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\markboth{\hfil Hardy-Sobolev operator with indefinite weights 
\hfil EJDE--2002/33}{EJDE--2002/33\hfil K. Sreenadh \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 33, pp. 1--12. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  On the eigenvalue problem for the Hardy-Sobolev operator 
  with indefinite weights 
 %
\thanks{ {\em Mathematics Subject Classifications:} 35J20, 35J70, 35P05, 35P30.
\hfil\break\indent
{\em Key words:} $p$-Laplcean, Hardy-Sobolev operator, Fu\v{c}ik spectrum,
 Indefinite weight.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted October 23, 2001. Published April 2, 2002.} }
\date{}
%
\author{K. Sreenadh}
\maketitle

\begin{abstract}
  In this paper we study the eigenvalue problem
  $$
  -\Delta_{p}u-a(x)|u|^{p-2}u=\lambda |u|^{p-2}u,
  \quad u\in W^{1,p}_{0}(\Omega),
  $$
  where $1<p\le N$, $\Omega$ is a bounded domain containing 
  $0$ in $\mathbb{R}^N$, $\Delta_{p}$ is the $p$-Laplacean, and 
  $a(x)$ is a function related to Hardy-Sobolev inequality.
  The weight function $V(x)\in L^{s}(\Omega)$  may change sign 
  and has nontrivial positive part. We study the simplicity, 
  isolatedness of the first eigenvalue, nodal domain properties.
  Furthermore we show the existence of a nontrivial curve in the 
  Fu\v{c}ik spectrum.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

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\catcode`@=12

\section{Introduction}

Let $\Omega $ be a bounded domain containing $0$ in $\mathbb{R}^N$.
Then the Hardy-Sobolev inequality for $1<p<N$ states that
\begin{equation}
\label{eq:a1}
\int_{\Omega}|\nabla u|^pdx \ge \Big(\frac{N-p}{p}\Big)^p
\int_{\Omega}\frac{|u|^p}{|x|^p}dx
\end{equation}
for all $u\in W^{1,p}_{0}(\Omega)$. It is known that $(\frac{N-p}{p})^p$
is the best constant in (\ref{eq:a1}). In a recent
work Adimurthi, Choudhuri and Ramaswamy \cite{ACM} improved the above
inequality. In particular, when $p=N$ their inequality reads
\begin{equation}
\label{eq:a2}
\int_{\Omega}|\nabla u|^N dx \ge \Big(\frac{N-1}{N}\Big)^N
\int_{\Omega}\frac{|u|^N}{|x|^N(\log\frac{R}{|x|})^N}dx,\quad \forall u\in W^{1,N}_{0}(\Omega),
\end{equation}
where $R>e^{2/N}\sup_{\Omega}|x|$.
Subsequently it was shown in \cite{AS} that $(\frac{N-1}{N})^N$
is the best constant in (\ref{eq:a2}).
In view of the above two inequalities we define the Hardy-Sobolev
Operator $L_{\mu}$ on $W^{1,p}_{0}(\Omega)$ as
\[
L_{\mu}u:=-\Delta_{p}u-\mu a(x)|u|^{p-2}u
\]
where
$$
a(x)=\begin{cases}
1/|x|^p & 1<p<N \\
1/(|x|\log\frac{R}{|x|})^N & p=N
\end{cases}$$
and $0\le \mu< (\frac{n-p}{p})^p$ or $(\frac{N-1}{N})^N$ depends on the
value of $p$.
Here  $\Delta_{p}u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ denotes the
$p$-Laplacean. In the present work we consider the following eigenvalue
problem:
\begin{equation} \label{eq:a3}
\begin{gathered}
L_{\mu}u=\lambda V(x) |u|^{p-2}u \quad\text{in }\Omega\\
u=0\quad \text{on } \partial \Omega.
\end{gathered}
\end{equation}
We assume that $V\in L^{1}_\text{loc}(\Omega), \; V^{+}=V_{1}+V_{2} \ncong 0$ with $V_{1}\in L^{\frac{N}{p}}(\Omega)$ and $V_{2}$ is such that
\[\lim_{x\to y,\;
        x\in \Omega}|x-y|^pV_{2}(x)=0 \quad \forall y\in \overline{\Omega} \quad\text{for} \;\;p<N\]
\begin{equation}
\label{eq:a4}
\lim_{x\to y,\;x\in \Omega}|x-y|^p(\log\frac{R}{|x-y|})^p
V_{2}(x)=0 \quad \forall y\in \overline{\Omega} \quad\text{for} \;\;p=N.
\end{equation}
where $V^{+}(x)=\max \{V(x),0\}$.
We also assume
\begin{enumerate}
\item[(H)] There exists $r>\frac{N}{p}$ and a closed subset $S$
of measure zero in $\mathbb{R}^N$ such that $\Omega\backslash S$
is connected and $V\in L^{r}_\text{loc}(\Omega\backslash S)$.
\end{enumerate}
We define the functional $J_{\mu}$ on $W^{1,p}_{0}(\Omega)$ as
$$
J_{\mu}(u):=\int_{\Omega}|\nabla u|^p-\mu \int_{\Omega}a(x)|u|^{p-2}u.
$$
Then $J_{\mu}$ is $C^{1}$ on $W^{1,p}_{0}(\Omega)$. Our goal here is to
study the eigenvalue problem and some main properties (simplicity,
isolatedness) of
\[\lambda_{1}:=\inf\left\{J_{\mu}(u); u\in W^{1,p}_{0}(\Omega) \quad\text{and } \int_{\Omega}V|u|^pdx=1\right\}\]
We use the following results in Section 2.

\begin{proposition}[\cite{BM}] \label{prop1.1}
 Let $\Omega \subset \mathbb{R}^{n}$ is bounded domain
and suppose $(u_{n}) \in W^{1,p}(\Omega)$ such that $u_{n}\to u$ weakly in $W^{1,p}_{0}(\Omega)$satisfies
\[-\Delta_{p} u_{n}=f_{n}+g_{n} \;\text{in} \;{\mathcal{D}}^{\prime}(\Omega)\]
where $f_{n} \to f$ in $W^{-1,p'}$ and $g_{n}$ is a
bounded sequence of Radon measures, i.e.,
\[\left<g_{n},\phi\right> \le C_{K} \;\|\phi\|_{\infty}\; \; \] for all $\phi$ in $C_{c}^{\infty}(\Omega)$ with support in $K$.
Then there exists a subsequence $(u_{n})$ of $(u_{n})$
such that $\nabla u_{n}(x) \to \nabla u(x)$ a.e. in $\Omega$.
\end{proposition}

\begin{proposition}[(Brezis-Lieb\cite{BL})] \label{prop1.2}
Suppose $f_{n}\to f $ a.e.
and $\|f_{n}\|_{p} \le C <\infty $ for all n and for some $0<p<\infty$. Then
\[\lim_{n\to \infty} \{\|f_{n}\|_{p}^{P}-\|f_{n}-f\|_{p}^p\}=\|f\|_{p}^p.
\]
\end{proposition}

In section 2 we study the eigenvalue problem for $L_{\mu}$ and
show that the first eigenvalue is simple and the eigenfunctions
corresponding to other eigenvalues changes sign.
In section 3 we study the existence of nontrivial curve in
the Fu\v{c}ik spectrum of $L_{\mu}$. Finally in the last section we
study some nodal domain properties of $L_{\mu}$ with a stronger assumption on $V$ that
$V\in L^{r}(\Omega)$ for some $r>\frac{N}{p}$. \par We now provide a brief account of what is known about the problems
of type (\ref{eq:a3}). In case of $\mu=0,$ the above properties are well
known when $V$ is bounded(see\cite{A}). For indefinite weights with
different integrability conditions see\cite{AH} and \cite{SW}. In \cite{SW} the
problem of simplicity and sign changing nature of other eigen functions
are left open. In Theorem \ref{thm2.1} below we prove the above properties. In a
recent work Cuesta \cite{C} proved above properties with stronger assumption
that $V\in L^{s}(\Omega)$ for some $s>\frac{N}{p}$. When $\mu\ne 0$ and
$V=1$ the above properties are studied in \cite{S},\cite{Sr}.


\section{Eigenvalue Problem}

In this section we show that the first eigenvalue is simple and the
eigenfunctions corresponding to other eigenvalues changes sign. We
prove the following theorem.

\begin{theorem} \label{thm2.1}
The first eigenvalue, $\lambda_{1}$, is simple and
the eigenfunctions corresponding to the other eigenvalues changes sign.
\end{theorem}

The next theorem is proven with the help of a deformation lemma for $C^{1}$
 manifolds.

\begin{theorem} \label{thm2.2}
There exists a sequence $\{\lambda_{n}\}$ of eigenvalues of $L_{\mu}$
such that $\lambda_{n}\to \infty$.
\end{theorem}
Let us define the operators
\begin{gather*}
L(u,v):=|\nabla u|^p-(p-1)\frac{u^p}{v^p}|\nabla v|^p
-p\frac{u^{p-1}}{v^{p-1}}\nabla u |\nabla v|^{p-2} \nabla v \\
R(u,v):=|\nabla u|^p-|\nabla v|^{p-2}\nabla v.\nabla \big(\frac{u^p}{v^{p-1}}
\big)
\end{gather*}
Then $R(u,v)=L(u,v)\ge 0$ for all $u,v \in C^{1}(\Omega\backslash\{0\})
\cap W^{1,p}(\Omega)$ with $u\ge 0, v>0$ and equal to 0 if and only if
$u=kv$ for some constant $k$ \cite[Theorem 1.1]{AH}.
We need following lemmas to prove our results.

\begin{lemma} \label{lm2.3}
 The mapping $u\longrightarrow \int_{\Omega}V^{+}|u|^pdx $ is weakly
 continuous.
\end{lemma}

\paragraph{Proof:}
In case the $1<p<N$, the proof follows as in \cite{SW}. Here we give
the proof when $p=N$. Clearly $u\to \int_{\Omega}V_{1}|u|^p$
is weakly continuous. Since $\overline{\Omega}$ is compact, there is a
finite covering of $\overline{\Omega}$ by closed balls $B(x_{i},r_{i})$
such that, for $1\le i\le k$,
\begin{equation}
\label{eq:b1}
|x-x_{i}|\le r_{i} \implies |x-x_{i}|^N(\log\frac{R}{|x-x_{i}|})^N V_{2}(x)
\le \epsilon.
\end{equation}
There exists $r>0$ such that, for $1\le i\le k,$
\[|x-x_{i}|\le r \implies |x-x_{j}|^N(\log\frac{R}{|x-x_{i}|})^NV_{2}(x)\le
\epsilon/k.\]
Define $A:=\cup_{j=1}^{k} B(x_{j},r)$. Then by inequality (\ref{eq:a2})
\begin{equation}
\label{eq:b2}
\int_{A}V_{2}|u_{n}|^Ndx\le \epsilon c^N,\quad \int_{A}V_{2}|u|^Ndx\le
\epsilon c^N
\end{equation}
where $c=\frac{N}{N-1}\sup_{n} \|u_{n}\|$. It follows from (\ref{eq:b1})
that $V_{2}\in L^{1}(\Omega\backslash A)$
so that
\begin{equation}
\label{eq:b3}
\int_{\Omega\backslash A} V_{2} |u_{n}|^N dx \longrightarrow
\int_{\Omega\backslash A} V_{2} |u|^Ndx
\end{equation}
Now the conclusion follows from (\ref{eq:b2}) and (\ref{eq:b3}).
\hfill$\Box$\smallskip

Define $M:=\left\{ u\in W^{1,p}_{0}(\Omega); \;\;\int_{\Omega}V|u|^p=1\right\}$

\begin{lemma} \label{lm2.4}
The eigenvalue $\lambda_{1}$ is attained.
\end{lemma}

\paragraph{Proof:}
 Let $u_{n}$ be a sequence in $M$ such that
$J_{\mu}(u_{n})\to \lambda_{1}$. Since $W^{1,p}_{0}(\Omega)$ is
reflexive, there exists a subsequence $\{u_{n}\}$ of $\{u_{n}\}$ such
that $u_{n}\to u$ weakly in $W^{1,p}_{0}$ and a.e. in $\Omega$.
Now for $n\in \mathbb{N} $ choose $u_{n}$ such that
$J_{\mu}(u_{n}) \le \inf_{M}J_{\mu}+\frac{1}{n^{2}}$.
Now by The Ekeland Variational Principle, there exists a sequence $\{v_{n}\}$
such that
\begin{gather*}
J_{\mu}(v_{n})\le J_{\mu}(u_{n})\\
\|u_{n}-v_{n}\|\le \frac{1}{n}\\
J_{\mu}(v_{n})\le J_{\mu}(u)+\frac{1}{n} \|v_{n}-u\| \quad \forall u\in M
\end{gather*}
Now standard calculations from above three equations, as in \cite{De}, gives
\begin{equation}
\label{eq:b4}
\big| J_{\mu}'(v_{n})w-J_{\mu}(v_{n})\int_{\Omega}V |v_{n}|^{p-2}v_{n}w
\big|\le C \frac{1}{n} \|w\|.
\end{equation}
By Proposition \ref{prop1.1}, there exists a subsequence of $\{v_{n}\}$, which we still denote by  $\{v_{n}\}$  such that $v_{n} \to v $ weakly in $W^{1,p}_{0}(\Omega)$ and $\nabla v_{n}\to
\nabla v $ a.e. in $\Omega$. Since $|\nabla v_{n}|^{p-2}\nabla v_{n}$ is bounded in $(L^{p'}(\Omega))^N, 1/p+1/p'=1$, and $\nabla v_{n}\to \nabla v$ a.e. in $\Omega$, we have
\begin{align*}
&|\nabla v_{n}|^{p-2}\nabla v_{n} \to |\nabla v|^{p-2}\nabla v
\quad \text{a.e. in }\; \Omega\\
&|\nabla v_{n}|^{p-2}\nabla v_{n} \to |\nabla v|^{p-2}\nabla v
\quad \text{weakly in}\; (L^{p'}(\Omega))^N
\end{align*}
which allows us to pass the  limit as $n\to \infty$ in (\ref{eq:b4}),
obtaining
\[-\Delta_{p}v-a(x)|v|^{p-2}v-\lambda_{1}|v|^{p-2}v=0 \quad \text{in} \;\;{\cal{D}}'(\Omega).\]
Observe that
\[\int_{\Omega}V^{-}|v_{n}|^pdx=\int_{\Omega}V^{+}|v_{n}|^pdx -1 \to \int_{\Omega}V^{+} |v|^pdx-1\]
as $n\to \infty$. Now using Fatau's lemma we can conclude that
$v\ncong 0$. \hfill$\Box$

\begin{lemma} \label{lm2.5}
The eigenvalue $\lambda_{1}$ is simple.
\end{lemma}

\paragraph{Proof:} This is an adaptation from a proof in \cite{AH}.
Let $\{\psi_{n}\}$ be a sequence of functions such that
$\psi_{n}\in C_{c}^{\infty}(\Omega), \psi_{n}\ge 0, \psi_{n}\to \phi_{1}$ in
$W^{1,p},$ a.e. in $\Omega $ and $\nabla \psi_{n} \to \nabla \phi_{1}$ a.e.
in $\Omega$. Then we have
\begin{equation}
\label{eq:b5} \begin{aligned}
0=&\int_{\Omega} \left( |\nabla \phi_{1}|^p-(\mu a(x)+\lambda_{1}V)
\phi_{1}^p\right)dx\\
=&\lim_{n \to \infty} \int_{\Omega}
\left( |\nabla \psi_{n}|^p-(\mu a(x)+V\lambda_{1})\psi_{n}^p\right)dx.
\end{aligned}\end{equation}
Consider the function $w_{1}:=\psi_{n}^p/(u_{2}+\frac{1}{n})^{p-1}$.
Then $w_{1}\in W^{1,p}_{0}(\Omega)$. So testing the equation
satisfied by $u_{2}$ with  $w_{1}$ we get,
\begin{equation}
\label{eq:b6}
\int_{\Omega}(\lambda_{1}V+\mu a(x)) \psi_{n}^p
\big(\frac{u_{2}}{u_{2}+\frac{1}{n}}\big)^{p-1}=\int_{\Omega}
|\nabla u_{2}|^{p-2}\nabla u_{2}.\nabla
\big(\frac{\psi_{n}^p}{(u_{2}+\frac{1}{n})^{p-1}}\big)
\end{equation}
Now from (\ref{eq:b5}) and (\ref{eq:b6}) we obtain
\begin{equation*}
\begin{split}
0&=\lim_{n\to \infty}\int_{\Omega}\left(|\nabla \psi_{n}|^p-|\nabla u_{2}|^{p-2}
 \nabla u_{2}.\nabla \big(\frac{\psi_{n}^p}{(u_{2}+\frac{1}{n})^{p-1}}\big)\right)\\
&=\lim_{n\to \infty}\int_{\Omega} L(\psi_{n},u_{2})\ge
\int_{\Omega}L(\phi_{1},u_{2})\ge 0
\end{split}
\end{equation*}
by Fatau's lemma.
Now by assumption (H), $\phi_{1},u_{2}$ are in $C^{1}(\Omega\backslash
S\cup \{0\})$ \cite{D,T}. Therefore $\phi_{1}=k u_{2}$ for some
constant $k$. \hfill$\Box$

\paragraph{Proof of Theorem \ref{thm2.1}, completed:}
Let $\phi_{1}, u$ be the eigenfunctions corresponding to $\lambda_{1}$ and
$\lambda$ respectively. Then $\phi_{1}, u$ satisfies
\begin{gather}
\label{eq:b7}
-\Delta_{p}\phi_{1}-\mu a(x)\phi_{1}^{p-1}=\lambda_{1}V(x)\phi_{1}^{p-1} \quad \text{in}\;\; {\mathcal{D}}^{\prime}(\Omega),\\
\label{eq:b8}
-\Delta_{p}u-\mu a(x) |u|^{p-2}u=\lambda V(x)|u|^{p-2}u\quad \text{in}\;\; {\mathcal{D}}^{\prime}(\Omega)
\end{gather}
respectively. Suppose $u$ does not change sign. We may assume $u\ge 0$ in
$\Omega$. Let $\{\psi_{n}\}$ be a sequence in $C_{c}^{\infty}$ such that
$\psi_{n}\to \phi_{1}$ as $n\to \infty$. Now consider the test functions
$w_{1}=\phi_{1}, w_{2}=\frac{\psi_{n}^p}{(u+\frac{1}{n})^{p-1}}$.
Then $w_{1}, w_{2}\in W^{1,p}_{0}(\Omega)$.
Testing (\ref{eq:b7}) with $w_{1}$ and (\ref{eq:b8}) with $w_{2}$ we get
\begin{equation}
\label{eq:b9}
\int_{\Omega}|\nabla \phi_{1}|^pdx-\int_{\Omega}(\lambda_{1}V(x)+\mu a(x))\phi_{1}^pdx=0
\end{equation}
\[\int_{\Omega}|\nabla u|^{p-2}\nabla u.\nabla \Big(\frac{\psi_{n}^p}
{(u+\frac{1}{n})^{p-1}}\Big)dx-\int_{\Omega}(\lambda V(x)+\mu a(x))\psi_{n}^p
\big(\frac{u}{u+\frac{1}{n}}\big)^{p-1}dx=0
\]
Since $R(u,v)\ge0$, we get
\begin{equation}
\label{eq:b10}
\int_{\Omega}|\nabla \psi_{n}|^pdx-\int_{\Omega}(\lambda V(x)+\mu a(x)
\psi_{n}^p \Big(\frac{u}{u+\frac{1}{n}}\Big)^{p-1}dx\ge 0.
\end{equation}
Subtracting (\ref{eq:b9}) from (\ref{eq:b10}) and taking the limit
as $n\to \infty$ we get,
\[(\lambda-\lambda_{1})\int_{\Omega}V(x)\phi_{1}^p\le 0\]
This is a contradiction to the fact that $\lambda>\lambda_{1}$. 
 \hfill $\Box$

\paragraph{Proof of Theorem \ref{thm2.2}:}
Let $\tilde{J}_{\mu}$ be the restriction of $J_{\mu}$ to the set $M$. Define
\[\lambda_{k}=\inf_{\gamma(A)\ge n} \sup_{u\in A} J_{\mu}(u) \]
where $A$ is a closed subset of $M$ such that $A=-A$, and $\gamma(A)$ is
the {\it {Krasnosel'ski\v{i} genus}} of $A$. Now we show that
$\tilde{J_{\mu}}$ satisfies (P.S.) condition at level $\lambda_{k}$.
Let $\{u_{n}\}$ be a sequence in $M$ such that $J_{\mu}(u_{n})\to \lambda_{k}$
and
\begin{equation}
\label{eq:II1}
\langle J_{\mu}(u_{n}),\phi\rangle -J_{\mu}(u_{n})\int_{\Omega}|u_{n}|^{p-2}u_{n}\phi Vdx =o(1).
\end{equation}
Since $u_{n}$ is bounded, there exists a subsequence $\{u_{n}\} ,u$ such that 
$u_{n}\to u$ weakly in $W^{1,p}_{0}(\Omega)$. Since $\lambda_{k}>0$ 
we may assume that $J_{\mu}(u_{n})\ge 0$. Using Lemma \ref{lm2.3}
 and (\ref{eq:II1}), we get
\[\langle J_{\mu}(u_{n})-J_{\mu}(u), u_{n}-u\rangle +J_{\mu}(u_{n})
\int_{\Omega}\left[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u\right](u_{n}-u)
V^{-}dx =o(1).\]
But
\[\int_{\Omega} \left[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u\right][u_{n}-u]V^{-}\ge 0.\]
By Propositions \ref{prop1.1} and \ref{prop1.2}, we have
\begin{gather*}
\|u_{n}-u\|_{1,p}=\|u_{n}\|_{1,p}-\|u\|_{1,p}+o(1)\\
\|\frac{u_{n}-u}{|x|}\|_{0,p}=\|\frac{u_{n}}{|x|}\|_{0,p}-\|\frac{u}{|x|}
\|_{0,p}+o(1)
\end{gather*}
Therefore
\begin{align*}
o(1)=& \langle J_{\mu}(u_{n})-J_{\mu}(u), (u_{n}-u)\rangle \\
 &+J_{\mu}(u_{n})\int_{\Omega}[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u](u_{n}-u) V^{-}dx\\
\ge& \int_{\Omega}|\nabla u_{n}-\nabla u|^p-\int_{\Omega}\mu a(x)|u_{n}-u|^p +o(1)\\
\ge& C \|u_{n}-u\|_{1,p}+o(1).
\end{align*}
Now by the classical critical point theory for $C^{1}$ manifolds
\cite{Sz}, it follows that $\lambda_{k}$'s are critical points of
$J_{\mu}$ on $M$. Since
$\lambda_{k}\ge c \lambda_{k}^{0}$, where $\lambda_{k}^{0}$ are eigenvalues
of $L_{0}$, we have $\lambda_{k}\to \infty$. \hfill$\Box$

\section{Fu\v{c}ik Spectrum}

In this section we study the existence of a non-trivial
curve in the Fu\v{c}ik spectrum $\sum_{p,\mu }$ of $L_{\mu}$.
The Fu\v{c}ik spectrum of $L_{\mu}$ is defined as the set of $(\alpha,\beta)\in \mathbb{R}^{2}$ such that
\begin{gather*}
L_{\mu}u=\alpha V (u^{+})^{p-1}+\beta V (u^{-})^{p-1}\quad \text{in } \Omega, \\
u=0\quad \text{on }\partial \Omega,
\end{gather*}
has a nontrivial solution $u \in W^{1,p}_{0}(\Omega)$.
The variational approach that we follow here is same as that of
\cite{CDG,Sr}. We prove the following statement.

\begin{theorem} \label{thm3.1}
There exists a nontrivial curve ${\mathcal{C}}$ in $\sum_{p,\mu}$.
\end{theorem}
Let us consider the functional
\[J_{s}(u)=\int_{\Omega} |\nabla u|^p - \int_{\Omega} \mu a(x)|u|^p-s
\int_{\Omega} Vu^{+^p}
\]
$J_{s}$ is a $C^{1}$functional on $W^{1,p}_{0}(\Omega)$. We are
interested in the critical points of the restriction $\tilde{J_{s}}$
of $J_{s}$ to $M$.
By Lagrange multiplier rule, $u\in M$ is a critical point of
$\tilde{J_{s}}$ if and only if there exist $t \in \mathbb{R}$ such that
$J_{s}'(u)=t.I'(u),$ i.e., for all $v \in W^{1,p}_{0}$ we have
\begin{equation}
\int_{\Omega} |\nabla u|^{p-2} \nabla u \nabla v -\int_{\Omega}\mu a(x)|u|^{p-2} uv
-s\int_{\Omega} Vu^{+^{p-1}}v= t \int_{\Omega} V|u|^{p-2} uv\,.
(\Omega)
\end{equation}
This implies that
\begin{gather*}
-\Delta_{p}u-\mu a(x) |u|^{p-2}u=(s+t)V(x)(u^{+})^{p-1}-tV(x) (u^{-})^{p-1}
\quad \text{in }\Omega \\
 u=0\quad \text{on } \partial \Omega
\end{gather*}
holds in the weak sense.  i.e., $(s+t,t) \in \sum_{p,\mu}$,
taking $v=u$ in (3.1), we get $t$ as a critical value of $\tilde{J_{s}}$. Thus
the points in $\sum_{p,\mu}$ on the parallel to the
diagonal passing through (s,0) are exactly of the form $(s+\tilde{J_{s}}
(u), \tilde{J_{s}}(u))$ with $u$ a critical point of  $\tilde{J_{s}}$.

A first critical point of  $\tilde{J_{s}}$ comes from global
minimization . Indeed
\[\tilde{J_{s}}(u) \ge \lambda_{1} \int_{\Omega} |u|^p-s \int_{\Omega} u^{+^p}
\ge \lambda_{1}-s \]
for all u $\in M$, and $\tilde{J_{s}}(u)= \lambda_{1}-s $ for $u=\phi_{1}$.

\begin{proposition} \label{prop3.2}
The function $\phi_{1}$ is a global minimum of  $\tilde{J_{s}}$ with
$ \tilde{J_{s}}(\phi_{1})=\lambda_{1}-s$, the corresponding point
in $\sum_{p,\mu}$ is $(\lambda_{1}, \lambda_{1}-s)$ which lies on
the vertical line through $(\lambda_{1}, \lambda_{1})$.
\end{proposition}

\begin{lemma} \label{lm3.3}
Let 0$\ne v_{n} \in W^{1,p}_{0}$ satisfy $v_{n}\ge 0$ a.e and
$|v_{n}>0| \to 0$, then
$\int_{\Omega}[|\nabla v_{n}|^p-\mu a(x) |v_{n}|^p]dx/
\int_{\Omega}V |v_{n}|^p \to +\infty$.
\end{lemma}

\paragraph{Proof:} Let $w_{n}=v_{n}/ \|v_{n}\|_{V,p}$ and assume by
contradiction that $\int_{\Omega} |\nabla w_{n} |^p-\int_{\Omega}
\mu a(x)|w_{n}|^p$ has a bounded subsequence. By (1.1) or (1.2),
 we get $w_{n}$  bounded in $W^{1,p}_{0}(\Omega)$. Then for a further
subsequence, $w_{n}\to w $ in  $L^p(\Omega,V^{+})$.
Now observe that
\[\int_{\Omega}V^{-}(x) |w|^p\le \lim_{n\to \infty}\int_{\Omega}
V^{-}|w_{n}|^p=\lim_{n\to \infty}\int_{\Omega}V^{+}|w_{n}|^p-1
=\int_{\Omega}V^{+}|w|^p-1.\]
Then $w\ge 0$ and $\int_{\Omega} V^{+}(x)w^p\ge 1$. So for some
$\epsilon >0, \delta =|w>\epsilon|>0$, we
deduce that $|w_{n}> \epsilon/2|> \frac{\delta}{2}$ for $n$ sufficiently
large, which contradicts the assumption $|v_{n}>0|\to 0$.
\hfill$\Box$

A second critical point of $\tilde{J}_{s}$ comes next.

\begin{proposition} \label{prop3.4}
  $- \phi_{1}$ is a strict local minimum of
$\tilde{J_{s}}$, and $ \tilde{J_{s}}(-\phi_{1})=\lambda_{1} $, the
corresponding point in $\sum_{p}$ is $(\lambda_{1}+s,\lambda_{1})$.
\end{proposition}

\paragraph{Proof:}
We follow the ideas in \cite[Prop. 2.3]{CDG}. Assume by contradiction
that there exist  a sequence
$u_{n} \in M$ with  $u_{n} \ne -\phi_{1},\; u_{n} \to -\phi_{1}$
in $W^{1,p}_{0}(\Omega)$ and  $\tilde{J_{s}}(u_{n})\le \lambda_{1}$.

Claim: $u_{n}$ changes sign for $n$ sufficiently large.
Since $u_{n} \to -\phi_{1}, u_{n}$, it must follow that $u_n\le 0 $ some
where. If $u_{n} \le 0$ a.e., in $\Omega$, then
\[ \tilde{J_{s}}(u_{n})=\int_{\Omega} |\nabla u_{n}|^p- \int_{\Omega}\mu a(x)
|u_{n}|^p > \lambda_{1}
\]
since $u_{n} \ne \pm\phi_{1}$, and this contradicts
$ \tilde{J_{s}}(u_{n})\le \lambda_{1}$. This completes the proof of claim.
Let $r_{n}=[ \int_{\Omega} |\nabla u_{n}^{+}|^p-\int_{\Omega}
\mu a(x) |u_{n}^{+}|^p]/ \int_{\Omega} Vu_{n}^{+{p}},$ we have
\begin{align*}
 \tilde{J_{s}}(u_{n})=&\int_{\Omega} |\nabla u_{n}^{+}|^p
 +\int_{\Omega} |\nabla u_{n}^{-}|^p-\int_{\Omega} \mu a(x) |u_{n}^{+}|^p \\
   &-\int_{\Omega} \mu a(x) |u_{n}^{-}|^p-s \int_{\Omega} V|u_{n}^{+}|^p \\
\ge& (r_{n}-s) \int_{\Omega}V u_{n}^{+^p}+ \lambda_{1}
\int_{\Omega}V u_{n}^{-^p}
\end{align*}
on the other hand
\[\tilde{J_{s}}(u_{n})\le \lambda_{1}=\lambda_{1} \int_{\Omega} Vu_{n}^{+^p} +\int_{\Omega}V u_{n}^{-^p} \]
combining the two inequalities, we get
$r_{n} \le \lambda_{1}+s$. Now since, $u_{n} \to -\phi_{1} $ in
$L^p(\Omega),\; |u_{n}>0|\to 0$.
The Lemma \ref{lm3.3} then implies $r_{n}\to +\infty$ , which contradicts
 $r_{n} \le \lambda_{1}+s$. \hfill$\Box$ \smallskip

Now as in the proof of Theorem \ref{thm2.2}, one can show that $\tilde{J}_{s}$
satisfies the P.S. condition at any positive level.

\begin{lemma} \label{lm3.5}
Let $\epsilon_{0}>0 $ be such that
\begin{equation}
\tilde{J_{s}}(u)>\tilde{J_{s}}(-\phi_{1}) \quad \forall
u\in B(-\phi_{1},\epsilon_{0}) \cap M
\end{equation}
with $u \ne -\phi_{1},  B \subset W^{1,p}_{0}$.  Then for any
$0<\epsilon <\epsilon_{0}$
\begin{equation}
\inf \{ \tilde{J_{s}}(u); u\in M \quad\text{and}\quad
 \|u-(-\phi_{1})\|_{1,p}=\epsilon \} > \tilde{J_{s}}(-\phi_{1}).
\end{equation}
\end{lemma}

The proof of this lemma follows from the Ekeland variational principle.
Therefore, we omit it. For details we refer the reader to \cite{CDG}.
Let
\[\Gamma=\{ \gamma\in C([-1,1];M) :\gamma(-1)=-\phi_{1},
\gamma(1)=\phi_{1}\} \ne \varnothing
\]
and the geometric assumptions of Mountain-pass Lemma are satisfied by
previous Lemma. Therefore, there exists $u\in W^{1,p}_{0}$ such that
$\tilde{J}_{s}'(u) =0$ and $J_{s}(u)=c$, where $c$ is given by
\begin{equation}
c(s)=\inf_{\Gamma} \sup_{\gamma} J_{s}(u).
\end{equation}
 Proceeding in this manner for each  $s\ge 0$ we get a non-trivial
 curve ${\mathcal{C}}$: $s\in \mathbb{R}^{+}\to (s+c(s),c(s))
\in \mathbb{R}^{2}$ in $\sum_{p,\mu}$, which completes the proof of
Theorem \ref{thm3.1}.

\section{Nodal Domain Properties}

In this section we show that $\lambda_{1}$ is isolated in the
spectrum under the assumption on $V$ that
$V\in L^{s}(\Omega)$ for some $s>\frac{N}{p}$.
By the regularity results in \cite{T,D} the solutions of
(\ref{eq:a3})
are $C^{1}(\Omega\backslash\{0\})$. In \cite{S} it is shown that the positive
solutions of (\ref{eq:a3}) when $V=1$ tends to $+\infty$ as $|x|
\to 0$. We prove the following theorem.

\begin{theorem} \label{thm4.1}
The eigenvalue $\lambda_{1}$ is isolated in the spectrum provided that
$V\in L^{s}(\Omega)$ for some $s>\frac{N}{p}$. Moreover, for
$v$ an eigenfunction corresponding to an eigenvalue $\lambda
\ne \lambda_{1}$ and  $O$ be a nodal domain of $v$, then
\begin{equation}
\label{eq:t2}
|O|\ge (C\lambda \|V\|_{s})^{-\gamma}
\end{equation}
where $\gamma=\frac{sN}{sp-N}$ and $C$ is a constant depending
only on $N$ and $p$.
\end{theorem}

\begin{lemma} \label{lm4.2}
Let $u\in C(\Omega\backslash\{0\})\cap W^{1,p}_{0}(\Omega)$
and let $O$ be a component of $\{x\in \Omega; u(x)>0\}$.
Then $u \vline_{O} \in W^{1,p}_{0}(O)$
\end{lemma}

\paragraph{Proof:} case (i): $1<p<N$.\\
Let $u_{n} \in C_{c}(\Omega)\cap W^{1,p}_{0}(\Omega)$ such that
$u_{n}\to u$ in $W^{1,p}_{0}(\Omega)$. Then $u_{n}^{+}\to u^{+}$
in $W^{1,p}_{0}(\Omega)$. Let $v_{n}=\min(u_{n},u)$ and let
$\psi_{r}\in C(\Omega)$ be a cutoff function such that
$$
\psi_{r}(x)=
 \begin{cases}
0 &\text{if } |x|\le r/2 \\
1 &\text{if } |x|\ge r
\end{cases}$$
and $|\nabla \psi_{r}(x)| \le \frac{C}{r}$  for some constant $C$.
Now consider the sequence $w_{n,r}(x)=\psi_{r}v_{n}(x)\vline_{O}$. Since
$\psi_{r}v_{n}\in C(\overline{\Omega})$, we have
$w_{n,r} \in C(\overline{O})$ and vanishes on the boundary $\partial O$.
Indeed for $x\in \partial O$ and $x=0$ then $\psi_{r}=0$ and so
$w_{n,r}=0$. If $x\in \partial O \cap \Omega$ and
$x\ne 0$ then $u(x)=0$(since $u$ is continuous except at 0) and so
$v_{n}(x)=0$. If $x\in \partial \Omega$ then $u_{n}(x)=0$ and hence
$v_{n}(x)=0$. So in all the cases $w_{n,r}(x)=0$ for $x \in \partial O$.
Therefore $w_{n,r}\in W^{1,p}_{0}(O)$ and
\begin{align*}
\int_{\Omega}|\nabla(w_{n,r})-\nabla(\psi_{r}u)|^p
=&\int_{O}|(\nabla \psi_{r}) v_{n}+\psi_{r} \nabla v_{n}
-(\nabla \psi_{r})u -\psi_{r} \nabla u|^pdx\\
\le& \|\nabla \psi_{r}v_{n}-\nabla \psi_{r}u\|_{L^p(O)}^p
+\|\psi_{r}\nabla v_{n}- \psi_{r} \nabla u\|_{L^p(O)}^p
\end{align*}
which approaches $0$ as $n\to \infty$. i.e.,
$w_{n,r}\to \psi_{r}u\vline_{O}$ in $W^{1,p}_{0}(O)$.
Now
\[\int_{O}|\nabla \psi_{r}u +\psi_{r} \nabla u -u|^p \le
 \int_{O}|\psi_{r} \nabla u -\nabla u|^p+\int_{O\cap \{r/2<|x|<r\}}|
 \nabla \psi_{r}|^p u
\]
which approaches $0$ as $r\to 0$ (by (\ref{eq:a1}). Therefore,
 $u\vline_{O}\in W^{1,p}_{0}(O)$.\\
\textbf{case(ii):} $p=N$.
In this case we use the following cut-off function which are introduced
in \cite{S}
$$\psi_{r}(x)=
\begin{cases}
0 &\text{if } |x|\le r\\
2\log\big(\frac{r}{|x|}\big)/\log(r) &\text{if } r\le |x|\le r^{1/2}\\
1 &\text{if } |x|\ge r^{1/2}.
\end{cases}$$
and we can proceed as in the previous case. \hfill $\Box$

\paragraph{Proof of Theorem \ref{thm4.1}:}
The proof follows as in \cite{A,C}.
Let $\mu_{n}$ be a sequence of eigenvalues
such that $\mu_{n}>\lambda_{1} $ and $\mu_{n}\to \lambda_{1}$. Let
the corresponding eigenfunctions $u_{n}$ converge to $\phi_{1}$.
such that $\|u_{n}\|_{L^p(V)}=1$. i.e., $u_{n}$ satisfies
\begin{equation}
\label{eq:b11}
-\Delta_{p}u_{n}-\mu a(x)|u_{n}|^{p-2}u_{n}
=\lambda_{n}V(x)|u_{n}|^{p-2}u_{n}.
\end{equation}
Testing (\ref{eq:b11}) with $u_{n}$ and applying weighted
Hardy-Sobolev inequallity we get $u_{n}$
to be bounded. Therefore by Proposition \ref{prop1.1},
there exists a subsequence $(u_{n})$ of $(u_{n})$ such that
$u_{n}\to u $ weakly in $W^{1,p}_{0}(\Omega)$, strongly in
 $L^p(\Omega)$ and $\nabla u_{n} \to \nabla u $ a.e in
$\Omega$. Taking limit $n\to \infty$ in (\ref{eq:b11}) we get
\[-\Delta_{p}u-\mu a(x)|u|^{p-2}u=\lambda_{1}V(x)|u|^{p-2}u \quad \text{in}\;\;  {\mathcal{D}}^{\prime}(\Omega).\]
Therefore $u=\pm \phi_{1}$. By Theorem \ref{thm2.1}, $u_{n}$ changes sign.
Without loss of generality, we can assume that $u=+\phi_{1}$, then
\begin{equation}
\label{eq:b12}
|\{x;u_{n}<0\}|\to 0.
\end{equation}
Testing (\ref{eq:b11}) with $u_{n}^{-}$, we get
\[\int_{\Omega} |\nabla u_{n}^{-}|^p -\int_{\Omega}\mu a(x) u_{n}^{-^p}=\int_{\Omega}\lambda_{n} V(x)u_{n}^{-^p}\]
By Hardy-Sobolev and Sobolev inequalities, we get
\[C_{1}\|u_{n}\|_{1,p}^p\le C \int_{\Omega^{-}}V(x)|u_{n}|^p\le C\|V\|_{s} \|u_{n}\|_{p^{*}}^p |\Omega_{n}^{-}|^{\gamma}\le C_{3}\|u_{n}\|_{1,p}^p |\Omega_{n}^{-}|^{\gamma}\|V\|_{s},\]
for some positive $\gamma >0$. This implies that
\[ |\Omega_{n}^{-}|\ge C_{4}^{1/\gamma}, \quad \Omega_{n}^{-}=\{x\in \Omega; u_{n}<0\}.\]
This contradicts (\ref{eq:b12}).

Next we prove the estimate (\ref{eq:t2}). Assume that $v>0$ in $O$, the
case $v<0$ being treated similarly. We observe by Lemma \ref{lm4.2}, that
$v\vline_{O}\in W^{1,p}_{0}(O)$. Hence the function defined as
$w(x)=v(x)$ if $x\in O$ and $w(x)=0$ if $x\in \Omega\backslash O$
belongs to $W^{1,p}_{0}(\Omega)$. Using $w$ as test function in the
equation satisfied by $v$, we find
\[\int_{O}|\nabla v|^pdx-\int_{\Omega}\mu a(x)|v|^pdx
=\lambda \int_{O}V|v|^pdx
\le \lambda \|V\|_{s}\|v\|_{p^{*},O}|O|^{\frac{p^{*}-s'p}{s'p^{*}}}
\]
by Holder inequality. On the other hand by Sobolev and Hardy-Sobolev
inequalities we have that
$\int_{O}|\nabla v|^pdx \ge C \|v\|_{p^{*},O}^p$ for some constant
 $C=C(N,p)$. Hence
\[C\le \lambda \|V\|_{s}|O|^{\frac{p^{*}-s'p}{s'p^{*}}}\]
\quad\hfill$\Box$

\begin{corollary}
 Each eigenfunction has a finite number of nodal domains.
\end{corollary}

\paragraph{Proof:}
Let $O_{j}$ be a nodal domain of an eigenfunction
associated to some positive eigenvalue $\lambda$. It follows from
(\ref{eq:t2}) that
\[|\Omega|\ge \sum_{j}|O_{j}|\ge (C\lambda \|V\|_{s})^{-\gamma}\sum_{j} 1
\]
and the proof follows.

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\noindent\textsc{Konijeti Sreenadh}\\
 Department of Mathematics\\
 Indian Institute of Technology\\
 Kanpur 208016, India.\\
e-mail: snadh@iitk.ac.in
\end{document}