\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 08, pp. 1--6.\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-2012/08\hfil Simplicity and stability] {Simplicity and stability of the first eigenvalue of a $(p;q)$ Laplacian system} \author[G. A. Afrouzi, M. Mirzapour, Q. Zhang \hfil EJDE-2012/08\hfilneg] {Ghasem A. Afrouzi, Maryam Mirzapour, Qihu Zhang} % in alphabetical order \address{Ghasem Alizadeh Afrouzi \newline Department of Mathematics, Faculty of Mathematical Sciences \\ University of Mazandaran, Babolsar, Iran} \email{afrouzi@umz.ac.ir} \address{Maryam Mirzapour \newline Department of Mathematics, Faculty of Mathematical Sciences \\ University of Mazandaran, Babolsar, Iran} \email{mirzapour@stu.umz.ac.ir} \address{Qihu Zhang \newline Department of Mathematics and Information Science \\ Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China} \email{zhangqh1999@yahoo.com.cn} \thanks{Submitted November 3, 2011. Published January 12, 2012.} \subjclass[2000]{35J60, 35B30, 35B40} \keywords{Eigenvalue problem; quasilinear operator; simplicity; stability} \begin{abstract} This article concerns special properties of the principal eigenvalue of a nonlinear elliptic system with Dirichlet boundary conditions. In particular, we show the simplicity of the first eigenvalue of \begin{gather*} -\Delta_p u = \lambda |u|^{\alpha-1}|v|^{\beta-1}v \quad \text{in } \Omega,\\ -\Delta_q v = \lambda |u|^{\alpha-1}|v|^{\beta-1}u \quad \text{in } \Omega,\\ (u,v)\in W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega), \end{gather*} with respect to the exponents $p$ and $q$, where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Preliminaries} Eigenvalue problems for $p$-Laplacian operators subject to Zero Dirichlet boundary conditions on a bounded domain have been studied extensively during the past two decades, and many interesting results have been obtained. Most of the investigations have relied on variational methods and deduced the existence of a principal eigenvalue as a consequence of minimization results of appropriate functionals. In this article, we study the eigenvalue system \begin{equation} \label{e1} \begin{gathered} -\Delta_p u = \lambda |u|^{\alpha-1}|v|^{\beta-1}v \quad \text{in } \Omega,\\ -\Delta_q v = \lambda |u|^{\alpha-1}|v|^{\beta-1}u \quad \text{in } \Omega,\\ (u,v)\in W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega), \end{gathered} \end{equation} where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain, $p,q>1$ and $\alpha,\beta$ are real numbers satisfying \begin{equation} \label{eH} \alpha>0,\quad \beta>0, \quad \frac{\alpha}{p}+\frac{\beta}{q}=1. \end{equation} We mention that problem \eqref{e1} aries in several fields of application. For instance, in the case where $p>2$, problem \eqref{e1} appears in the study of non-Newtonian fluids, pseudoplastics for $10$. We call it positive eigenvector. \begin{definition} \label{def1} \rm An open subset $\Omega$ of $\mathbb{R}^{N}$ is said to have the segment property if for any $x\in \partial\Omega$, there exists an open set $G_{x}\in \mathbb{R}^{N}$ with $x\in G_x$ and a pair $y_x$ of $\mathbb{R}^{N}\backslash\{0\}$ such that if $z\in \overline{\Omega}\cap G_x$ and $t\in (0,1)$, then $z+ty_x\in \Omega$. \end{definition} This property allows us to push the support of a function $u$ in $\Omega$ by a translation . The following results play an important role in the proof of Theorem 2. \begin{lemma} \label{lem1} Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ having the segment property. If $u\in W^{1,p}(\Omega)\cap W_{0}^{1,s}(\Omega)$ for some $s\in (1,p)$, then $u\in W_{0}^{1,p}(\Omega)$. \end{lemma} \section{Simplicity} Firstly we introduce \begin{equation} \label{e3} \begin{aligned} A_{p}(u,\varphi) &= \int_{\Omega}|\nabla u|^{p}dx +(p-1)\int_{\Omega}|\nabla \varphi|^{p}\Big( \frac{|u|}{\varphi}\Big)^{p}dx \\ &\quad -p \int_{\Omega}\frac{|u|^{p-2}u}{\varphi^{p-1}} |\nabla \varphi|^{p-2}\nabla \varphi\nabla u dx \\ &= \int_{\Omega}|\nabla u|^{p}dx +\int_{\Omega}\frac{\Delta_p\varphi}{\varphi^{p-1}}|u|^{p}dx. \end{aligned} \end{equation} \begin{lemma}[\cite{E1}] \label{lem2} For all $(u,\varphi)\in (W_{0}^{1,p}(\Omega) \cap C^{1,\gamma}(\Omega))^{2}$ with $\varphi>0$ in $\Omega$ and $\gamma\in(0,1)$, we have $A_{p}(u,\varphi)$; i.e., $$ \int_{\Omega}|\nabla u|^{p}dx\geq\int_{\Omega} \frac{\Delta_p\varphi}{\varphi^{p-1}}|u|^{p}dx, $$ and if $A_{p}(u,\varphi)=0$ there is $c\in \mathbb{R}$ such that $u=c\varphi$. \end{lemma} \begin{proof} Using Young's inequality (since $\frac{1}{p}+\frac{p-1}{p}=1)$ we can write, for $\epsilon>0$, \begin{equation} \label{e4} \begin{split} \nabla u|\nabla \varphi|^{p-2}\nabla \varphi \frac{u|u|^{p-2}}{\varphi^{p-1}} &\leq |\nabla u||\nabla \varphi|^{p-1}\Big(\frac{|u|}{\varphi} \Big)^{p-1} \\ &\leq \frac{\epsilon^{p}}{p}|\nabla u|^{p}+\frac{p-1}{p\epsilon^{p}} |\frac{u}{\varphi}|^{p}|\nabla \varphi|^{p}. \end{split} \end{equation} By choosing $\epsilon=1$ and integration over $\Omega$, we have \begin{equation} \label{e5} p\int_{\Omega}|\nabla \varphi|^{p-2}\nabla \varphi\nabla u \Big(\frac{u|u|^{p-2}}{\varphi^{p-1}}\Big)dx \leq\int_{\Omega}|\nabla u|^{p}dx+(p-1)\int_{\Omega} |\frac{u}{\varphi}|^{p}|\nabla \varphi|^{p}dx. \end{equation} Therefore, we conclude that $A_{p}(u,\varphi)\geq0$. If $A_{p}(u,\varphi)=0$, then we obtain \begin{equation} \label{e6} p\int_{\Omega}|\nabla \varphi|^{p-2} \nabla \varphi\nabla u\Big(\frac{u|u|^{p-2}}{\varphi^{p-1}}\Big)dx =\int_{\Omega}|\nabla u|^{p}dx+(p-1) \int_{\Omega}|\frac{u}{\varphi}|^{p}|\nabla \varphi|^{p}dx. \end{equation} Letting $\epsilon=1$ in \eqref{e4}, we obtain \begin{equation} \label{e7} \int_{\Omega}\nabla u|\nabla \varphi|^{p-2}\nabla \varphi \frac{u|u|^{p-2}}{\varphi^{p-1}}dx =\int_{\Omega}|\nabla u||\nabla \varphi|^{p-1} \Big(\frac{|u|}{\varphi}\Big)^{p-1}dx. \end{equation} Combining the two inequalities, we deduce that $|\nabla u|=|\big(\frac{u}{\varphi}\big)\nabla \varphi|$, it follows that $\nabla u=\eta\big(\frac{u}{\varphi}\big)\nabla \varphi$, where $|\eta|=1$. On the other hand, $A_{p}(u,\varphi)=0$ implies that $\eta=1$ and $\nabla(\frac{u}{\varphi})=0$; that is, there is $c\in \mathbb{R}$ such that $u=c\varphi$. This completes the proof. \end{proof} \begin{theorem} \label{thm1} Let $\lambda_{1}(p,q)$ be defined by \eqref{e2}, then $\lambda_{1}(p,q)$ is simple. \end{theorem} \begin{proof} Let $(u,v)$ and $(\varphi,\psi)$ be two eigenvectors associated with $\lambda_{1}(p,q)$. We show that there exist real numbers $k_1$, $k_2$ such that $u=k_{1}\varphi$ and $v=k_{2}\psi$. Using Young's inequality, by \eqref{eH} and the definition of $\lambda_{1}(p,q)$, we can write \begin{align*} J(\varphi,\psi) &= \lambda_{1}(p,q)\Lambda(\varphi,\psi)\\ &\leq \lambda_{1}(p,q)\int_{\Omega}u^{\alpha}v^{\beta} \frac{|\varphi|^{\alpha}|\psi|^{\beta}}{u^{\alpha}v^{\beta}}dx\\ &\leq \lambda_{1}(p,q)\int_{\Omega}u^{\alpha}v^{\beta} \Big[\frac{\alpha}{p}\frac{|\varphi|^{p}}{u^{p}}+\frac{\beta}{q}\frac{|\psi|^{q}}{v^{q}} \Big]dx\\ &\leq \lambda_{1}(p,q)\int_{\Omega}\Big[\frac{\alpha}{p} \frac{u^{\alpha-1}v^{\beta}}{u^{p-1}}|\varphi|^{p} +\frac{\beta}{q}\frac{u^{\alpha}v^{\beta-1}}{v^{q-1}}|\psi|^{q} \Big]dx\\ &\leq \frac{\alpha}{p}\int_{\Omega}\frac{-\Delta_{p}u}{u^{p-1}} |\varphi|^{p}dx+\frac{\beta}{q}\int_{\Omega} \frac{-\Delta_{q}v}{v^{q-1}}|\psi|^{q}dx. \end{align*} Due to Lemma \ref{lem2}, we obtain $$ J(u,v)=\frac{\alpha}{p}\int_{\Omega}\frac{-\Delta_{p}u}{u^{p-1}} |\varphi|^{p}dx+\frac{\beta}{q}\int_{\Omega} \frac{-\Delta_{q}v}{v^{q-1}}|\psi|^{q}dx. $$ Thus $$ \int_{\Omega}|\nabla \varphi|^{p}dx =\int_{\Omega}\frac{-\Delta_{p}u}{u^{p-1}}|\varphi|^{p}dx,\quad \int_{\Omega}|\nabla \psi|^{q}dx =\int_{\Omega}\frac{-\Delta_{q}v}{v^{q-1}}|\psi|^{q}dx. $$ By Lemma \ref{lem2}, there exist real numbers $k_1$ and $k_2$ such that $u=k_{1}\varphi$ and $v=k_{2}\psi$ and the theorem follows. \end{proof} \section{Stability} \begin{theorem} \label{thm2} Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ having the segment property. Then, the function $(p,q)\to \lambda_{1}(p,q)$ is continuous from $I_{\alpha,\beta}$ to $\mathbb{R}^{+}$, where $$ I_{\alpha,\beta}=\{(p,q)\in (1,+\infty)\times(1,+\infty) \text{ such that \eqref{eH} is satisfied}\}. $$ \end{theorem} \begin{proof} Let $(t_n)_{n\geq1}$, $t_n=(p_n,q_n)$ be a sequence in $I_{\alpha,\beta}$ converging at $t=(p,q)\in I_{\alpha,\beta}$. We will prove that $$ \lim_{n\to\infty}\lambda_{1}(p_n,q_n)=\lambda_{1}(p,q). $$ Indeed, let $(\varphi,\psi)\in C_{0}^{\infty}(\Omega)\times C_{0}^{\infty}(\Omega)$ such that $\Lambda(\varphi,\psi)>0$; hence, $$ \lambda_{1}(p_n,q_n)\leq \frac{\frac{\alpha}{p_n}\|\nabla \varphi\|_{p_n}^{p_n} +\frac{\beta}{q_n}\|\nabla \psi\|_{q_n}^{q_n}}{\Lambda(\varphi,\psi)}, $$ since $\lambda_{1}(p_n,q_n)$ is the infimum. Letting $n$ tend to infinity, we deduce from Lebesgue's theorem $$ \limsup_{n\to\infty}\lambda_{1}(p_n,q_n) \leq\frac{\frac{\alpha}{p}\|\nabla \varphi\|_{p}^{p} +\frac{\beta}{q}\|\nabla \psi\|_{q}^{q}}{\Lambda(\varphi,\psi)}. $$ Then \begin{equation}\label{e8} \limsup_{n\to\infty}\lambda_{1}(p_n,q_n)\leq\lambda_{1}(p,q). \end{equation} On the other hand, let $\{(p_{n_{k}},q_{n_{k}})\}_{k\geq1}$ be a subsequence of $(t_{n})_{n}$ such that $$ \lim_{k\to\infty}\lambda_{1}(p_{n_{k}},q_{n_{k}}) =\liminf_{n\to\infty}\lambda_{1}(p_n,q_n). $$ Let us fix $\epsilon_{0}>0$ small enough, so that for all $\epsilon\in (o,\epsilon_{0})$, we have \begin{gather}\label{e9} 1<\min(p-\epsilon,q-\epsilon), \\ \label{e10} \max(p+\epsilon,q+\epsilon)<\min((p-\epsilon)^{*},(q-\epsilon)^{*}). \end{gather} For each $k\in \mathbb{N}$, let $(u_{(p_{n_{k}},q_{n_{k}})},v_{(p_{n_{k}},q_{n_{k}})}) \in W_{0}^{1,p_{n_{k}}}(\Omega)\times W_{0}^{1,q_{n_{k}}}(\Omega)$ be a principal eigenfunction of $(S_{p_{n_{k}},q_{n_{k}}})$ related with $\lambda_{1}(p_{n_{k}},q_{n_{k}})$. Then, by Holder's inequality, for $\epsilon\in (0,\epsilon_0)$, the following inequalities hold: \begin{gather}\label{e11} \|\nabla u_{(p_{n_{k}},q_{n_{k}})}\|_{p-\epsilon} \leq\|\nabla u_{(p_{n_{k}},q_{n_{k}})}\|_{p_{n_{k}}} |\Omega|^{\frac{p_{n_{k}}-p+\epsilon}{p_{n_{k}}(p-\epsilon)}}, \\ \label{e12} \|\nabla v_{(p_{n_{k}},q_{n_{k}})}\|_{q-\epsilon} \leq\|\nabla v_{(p_{n_{k}},q_{n_{k}})}\|_{q_{n_{k}}} |\Omega|^{\frac{q_{n_{k}}-q+\epsilon}{q_{n_{k}}(q-\epsilon)}}. \end{gather} Combining these two inequalities and using the variational characterization of $\lambda_1$, we have \begin{gather}\label{e13} \|\nabla u_{(p_{n_{k}},q_{n_{k}})}\|_{p-\epsilon} \leq\Big{\{}\frac{p_{n_{k}}\lambda_{1} (p_{n_{k}},q_{n_{k}})}{\alpha}\Big{\}}^{\frac{1}{p_{n_{k}}}} |\Omega|^{\frac{p_{n_{k}}-p+\epsilon}{p_{n_{k}}(p-\epsilon)}} \\ \label{e14} \|\nabla v_{(p_{n_{k}},q_{n_{k}})}\|_{q-\epsilon} \leq\Big{\{}\frac{q_{n_{k}}\lambda_{1}(p_{n_{k}},q_{n_{k}})}{\beta} \Big{\}}^{\frac{1}{q_{n_{k}}}}|\Omega|^{\frac{q_{n_{k}}-q+\epsilon} {q_{n_{k}}(q-\epsilon)}}. \end{gather} Therefore, via \eqref{e9} and \eqref{e10}, for a subsequence \begin{gather*} (u_{(p_{n_{k}},q_{n_{k}})},v_{(p_{n_{k}},q_{n_{k}})}) \rightharpoonup(u,v)\quad \text{weakly in } W_{0}^{1,p-\epsilon}(\Omega)\times W_{0}^{1,q-\epsilon}(\Omega), \\ (u_{(p_{n_{k}},q_{n_{k}})},v_{(p_{n_{k}},q_{n_{k}})})\to (u,v)\quad \text{strongly in } L^{p+\epsilon}(\Omega)\times L^{q+\epsilon}(\Omega) \end{gather*} Passing to the limit in \eqref{e13} and \eqref{e14}, respectively as $k\to\infty$ and as $\epsilon\to\infty$, we have \begin{gather*} \|\nabla u\|_{p}^{p}\leq\frac{p}{\alpha} \lim_{k\to\infty}\lambda_{1}(p_{n_{k}},q_{n_{k}})<\infty,\\ \|\nabla v\|_{q}^{q}\leq\frac{q}{\beta} \lim_{k\to\infty}\lambda_{1}(p_{n_{k}},q_{n_{k}})<\infty. \end{gather*} Then $$ u\in W_{0}^{1,p-\epsilon}(\Omega)\cap W^{1,p}(\Omega) =W_{0}^{1,p}(\Omega),\quad v\in W_{0}^{1,q-\epsilon}(\Omega)\cap W^{1,q}(\Omega) =W_{0}^{1,q}(\Omega), $$ because $\Omega$ satisfies the segment property. On the other hand, from the variational characterization of $\lambda_{1}(p_{n_{k}},q_{n_{k}})$, \eqref{e11}, \eqref{e12}, and using the weak lower semi continuity of the norm; it follows that \begin{equation}\label{e15} \frac{1}{|\Omega|^{\frac{\epsilon}{p-\epsilon}}} \frac{\alpha}{p}\|\nabla u\|_{p-\epsilon}^{p} +\frac{1}{|\Omega|^{\frac{\epsilon}{q-\epsilon}}} \frac{\beta}{q}\|\nabla v\|_{q-\epsilon}^{q} \leq\lim_{k\to+\infty}\lambda_{1}(p_{n_{k}},q_{n_{k}}). \end{equation} Letting $\epsilon\to 0^{+}$ in \eqref{e15}, the Fatou lemma yields $$ \frac{\alpha}{p}\|\nabla u\|_{p}^{p} +\frac{\beta}{q}\|\nabla v\|_{q}^{q} \leq\lim_{k\to \infty}\lambda_{1}(p_{n_{k}},q_{n_{k}}). $$ Since $\Lambda(u_{(p_{n_{k}},q_{n_{k}})},v_{(p_{n_{k}},q_{n_{k}})})=1$ via compactness of $\Lambda$, $(u,v)$ is admissible in the variational characterization of $\lambda_{1}(p,q)$; hence $$ \lambda_{1}(p,q)\leq\lim_{k\to \infty}\lambda_{1}(p_{n_{k}},q_{n_{k}}) =\liminf_{n\to\infty}\lambda_{1}(p_n,q_n). $$ This and \eqref{e8} will complete the proof. 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