\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small 2005-Oujda International Conference on Nonlinear Analysis. \newline {\em Electronic Journal of Differential Equations}, Conference 14, 2006, pp. 155--162.\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 2006 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{155} \begin{document} \title[\hfilneg EJDE/Conf/14 \hfil Nodal line of the p-Laplacian] {Note on the nodal line of the p-Laplacian} \author[A. R. El Amrouss, Z. El Allali, N. Tsouli \hfil EJDE/Conf/14 \hfilneg] {Abdel R. El Amrouss, Zakaria El Allali, Najib Tsouli} % in alphabetical order \address{Abdel R. El Amrouss \newline D\'epartement de Math\'ematiques et Informatique Facult\'e des Sciences, Universit\'e Mohamed 1er, Oujda, Maroc} \email{amrouss@sciences.univ-oujda.ac.ma} \address{Zakaria El Allali\newline D\'epartement de Math\'ematiques et Informatique Facult\'e des Sciences, Universit\'e Mohamed 1er, Oujda, Maroc} \email{zakaria@sciences.univ-oujda.ac.ma} \address{Najib Tsouli \newline D\'epartement de Math\'ematiques et Informatique Facult\'e des Sciences, Universit\'e Mohamed 1er, Oujda, Maroc} \email{tsouli@sciences.univ-oujda.ac.ma} \date{} \thanks{Published September 20, 2006.} \subjclass[2000]{58E05, 35J65, 56J20} \keywords{Nonlinear eigenvalue problem; p-Laplacian; nodal line} \begin{abstract} In this paper, we prove that the length of the nodal line of the eigenfunctions associated to the second eigenvalue of the problem $$ -\Delta_p u = \lambda \rho (x) |u|^{p-2}u \quad \mbox{in } \Omega $$ with the Dirichlet conditions is not bounded uniformly with respect to the weight. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}{Remark}[section] \section{Introduction} In this paper we consider the nonlinear elliptic boundary-value problem \begin{equation} \begin{gathered} -\Delta_p u = \lambda \rho (x) |u|^{p-2}u \quad \mbox{in } \Omega \\ u = 0 \quad \mbox{on } \partial \Omega , \end{gathered} \label{POr} \end{equation} where $\Delta_p u = \mathop{\rm div}(|\nabla u|^{p-2} \nabla u)$ is the p-Laplacian operator, $\Omega$ is a bounded and smooth domain in $\mathbb{R}^N$ ($10\} . $$ Several authors have been studied the spectrum $\sigma(-\Delta _p)$ of p-Laplacian, precisely around of the first and the second eigenvalue. In particular Anane \cite{AN} proved that the spectrum $\sigma(-\Delta _p)$ contains a positive non-decreasing sequence of eigenvalues $(\lambda_n)_{n\in \mathbb{N}^*}$ such that $\lambda_n \to +\infty$ by using the Ljusternik-Schnirelmann, where $$ \lambda_n^{-1}=\lambda_n(\Omega,\rho)^{-1}=\sup\limits_{K\in \mathcal{A}_n}\inf\limits_{v\in K}\int_{\Omega}\rho(x) |v|^p dx $$ and $$ \mathcal{A}_n=\{K \subset W^{1,p}_0(\Omega) : K \text{ is symmetrical compact and } \gamma(K)\geq n\}. $$ Moreover, he showed that the first eigenvalue is simple and isolated, and that the first eigenfunction corresponding to $\lambda_1$ does not change the sign in $\Omega$. In \cite{AN-TS} they have showed that the second eigenvalue of the spectrum $\sigma(-\Delta_p)$ is exactly $\lambda_2$. The complete determination of this spectrum remains unanswered question. It is useful to announce that in the linear case ($p=2$), the spectrum is perfectly given \cite{DE,OT}. Let us consider a solution $(u,\lambda)$ of problem $P(\Omega ,\rho)$. We denote by $$ \mathcal{Z}(u)=\{x\in \Omega : u(x)=0\} $$ the nodal set of $u$, $\mathcal{N}(u)$ is the number of connected components of $\Omega\backslash \mathcal{Z}(u)$, $\mathcal{C}(u)$ is the set of connected components of $\Omega\backslash \mathcal{Z}(u)$, $$ \mathcal{N}(\lambda)=\max \{\mathcal{N}(u) \mid (u,\lambda)\ \text{solution of\ } P(\Omega , \rho) \} \quad \text{and} \quad \mathcal{N}(\lambda_n)= \mathcal{N}(n). $$ Recently Cuesta, de Figueiredo and Gossez proved that $\mathcal{N}(2)=2$ \cite{C-DF-GO}. The main result in this paper is the generalization of the work of Kappeler and Ruf \cite{K-R}, in which they affirmed that the length of the nodal lines is not bounded uniformly with respect to the weights in dimension $N=2$ and $p=2$. In this work, we locate in the case $p>N$ and we prove that for all real number $L>0$, there exists a weights $\rho \in L^{\infty}(\Omega)$ and an eigenfunction associated to the second eigenvalue of \eqref{POr} such that the length of $\mathcal{Z}(u)$ is largest than $L$. The proof is relatively simpler than that given by Kappeler and Ruf; in which they use the uniform convergence of the gradient. \section{On the measure of nodal sets} In this section, we will extend the result of Kappeler and Ruf \cite{K-R} in the case $p\neq 2$. \subsection{Main result} We consider the case $p>N$. Let $\Gamma$ be a surface of class $C^1$ which subdivides $\Omega$ in two nodal components $\Omega_1$ and $\Omega_2$ such that \begin{equation} \mu_1 \leq \nu_1 \label{ENp1} \end{equation} where $\mu_1$ (respectively $\nu_1$) is the first eigenvalue of $P(\Omega_1,1)$ (respectively $P(\Omega_2,1)$. Let $v_1$ (respectively $w_1$) the associated eigenfunction. For $n\in \mathbb{N}^*$, let \begin{gather*} \Omega'_n = \big\{ x\in \Omega_1 : \mathop{\rm dist} (x,\Gamma) < \frac {1}{n+1} \big\},\\ \Omega_n = \Omega_1\backslash \Omega'_n \end{gather*} where $\Gamma \subset \partial \Omega'_n$ of class $C^1$. Then, we denote by $v^n_1$ the eigenfunction associated to $\mu^n_1$ the first eigenvalue of \eqref{POr} with $\rho=1$. Let $(a_n)_{n\in \mathbb{N}^*}$ be a sequence of decreasing positive real numbers such that \begin{equation} a_n=\frac{\mu_1^n}{\nu_1} \label{ENp2} \end{equation} which tends to the limit $a\in \mathbb{R}^*_+$ ($0< a=\frac{\mu_1}{\nu_1}\leq 1$). Let $(\rho_n)_{n\in \mathbb{N}^*}$ be a sequence of weight functions defined by \begin{equation} \rho_n(x)= - r_n 1_{\Omega'_n}(x)+a_n 1_{\Omega_n}(x)+ 1_{\Omega_2}(x) , \label{ENp3} \end{equation} for all $x\in \Omega $, where $r_n>0$ such that $\displaystyle\lim _{n \to +\infty} \frac{c_nd_n}{r_n^{(p-1)/p^2}}=0$ with $c_n$ and $d_n$ are strictly positive constants of immersion and interpolation. Let us denote $u_2^n$ the eigenfunction associated to the second eigenvalue $\lambda_2^n$ of \eqref{POr}. \begin{theorem} There exists a subsequence of $(u_2^n)_{n\in \mathbb{N}^*}$ still denoted by $(u_2^n)_{n\in \mathbb{N}^*}$ such that \begin{itemize} \item[(i)] The sequence $(u_2^n)_{n\in \mathbb{N}^*}$ converges weakly to $(\alpha \overline{v_1}+\beta\overline{w_1})$ in $W^{1,p}_0(\Omega)$, for some scalars $\alpha$ , $\beta$ not all null, where $\overline{v_1}$ (respectively $\overline{w_1}$ ) is the extension of $v_1$ (respectively $w_1$) by zero in $\overline{\Omega}$. \item[(ii)] If $U$ and $V$ are two opens of $\mathbb{R}^N$ such that $\overline{U} \subset \Omega_1$ and $\overline{V} \subset \Omega_2$. Then for $n $ enough large, we have $$\overline{U} \cap \mathcal{Z}(u_2^n) = \overline{V} \cap \mathcal{Z}(u_2^n) = \emptyset $$ and $u_2^n$ change the sign on $U \cup V$ \end{itemize} \label{ENpm} \end{theorem} To prove this result, we need the following preliminary lemmas. \begin{lemma} The following inequalities are true independently of $(r_n)_{n\in \mathbb{N}^*}$: \begin{itemize} \item[(i)] $0< b^{-1} \lambda_2(\Omega , 1) \leq \lambda_2^n \leq \nu_1$ , where $b=\begin{cases} 1 & \mbox{if } \mu_1 < \nu_1\\ a_1 &\mbox{if } \mu_1=\nu_1, \end{cases} $ \item[(ii)] $\|\Delta_p u_2^n\|^{p'}_{L^{p'}(\Omega_2)} \leq M \nu_1^{p'}$, \item[(iii)] $\|\Delta_p u_2^n\|^{p'}_{L^{p'}(\Omega_n)} \leq M (a_1 \nu_1)^{p'}$, where $M$ is the Sobolev-Poincar\'e constant. \end{itemize} \label{ENp4} \end{lemma} \begin{proof} (i) Let $F_2$ be the vector subspace of $W^{1,p}_0(\Omega)$ spanned by $\{\overline{v_1^n},\overline{w_1}\}$, where $ \overline{v_1^n}$ (resp. $\overline{w_1}$) is the extension by zero of $ v_1^n$ (resp. $w_1$) in $\Omega\setminus \Omega_n$ (resp. $\Omega\setminus \Omega_1$). Let $S_2$ denote the unit sphere of $F_2$. For all $v\in S_2$ such that $v=\alpha \overline{v_1^n} + \beta \overline{w_1}$, we have $$ |\alpha |^p + |\beta |^p =1 \quad \mbox{and \ } \int_{\Omega }\rho_n(x) |v(x) |^p dx = |\alpha |^p a_n \frac{1}{\mu^n_1} + |\beta |^p\nu_1^{-1}. $$ Using (\ref{ENp3}), we get $$ \int_{\Omega} \rho_n(x) |v(x)|^p dx = \frac{1}{\nu_1} $$ In particular $$ \frac{1}{\nu_1} \leq \inf\limits_{v \in S_2} \{\int_{\Omega} \rho_n(x) v(x) dx \} \leq \frac{1}{\lambda^n_2}. $$ Since $\rho_n(x) \leq b$, $$\frac{\lambda_2(\Omega,b)}{b}= \lambda_2(\Omega,b) \leq \lambda_2^n(\Omega,\rho_n)= \lambda_2^n$$ \noindent (ii) It is sufficient to notice that $$ -\Delta_p u^n_2 = \lambda^n_2 |u^n_2|^{p-2}u^n_2 \quad \mbox{a.e. on } \Omega_2 $$ and by using the Sobolev-Poincar\'e inequality, we have $$ \int_{\Omega_2}|-\Delta_p u^n_2 |^{p'} \leq M(\lambda_2^n)^{p'}\|\nabla u^n_2\|^p_{L^p(\Omega)} \leq M \nu^{p'}_1. $$ \noindent (iii) Using $-\Delta_p u^n_2 = \lambda^n_2 a_n |u^n_2|^{p-2}u^n_2$ a.e. on $\Omega_n$ and with the same argument as above, one gets $$ \int_{\Omega_n}|-\Delta_p u^n_2 |^{p'} \leq M (a_1 \nu_1)^{p'}. $$ \end{proof} \begin{remark} \begin{itemize} \item[(1)] By the lemma \ref{ENp4} we can choose the sequence $(\rho)_{n\in \mathbb{N}^*}$ such that $(\lambda_2^n)_{n\in \mathbb{N}^*}$ converges to the positive limit $\lambda_2$. For all $p>N$ , $u_2^n$ converges weakly in $W^{1,p}_0(\Omega)$ and strongly in $C(\overline{\Omega})$ to $u_2 \in W^{1,p}_0(\Omega)$. \item[(2)] $\rho_n(x) \leq b$ \quad for all $x\in \Omega$. \item[(3)] $u_2 \neq 0$ in $L^p(\Omega)$. \end{itemize} \label{ENp5} \end{remark} \begin{lemma} With the above notation, $\lambda_2 = \nu_1$ \label{ENp4'} \end{lemma} \begin{proof} To prove this lemma, we proceed in three steps:\\ \textbf{First step:} We show that \begin{equation} -\Delta_p u_2 = \lambda_2 a |u_2|^{p-2}u_2 \quad \mbox{a.e. on } \Omega_1 \label{ENp6} \end{equation} Indeed; for $m \geq 1$ and by the lemma \ref{ENp4} , we have $$ \|\Delta_p u_2^n\|^{p'}_{L^{p'}(\Omega_m)} \leq \|\Delta_p u_2^n\|^{p'}_{L^{p'}(\Omega_n)} \leq M (a_1\nu_1)^{p'} \quad \forall n \geq m $$ hence $$ \|\Delta_p u_2^n\|^{p'}_{(W^{1,p}(\Omega_m))'} \leq M (a_1\nu_1)^{p'} \quad \forall n \geq m . $$ It follows that there exists a subsequence, still denoted by $(-\Delta_p u_2^n)$, such that $ -\Delta_p u_2^n \rightharpoonup T_m $ weakly in the sapce $(W^{1,p}(\Omega_m))'$. By remark \ref{ENp5}, $u^n_2 \rightharpoonup u_2$ weakly in $W^{1,p}(\Omega_m)$. Since $$ -\Delta_p u^n_2 = \lambda^n_2 a_n |u^n_2|^{p-2}u^n_2 \quad \mbox{a.e. on $\Omega_m$ for all }n \geq m, $$ we have $$ \lim _{n \to +\infty} \langle -\Delta_p u^n_2 , u^n_2\rangle _m = \langle T_m,u_2\rangle _m $$ where $\langle \cdot, \cdot \rangle_m$ is the duality bracket between $W^{1,p}(\Omega_m)$ and its dual $(W^{1,p}(\Omega_m))'$. However, $-\Delta_p$ is an operator of type (M), consequently $T_m= -\Delta_p u_2$ is in the space $(W^{1,p}(\Omega_m))'$. We deduce that $$ -\Delta_p u_2 = \lambda_2 a |u_2|^{p-2}u_2 \quad \mbox{a.e. on } \Omega_m\ \forall m\geq 1 $$ hence (\ref{ENp6}). Similarly, we prove that \begin{equation} -\Delta_p u_2 = \lambda_2 |u_2|^{p-2}u_2 \quad \mbox{a.e. on } \Omega_2 \label{ENp6'} \end{equation} \noindent\textbf{Second step:} We show that $$ u_{2_{/ \partial \Omega _i }} = 0 \quad \mbox{in } L^p(\partial\Omega _i) \mbox{ for } i=1,2. $$ Indeed, it follows from (\ref{ENp2}) and since $\partial \Omega '_n $ is of $C^1$ and $(\Omega \cap \partial \Omega_2) \subset \partial \Omega'_n$, we have \begin{equation} \|u^n_{2_{/ \Omega \cap \partial \Omega_2 }}\|_{L^p(\Omega \cap \partial \Omega_2)} \leq c_n d_n \|u^n_2\|^{\sigma}_{W^{1,p}(\Omega '_n)} \|u^n_2\|^{1-\sigma}_{L^p(\Omega '_n)} , \label{ENp7} \end{equation} where $c_n$ is the constant of the immersion $W^{\sigma , p}(\Omega '_n)\hookrightarrow L^p(\partial \Omega '_n)$; $\sigma=\frac{1}{p}$ and $d_n$ is the constant of the interpolation of the inequality $$ \|u\|_{W^{\sigma , p}(\Omega'_n)} \leq d_n \|u\|^{\sigma }_{W^{1 ,p}(\Omega'_n)} \|u\|^{1-\sigma }_{L^p(\Omega'_n)} \quad \mbox{for all \ } u\in W^{\sigma , p}(\Omega'_n) . $$ The two norms of the second member in (\ref{ENp7}) can be estimated as follows: \begin{equation} \|u_2^n\|^p_{W^{1 , p}(\Omega'_n)} \leq \|u_2^n\|^p_{L^p(\Omega'_n)} + \|\nabla u_2^n\|^p_{L^p(\Omega'_n)} \leq M+1 ,\label{ENp8} \end{equation} where $M$ is the constant of the Sobolev-Poincar\'e of lemma \ref{ENp4}. Moreover, since $(u_2^n,\lambda_2^n)$ is a solution of $P(\Omega , \rho_n)$, by (\ref{ENp3}) and lemma \ref{ENp4}, we get $$ r_n \int_{\Omega'_n}|u_2^n|^p dx \leq \int_{\Omega_2} |u_2^n|^p dx + a_n \int_{\Omega_n} |u_2^n|^p dx\,. $$ Since $b\geq1$, we deduce that \begin{equation} \int_{\Omega'_n}|u_2^n|^p dx \leq \frac{b}{r_n} \int_{\Omega}|u_2^n|^p dx \leq \frac{b}{r_n} M .\label{ENp9} \end{equation} Thus, by (\ref{ENp7}), (\ref{ENp8}) and (\ref{ENp9}), we have $$ \|u^n_{2_{/ \Omega \cap \partial \Omega_2 }}\|_{L^p(\Omega \cap \partial \Omega_2)} \leq c_n d_n (M+1)^{\frac{\sigma}{p}}(\frac{bM}{r_n})^{\frac{1-\sigma }{p}} . $$ However, $\lim_{n \to +\infty } \frac{c_nd_n}{r_n^{(1-\sigma)/p}} =0$ and $u_2 \in W^{1,p}_0 (\Omega)$, consequently $u_{2_{/ \partial \Omega_2}} =0$ in $L^p(\partial \Omega_2)$. Similarly, we have $u_{2_{/ \partial \Omega_1}} =0 $ in $L^p(\partial \Omega_1)$ because $ (\Omega \cap \partial \Omega_1) \subset \partial \Omega'_n $. \noindent\textbf{Third step:} We establish that $\lambda_2 = \nu_1 = \eta_1$. Indeed, since $u_2=0$ in $L^p(\partial \Omega_1)$ and $L^p(\partial \Omega_2)$ with $u_2\in W^{1,p}_0(\Omega)$, we have $u_2\in W^{1,p}_0(\Omega_1)$ and $u_2\in W^{1,p}_0(\Omega_2)$. Moreover, if we use (\ref{ENp6}), (\ref{ENp6'}) and the remark \ref{ENp5}, then $(u_{2_{/\Omega_1}} , \lambda_2)$ and $(u_{2_{/\Omega_2}} , \lambda_2)$ are respectively solutions of problems \eqref{POr} with $\Omega=\Omega_1$ and with $\Omega=\Omega_2$. We have by lemma \ref{ENp4}, $$ \lambda_2 = \lim_{n \to +\infty} \lambda_2^n \leq \nu_1, $$ where $\nu_1$ is the first eigenvalue of \eqref{POr} with $\Omega=\Omega_2$. We conclude that $\lambda_2 = \nu_1 = \eta_1$. \begin{lemma} The sequence $(f_n)_{n\in \mathbb{N}^*}$ admits a subsequence which converges weakly in $W^{1,p}_0(\Omega)$ to $\overline{v_1}$, where $$ f_n=\frac{(u_2^n)^+}{(\frac{a}{p})^{1/p}\|(u_2^n)^+\|_{L^p(\Omega)}}. $$ with $(u_2^n)^+ =max \{0, u_2^n \}$. \label{ENp4''} \end{lemma} \begin{proof} It is known that $$ \int_{\Omega} |\nabla (u_2^n)^+|^p dx = \lambda_2^n \int_{\Omega} \rho_n | (u_2^n)^+|^p dx\,. $$ If we multiply by $\big((\frac{a}{p})^{1/p} \|(u_2^n)^+\|_{L^p(\Omega)}^p\big)^{-p}$, then \begin{equation} \label{ENp10} \int_{\Omega}|\nabla f_n|^p dx = \lambda_2^n \int_{\Omega} \rho_n | f_n|^p dx \leq \lambda_2^n b \int_{\Omega} | f_n|^p dx = b \lambda_2^n \frac{p}{a}. \end{equation} Thus, by lemmas \ref{ENp4} and \ref{ENp4'}, we have $$ \int_{\Omega}|\nabla f_n|^p dx \leq \frac{p}{a} b \lambda_2 < +\infty \quad \forall n\in \mathbb{N}^* . $$ So, for a subsequence of the sequence $(f_n)_{n\in \mathbb{N}^*}$, still denoted $(f_n)_{n\in \mathbb{N}^*}$, we have $f_n \rightharpoonup f$ weakly in $W^{1,p}_0(\Omega)$, then $f_n \to f$ strongly in $C(\overline{\Omega})$ with $p>N$. Since $\|f_n\|_{L^p(\Omega)}=(\frac{p}{a})^{1/p}$ then $\|f\|_{L^p(\Omega)} \neq 0$. Hence $f \neq 0$ on $\Omega $, since $u_{2_{/\Omega_2}} = \beta w_1 < 0$, \begin{equation} f=0 \quad \mbox{ on } \Omega_2 \label{ENp11} \end{equation} a fortiori $f=0$ on $\partial\Omega_2 $ and $f=0$ on $\partial\Omega_1$. It results that $f\in W_0^{1,p}(\Omega_1)$. According to (\ref{ENp10}), we have \begin{align*} \int_{\Omega}|\nabla f_n|^p dx & = \lambda_2^n \Big( a_n \int_{\Omega}| f_n|^p dx - r_n \int_{\Omega'_n}| f_n|^p dx + \int_{\Omega_2}| f_n|^p dx \Big)\\ & \leq \lambda_2^n \Big( a_n \int_{\Omega_n}| f_n|^p dx + \int_{\Omega_2}| f_n|^p dx \Big). \end{align*} Hence, $\displaystyle\liminf_{n\to +\infty} \int_{\Omega}|\nabla f_n|^p dx \leq \lambda_2\big( a \int_{\Omega_1}|f|^p dx + \int_{\Omega_2}|f|^p dx \big)$. From (\ref{ENp11}), we deduce that $$ \lim_{n\to +\infty} \inf \int_{\Omega}|\nabla f_n|^p dx \leq \lambda_2 a \int_{\Omega}| f|^p \,dx. $$ Thus $$ \int_{\Omega_1}|\nabla f|^p dx \leq \lambda_2 a \int_{\Omega}| f|^p dx = \lambda_2 p . $$ We have $\lambda_2 = \eta_1$ being the first eigenvalue of \eqref{POr} with $\Omega=\Omega_1$ and $\rho=a$; consequently $$ \lambda_2=\frac{1}{p}\int_{\Omega_1}|\nabla f |^p dx \quad \mbox{and}\quad f=\overline{v_1}. $$ \end{proof} \subsection{Proof of the main result} (i) From lemma \ref{ENp4'}, $u_2$ is an eigenfunction associated to $\nu_1$ (resp. $\eta_1$). So, there exists $\alpha , \beta \in \mathbb{R}^n$ such that $$ u_2= \alpha v_1 + \beta w_1 \quad \mbox{with } |\alpha |^p+|\beta |^p > 0. $$ (ii) We distinguish two possible cases: \noindent\textbf{First case:} If $\alpha \neq 0$ and $\beta \neq 0$, we can assume that $\alpha>0$ and $\beta<0$ (the other cases will be treated in the same way ). As $u_2= \alpha v_1$ on $\Omega_1$ and $v_1$ is a positive eigenfunction of class $C^1$ on $\Omega_1$, then $\exists x_0\in \overline{U}$ such that $\min \{u_2(x) : x\in \overline{U} \} = u_2(x_0)>0$. By lemma \ref{ENp4}, $u_2^n$ converges uniformly $(p>N)$ to $u_2$ in $\overline{\Omega}$, consequently for $\epsilon =u_2(x_0)>0$ there exists $n_0(\overline{U})\in \mathbb{N}$ such that for all $n\geq n_0(\overline{U})$, we have $$ u_2^n(x)>\frac{\epsilon }{2} \quad \forall x \in \overline{U} $$ i.e. $(\overline{U} \cap\mathcal{Z}(u_2^n) ) = \emptyset $ for all $n\geq n_0(\overline{U})$. It is the same for $(\overline{V} \cap\mathcal{Z}(u_2^n) ) = \emptyset $ for all $n\geq n_0(\overline{V})$. We announce here that according to the lemma \ref{ENp4} the case where $\alpha \beta >0$ does not intervene. \noindent\textbf{Second case:} If $\alpha=0 $ or $\beta = 0$. We consider now the case where $\alpha=0 $ and $\beta < 0$. The other cases will be treated in the same way. By lemma \ref{ENp4''}, there exists a subsequence, still denoted $(f_n)_{n\in \mathbb{N}}$, which converges uniformly to $f=\overline{v_1}$ in $\overline{\Omega}$. Moreover $\overline{v_1}> 0$ in $\overline{U}$, and there exists $x_0 \in \overline{U}$ such that $$ f(x_0)= \min \{ f(x)=\overline{v_1}(x) : x\in \overline{U} \} >0 . $$ Thus, for $\epsilon = f(x_0)>0$, there exists $n_0(\overline{U}) \in \mathbb{N}^*$ such that for all $n\geq n_0(\overline{U})$ we have $$ f_n(x)>\frac{\epsilon}{2} \quad \forall x\in \overline{U}. $$ i.e for all $n \geq n_0(\overline{U})$, $\overline{U} \cap \mathcal{Z}(u_2^n) = \emptyset $. Therefore, since $\beta < 0$, it is the same for $\overline{V} \cap \mathcal{Z}(u_2^n) = \emptyset$ for all $n\geq n_0(\overline{V})$ . We remark here that by (i) of the lemma \ref{ENp4} the case where $\alpha=\beta=0$ does not intervene. \end{proof} \subsection{Consequences of the main result} \begin{corollary} If $\Gamma$ is a surface of class $C^1$ in $\Omega$ which subdivides $\Omega$ in two connected components, then for all neighborhood $[\Gamma]_{\epsilon}$ of $\Gamma$, there exists a weight $\rho_{\epsilon} \in L^{\infty}(\Omega)$ and an eigenfunction $u$ associated to the second eigenvalue of $P(\Omega , \rho_{\epsilon})$ such that $\mathcal{Z}(u) \subset [\Gamma]_{\epsilon}$; where $[\Gamma]_{\epsilon} = \{ x\in \Omega : d(x,\Gamma) \leq \epsilon\}$.\label{cor1} \end{corollary} \begin{proof} We distinguish two cases \noindent\textbf{First case:} $\overline{\Gamma} \cap \partial \Omega = \emptyset $. Let $\varepsilon >0$, we consider $U= \Omega_1 \backslash\ ([\partial \Omega ]_{\epsilon} \cup [\Gamma]_{\epsilon})$ and $V= \Omega_2 \backslash\ ([\partial \Omega ]_{\epsilon} \cup [\Gamma]_{\epsilon})$, where $[\partial \Omega ]_{\epsilon} = \{ x\in \Omega : d(x, \Omega) \leq \epsilon\}$. Since $\overline{\Gamma} \cap \partial \Omega =\emptyset$, we can choose $\epsilon$ enough small, so that $[\partial \Omega ]_{\epsilon} \cap [\Gamma]_{\epsilon} = \emptyset $. By theorem \ref{ENpm}, there exists $n\in \mathbb{N}^*$ such that $\mathcal{Z}(u^n_2) \subset [\partial \Omega ]_{\epsilon} \cup [\Gamma]_{\epsilon} $. We assume that $\mathcal{Z}(u^n_2) \cap [\partial \Omega ]_{\epsilon} \neq \emptyset $ then there exists a nodal component $D_{\epsilon}$ of $u_2^n$ included in $[\partial \Omega ]_{\epsilon}$. Thus $(u^n_{2_{/D_{\epsilon}}} , \lambda^n_2)$ is a solution of the problem $P(D_{\epsilon}, \rho_{n_{/D_{\epsilon}}})$ with $\lambda^n_2$ its first eigenvalue \cite{AN,TS}. By the remark \ref{ENp5}, we have $$ \lambda^n_2 = \lambda_1 (D_{\epsilon},\rho_{n_{/D_{\epsilon}}}) \geq \lambda_1 (D_{\epsilon},b) $$ we have $ \mathop{\rm meas}(D_{\epsilon}) \to 0 $ when $\epsilon \to 0$ , consequently $\lambda^n_2 = \lambda_1(D_{\epsilon}, \rho_n) \to +\infty$ when $\epsilon \to 0$ which is absurd with lemma \ref{ENp4}. So $\mathcal{Z} (u^n_2) \subset [\Gamma ]_{\epsilon}$. \noindent\textbf{Second case:} $\overline{\Gamma} \cap \partial \Omega \neq \emptyset$. Let $\epsilon > 0$, there exists a surface $\Gamma'_{\epsilon}$ of $C^1$ which subdivide $\Omega$ in two connected components such that $$ \Gamma'_{\epsilon} \subset [\Gamma]_{\epsilon} \quad \mbox{and}\quad \overline{\Gamma'_{\epsilon}} \cap \partial \Omega =\emptyset $$ Let $\eta >0$ (enough small) so that $ [\Gamma'_{\epsilon}]_{\eta} \subset [\Gamma]_{\varepsilon}$ and $[\partial \Omega]_{\eta} \cap [\Gamma'_{\epsilon}]_{\eta} = \emptyset$, finally we conclude the result by applying the proof of the first case with $\Gamma'_{\epsilon}$. \end{proof} \begin{remark} \rm The result of the corollary \ref{cor1} remains true even if $\Gamma$ is not of class $C^1$, only it is enough to approach $\Gamma$ by a surface $\Gamma'$ of class $C^1$ which located in $[\Gamma]_{\epsilon}$.\label{rem2} \end{remark} \begin{corollary} For all $L>0$, there exists $\rho \in L^{\infty}(\Omega)$ and an eigenfunction $u$ associated to the second eigenvalue of $P(\Omega , \rho)$ such that the length of $\mathcal{Z}(u)$ is larger than $L$. \end{corollary} \begin{proof} Let $L>0$, there exists a surface $\Gamma$ is of class $C^1$ in $\Omega$ which subdivide $\Omega$ in two connected components such that $$ \overline{\Gamma} \cap \partial \Omega = \emptyset \quad \mbox{and}\quad \mathop{\rm meas}(\Gamma) > L + 1. $$ For $\epsilon >0$ (enough small) we consider $[\Gamma]_{\epsilon}$ and $[\partial \Omega]_{\epsilon}$ two neighborhood of $\Gamma$ and $\partial \Omega$ respectively such that $$[\partial \Omega]_{\epsilon} \cap [\Gamma]_{\epsilon} = \emptyset$$ Denote by $U=\Omega_1 \backslash ([\partial \Omega]_{\epsilon} \cup [\Gamma]_{\epsilon})$ and $V=\Omega_2 \backslash ([\partial \Omega]_{\epsilon} \cup [\Gamma]_{\epsilon})$ two open. In virtue of the Theorem \ref{ENpm} and of the Corollary \ref{cor1}, $\exists n \in \mathbb{N}^*$ such that $$ U\cap \mathcal{Z}(u^n_2)= V\cap \mathcal{Z}(u^n_2) = \emptyset\quad\text{and}\quad \mathcal{Z}(u^n_2) \subset [\Gamma]_{\epsilon}. $$ Let us suppose that for an infinity of $\epsilon >0$, $u_2^n$ admits a nodal component $D_{\epsilon}$ included in $[\Gamma]_{\epsilon}$. So $$ \lambda_2^n= \lambda_1(D_{\epsilon} , \rho_{n_{/D_{\epsilon}}}) \geq \lambda_1(D_{\epsilon} , b). $$ Since $\lim _{\epsilon \to 0} \mathop{\rm meas}(D_{\epsilon}) = 0$, it follows that $\lambda_2^n \geq \lim _{\epsilon \to 0} \lambda_1(D_{\epsilon} , b) = +\infty$ which is absurd with the lemma \ref{ENp4} . Thus, for $\epsilon$ enough small, there exists $n \in \mathbb{N}$ such that $\mathcal{Z}(u^n_2)$ is a closed surface in $[\Gamma ]_{\epsilon}$ with $\mathcal{Z}(u^n_2) = \partial W$ where $W$ is an open containing $\Omega_i^{\epsilon}$ which is an open included in $\Omega_i$ such that $\partial \Omega_i^{\epsilon} \subset \partial [\Gamma]_{\epsilon}$. So if $\mathop{\rm meas}(\Gamma) > L+1$, then $\exists \epsilon > 0$ (enough small) and $\exists n\in \mathbb{N}^*$ such that $\mathop{\rm meas} (\mathcal{Z}(u_2^n) > L$ for $i=1,2$. \end{proof} \begin{thebibliography}{00} \bibitem{AN} A. Anane, \emph{Simplicit\'e et isolation de la premi\`ere valeur propre du p-Laplacien avec poids}, C. R. Acad Sci. Paris t.305 S\'erie I (1987), 725-728. \bibitem{AN-TS} A. Anane and N.Tsouli, \emph{On the second eigenvalue of the p-Laplacian,} Pitman Research Notes in Mathematics Series, 343, 1-9. \bibitem{C-DF-GO} M. Cuesta D. de Figueiredo and J.P. Gossez; \emph{A nodal domain property for the p-Laplacian,} C. R. Acad. Sci. Paris, t. 330, Serie I (2000), 669-673. \bibitem{DE} D. G. De Figuerido, \emph{The Dirichlet problem for non linear elliptic equation: a Hilbert space approch,} in lecture note in math, No 446, Springer, Berlin (1975), 144-165. \bibitem{K-R} T. Kappeler and B. Ruf, \emph{On the nodal line of the second eigenfunction of elliptic operators in two dimension,} Journal Rvine und. Ange Maths. (396), 1-13. \bibitem{OT} M. Otani, \emph{A remark on certain nonlinear elliptic equation,} Proc. Fac. Sci. Tokai Univ. No. 19 (1984), 23-28. \bibitem{TS} N. Tsouli, \emph{Etude de l'ensemble nodal des fonctions propres et de la non-r\'esonance pour l'op\'erateur p-Laplacien,} Th\`ese d'\'etat, Universit\'e Mohamed I, Oujda (1996). \end{thebibliography} \end{document}