\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 $11\}$. 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 $$\label{Hol} \big|\int_\Omega uv\,dx\big|\leq\Big(\frac{1}{p^-}+ \frac{1}{{p'}^-}\Big)|u|_{p(\cdot)}|v|_{p'(\cdot)}$$ 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 $11$, 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 $$\label{fef1} \int_{\mathbb{R}^{N}}|u(x)|^{p}|f(x)|\,dx\leq C\int_{\mathbb{R}^{N}}|\nabla u(x)|^{p}\,dx\quad \forall u\in C^{\infty}_0(\mathbb{R}^{N}).$$ in the case $p=2$, assuming $f$ in the Morrey's space $L^{r,N-2r}(\mathbb{R}^{N})$, with $1 1$. 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 $$\label{e6} Hu:=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u) +V(x)|u|^{p(x)-2}u=0,\quad x \in \Omega,$$ $u\in W^{1.p(x)}_{\rm loc}(\Omega)$, $10 \] 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$10\big\}$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$r0\big\}$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>0and $\pm\eta = \begin{cases} -\eta, & \text{if }|u|\le 1,\\ \eta, & \text{if }|u|>1. \end{cases}$ 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*} ForI_3we 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*} Let0<\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*} SinceC>0$, this completes the proof. \end{proof} \begin{theorem}\label{e11} Let$p: \Omega\to (1,N)$be an exponent with$1 0$, 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}. 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Sb. 183 (1992), 47-84. \end{thebibliography} \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}