\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 $1
1\}$. 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 $q(x)
0$ 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 $1
1$, 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 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 $1 0\big\}$ 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 $r 0$, 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}