes of infinite order (For definitions see section 3.) at one point $x_0\in \Omega$,
then $u$ must be identically zero in $\Omega$. A positive answer to
Simon 's conjeture was given by Fabes,Garofalo and Lin  for
radial potential $V$.

At the same time Chanilo and Sawyer in \cite{cs} proved the unique
continuation property for solutions of the inequality $|\Delta
u|\leq|V||u|$, assuming $V$ in the Morrey spaces
$L^{r,N-2r}(\mathbb{R}^{N})$ for $r>\frac{N-1}{2}$. Jarison and
Kening proved the continuation unique for Schr\"{o}dinger operator
\cite{jk}.The same work is done Gossez and Figueiredo, but for linear
elliptic operator in the case $V\in L^{\frac{N}{2}}(\Omega)$, $N>2$,
\cite{fg}. Also, Loulit extended this property to $N=2$ by introducing
Orlicz's space \cite{lo}. In this paper  we extended  to Variable Exponent
Space a result of Zamboni \cite{za1} to the solution of a quasilinear
elliptic equation
\begin{gather}\label{e2}
  \operatorname{div}(|\nabla u|^{p(x)-2}\nabla  u)+V(x)|u|^{p(x)-2}u=0\quad
   \text{in }\Omega,
\end{gather}
where $1<p(x)<N$, $V\in L^{\frac{N}{p(x)}}(\Omega)$.

\section{Fefferman's type inequality}

For every $u\in W^{1,p(\cdot)}_0(\Omega)$ the norm Poincar\'e
 inequality
 $$
 |u|_{L^{p(\cdot)}(\Omega)}
 \leq c \operatorname{diam}(\Omega)|\nabla u|_{L^{p(\cdot)}}
$$
 $c=C(N,\Omega,c\,log(p))$ holds (we refer to \cite{har} for
 notation and proofs). Nevertheless, the modular
 inequality
 \begin{equation}\label{e3}
 \int_\Omega|u|^{p(x)}dx\leq C\int_\Omega |\nabla u|^{p(x)}dx,\quad
 \forall u\in W^{1,p(\cdot)}_0(\Omega)
 \end{equation}
 not always holds (see \cite[ Thm. 3.1]{fa2}).
 It is known that \eqref{e3} holds if, for instance: i) $N>1$, and the function $f(t):=p(x_o+tw)$ is monotone \cite[Thm.3.4]{fa2} with $x_o+tw$ with an appropriate setting in $\Omega;$ ii) if there exists a function $\xi\geq 0$ such that $\nabla p\cdot\nabla \xi\geq 0$, $\|\nabla \xi\|\neq 0$ \cite[Thm. 1]{all}; iii) If there exists $a:\Omega\to\mathbb{R}^N$ bounded such that $div\, a(x)\geq a_0>0$ for all $x\in\bar{\Omega}$ and $a(x)\cdot\nabla p(x)=0$ for all $x\in \Omega$, \cite[Thm. 1]{mi3}.
To the best of our knowledge necessary and sufficient conditions in order to ensure that
$$\inf_{u\in W^{1,p(\cdot)}(\Omega)/\{0\}}\frac{\int_\Omega |\nabla u|^{p(x)}}{\int_\Omega |u|^{p(x)}}>0$$
has not been obtained yet, except in the case $N=1$, \cite[Thm. 3.2]{fa2}. The following definition is in order.

\begin{definition} \rm
We say that $p(\cdot)$ belongs to the Modular Poincar\'e
Inequality Class, $MPIC(\Omega)$, if there exists necessary
conditions to ensure that
$$
\int_\Omega |u|^{p(x)}\leq C\int_\Omega |\nabla u|^{p(x)},\quad
\forall u\in W^{1,p(\cdot)}_0(\Omega)
$$
$C=C(N,\Omega, c_{log}(p))>0$ holds.
\end{definition}

Fefferman \cite{fe} proved the inequality
\begin{equation}\label{fef1}
\int_{\mathbb{R}^{N}}|u(x)|^{p}|f(x)|\,dx\leq C\int_{\mathbb{R}^{N}}|\nabla
u(x)|^{p}\,dx\quad \forall u\in C^{\infty}_0(\mathbb{R}^{N}).
\end{equation}
in the case $p=2$, assuming $f$ in the Morrey's space
$L^{r,N-2r}(\mathbb{R}^{N})$, with $1<r\leq\frac{N}{2}$.
Later in \cite{sch} Schechter showed the same result taking $f$ in the
Stummel-Kato class $S(\mathbb{R}^{N})$.
Chiarenza and Frasca \cite{cf} generalized Fefferman's result
proving \eqref{fef1} under the assumption
$f\in L^{r,N-pr}(\mathbb{R}^{N})$, with $1<r<\frac{N}{p}$ and $1<p<N$.
Zamboni \cite{za1} generalized Schecter's result proving \eqref{fef1}
under the assumption $f\in \tilde{M}_{p}(\mathbb{R}^{N})$,
with $1<p<N$. We stress out that is not possible to compare
the assumptions $f\in L^{r,N-pr}(\mathbb{R}^{N})$ the Morrey
class and $f\in S(\mathbb{R}^N)$, the Stumel-Kato class.
The theory for a variable exponent spaces is a growing area but
Modular Fefferman type inequalities are more scarce than
Poincar\'e inequalities in variable exponent setting.
In the following theorem we provide a basic Fefferman's type
result, for variable exponent spaces.

\begin{theorem} \label{tff}
Let $p$ be a Log-H\"older continuous exponent with $1<p(x)<N$,
and $p\in MPIC(\Omega)$.
Let $V\in L^{1}_{\rm loc}(\Omega)$ with $0<\varepsilon<V(x)$ a.e..
Then there exist a positive constant $C=C(N,\Omega,c_{log}(p))$
such that
\[
\int_{\Omega}V(x)|u(x)|^{p(x)}\,dx\leq C \int_{\Omega}|\nabla
                 u(x)|^{p(x)}\,dx
\]
for any  $u\in W^{1,p(x)}_0(\Omega)$.
\end{theorem}

\begin{proof}
Let $u\in W^{1,p(x)}_0(\Omega)$ supported in $B(x_0,r)$.
Given that $V\in L^1_{\rm loc}(\Omega)$ the function
$$
w(x):=\Big( \int_{x_1^0}^{x_1}V(\xi_1,x_2,\dots,x_n)d\xi_1,
\dots,\int_{x_N^0}^{x_N}V(x_1,\dots,x_{N-1},\xi_N)d\xi_N \Big),
$$
where $x_0=(x_1^0,\dots,x_N^0)$ and $x=(x_1,\dots,x_N)\in B(x_0,r)$,
is well defined. Notice that
$ \int_{x_i^0}^{x_i}V(x_1,\dots,\xi_i,\dots,x_n)d\xi_i\in
\mathcal{C}[x_i^0,x_i]$ for $i=1,\dots,N$
\cite[Lemme VIII.2]{br}.
So that $\operatorname{div}w(x)=NV(x)$. Moreover $$|V(x)|_{L^1(B(x_0,r))}\geq\int_{x_1^0}^{x_1}\cdots\int_{x_N^0}^{x_N}V(\xi)\,d\xi_n\cdots d\xi_1$$ where $\xi=(\xi_1,\dots,\xi_N)$. Therefore, $|w(x)|\leq \sqrt{N}|V(x)|_{L^1(B(x_0,r))}$.

A direct calculation leads to
\begin{align*}
\operatorname{div}(|u|^{p(x)}w(x))
&=|u(x)|^{p(x)}\operatorname{div}\,w(x)+p(x)|u|^{p(x)-2}
 u\nabla u\cdot w(x)\\
&\quad +|u|^{p(x)}\log u\nabla p(x)\cdot w(x).
\end{align*}
Now the Divergence Theorem implies $\int_{B(x_0,r)}\operatorname{div}\,(|u|^{p(x)}w(x))=0$, and so
\begin{align*}
\int_{B(x_0,r)}|u(x)|^{p(x)}\operatorname{div}w(x)dx
&\leq p^+\int_{B(x_0,r)}|u(x)|^{p(x)-1}|\nabla u(x)||w(x)|dx \\
&\quad +\int_{B(x_0,r)}|u(x)|^{p(x)}\log |u(x)||\nabla p(x)||w(x)|dx.
\end{align*}
Set
$$
I_1:=p^+\int_{B(x_0,r)}|u(x)|^{p(x)-1}|\nabla u(x)||w(x)|dx
$$
 and
$$
I_2:=\int_{B(x_0,r)}|u(x)|^{p(x)}\log |u(x)||\nabla p(x)||w(x)|dx.
$$

 Now we estimate $I_{2}$ by  distinguishing the case when
$|u(x)|\leq 1$ and $|u(x)|> 1$. Notice that  the relations
\begin{gather}\label{e4}
  \sup_{0\leq t\leq 1}t^{\eta}|\log t|<\infty,\\
\label{e5}
\sup_{t>1}t^{-\eta}\log t<\infty
\end{gather}
hold for $\eta >0$.
Let $\Omega_{1}=:\{x \in B_{r}: |u(x)|\leq 1  \} $ and
$\Omega_{2}=:\{x \in B_{r}:  |u(x)|>1  \} $, then by \eqref{e4}
and \eqref{e5} we have
\[
I_2\leq C_1\int_{\Omega_1}|w(x)||u(x)|^{p(x)-\eta_1}dx
+C_2\int_{\Omega_2}|w(x)||u(x)|^{p(x)+\eta_2}dx.
\]
We can choose $k\in\mathbb{N}$ such that $p(x)-1/k\geq p^-$.
Since $u\in L^{p^-}(B(x_0,r))$ and in $\Omega_1$, $|u(x)|\leq 1$
we have
$$
|u(x)|^{p(x)-1/n}\leq|u(x)|^{p^-},
$$
for $n>k$.
The Lebesgue Dominated Convergence Theorem implies
$$
\lim_{n\to\infty}\int_{\Omega_1} |u(x)|^{p(x)-1/n}dx
= \int_{\Omega_1} |u(x)|^{p(x)}dx.
$$
For $\Omega_2$ we can choose $k'$ such that
$p(x)+1/k'\leq (p(x))^*=Np(x)/(N-p(x))$.
So
$$
|u(x)|^{p(x)+1/n}\leq|u(x)|^{(p(x))^*},
$$
 for $n>k'$, and $x\in\Omega_2$.
Since $u\in L^{(p(x))^*}(B(x_0,r))$ \cite[Thm. 8.3.1]{dil}
we may use the Lebesgue Theorem again to obtain
$$
\lim_{n\to\infty}\int_{\Omega_2} |u(x)|^{p(x)+1/n}dx
= \int_{\Omega_2} |u(x)|^{p(x)}dx.
$$
Given that $p\in MPI(\Omega)$, we have
$$
I_2\leq C\int_{B(x_0,r)}|u|^{p(x)}dx
\leq C\int_{B(x_0,r)}|\nabla u|^{p(x)}dx.
$$
Now we estimate $I_1$ by using the modular Young's inequality
 \cite[Theorem 3.2.21]{har},
$$
I_1\leq p^+C_1\int_{B(x_0,r)}|w(x)|^{p(x)/(p(x)-1)}|u(x)|^{p(x)}
+p^+C_2\int_{B(x_0,r)}|\nabla u(x)|^{p(x)}.
$$
Again, since $p\in MPI(\Omega)$ we obtain
$$
I_1\leq C\int_{B(x_0,r)}|\nabla u|^{p(x)}dx.
$$
Finally, recalling that $\operatorname{div}\,w(x)=NV(x)$ we obtain
$$
N\int_{B(x_o,r)}V(x)|u(x)|^{p(x)}
\leq C\int_{B(x_0,r)}|\nabla u(x)|^{p(x)}dx,
$$
which leads to the claim of the theorem.
\end{proof}

\section{Unique Continuation}

Consider the equation
\begin{equation}\label{e6}
  Hu:=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)
+V(x)|u|^{p(x)-2}u=0,\quad  x \in \Omega,
\end{equation}
$u\in W^{1.p(x)}_{\rm loc}(\Omega)$, $1<p(x)<N$,
$V\in L^{\frac{N}{p(x)}}(\Omega)$.
A weak solution of \eqref{e6} is a function
$u\in W^{1.p(x)}_{\rm loc}(\Omega)$
such that
\begin{equation}\label{e7}
\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u\cdot\nabla\varphi dx+
\int_{\Omega} V(x)|u|^{p(x)-2}u\cdot\varphi dx=0,
\end{equation}
 for all $\varphi\in W^{1,p(x)}_0(\Omega)$.

 Note that $L^{\frac{N}{p(x)}}(\Omega)$ implies
$V\in L^1(\Omega)$ by \cite[Theorem 3.3.1]{har}.
The main interest of this section is to prove some unique
continuation results for solution of \eqref{e6} according
to the following  definitions.

\begin{definition} \rm
A function $u \in L^{p(x)}_{\rm loc}(\Omega)$ vanishes of
infinite order in the
$p(x)$-mean at a point $x_0\in \Omega$ if , for each
$k\in \mathbb{N}$
\begin{equation}\label{e8}
 \lim_{R\to 0}\frac{1}{R^{k}}\int_{|x-x_0|< R}|u|^{p(x)}dx=0.
\end{equation}
\end{definition}

\begin{definition} \rm
The operator $H$ has the unique continuation property in $\Omega$ if
the only solution to $Hu=0$ such that $u$  vanishes of infinity
order in the $p(x)$-mean at a point $x_0\in \Omega$  is $u$ must
be identically zero in $\Omega$.
\end{definition}

\begin{lemma}[\cite{za1}]\label{e9}
Assume $w\in L^{1}{loc}{\Omega}$, $w \ge 0$ almost everywhere
in $\Omega$, $w\not\equiv 0$. If there exists $C$ such that
\[
 \int_{B(x_0,2r)}w(x)\,dx \leq C \int_{B(x_0,r)}w(x)\,dx,\quad
 \forall r >0
\]
Then $w(x)$ has no zero of infinity order in $\Omega$.
\end{lemma}

Recall that $\Omega\subset \mathbb{R^{N}}$ is a bounded open set.
We want to prove estimates independent of $p^{+}$ for bounded
solutions. For this purpose we assume throughout this section
that $1<p^{-}\le p^{+}<\infty$ and $p$ is Lipschitz continuous.
In particular, $p$ is Log-H\"older continuous. The new
feature in the estimate is the choice of a test function which
include the variable exponent.
This has both advantages and disadvantages: we need to assume
that $p$ is differentiable almost everywhere, but, on the other hand,
we avoid terms involving $p^{+}$, which
would be impossible to control later, see\cite{har}.


 In this section we prove the unique continuation property for
the operator $Hu$,  defined in \ref{e6} extending in some sense
the results obtained by Zamboni \cite{za1} to variable exponent spaces.
To prove this property we need the following Lemma.

\begin{lemma}(Caccioppoli estimate)\label{e10}
Let $p: \Omega\to (1,N)$ be an exponent with
$1<p^{-}\le p^{+}<\infty$ and such that $p\in MPI(\Omega) $
is Lipschitz continuous. Let $u$ be a non negative solution of
\eqref{e6} in $\Omega$ and $\eta:\Omega\to [0,1]$ be a Lipschitz
function with compact support in $\Omega$ satisfying
$\eta\log\frac{1}{\eta}\le a|\nabla\eta|$ a.e. in
$\big\{\eta>0\big\}$ for  some constant $a>0$. Then
\[
  \int_{\Omega}|\nabla\log u|^{p(x)}\eta^{p(x)}\,dx \le C\int_{\Omega}|\eta|^{p(x)}\,dx
\]
for non-negative Lipschitz function $\eta\in C_0^{\infty}$.
\end{lemma}

\begin{proof}
Let $x_0\in\Omega$, Let $B(x_0,h)$ be a ball such that $B(x_0,2h)$
is contained in $\Omega$. Consider any ball $B(x_0, r)$ with $r<h$.
Let $\eta\in C_0^{\infty}$ with compact support in $B(x_0,2r)$
such that $\eta\log\frac{1}{\eta}\le a|\nabla\eta|$ a.e.
 in $\big\{x\in B_{2r}: \eta>0\big\}$ for  some
constant $a>0$, and $\eta =1$ in $B_{r}$ and
$|\nabla\eta|\le\frac{C}{r}$. Then
using
\[
  \varphi (x) =|u(x)|^{1-p(x)}\eta^{p(x)}
\]
as test function in \eqref{e7} we obtain
\begin{align*}
0 &= \int_{B_{2r}}(1-p(x))\eta^{p(x)}|\nabla u|^{p(x)}|u|^{-p(x)}\,dx\\
&\quad - \int_{B_{2r}}\eta^{p(x)}|\nabla u|^{p(x)-2}\nabla u\cdot\nabla p(x)|u|^{1-p(x)}\log u\\
&\quad +\int_{B_{2r}}p(x)\eta^{p(x)-1}\nabla u\cdot\nabla \eta |\nabla u|^{p(x)-2}|u|^{1-p(x)}\,dx\\
&\quad +\int_{B_{2r}}|\nabla u|^{p(x)-2}\nabla u\cdot\nabla p(x) |u|^{1-p(x)}\eta^{p(x)}\log\eta\,dx\\
&\quad +\int_{B_{2r}}V|u|^{p(x)-2}u\eta^{p(x)}|u|^{1-p(x)}\,dx;
\end{align*}
therefore,
\[
(p^{-}-1)\int_{B_{2r}}\eta^{p(x)}|\nabla\log u |^{p(x)}\,dx
\le |I_{1}|+|I_{2}|+|I_{3}|+|I_4|,
\]
where
\begin{gather*}
 I_1:=-\int_{B_{2r}}\eta^{p(x)}|\nabla u|^{p(x)-2}
\nabla u\cdot\nabla p(x)|u|^{1-p(x)}\log u\,dx,\\
I_2:=\int_{B_{2r}}p(x)\eta^{p(x)-1}\nabla u\nabla \eta
 |\nabla u|^{p(x)-2}|u|^{1-p(x)}\,dx,\\
I_3:=\int_{B_{2r}}|\nabla u|^{p(x)-2}\nabla u\nabla p(x)
 |u|^{1-p(x)}\eta^{p(x)}\log\eta\,dx,\\
I_4:=\int_{B_{2r}}V|u|^{p(x)-2}u\eta^{p(x)}|u|^{1-p(x)}\,dx\,.
\end{gather*}
Now we estimate $ I_{1}$, $I_{2}$, $I_{3}$ and $I_4$.
We have
\begin{align*}
 |I_1|& \leq \int_{B_{2r}}\eta^{p(x)}|\nabla p(x)|
 |\nabla u|^{p(x)-1}|u|^{1-p(x)}\log u\,dx\\
      & \leq \int_{B_{2r}}\eta^{p(x)}|\nabla p(x)|
 |\nabla u|^{p(x)-1}|u|^{1-p(x)} |u|^{\pm\eta}\,dx\,,
\end{align*}
where $\eta>0$ and
\[
\pm\eta = \begin{cases}
   -\eta,   &   \text{if }|u|\le 1,\\
 \eta,    &   \text{if }|u|>1.
 \end{cases}
\]
Using  the Lebesgue Dominated Convergence Theorem as in the proof
 of Theorem \ref{tff} and Young's inequality we obtain
\begin{align*}
I_1& \leq \int_{B_{2r}}\eta^{p(x)}|\nabla p(x)|
 |\nabla u|^{p(x)-1}|u|^{1-p(x)} \,dx\\
& \leq \varepsilon C_p\int_{B_{2r}}\eta^{p(x)}
 |\nabla \log u|^{p(x)}dx+\varepsilon C_p\int_{B_{2r}}
\big(\frac{1}{\varepsilon }\big)^{p(x)-1}\eta^{p(x)}dx\,.
\end{align*}
On the other hand,
\begin{align*}
 |I_{2}|
&\le p^{+}|\int_{B_{2r}}\eta^{p(x)-1}\nabla u\cdot\nabla \eta |\nabla u|^{p(x)-2}|u|^{1-p(x)}\,dx|\\
&\le p^{+}\int_{B_{2r}}\eta^{p(x)-1}|\nabla u | |\nabla \eta| |\nabla u|^{p(x)-2}|u|^{1-p(x)}\,dx\\
&= p^{+}\int_{B_{2r}}\eta^{p(x)-1}|\nabla \eta| |\nabla u |^{p(x)-1}|u|^{1-p(x)}\,dx\\
&=  p^{+}\int_{B_{2r}}|\nabla \eta|\eta^{p(x)-1}|\nabla\log u |^{p(x)-1}\,dx\\
&\le p^{+}\int_{B_{2r}}\left(\frac{1}{\varepsilon}\right)^{p(x)-1}
|\nabla \eta|^{p(x)}dx+p^+\varepsilon\int_{B_{2r}}|\eta|^{p(x)}
|\nabla\log u |^{p(x)}\,dx\,.
 \end{align*}
For $I_3$ we have
 \begin{align*}
|I_{3}|&=
 \big|\int_{B_{2r}}|\nabla u|^{p(x)-2}\nabla u\cdot\nabla p(x)
  |u|^{1-p(x)}\eta^{p(x)}|\log\eta|\,dx\big|\\
&\le  \int_{B_{2r}}|\nabla u|^{p(x)-2}|\nabla u| |\nabla p(x)| |u|^{1-p(x)}\eta^{p(x)}|\log\eta|\,dx\\
&\le L \int_{B_{2r}}|\nabla u |^{p(x)-1}|u|^{1-p(x)}\eta^{p(x)-1}\eta |\log\eta|\,dx\\
&=L \int_{B_{2r}}\eta^{p(x)-1}|\nabla\log u |^{p(x)-1}\eta \log\frac{1}{\eta}\,dx\\
&\le aL \int_{B_{2r}}|\nabla\eta|\eta^{p(x)-1}|\nabla\log u |^{p(x)-1}\,dx\\
&\le aL\int_{B_{2r}}\left(\frac{1}{\varepsilon}\right)^{p(x)-1}
|\nabla \eta|^{p(x)}dx+aL\varepsilon\int_{B_{2r}}|\eta|^{p(x)}
|\nabla\log u |^{p(x)}\,dx
 \end{align*}
and
 \begin{align*}
 I_{4}&\le
 \int_{B_{2r}}V|u|^{p(x)-2}u\eta^{p(x)}|u|^{1-p(x)}\,dx\\
&\le \int_{B_{2r}}V|u|^{p(x)-2}|u| \eta^{p(x)}|u|^{1-p(x)}\,dx\\
&= \int_{B_{2r}}V\eta^{p(x)}\,dx;
\end{align*}
therefore,
\begin{align*}
&(p^{-}-1)\int_{B_{2r}}\eta^{p(x)}|\nabla\log u |^{p(x)}\,dx\\
&\le  (p^{+}+aL)\varepsilon\int_{B_{2r}}\eta^{p(x)}|\nabla\log u
 |^{p(x)}\,dx + \int_{B_{2r}}V\eta^{p(x)}\,dx \\
&\quad +(p^{+}+aL)\int_{B_{2r}}
\big(\frac{1}{\varepsilon}\big)^{p(x)-1}|\nabla \eta|^{p(x)}\,dx
\end{align*}
Let $0<\epsilon\le1$ such that
$\epsilon<\min\big\{1,\frac{p^--1}{2(p^++aL)}\big\}$.
Since $(\frac{1}{\varepsilon})^{p(x)-1}\le\frac{1}{\varepsilon})
^{p^+-1}$, we obtain
 \[
 \int_{B_{2r}}\eta^{p(x)}|\nabla\log u |^{p(x)}\,dx
\le  C\int_{B_{2r}}|\nabla\eta |^{p(x)}\,dx
+ \int_{B_{2r}}V\eta^{p(x)}\,dx
 \]
and by Theorem \ref{tff}, we have
 \begin{align*}
 \int_{B_{2r}}\eta^{p(x)}|\nabla\log u |^{p(x)}\,dx
&\le  C\int_{B_{2r}}|\nabla\eta |^{p(x)}\,dx +C\int_{B_{2r}}|\nabla\eta|^{p(x)}\,dx\\
&\le  C\left( p^{+},a,L,\Omega\right)\int_{B_{2r}}|\nabla\eta|^{p(x)}\,dx\\
&=  C\int_{B_{2r}}|\nabla\eta|^{p(x)}\,dx
\end{align*}
Since $C>0$, this completes the proof.
\end{proof}

\begin{theorem}\label{e11}
Let $p: \Omega\to (1,N)$ be an exponent with $1<p^{-}\le p^{+}<\infty$
and such that $p\in MPI(\Omega) $ is Lipschitz continuous.
Let $u\in W^{1,p(x)}(\Omega)$, $u\geq 0$, be a solution of \eqref{e6},
then $u$ has no zero of infinite order in $\Omega$,
 for all $V\in L^{\frac{N}{p(x)}}(\Omega)$.
\end{theorem}

\begin{proof}
Let $\varphi (x)$  as in the proof of Lemma \ref{e10} then,
we have
 \[
 \int_{B_{2r}}\eta^{p(x)}|\nabla\log u |^{p(x)}\,dx
\le C\int_{B_{2r}}|\nabla\eta|^{p(x)}\,dx.
 \]
And,  since $p(x)$ is Log-H\"older,
$r^{-p(x)}\le Cr^{-p(x_0)}$ for all $x_0\in B_{2r}$,
 by \eqref{*l}, we have
\begin{align*}
\int_{B_{2r}}|\nabla\eta|^{p(x)}\,dx &\le
\int_{B_{2r}}\big(\frac{C}{r}\big)^{p(x)}\,dx\\
&\le \frac{C}{{r}^{p(x_0)}}\int_{B_{2r}}\,dx\\
&\le Cr^{-p(x_0)}|B_{2r}|\\
&\le Cr^{N-p(x_0)};
 \end{align*}
 therefore,
\[
 \int_{B_{2r}}\eta^{p(x)}|\nabla\log u |^{p(x)}\,dx\le Cr^{N-p(x_0)}
\]
 and hence
\[
 \int_{B_{r}}|\nabla\log u |^{p(x)}\,dx\le Cr^{N-p(x_0)}
\]
since $\eta=1$ in $B_{r}$.
Now by the Poincar\'e inequality \cite[Proposition 8.2.8]{dil},
\[
 \int_{B_{r}}\Big(\frac{|v-v_{B_{r}}|}{r}\Big)^{p(x)}\,dx
\le C \int_{B_{r}} |\nabla v|^{p(x)}\,dx + C|B_{r}|
\]
for all $v\in W^{1,p(x)}(B_{r})$.
We apply this to the function $v:= \log u$:
\begin{align*}
\slashint_{B_{r}}\Big(\frac{|\log u-(\log u)_{B_{r}}|}{r}\Big)^{p(x)}
&\le C\slashint_{B_{r}}|\nabla\log u |^{p(x)}\,dx +C\\
&\le C r^{-p(x_0)}
\end{align*}
by Log-H\"{o}lder continuity of $p(x)$, we have
\[
 \frac{1}{r^{p(x_0)}}\slashint_{B_{r}}|\log u-(\log u)_{B_{r}}
|^{p(x)}\,dx
\le \slashint_{B_{r}}\Big(\frac{|\log u-(\log u)_{B_{r}}|}{r}
\Big)^{p(x)}\,dx
\le  C r^{-p(x_0)};
\]
thus
\[
\slashint_{B_{r}}|\log u-(\log u)_{B_{r}}|^{p(x)}\,dx
\le C r^{-p(x_0)}r^{p(x_0)}= C,
\]
and since
\[
 \slashint_{B_{r}}|\log u-(\log u)_{B_{r}}|\,dx
\le \slashint_{B_{r}}|\log u-(\log u)_{B_{r}}|^{p(x)} + 1\,dx\le C,
\]
it follows that $\log u\in BMO(B_{r})$ uniformly, see \cite{gr}.
The measure theoretic John-Nirenberg \cite{jn}
implies that there exist positive constants $\alpha$ and
$C$ depending on the BMO-norm such that
\[
\slashint_{B_{r}} e^{\alpha|  f -f_{B_{r}} |}\,dx\le C,
\]
where $f := \log u$. Using this we can conclude that
\begin{align*}
 \slashint_{B_{r}} e^{\alpha f}\,dx\slashint_{B_{r}}
 e^{-\alpha f}\,dx
&= \slashint_{B_{r}} e^{\alpha (f-f_{B_{r}})}\,dx\slashint_{B_{r}}
 e^{-\alpha (f-f_{B_{r}})}\,dx\\
&\le  \Big(\slashint_{B_{r}} e^{\alpha|  f -f_{B_{r}}|}\,dx\Big)^{2}
\le C
\end{align*}
which implies
\[
 \int_{B_{r}} e^{\alpha f}\,dx\int_{B_{r}} e^{-\alpha f}\,dx
\le C |B_{r}|^{2}.
\]
So
\[
 \int_{B_{r}}|u|^{\alpha}\,dx\int_{B_{r}}|u|^{-\alpha}\,dx
\le C |B_{r}|^{2};
\]
that is, $|u|^{\alpha}$ belongs to the Muckenhoupt class $A_{2}$
for $\alpha > 0$, see \cite{gr}.
Now it is well known that $A_{2}$ implies the doubling property
for $|u|^{\alpha}$, that is the assumption of Lemma\eqref{e9}.
So the conclusion follows for $|u|^{\alpha}$  and hence also
for $u$.
\end{proof}

\subsection*{Acknowledgements}
The authors want to thank Peter H\"ast\"o
for the careful reading of a draft of this article, and for
his suggestions. Johnny Cuadro was supported by a CONACYT M\'exico's
Ph. D. Scholarship.

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\section*{Addendum posted on October 14, 2012}

The authors want to correct the following misprints:

\noindent Page 3, line 4:  the inclusion is just continuous. 

\noindent Page 6, Definition 3.1 must say:

\noindent \textbf{Definition 3.1}
Assume $w\in L^1_{\rm loc}(\Omega)$, $w\geq 0$ almost everywhere in $\Omega$.
We say that $w$ has a zero of infinite order at $x_0\in\Omega$ if 
$$
\lim_{\sigma\to 0}\frac{\int_{B(x_0,\sigma)}w(x)\,dx}
 {|B(x_0,\sigma)|^k}= 0,\quad\forall k>0. 
$$

\noindent Page 6, Definition 3.2 must say:

\noindent \textbf{Definition 3.2} 
The operator $H$ has the strong unique continuation property in $\Omega$ if
the only solution to $Hu=0$ such that $u$  vanishes of infinity
order at a point $x_0\in \Omega$ is $u\equiv 0$ in $\Omega$.

\noindent Page 7, in Lemma  3.3 must say:  $w\in L^{1}_{\rm loc}(\Omega)$. 

\noindent Page 7, in Lemma 3.4: The constant  $C$ is missing.

\noindent Page 9, Theorem 3.5 should include:  ``$w\not\equiv 0$ a.e.'' 

\noindent Page 9, In Theorem 3.5: The constant  $C$ is missing.

End of addendum.

\end{document}