\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small 2004-Fez conference on Differential Equations and Mechanics \newline {\em Electronic Journal of Differential Equations}, Conference 11, 2004, pp. 129--134.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{129} \begin{document} \title[\hfilneg EJDE/Conf/11 \hfil Principal eigenvalue] {Stability of the principal eigenvalue of a nonlinear elliptic system} \author[A. El Khalil, M. Ouanan, A. Touzani \hfil EJDE/Conf/11 \hfilneg] {Abdelouahed El Khalil, Mohammed Ouanan, \\ Abdelfattah Touzani} % in alphabetical order \address{Abdelouahed El Khalil \hfill\break Department of Mathematics and Industrial Genie\\ Ecole Polytechnic School Montreal, Montreal, Canada} \email{lkhlil@hotmail.com} \address{Mohammed Ouanan \hfill\break Department of mathematics\\ Faculty of sciences Dhar-Mahraz \\ P.O. Box 1796 Atlas \\ Fez 30000, Morocco} \email{m\_ouanan@hotmail.com} \address{Abdelfattah Touzani \hfill\break Department of mathematics\\ Faculty of sciences Dhar-Mahraz \\ P.O. Box 1796 Atlas \\ Fez 30000, Morocco} \email{atouzani@iam.net.ma} \date{} \thanks{Published October 15, 2004.} \subjclass[2000]{35P20, 35J70, 35J20} \keywords{p-Laplacian operator; principal eigenvalue; stability; \hfill\break\indent segment property; dependence on the domain} \begin{abstract} This paper concerns some special properties of the principal eigenvalue of nonlinear elliptic systems with Dirichlet boundary conditions. We study the stability with respect to the exponents $p$ and $q$; and the dependence on the domain variations. \end{abstract} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \maketitle \section{Introduction} In this work, we survey two results concerning the nonlinear elliptic system \begin{equation} \label{Spq} \begin{gathered} -\Delta_p u = \lambda |u|^{\alpha}|v|^{\beta}v \quad \mbox{in } \Omega\\ -\Delta_q v = \lambda |u|^{\alpha}|v|^{\beta}u \quad \mbox{in } \Omega \\ (u,v)\in W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega), \end{gathered} \end{equation} where $\Omega\subset \mathbb{R}^N$, ( $N\geq 3$), is a bounded domain not necessary regular; $\alpha, \beta, p, q$ are reel numbers such that $p> 1$, $q >1$, $\alpha \geq 0$, $\beta \geq 0$ satisfying \begin{equation} \frac{\alpha+1}{p}+\frac{\beta+1}{q}=1. \label{e1.1} \end{equation} The principal eigenvalue $\lambda_1(p,q)$ of \eqref{Spq} is defined as \begin{equation} \lambda_1= \inf \big\{ A(u,v): (u,v)\in W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega),\, B(u,v)=1 \big\}, \label{e1.2} \end{equation} where \begin{gather*} A(u,v)=\frac{\alpha+1}{p}\int_\Omega |\nabla u|^p\,dx+\frac{\beta+1}{q} \int_\Omega |\nabla v|^q\,dx,\\ B(u,v)= \int_\Omega |u|^{\alpha}|v|^{\beta}uv\,dx. \end{gather*} It is known (see \cite{OU}) that this eigenvalue is simple, isolated, and can be expressed as \eqref{e1.2}. It is also known that the associated principal eigenfunction $(u,v)$ can be chosen strictly positive in $\Omega$. This eigenfunction is a regular pair of functions where the infimum of the energy functional $A$ is attained on the $C^1$-manifold $\{(u,v)\in W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega):B(u,v)=1\}$. \par The operator $-\Delta_p$ arises in problems from pure mathematics, such as the theory of quasiregular and quasiconformal mapping, as well as in problems from a variety of applications, e.g. non-Newtonian fluids, reaction diffusion problems, flow through porous media, nonlinear elasticity, glaciology, petroleum extraction, astronomy, etc.\par The main purpose of this work is to prove that the mapping $(p,q)\mapsto\lambda_1(p,q)$ is continuous (stable) on the set $$ I_{\alpha,\beta}= \{(p,q)\in (]1,+\infty[)^2 \mbox{ such that \eqref{e1.1} is satisfied}\}, $$ for any bounded domain $\Omega$ having the segment property. Recall that an open subset $\Omega $ of $\mathbb{R}^N$ is said to have the segment property if, given any $x\in \partial \Omega $, there exist an open set $G_x$ in $\mathbb{R}^N$ with $x\in G_x$ and $y_x$ of $\mathbb{R}^N\setminus \{ 0\} $ such that, if $z\in \overline{\Omega }\cap G_x$ and $t\in ]0,1[$, then $z+ty_x \in \Omega$.\par This property allows us by a translation to push the support of a function $u$ in $\Omega $. This class of domains for which the boundary is sufficiently regular to guarantee that $$ W^{1,t}(\Omega )\cap_{11$. See \cite{El-Li-To} for more details about this property. \par The difficulty is that as the exponents $(p,q)$ change, the appropriate spaces $L^p(\Omega)$ and $L^q(\Omega)$ also change. In the case of equation $\Delta_p u+\lambda |u|^{p-2}u=0$, the same result is studied by Lindqvist \cite{L1} but in a regular bounded domain. El Khalil et al.~\cite{El-Li-To} studied the equation $\Delta_p u+\lambda g|u|^{p-2}u=0$ with $g\in L_{\rm loc}^{\infty}(\Omega)\cap L^r(\Omega)$ and $\Omega$ having the segment property. Note that the stability was partially studied in \cite{E-M-O}.\\ For the dependence of $\lambda_1$ upon variations of the domain, instead of the exponents $p$ and $q$, we prove that $\lambda_1^{(p,q)}(\Omega_j)\to\lambda_1^{(p,q)}(\Omega)$ as $j\to+\infty$, where $ \Omega ={ \cup_{j\in \mathbb{N}^*}\Omega_j}$. In the case of only one equation, see \cite{An} for contracting domain, and see \cite{L2} for variations of the domain. \section{Stability} \begin{theorem} Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ having the segment property. Then, the function $(p,q)\mapsto \lambda_1(p,q)$ is continuous from $I_{\alpha,\beta}$ to $\mathbb{R}^+$. where $$ I_{\alpha,\beta}= \{(p,q)\in (]1,+\infty[)^2 \mbox{ such that \eqref{e1.1} is satisfied }\}. $$ \end{theorem} \begin{proof} Let $(t_n)_{n\geq 1}$, $t_n=(p_n,q_n)$ be a sequence in $I_{\alpha,\beta}$ converging at $ t=(p,q)\in I_{\alpha,\beta}$. We will prove that $$ \lim_{n\to+\infty}\lambda_1(p_n,q_n)=\lambda_1(p,q). $$ Indeed, let $(\phi,\psi)\in C_0^{\infty}(\Omega)\times C_0^{\infty}(\Omega)$ such that $B(\phi,\psi)>0$; hence, $$ \lambda_1(p_n,q_n)\leq\frac{\frac{\alpha+1}{p_n}\|\nabla \phi\|_{p_n}^{p_n} +\frac{\beta+1}{q_n}\|\nabla \psi\|_{q_n}^{q_n}}{B(\phi,\psi)}, $$ since $\lambda_1(p_n,q_n)$ being the infimum. Letting $n$ tend to infinity, we deduce from Lebesgue's theorem \begin{equation} \limsup_{n\to+\infty}\lambda_1(p_n,q_n)\leq\frac{\frac{\alpha+1}{p} \|\nabla \phi\|_{p}^{p}+\frac{\beta+1}{q}\|\nabla \psi\|_{q}^{q}}{B(\phi,\psi)}. \label{e2.1} \end{equation} Then, \begin{equation} \limsup_{n\to+\infty}\lambda_1(p_n,q_n)\leq \lambda_1(p,q). \label{e2.2} \end{equation} On the other hand, let $\{(p_{n_k},q_{n_k})\}_{k\geq 1}$ be a subsequence of $(t_n)_n$ such that $$ \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 $\varepsilon_0 >0$ small enough, so that for all $\varepsilon\in (0,\varepsilon_0)$, we have \begin{gather} 1<\min (p-\varepsilon,q-\varepsilon), \label{e2.3}\\ \max(p+\varepsilon,q+\varepsilon)<\min((p-\varepsilon)^*,(q-\varepsilon)^*). \label{e2.4} \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 H\"older's inequality, for $\varepsilon \in (0,\varepsilon_0)$, the following inequalities hold \begin{gather} \|\nabla u_{(p_{n_k},q_{n_k})}\|_{p-\varepsilon}\leq \|\nabla u_{(p_{n_k},q_{n_k})}\|_{p_{n_k}}|\Omega|^{\frac{p_{n_k}-p +\varepsilon}{p_{n_k}(p-\varepsilon)}}, \label{e2.5} \\ \|\nabla v_{(p_{n_k},q_{n_k})}\|_{q-\varepsilon}\leq \|\nabla v_{(p_{n_k},q_{n_k})}\|_{q_{n_k}}|\Omega|^{\frac{q_{n_k}-q +\varepsilon}{q_{n_k}(q-\varepsilon)}}. \label{e2.6} \end{gather} Combining these two inequalities and using the variational characterization of $\lambda_1$, we have \begin{gather} \|\nabla u_{(p_{n_k},q_{n_k})}\|_{p-\varepsilon} \leq \big\{\frac{p_{n_k}\lambda_1(p_{n_k},q_{n_k})}{\alpha+1}\big\}^{\frac{1}{p_{n_k}}} |\Omega|^{\frac{p_{n_k}-p+\varepsilon}{p_{n_k}(p-\varepsilon)}}\label{e2.7}\\ \|\nabla v_{(p_{n_k},q_{n_k})}\|_{q-\varepsilon} \leq \big\{ \frac{q_{n_k}\lambda_1(p_{n_k},q_{n_k})}{\beta+1}\big\} ^{\frac{1}{q_{n_k}}}|\Omega|^{\frac{q_{n_k}-q+\varepsilon}{q_{n_k} (q-\varepsilon)}}. \label{e2.8} \end{gather} Therefore, via \eqref{e2.3} and \eqref{e2.4}, for a subsequence \begin{gather*} (u_{(p_{n_k},q_{n_k})},v_{(p_{n_k},q_{n_k})})\rightharpoonup (u,v) \quad\mbox{weakly in }W_0^{1,p-\varepsilon}(\Omega)\times W_0^{1,q-\varepsilon} (\Omega),\\ (u_{(p_{n_k},q_{n_k})},v_{(p_{n_k},q_{n_k})})\to(u,v) \quad \mbox {strongly in } L^{p+\varepsilon}(\Omega)\times L^{q+\varepsilon}(\Omega). \end{gather*} Passing to the limit in \eqref{e2.7} and \eqref{e2.8}, respectively as $k\to\infty $ and as $\epsilon \to0^+ $, we have \begin{gather*} \|\nabla u\|_p^p\leq \frac{p}{\alpha+1}\lim_{k\to+\infty}\lambda_1(p_{n_k},q_{n_k}) < +\infty,\\ \|\nabla v\|_q^q\leq \frac{q}{\beta+1}\lim_{k\to+\infty}\lambda_1(p_{n_k},q_{n_k}) < +\infty. \end{gather*} Then, $$ u\in W_0^{1,p-\varepsilon}(\Omega)\cap W^{1,p}(\Omega)=W_0^{1,p}(\Omega),\quad v\in W_0^{1,q-\varepsilon}(\Omega)\cap W^{1,q}(\Omega)=W_0^{1,q}(\Omega), %\eqno{(2.9)} $$ because $\Omega$ satisfies the segment property. On the other hand, from the variational characterization of $\lambda_1(p_{n_k},q_{n_k})$, \eqref{e2.5}, \eqref{e2.6}, and using weak lower semi continuity of the norm; it follows that $$ \frac{1}{|\Omega|^{\frac{\varepsilon}{p-\varepsilon}}}\frac{\alpha+1}{p} \|\nabla u\|_{p-\varepsilon}^p +\frac{1}{|\Omega|^{\frac{\varepsilon}{q-\varepsilon}}}\frac{\beta+1}{q} \|\nabla v\|_{q-\varepsilon}^q\leq \lim_{k\to+\infty}\lambda_1(p_{n_k},q_{n_k}). %\eqno{(2.10)} $$ Letting $\varepsilon \to 0^+$ in (2.10), the Fatou lemma yields, $$ \frac{\alpha+1}{p}\|\nabla u\|_p^p+\frac{\beta+1}{q}\|\nabla v\|_q^q \leq \lim_{k\to+\infty}\lambda_1(p_{n_k},q_{n_k}).%\eqno{(2.11)} $$ Since $ B(u_{(p_{n_k},q_{n_k})}, v_{(p_{n_k},q_{n_k})})=1$ via compactness of $B$, $(u,v)$ is admissible in the variational characterization of $\lambda_1(p,q)$; hence $$ \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}).%\eqno{(2.12)} $$ This and \eqref{e2.2} complete the proof. \end{proof} \begin{remark} \rm Observe that the segment property is used only for to prove $$\lambda_1(p,q)\leq \liminf_{n\to+\infty}\lambda_1(p_{n},q_{n}). $$ \end{remark} \section{Dependence with variations domain} \begin{theorem} \label{thm3.1} Let $\Omega_1\subset \Omega_2\subset \Omega_3\subset\dots $ be an exhaustion of $\Omega $ such that $$ \Omega =\bigcup_{j\in \mathbb{N}^*}\Omega_j . $$ Then \begin{gather} \lim _{j\to+\infty } \lambda_1^{(p,q)}(\Omega_j) = \lambda _1^{(p,q)}(\Omega ), \label{e3.1} \\ \lim_{j\to+\infty }\int _\Omega |\nabla \widetilde{u_j} -\nabla u|^p\,dx =\lim _{j\to+\infty }\int _\Omega |\nabla \widetilde{v_j} -\nabla v|^q\, dx =0, \label{e3.2} \end{gather} where $(u,v)$ is a positive principal eigenfunction of \eqref{Spq} related to $ \lambda_1^{(p,q)}(\Omega) $; $(u_j,v_j)$ is the principal eigenfunction related to $\lambda_1^{(p,q)}(\Omega_j)$ and $(\widetilde{u_j},\widetilde{v_j}) $ is the extended by $0$ in $\Omega \backslash \Omega_j$ of $(u_j,v_j)$. \end{theorem} \begin{proof} To prove \eqref{e3.1}, we note that $$ \lambda_1^{(p,q)}(\Omega_1)\geq \lambda_1^{(p,q)}(\Omega_2)\geq \dots \geq \lambda_1^{(p,q)}(\Omega). $$ For each $\varepsilon >0$, there is a pair of function $(\varphi_\varepsilon, \psi_\varepsilon) \in C_0^\infty (\Omega ) \times C_0^\infty (\Omega ) $ such that \begin{gather*} \int_\Omega |\varphi_\varepsilon |^{\alpha}|\psi_\varepsilon |^{\beta} \varphi_\varepsilon\psi_\varepsilon \, dx>0 \\ \lambda_1^{(p,q)}(\Omega )> \frac{\frac{\alpha+1}{p} { \int }_\Omega |\nabla\varphi_\varepsilon |^p\,dx+\frac{\beta+1}{q}{ \int} _\Omega |\nabla \psi _\varepsilon |^q\, dx }{{ \int _\Omega} |\varphi _\varepsilon |^{\alpha}| \psi _\varepsilon |^{\beta}\varphi _\varepsilon \psi _\varepsilon\, dx} -\varepsilon, %\eqno{(3.3)} \end{gather*} since $\lambda_1^{(p,q)}(\Omega)$ is the infimum. The support of $\varphi _\varepsilon$ and $\psi _\varepsilon$ being compact sets, are covered a finite number of the sets $\Omega_j$; hence, there exist $j_0\in \mathbb{N}^*$ such that $$ \mathop{\rm supp}\varphi _\varepsilon \subset \Omega _j \quad \mbox{and}\quad \mathop{\rm supp}\psi _\varepsilon \subset \Omega _j\quad \forall j\geq j_0. $$ Then, \begin{align*} \lambda_1^{(p,q)}(\Omega_j) & \leq \frac{\frac{\alpha+1}{p} { \int }_{\Omega_j} |\nabla\varphi_\varepsilon |^p\,dx+\frac{\beta+1}{q} {\int }_{\Omega_j} |\nabla \psi _\varepsilon |^q\, dx }{{ \int _{\Omega_j}} |\varphi _\varepsilon |^{\alpha}|\psi _\varepsilon |^{\beta}\varphi _\varepsilon \psi _\varepsilon\, dx}\\ & = \frac{\frac{\alpha+1}{p} { \int }_\Omega |\nabla\varphi_\varepsilon |^p\,dx +\frac{\beta+1}{q}{ \int} _\Omega |\nabla \psi _\varepsilon |^q\, dx } {{ \int _\Omega} |\varphi _\varepsilon |^{\alpha}|\psi _\varepsilon | ^{\beta}\varphi _\varepsilon\psi _\varepsilon\, dx}\\ &\leq \lambda_1^{(p,q)}(\Omega)+\epsilon \end{align*} for all large $j$. It is clear that $\lambda_1^{(p,q)}(\Omega)\geq { \lim_{j\to+\infty}\lambda_1^{(p,q)}(\Omega_j)}$. which proves the desired result. For the strong convergence \eqref{e3.2}, we proceed as follows: First, we have $(\widetilde{u_j},\widetilde{v_j})\in W_0^{1,p}(\Omega )\times W_0^{1,q}(\Omega)$; $\widetilde{u_j} \geq 0$, $\widetilde{v_j}\geq 0$ a.e in $\Omega$; $$ \int _\Omega |\widetilde{u_j} |^{\alpha}|\widetilde{v_j} |^{\beta} \widetilde u_j\widetilde v_j\,dx=1, $$ because $$ \int _\Omega |\widetilde{u_j} |^{\alpha}|\widetilde{v_j} |^{\beta}\widetilde{u_j}\widetilde{v_j}\, dx =\int _{\Omega _j}|u_j|^{\alpha}|v_j|^{\beta}u_jv_j\,dx=1; $$ and $\nabla \widetilde{u_j} =\widetilde{\nabla u_j} $ and $ \nabla \widetilde{v_j} =\widetilde{\nabla v_j}$ a.e. in $\Omega$. Hence, $(\widetilde{u_j},\widetilde{v_j})$ is admissible function in definition of $\lambda_1^{(p,q)}(\Omega )$. Which implies $$ \lambda_1^{(p,q)}(\Omega ) \leq \frac{\alpha+1}{p}\int_\Omega |\nabla \widetilde{u_j}|^p\,dx+\frac{\beta+1}{q}\int_\Omega |\nabla \widetilde{v_j}|^q\,dx =\lambda_1^{(p,q)}(\Omega_j) . %\eqno{(3.5)} $$ Therefore, via \eqref{e3.1} and for a subsequence, \begin{gather*} (\widetilde{u_j},\widetilde{v_j})\rightharpoonup (\varphi,\psi) \quad \mbox{weakly in }W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega),\\ (\widetilde{u_j},\widetilde{v_j})\to(u,v) \quad \mbox{ strongly in }L^{p}(\Omega)\times L^{q}(\Omega),\\ \varphi\geq 0,\quad \psi \geq 0 \quad \mbox{a.e in }\Omega \end{gather*} Since $B$ is compact, we have $ \int_\Omega \varphi^{\alpha+1}\psi^{\beta+1}dx=1$. Then $(\varphi,\psi)$ is admissible function in definition of $\lambda_1^{(p,q)}(\Omega)$. We deduce that \begin{align*} \lambda_1^{(p,q)}(\Omega ) & \leq \frac{\alpha+1}{p}\int_\Omega |\nabla \phi |^p\, dx+ \frac{\beta+1}{q}\int_\Omega |\nabla \psi|^q\,dx\\ & \leq \liminf _{j\to\infty } \Big(\frac{\alpha+1}{p}\int_\Omega |\nabla \widetilde{u_j}|^p\,dx +\frac{\beta+1}{q}\int_\Omega |\nabla \widetilde{v_j}|^q\,dx \Big) \\ & = \liminf _{j\to\infty }\lambda _1^{(p,q)}(\Omega _j). \end{align*} Then $$ \lambda_1^{(p,q)}(\Omega )=\frac{\alpha+1}{p} \int_\Omega |\nabla \phi |^p\, dx +\frac{\beta+1}{q} \int_\Omega |\nabla \psi |^q\,dx $$ which implies $(\varphi,\psi)$ is positive eigenfunction related to $\lambda_1^{(p,q)}(\Omega)$ that it is simple (see \cite{OU}) and ${ \int_\Omega |\varphi|^{\alpha}|\psi|^{\beta}\varphi \psi dx=1}$, hence $\phi \equiv u$ and $\psi\equiv v$ . Also, $(u,v)$ is independent of the subsequence. Then $\{ (\widetilde{u_j},\widetilde{v_j})\} _j $ converge to $(u,v)$ least in $L^p(\Omega )\times L^q(\Omega)$. For the strong convergence \eqref{e2.3}, we use Clarckson inequality and distinguish three possible cases for $(p,q)\in I_{\alpha,\beta}$. \noindent\textbf{case 1: $p\geq 2$ and $q\geq 2$.} For all $j\geq j_0$, we have \begin{align*} &{\frac{\alpha+1}{p} \int _\Omega \big| \frac{\nabla \widetilde{u_j}-\nabla u}{2} \big| ^p\,dx}+ \frac{\beta+1}{q}\int _\Omega \big| \frac{\nabla \widetilde{v_j}-\nabla v}{2}\big| ^q\,dx \\ &\leq \frac{\alpha+1}{p}\Big[-\int_\Omega \big| \frac{\nabla \widetilde{u_j} +\nabla u}{2}\big| ^p\,dx +\frac{1}{2}\int_\Omega |\nabla \widetilde{u_j}|^p\,dx +\frac{1}{2}\int_\Omega |\nabla u|^p\,dx\Big] \\ &\quad + \frac{\beta+1}{q} \Big[-\int _\Omega \big| \frac{\nabla \widetilde{v_j}+\nabla v}{2}\big| ^p\,dx +\frac{1}{2} \int_\Omega |\nabla \widetilde{v_j}|^q\,dx +\frac{1}{2} \int _\Omega |\nabla v|^q\,dx\Big]. \end{align*} Also, we have \begin{align*} &\lambda_1^{(p,q)}(\Omega)\int_\Omega \big(\frac{\widetilde{u_j}+u}{2}\big)^{\alpha+1} \big(\frac{\widetilde{v_j}+v}{2}\big)^{\beta+1}dx \\ &\leq \frac{\alpha+1}{p}\int_\Omega \big| \frac{\nabla \widetilde{u_j} +\nabla u}{2}\big|^p\,dx+\frac{\beta+1}{q}\int_\Omega \big| \frac{\nabla \widetilde{v_j}+\nabla v}{2}\big|^q\,dx. \end{align*} Combining the last inequality and using Lebegue's Theorem; we obtain $$ \limsup_{j\to+\infty }\big\{ \frac{\alpha+1}{p}\int_\Omega \big|\frac{\nabla \widetilde{u_j}-\nabla u}{2} \big| ^p\,dx+\frac{\beta+1}{q}\int_\Omega \big|\frac{\nabla \widetilde{v_j}-\nabla v}{2}\big| ^q\,dx \big\}\leq 0, $$ \noindent\textbf{Case 2: $1