\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small {\em 
Electronic Journal of Differential Equations}, 
Vol. 2007(2007), No. 35, pp. 1--7.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/35\hfil Eigencurves of the $p$-Laplacian]
{Eigencurves of the $p$-Laplacian with  weights and their
asymptotic behavior}

\author[A. Dakkak, M. Hadda\hfil EJDE-2007/35\hfilneg]
{Ahmed Dakkak, Mohammed Hadda}  % in alphabetical order

\address{Ahmed Dakkak \newline
Department of Mathematics, University Moulay Ismail, Faculty of
Sciences and Techniques, B. P. 509 - Boutalamine, 
Errachidia, Morocco} 
\email{dakkakahmed@hotmail.com}

\address{Mohammed Hadda \newline
Department of Mathematics, University Moulay Ismail, Faculty of
Sciences and Techniques, B. P. 509 - Boutalamine, 
Errachidia, Morocco} 
\email{mohammedhad@hotmail.com}

\thanks{Submitted July 21, 2006. Published February 27, 2007.}
\subjclass[2000]{35J20, 35J70, 35P05, 35P30} 
\keywords{Nonlinear eigenvalue problem;  eigencurves; $p$-Laplacian;
\hfil\break\indent indefinite weight; asymptotic behavior}

\begin{abstract}
 In this paper we study the existence of the
 eigencurves of the $p$-Laplacian with indefinite weights. We obtain also
 their variational formulations and asymptotic behavior.
\end{abstract}

\maketitle 
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction and preliminaries}

We consider the nonlinear  eigenvalue problem
\begin{equation} \label{VP}
\begin{gathered}
-\Delta _pu  =  \lambda m(x)|u|^{p-2}u \quad  \text{in }\Omega \\
u  =  0 \quad \text{on }\partial \Omega \,,
\end{gathered}
\end{equation}
where $\Omega $ is a smooth bounded domain in $\mathbb{R}^N$,
$-\Delta _pu=-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ is the
 $p $-Laplacian, $1<p<\infty $,
$m\in \mathcal{L}^\infty (\Omega )$ is a weight function which can
change sign and verifies
$$
\mathop{\rm meas}\{x\in \Omega: m(x)>0\}\neq 0 \,.
$$
We denote
\[
\mathcal{M}^{+}(\Omega )=\big\{m\in \mathcal{L}^\infty (\Omega ):
\mathop{\rm meas}\{x\in \Omega : m(x)>0\}\neq 0\big\}\,.
\]
We say that $\lambda $ is a eigenvalue of $p$-Laplacian with
weight $m$, when the problem \eqref{VP} has at least a nontrivial
solution $u$ in $\mathcal{W}_0^{1,p}(\Omega )$. The set of
positive eigenvalues constitutes the spectrum $\sigma ^{+}(-\Delta
_p,m,\Omega )$ of $p$-Laplacian with weight $m$ in the domain
$\Omega $. This spectrum contains an infinite sequence given by
$\lambda _1<\lambda _2\leq \dots \leq \lambda _n\to +\infty $ and
formulated as follows
\begin{equation}\label{1.1}
\frac 1{\lambda _n}=\frac 1{\lambda _n(m)}=\sup_{K\in \Gamma
_n}\min_{u\in K}\int_\Omega m|u|^p \,,
\end{equation}
where $\Gamma _n$ is defined by
\[
\Gamma _n=\{K\subset S: K \mbox{ is  symetric, compact  and }
\gamma (K)\geq n\} ,
\]
$S=\{u\in \mathcal{W}_0^{1,p}(\Omega ):\int_\Omega |\nabla
u|^p=1\}$ is the sphere unity of $\mathcal{W}_0^{1,p}(\Omega )$
and $\gamma $ is the genus function. We may also define the
negative spectrum when $-m\in \mathcal{M}^{+}(\Omega )$ by
$-\sigma ^{+}(-\Delta _p,-m,\Omega )$ which contains an infinite
sequence $\lambda _{-1}>\lambda _{-2}\geq \dots \geq \lambda
_{-n}\to -\infty $ such that
\begin{equation}\label{1.2}
\lambda _{-n}=\lambda _{-n}(m):=-\lambda _n(-m) \quad.
\end{equation}
The variational characterization $(\ref{1.1})$ and the properties
of $\lambda _n$ depending on weight $m$ was the subject of several
works of which we cite for example \cite{A1,A2,AT,T}.

In this note, we study the following problem: Find all the real
numbers  $\alpha $, $\beta $ such that $\lambda_n(\alpha m_1+\beta
m_2)=1$.

 This last equation comes from the problem of eigencurves
of Sturm-Liouville. Several applications of these problems can be
found in the bifurcation domain and other, making reference
\cite{BH2}. In \cite{BH1} we find the properties related to the
first eigencurve such as concavity, differentiability and the
asymptotic behavior. The authors wished  to have information about
the other eigencurves, especially their asymptotic behavior. This
will be the object of our study. Let $m_1, m_2 \in
\mathcal{M}^{+}(\Omega )$ so that $\mathop{\rm ess\,inf}_{\Omega}
m_2>0$.
 we define the graph of the $n^{th}$ eigencurve by
\begin{equation}\label{1.3}
C_n=\{(\alpha ,\beta )\in \mathbb{R}^{2}:\lambda _n(\alpha
m_1+\beta m_2)=1\} \,,
\end{equation}
 We note that this definition differs from that given in
\cite{BH2}, which is
\begin{equation}\label{1.4}
\beta _n(\alpha )=\inf_{K\in \Gamma _n}\max_{u\in K}\frac{
\int_\Omega |\nabla u|^p-\alpha \int_\Omega m_1|u|^p}{\int_\Omega
m_2|u|^p} \quad.
\end{equation}

This paper is organized as follows. First, we are interested to
the existence of eigencurve $C_n$ . Then we show that
$(\alpha,\beta _n(\alpha ))\in $ $C_n$. This would allow us to
affirm the coincidence of the two definitions \eqref{1.3} et
\eqref{1.4} and also present the variational formulation of that
eigencurve. We would end up with the study of the asymptotic
behavior of the eigencurves $C_n$. And finally we affirm that all
eigencurves have the same asymptotic behavior.

\section{Existence of the eigencurve $C_n$}

We first recall the following

\begin{proposition}\label{prop2.1}
\begin{enumerate}
\item  Let $m, m' \in \mathcal{M}^{+}(\Omega )$.
If $m\leq m'$ (resp. $m< m'$), then $\lambda _n(m)\geq \lambda
_n(m')$ (resp. $\lambda _n(m)> \lambda _n(m')$).

\item $\lambda _n : m\mapsto \lambda _n(m)$ is continuous in
$(\mathcal{M}^{+}(\Omega ),\| .\| _\infty )$.
\end{enumerate}
\end{proposition}

For the proof, see for example \cite{T}. \\

Next we can establish the following

\begin{proposition}\label{prop2.2}
 Let $(m_k)_k$ be a sequence in $\mathcal{ M}^{+}(\Omega )$
such that $m_k\to m$ in $\mathcal{L}^\infty (\Omega )$. Then
$\lim_k  \lambda _n(m_k)=+\infty$ if and only if $m\leq 0$ almost
everywhere in  $\Omega$.
\end{proposition}

\begin{proof}
Let $(m_k)_k$ be a sequence in $\mathcal{ M}^{+}(\Omega )$ such
that $m_k$ converges to $m$ in $\mathcal{L}^\infty (\Omega )$.
Assume first that $\lim_k  \lambda_n(m_k)=+\infty$, we claim that
$m\leq 0$ almost everywhere in $\Omega$;
 otherwise
$$
\mathop{\rm meas}\{x\in \Omega : m(x)>0\}\neq 0
$$
it then follows that $\lim_k  \lambda _n(m_k)=\lambda_n(m)$, is a
finite, a contradiction.

Inversely, if $m\leq 0$ almost everywhere in $\Omega $, suppose
that $\lim_k  \lambda _n(m_k)$ is finite, then there exist
$\lambda >0$ such that
\begin{equation}\label{2.1}
\lambda _n(m_k)\leq \lambda \quad \mbox{for all } k \in \mathbb{N}
.
\end{equation}
We put $r=2\lambda/\lambda_n(2)$ and $\varepsilon=1/r$. Then there
exist $k=k(r)$, such that
$$
\|m_k-m\|_\infty <\varepsilon \,.
$$
We consider the following weights
$$
m_{k,r}(x)=\begin{cases}
m_k(x) & \text{if } x \in \Omega\setminus B_r\cap \Omega_k^-  \\
\frac{1}{r} & \text{if } x \in B_r\cap \Omega_k^-
\end{cases}
$$
and
$$
m_r(x)=\begin{cases}
m(x) & \text{if } x \in \Omega\setminus B_r\cap \Omega_k^-  \\
\frac{1}{r} & \text{if } x \in B_r\cap \Omega_k^-
\end{cases}
$$
where $B_r=B(x_k,\frac{1}{r})$ is a ball and $x_k \in \Omega
_k^-=\{x\in \Omega : m_k(x)<0\}$. It is clear that
$$
\|m_{k,r}-m_r\|_\infty \leq \|m_k-m\|_\infty
$$
so that
$$
m_{k,r}\leq m_r+\varepsilon \quad \mbox{almost everywhere  in }
\Omega\,.
$$
Observe that $m_k\leq m_{k,r}$. Since $m\leq 0$ almost everywhere
in $\Omega$, we have $ m_r\leq 1/r$ almost everywhere  in
$\Omega$, and
\begin{equation}\label{2.2}
m_k\leq\frac{1}{r}+\varepsilon \quad \mbox{almost everywhere in }
\Omega \,.
\end{equation}
It follows from \eqref{2.1} and \eqref{2.2} that
$$
\lambda\geq \lambda_n(m_k)\geq
\lambda_n(\frac{1}{r}+\varepsilon)\,.
$$
Since $\frac{1}{r}+\varepsilon=\frac{2}{r}$,
$$
2\lambda=r\lambda_n(2)=\lambda_n(\frac{1}{r}+\varepsilon)\leq
\lambda_n(m_k)\leq \lambda \;,
$$
which is a contradiction. So ${\lim_k } \lambda_n(m_k)=+\infty$.
\end{proof}

Now we can state the main theorem of this section.

\begin{theorem}\label{thm2.3}
Let $m_1, \, m_2\in \mathcal{M}^{+}(\Omega )$ be such that
 $\mathop{\rm ess\,inf}_{\,\Omega} m_2>0$. So for all
$\alpha \in \mathbb{R}$ there exist a unique real  $t_n(\alpha )$
 which satisfies $\lambda _n(\alpha m_1+t_n(\alpha )m_2)=1$.
\end{theorem}

\begin{proof}
Let $\alpha \in \mathbb{R}$. We consider the function
$f_{\alpha}:t\mapsto \lambda _n(\alpha m_1+tm_2)$. According to
the proposition \ref{prop2.1} we affirm that $f_\alpha $ is
decreasing continuous. Consequently $f_\alpha $ is injective. To
show that the equation $f_{\alpha} (t)=1$ has a solution (hence
only one ), we distinguish three cases.

\noindent\textbf{Case 1:} $ \lambda _{-n}(m_1)<\alpha <\lambda
_n(m_1)$ It is clear that when $\alpha =0$,  the unique real $t_0$
that verifies $\lambda _n(\alpha m_1+t_0m_2)=1$ is $t_0=\lambda
_n(m_2)$. Suppose that $\alpha $ is not nil, in this case we have
$$
f_\alpha (0)=\frac{\lambda _n(m_1)}{\alpha} \quad \mbox{if }
\alpha >0 \quad \mbox{and} \quad f_\alpha (0)=\frac{\lambda
_{-n}(m_1)}{\alpha} \quad \mbox{if } \alpha <0 \,;
$$
So that  $f_{\alpha}(0)>1$. Now $\frac {\alpha} {t} m_1+m_2 \to
m_2$ in $\mathcal{L}^\infty (\Omega )$  as $t \to +\infty $;
$$
\lim_{t\to +\infty } f_{\alpha} (t)=\lim_{t\to +\infty }\frac
{1}{t}  \lambda _n(\frac {\alpha} {t} m_1+m_2)=0 \,,
$$
so there exist a unique real $t_n(\alpha )\in ]0,+\infty [$ which
verifies $f_\alpha (t_n(\alpha ))=1 $.

\noindent\textbf{Case 2:} $\alpha >\lambda _n(m_1)$. In this case
$\alpha >0$ and
\begin{equation}\label{2.3}
0<f_\alpha (0)=\frac{\lambda_n(m_1)}{\alpha} <1 \,.
\end{equation}
Let $ \gamma _{\alpha} =\frac{-\alpha  \|m_1\|_\infty }{
\mathop{\rm ess\,inf}_{\,\Omega} \, m_2} $, it is easy to see that
$$
\alpha m_1+\gamma _{\alpha} m_2\leq 0 \quad \mbox{almost
everywhere  in }\Omega\;,
$$
so
$$
\gamma_{\alpha} \in\{t<0 : \alpha m_1(x)+t m_2(x)\leq 0 \quad
\mbox{a.e. }  x\in \Omega\}=A_{\alpha}
$$
we put $ \tau_{\alpha} = \sup  A_{\alpha}$. We prove that
$\tau_{\alpha} \in A_{\alpha}$.

We first show that $\tau_{\alpha}<0$. Since $f_{\alpha}(0)>0$, and
$f_{\alpha}$, is a continuous function then there exist $ \eta < 0
$ such that $f_{\alpha}(t)>0$ for all $t\in  [\eta ,0]$. so $
\lambda _n(\alpha  m_1+tm_2)>0$ for all $t\in  [\eta ,0]$; i.e.,
$$
\mathop{\rm meas}\{x\in \Omega : \alpha m_1(x)+t m_2(x)>0\}\neq 0
\quad \forall
 t\in [\eta ,0]\,;
$$
hence $ \tau_{\alpha}\leq\eta <0$. Moreover, for all $n\in
\mathbb{N}$, there exist $t_n\in A_{\alpha}$ such that
$\tau_{\alpha}-\frac{1}{n}<t_n$. It follows that
\begin{align*}
\alpha m_1(x)+\tau_{\alpha} m_2(x) &\leq \alpha m_1(x)+t_n
m_2(x)+\frac{1}{n} m_2(x) &\leq \frac{1}{n} \|m_2\|_\infty \quad
\mbox{a.e. }  x\in \Omega, \forall n\in \mathbb{N}
\end{align*}
thus we have $\alpha m_1+\tau_{\alpha} m_2\leq 0$ almost
everywhere
 in $\Omega$. Then Proposition \ref{prop2.2} implies
\begin{equation}\label{2.4}
\lim_{t\to \tau_{\alpha}} f_{\alpha} (t)=+\infty \,.
\end{equation}
It then follows from \eqref{2.3} and \eqref{2.4} that there exist
a unique real $ t_n(\alpha)\in ]\tau _{\alpha} ,0[ $ which
verifies $ f_{\alpha} (t_n(\alpha ))=1 $.

\noindent\textbf{case 3:}  $ \alpha < \lambda _{-n}(m_1)$. In this
case we have $\alpha <0$ and
$$
0<f_\alpha (0)=\frac{\lambda _{-n}(m_1)}{\alpha} <1 \,.
$$
Let $\delta _{\alpha} =\frac{\alpha  \|m_1\|_\infty } {\mathop{\rm
ess \,inf}_{\Omega} m_2}$,
$$
B_{\alpha}=\{t<0: \alpha m_1(x)+t m_2(x)\leq 0 \mbox{ a.e. }  x\in
\Omega\} \quad \mbox{and} \quad \rho_{\alpha}= \sup  B_{\alpha}.
$$
Obviously
$$
\alpha m_1+\delta _{\alpha} m_2\leq 0 \quad \mbox{almost
everywhere in }\Omega\,.
$$
The rest of the proof can be carried out in a similar manner to
that of the case 2. The proof is complete.
\end{proof}

\section{Variational formulation of the eigencurve $C_n$}

We consider the formula \eqref{1.4} of $\beta _n(\alpha )$. By the
Sobolev inequality and hypothesis $\mathop{\rm
ess\,inf}_{\,\Omega }m_2>0$, it is easy to see that
 $\beta _n(\alpha )$ is finite.
Our objective in this section is to show that the graph of $\beta
_n(\alpha )$ is exactly $C_n$.

\begin{theorem}\label{thm3.1}
We take again the notation of Theorem \ref{thm2.3}. So we have
$$
t_n(\alpha )=\beta _n(\alpha ) \quad \mbox{for all } \alpha \in
\mathbb{R} \,.
$$
\end{theorem}

\begin{proof} We have on one hand, according to \eqref{1.4}
for all $K \in \Gamma _n$, there is $u_K\in K$ such that
$$
\beta _n(\alpha )\leq \max_{u\in K}\frac{\int_\Omega |\nabla
u|^p-\alpha \int_\Omega m_1|u|^p}{\int_\Omega
m_2|u|^p}=\frac{\int_\Omega |\nabla u_K|^p-\alpha \int_\Omega
m_1|u_K|^p}{\int_\Omega m_2|u_K|^p},
$$
 then
$$
\alpha \int_\Omega m_1|u_K|^p+\beta _n(\alpha )\int_\Omega
m_2|u_K|^p\leq \int_\Omega |\nabla u_K|^p=1\,.
$$
So that
$$
\min_{u\in K}\int_\Omega (\alpha m_1+\beta _n(\alpha
)m_2)|u|^p\leq \alpha \int_\Omega m_1|u_K|^p+\beta _n(\alpha
)\int_\Omega m_2|u_K|^p\leq 1 ,
$$
 for all $K$ $\in \Gamma _n $, this implies
$$
\sup_{K\in \Gamma _n}\min_{u\in K}\int_\Omega (\alpha m_1+\beta
_n(\alpha )m_2)|u|^p\leq 1 \,.
$$
Since
$$
\frac 1{\lambda _n(\alpha m_1+\beta _n(\alpha )m_2)}=\sup_{K\in
\Gamma _n}\min_{u\in K}\int_\Omega (\alpha m_1+\beta _n(\alpha
)m_2)|u|^p ,
$$
it follows that
\begin{equation}\label{3.1}
\lambda _n(\alpha m_1+\beta _n(\alpha )m_2)\geq 1 \,.
\end{equation}
On the other hand, from Theorem \ref{thm2.3}, we have
$$
\lambda_n(\alpha m_1+t_n(\alpha )m_2)=1 \,,
$$
so for all $K \in \Gamma _n$, there is $u_K\in K$ such that
$$
\alpha \int_\Omega m_1|u_K|^p+t_n(\alpha )\int_\Omega m_2|u_K|^p=
\min_{u\in K}\int_\Omega (\alpha m_1+t_n(\alpha )m_2)|u|^p,
$$
and
$$
\min_{u\in K}\int_\Omega (\alpha m_1+t_n(\alpha )m_2)|u|^p\leq
\lambda _n(\alpha m_1+t_n(\alpha )m_2)=1 \,.
$$
Since $1=\int_\Omega |\nabla u_K|^p $,
$$
\alpha \int_\Omega m_1|u_K|^p+t_n(\alpha )\int_\Omega
m_2|u_K|^p\leq \int_\Omega |\nabla u_K|^p \,.
$$
This implies
$$
t_n(\alpha )\leq \frac{\int_\Omega |\nabla u_K|^p-\alpha
\int_\Omega m_1|u_K|^p}{\int_\Omega m_2|u_K|^p}\leq \max_{u\in
K}\frac{\int_\Omega |\nabla u|^p-\alpha \int_\Omega
m_1|u|^p}{\int_\Omega m_2|u|^p} ,
$$
for all $K\in \Gamma _n $, thus we deduce
$$
t_n(\alpha )\leq \inf_{K\in \Gamma _n}\max_{u\in K}
\frac{\int_\Omega |\nabla u|^p-\alpha \int_\Omega
m_1|u|^p}{\int_\Omega m_2|u|^p} =\beta _n(\alpha ) \,.
$$
Using the monotony of  $\lambda _n$ with respect to the weight, it
follows that
\begin{equation}\label{3.2}
\lambda _n(\alpha m_1+\beta _n(\alpha )m_2)\leq \lambda _n(\alpha
m_1+t_n(\alpha )m_2)=1 \,.
\end{equation}
From \eqref{3.1} and \eqref{3.2}, we obtain
$$
\lambda _n(\alpha m_1+\beta _n(\alpha )m_2)=1\,.
$$
Since $t_n(\alpha )$ is unique, we then conclude that $t_n(\alpha
)=\beta _n(\alpha )$.
\end{proof}

\section{Asymptotic behavior of $C_n$}

The fact of considering $\beta _n(\alpha )$ by its expression
given in the variational formulation \eqref{1.4} makes regrettably
the study of its asymptotic behavior difficult. So, our aim in
this section is to determine the asymptotic behavior with the help
of Theorem \ref{thm3.1} and the definition of $C_n$ (cf.
\eqref{1.3}).

\begin{theorem}\label{thm4.1}
Let $m_1, m_2\in \mathcal{M}^{+}(\Omega )$ be such that
$\mathop{\rm ess\,inf}_{\,\Omega} m_2>0 $. So we have the
following asymptotic behavior:
\begin{itemize}
\item[(i)] $\lim_{\alpha \to +\infty}
\beta _n(\alpha)/\alpha =-\mathop{\rm ess \,sup}_{\Omega}
m_1/m_2$,

\item[(ii)] $\lim_{\alpha \to -\infty} \beta _n(\alpha)/\alpha
 =-\mathop{\rm ess \,inf}_{\Omega} m_1/m_2$.
\end{itemize}
\end{theorem}

\begin{proof} To prove (i), we consider $\alpha>\lambda _n(m_1)$:
The formula
$$
\lambda _n(\alpha m_1+\beta _n(\alpha )m_2)=1
$$
then implies
$$
\lambda_n(m_1+\frac{\beta _n(\alpha )}\alpha m_2)=\alpha
$$
which is a finite quantity and positive, so \\
$$
m_1+\frac{\beta _n(\alpha )}{\alpha}  m_2\in
\mathcal{M}^{+}(\Omega ),
$$
thus there exist a subset $\Omega _\alpha $ such that
$$
\mathop{\rm meas} (\Omega _\alpha) \neq 0 \quad \mbox{and} \quad
m_1(x)+\frac{\beta _n(\alpha )}\alpha  m_2(x)>0 \quad \mbox{a.e. }
 x\in \Omega _\alpha \,.
$$
Hence
$$ -\frac{\beta _n(\alpha )}\alpha <%
\frac{m_1(x)}{m_2(x)} \quad \mbox{a.e. } x\in \Omega _\alpha \,;
$$
thus we have
$$
-\frac{\beta _n(\alpha )}\alpha <\mathop{ess \,sup}_{\,\Omega}
\frac{m_1}{m_2} \,.
$$
So we get
\begin{equation}\label{4.1}
\mathit{\limsup_{\alpha \to +\infty}}-\frac{\beta _n(\alpha
)}{\alpha} \leq \mathop{\rm ess \,sup}_{\,\Omega} \frac{m_1}{m_2}.
\end{equation}
 On the other hand, suppose that
$$
l=\liminf_{\alpha \to +\infty}- \frac{\beta _n(\alpha)}{\alpha}
\,.
$$
We choose a sequence $\alpha _k \to +\infty$, so that
$$
m_1+\frac{\beta _n(\alpha _k)}{\alpha _k}m_2 \to m_1-l\,m_2 \quad
\mbox{in }\mathcal{L}^\infty (\Omega )\,.
$$
Since
$$
\lambda_n(m_1+\frac{\beta _n(\alpha _k)}{\alpha _k}m_2)=\alpha
_k\to +\infty,
$$
according to Proposition \ref{prop2.2}, we obtain $m_1-lm_2\leq 0$
almost everywhere in $\Omega$, i.e.,
$$
\frac{m_1}{m_2}\leq l \quad \mbox{almost everywhere in }\Omega\,;
$$
so that
\begin{equation}\label{4.2}
\mathop{\rm ess \,sup}_{\Omega} \frac{m_1}{m_2}\leq
l={\liminf_{\alpha \to +\infty}}-\frac{\beta _n(\alpha )}{\alpha}
\,.
\end{equation}
Then \eqref{4.1} and \eqref{4.2} yield the result (i).

The proof of (ii) can be carried out as that of (i). This
concludes the proof .
\end{proof}

\subsection*{Remarks}
\begin{itemize}
\item[(i)] All eigencurves of the $p$-Laplacian have  the
same asymptotic behavior.

\item[(ii)] The asymptotic behavior of the first eigencurve of
the $p$-Laplacian is already established in \cite{BH1}, but their
method which uses the properties of the first eigenfunction is not
generalised  to  the higher orders.

\item[(iii)] The results  established in this
paper can also be generalised to eigencurves of order $\geq 2$ of
the $p$-Laplacian with Neumann condition.

\end{itemize}

\subsection*{Acknowledgment} The authors are grateful for the
referee's valuable suggestions.

\begin{thebibliography}{00}

\bibitem{A1} A. Anane,
\emph{Simplicit\'e et isolation de la premi\`ere valeur propre du
$p$-Laplacien avec poids,} C.R.A.S, Paris, t. 305 (1987), pp.
725-728.

\bibitem{A2} A. Anane,
\emph{Etude des valeurs propres et de la r\'esonance pour
l'op\'erateur $p$-Laplacien,} Th\`ese de Doctorat ,Universit\'e
Libre de Bruxelles, Belgique. (1988).

\bibitem{AT} A. Anane, N. Tsouli,
\emph{On the second eigenvalue of the $p$Laplacian,} Nonlinear
Partial Differential Equations, Pitman Research Notes 343 (1996),
1-9.

\bibitem{BH1} P. A. Binding, Y. X. Huang,
\emph{The principal eigenvalue of the $p$Laplacian,} Diff. Int.
Eqns. 8 (1995), 405-414.


\bibitem{BH2} P. A. Binding, Y. X. Huang, \emph{Bifurcation from
eigencurves of the $p$-Laplacian,} Diff. Int. Eqns. 8 (1995),
415-428.

\bibitem{T} N. Tsouli,
\emph{Etude de l'ensemble nodal des fonctions propres et de la
non-r\'esonance pour l'op\'erateur $p$-Laplacien,} T\`ese de
Doctorat d'Etat Es-Sciences, Facult\'e des Sciences Oujda Maroc.
(1995).

\end{thebibliography}

\end{document}