\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 64, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2011/64\hfil Weighted eigenvalue problems]
{Weighted eigenvalue problems for the $p$-Laplacian with
weights in weak Lebesgue spaces}
\author[T. V. Anoop\hfil EJDE-2011/64\hfilneg]
{T. V. Anoop} % in alphabetical order
\address{T. V. Anoop \newline
The Institute of Mathematical Sciences \\
Chennai 600113, India}
\email{tvanoop@imsc.res.in}
\thanks{Submitted November 11, 2011. Published May 17, 2011.}
\subjclass[2000]{35J92, 35P30, 35A15}
\keywords{Lorentz spaces; principal eigenvalue; radial symmetry;
\hfill\break\indent Ljusternik-Schnirelmann theory}
\begin{abstract}
We consider the nonlinear eigenvalue problem
\[
-\Delta_p u= \lambda g |u|^{p-2}u,\quad
u\in \mathcal{D}^{1,p}_0(\Omega)
\]
where $\Delta_p$ is the p-Laplacian operator, $\Omega$ is
a connected domain in $\mathbb{R}^N$ with $N>p$ and the weight
function $g$ is locally integrable. We obtain the existence
of a unique positive principal eigenvalue for $g$ such
that $g^+$ lies in certain subspace of weak-$L^{N/p}(\Omega)$.
The radial symmetry of the first eigenfunctions are obtained for
radial $g$, when $\Omega$ is a ball centered at the origin or
$\mathbb{R}^N$. The existence of an infinite set of eigenvalues
is proved using the Ljusternik-Schnirelmann theory on
$\mathcal{C}^1$ manifolds.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Porposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks
\section{Introduction}
For given $N\geq 2$, $1
1$,
when $g\equiv 1$ and the domain $\Omega$ bounded.
Later, Azorero and Alonso \cite{Azorero} identified infinitely
many eigenvalues of \eqref{eq1}, for $p\not =2$,
using the Ljusternik-Schnirelmann type minmax theorem.
Many authors have given sufficient conditions on $g$ for the
existence of a positive principal eigenvalue for \eqref{eq1},
when $\Omega=\mathbb{R}^N$, for example Brown et. al. \cite{BroCosFle}
and Allegretto \cite{All1} for $p=2$, Huang \cite{Huang},
Allegretto and Huang \cite{AllHua1} for the respective
generalization to $p\neq 2$. Fleckinger et al. \cite{FleGosThe},
studied the problem \eqref{eq1} for general $p$. All these earlier
results assume that either $g$ or $g^+$ should be in
$L^{N/p}(\mathbb{R}^N)$. In \cite{SWi},
Willem and Szulkin enlarged the class of weight functions
beyond the Lebesgue space $L^{N/p}(\mathbb{R}^N)$.
They obtained the existence of positive principal eigenvalue,
even for the weights whose positive part has a faster decay
than $1/|x|^p$ at infinity and at all the points in
the domain (see \eqref{eq:SzulinWillem}).
For $p=2$, there are some results available for the weights in
Lorentz spaces, for example, Visciglia in \cite{Visciglia}
looked at \eqref{eq1} in the context of generalized Hardy-Sobolev
inequality for the positive weights in certain Lorentz spaces.
Following this direction, Mythily and Marcello in \cite{ML-MR}
showed the existence of a unique positive principal eigenvalue
for \eqref{eq1}, when $g$ is in certain Lorentz spaces.
Anoop, Lucia and Ramaswamy \cite{ALM} unified the sufficient
conditions given in \cite{All1,BroCosFle,ML-MR,SWi} by showing
the existence of a positive principal eigenvalue for \eqref{eq1},
when $g^+$ lies in a suitable subspace of
weak-$L^{\frac{N}{2}}(\Omega)$. In this paper we obtain an analogous
result that unify the sufficient conditions given
in \cite{AllHua1,Huang,FleGosThe,SWi} for the existence of a
positive eigenvalue for \eqref{eq1} by considering weights
in a suitable subspace of the weak- $L^{N/p}(\Omega)$.
For $p=2$, the existence of a positive principal eigenvalue for
more general positive weights is obtained in \cite{Rozen} using
certain capacity conditions of Maz\textquoteright ja \cite{Mazja}
and in \cite{Te} using the concentration compactness lemma.
However, their eigenfunctions are only a distributional solutions
of \eqref{sol1} and the first eigenvalue lacks certain qualitative
properties. Indeed, here we obtain a unique positive principal
eigenvalue and an infinite set of eigenvalues for \eqref{eq1} for
the weights in a suitable subspace of the Lorentz space
$L({\frac{N}{p}},\infty)$.
Here we fix the solution space as $\mathcal{D}^{1,p}_0(\Omega)$, which fits very well with
the weak formulation of boundary value problems in the unbounded
domains. Furthermore, when $1
0 \}.
\end{equation}
Let \begin{gather}
M:=\{ u \, \in \mathcal{D}^{1,p}_0(\Omega):
\int_\Omega g |u|^p =1 \}, \\
J(u) := \frac{1}{p} \int_{\Omega}|\nabla u|^p
\end{gather}
If $R$ is $\mathcal{C}^1$, then we arrive at \eqref{eq1} as the
Euler-Lagrange equation corresponding to the critical
points of $R$ on $\mathcal{D}^+(g)$, with the critical values as
the eigenvalues of \eqref{eq1}. Moreover, there is a one
to one correspondence between the critical points of $R$
over $\mathcal{D}^+(g)$ and the critical points of $J$ over $M$.
Thus we look for the sufficient conditions on $g^+$ for the
existence of a critical points of $J$ on $M$.
As in \cite{ALM}, here we consider the space
\[
\mathcal{F}_{N/p}:= \text{closure of $\mathcal{C}_c^{\infty}(\Omega)$ in $L(N/p,\infty)$}
\]
Now we state one of our main results.
\begin{theorem}\label{Exist}
Let $\Omega$ be an open connected subset of $\mathbb{R}^N$
with $p\in(1,N)$ . Let $g\in L^1_{\rm loc}(\Omega)$ be
such that $g^+\in \mathcal{F}_{N/p}\setminus\{0\}$. Then
\begin{equation}
\lambda_1=\inf\{J(u):u\in M \}
\end{equation}
is the unique positive principal eigenvalue of \eqref{eq1}.
Furthermore, all the eigenfunctions corresponding to
$\lambda_1$ are of the constant sign and $\lambda_1$ is simple.
\end{theorem}
Note that $g^-$ is only locally integrable and hence the
map $G$ defined as $$G(u)=\int_{\Omega}g |u|^p$$ may not
even be continuous and hence $M$ may not even be closed in $\mathcal{D}^{1,p}_0(\Omega)$.
Nevertheless, we show that the weak limit of a minimizing
sequence of $J$ on $M$ lies in $M$.
In general the eigenfunctions are only in $W^{1,p}_{\rm
loc}(\Omega)$ and hence the classical tools for proving the
qualitative properties of $\lambda_1$ are not applicable, as they
require more regularity for the eigenfunctions. However,
Kawohl, Lucia and Prashanth \cite{KPL} developed a weaker version
of strong maximum principle for quasilinear operator analogous to
the result in \cite{BrezisPonce}.
Further, we discuss the sufficient conditions on $g$ for the
radial symmetry of the eigenfunctions corresponding $\lambda_1$,
when $\Omega$ is a ball centered at origin or $\mathbb{R}^N$. This
generalizes the result of Bhattacharya \cite{Bhattacharya}, who
proved the radial symmetry of the first eigenfunctions of
\eqref{eq1}, when $\Omega$ is a ball centered at origin and
$g\equiv 1$.
\begin{theorem}\label{radial}
Let $\Omega$ be a ball centered at origin or $\mathbb{R}^N$.
Let $g$ be nonnegative, radial and radially decreasing
measurable function. If $\lambda_1$ is an eigenvalue of \eqref{eq1},
then any positive eigenfunction corresponding to $\lambda_1$ is
radial and radially decreasing.
\end{theorem}
A sufficient condition on $g$, for the existence of infinitely
many eigenvalues of \eqref{eq1} is also discussed here. Let us
point out that a complete description of the set of all
eigenvalues of $p$-Laplacian is widely open for $p\neq 2$. The
question of discreteness, countability of the set of all
eigenvalues of $p$-Laplacian is not known, even in the simplest
case: $g\equiv 1$ and $\Omega$ is a ball. However there are
several methods that exhibit infinite number of eigenvalues goes
to infinity. For $p\neq 2$, the existence of infinitely many
eigenvalues is obtained in \cite{AllHua1,Huang, SWi}, using the
Ljusternik-Schnirelmann minimax theorem. In this direction we have
the following result under certain weaker assumptions on $g^+$.
\begin{theorem}\label{infinite}
Let $\Omega$ be an open connected subset of $\mathbb{R}^N$ with
$p\in(1,N)$ . Let $g\in L^1_{\rm loc}(\Omega)$ be such that
$g^+\in \mathcal{F}_{N/p}\setminus \{0 \}$. Then
\eqref{eq1} admits a sequence of positive eigenvalues going
to $\infty$.
\end{theorem}
The classical Ljusternik-Schnirelmann minimax theorem requires
a deformation homotopy that is available when $M$ is at least
a $\mathcal{C}^{1,1}$ manifold(i.e, transition maps are $\mathcal{C}^1$ and its
derivative is locally Lipschitz). The set $M$ that we are considering
here is $\mathcal{C}^1$ but generally not $\mathcal{C}^{1,1}$.
Szulkin \cite{Szulkin} developed the Ljusternik-Schnirelmann theorem
on $\mathcal{C}^1$ manifold using the Ekeland variational principle. We
use Szulkin's result to obtain an increasing sequence of
positive eigenvalues of \eqref{eq1} that going to infinity.
This paper is organized as follows. In Section 2, we recall
certain basic properties of the symmetric rearrangement of a
function and the Lorentz spaces. Section 3 deals with several
characterizations of the spaces $\mathcal{F}_d, d>1$. The examples of
functions belonging to $\mathcal{F}_{N/p}$ are also given in
Section 3. In Section 4, we present a proof of the existence and
other qualitative properties of the first eigenvalue like,
simplicity, uniqueness. The radial symmetry of the eigenfunctions
corresponding to $\lambda_1$ is discussed in Section 4. In
section 5, we discuss the Ljusternik-Schirelmann theory on $\mathcal{C}^1$
Banach manifold and give a proof for the existence of infinitely
many eigenvalues of \eqref{eq1}. Further extensions and the
applications of weighted eigenvalue problems for the $p$-Laplacian
are indicated in Section 6.
\section{Prerequisites}
\subsection{Symmetrization}
First, we recall the definition of the symmetrization of a function
and its properties. Then we state certain rearrangement inequalities
needed for the subsequent sections, for more details on
symmetrization we refer to \cite{Lieb,kesh1,EdEv}.
Let $\Omega$ be a domain in $\mathbb{R}^N$.
Given a measurable function $f$ on $\Omega$, we define
distribution function $\alpha_f$ and decreasing rearrangement
$f^*$ of $f$ as below
\begin{equation}
\alpha_f(s) : =
\big\vert \{x \in \Omega : |f(x)| >s\} \big\vert,
\quad
f^*(t) : =\inf \{s>0 : \alpha_f(s)\leq t \}.
\label{2.1}
\end{equation}
In the following proposition we summarize some useful properties
of distribution and rearrangements.
\begin{proposition}\label{properties-f*}
Let $\Omega$ be a domain and $f$ be a measurable function on $\Omega$.
Then
\begin{itemize}
\item[(i)]
$\alpha_f, f^*$ are nonnegative, decreasing and right continuous.
\item[(ii)]
$f^* (\alpha_f(s_0))\leq s_0$, $\alpha_f (f^*(t_0))\leq t_0$;
\item[(iii)] $f^*(t)\leq s$ if and only if $\alpha_f(s)\leq t$,
\item[(iv)] $f$ and $f^*$ are equimeasurable; i.e,
$\alpha_f(s)=\alpha_{f^*}(s)$ for all $s>0$.
\item[(v)] Let $c,s,t>0$ such that $c=s t^{1/p}$. Then
\begin{equation}\label{relation1}
t^{1/p}f^*(t) \leq c \quad \text{if and only if} \quad
s(\alpha_f(s))^{1/p} \leq c.
\end{equation}
\end{itemize}
\end{proposition}
\begin{proof}
For a proof of (i), (ii) and (iii),
see \cite[Propositions 3.2.2 and 3.2.3]{EdEv}.
Item (iv) follows from (iii) as follows
\[
\alpha_{f^*}(s)= |\{t:f^*(t)> s\}|= |\{t:t<\alpha_f(s)\}|=\alpha_f(s).
\]
(v) Taking $s=c t^{\frac{-1}{p}}$ in (iii) one deduces that
\[
t^{1/p}f^*(t) \leq c \quad \text{if and only if} \quad
\alpha_f(s) \leq t .
\]
Now as $t=(c/s)^p$, we obtain
\[
\alpha_f(s) \leq t \quad \text{if and only if} \quad
s (\alpha_f(s))^{1/p}\leq c.
\]
\end{proof}
Next we define Schwarz symmetrization of measurable sets and functions,
see \cite{Lieb} for more details.
\begin{definition} \label{def2.2} \rm
Let $A\subset \mathbb{R}^N$ be a Borel measurable set of finite
measure. We define $A_*$, the symmetric rearrangement of the
set $A$, to be the open ball centered at origin having the
same measure that of $A$. Thus
\[
A_*=\{x:|x|0$.
Then we define the \emph{symmetric decreasing rearrangement}
$f_*$ of $f$ on $\Omega_*$ as
\[
f_*(x)= \int_0^\infty \chi_{\{{|f|>s\}}_*}(x)ds
\]
Next we list a few inequalities concerning $f_*$ that we use
for proving the radial symmetry of the eigenfunctions
corresponding to the first eigenvalue. For a proof see
\cite[Section 3.3]{Lieb}.
\begin{proposition}\label{symmetric}
Let $\Omega$ be a ball centered at origin or $\mathbb{R}^N$.
Let $f$ be a nonnegative measurable function on $\Omega$
such that $\alpha_f(s) <\infty$ for each $s>0$.
\begin{itemize}
\item[(a)] If $f$ is radial and radially decreasing then $f=f_*$ a.e.
\item[(a)] Let $F: \mathbb{R}^+ \to \mathbb{R}$ be a nonnegative
Borel measurable function. Then
$$
\int_{\mathbb{R}^N} F(f_*(x))dx=\int_{\mathbb{R}^N} F(f(x))dx.
$$
\item[(b)] If $\Phi:\mathbb{R}^+ \to \mathbb{R}$ is nonnegative
and nondecreasing then
$$
(\Phi\circ f)_*=\Phi \circ f_* \quad \text{a.e.}
$$
\end{itemize}
\end{proposition}
\subsection{Lorentz Spaces}
In this section, we recall the definition and the main properties
of the Lorentz spaces. For more details on Lorentz spaces see
\cite{Adams,EdEv,RH}.
Given a measurable function $f$ and $p,q \in [1,\infty]$, we set
$$
\|f\|_{(p,q)} := \|t^{\frac{1}{p}-\frac{1}{q}} f^* (t)
\|_{q;(0,\infty)}
$$
and the Lorentz spaces are defined by
$ L(p,q) := \{ f : \|f\|_{(p,q)}<\infty \}$.
In particular for $q=\infty$, we obtain
$$
\|f\|_{(p,\infty)}=\sup_{t>0}t^{1/p}f^*(t).
$$
For $p> 1$, the weak-$L^p$ space is defined as
\[
\text{weak-}L^p:=\{ f:\sup_{s>0}s(\alpha_f(s))^{1/p}<\infty \}.
\]
The following lemma identifies the Lorentz space
$L(p,\infty)$ with the weak-$L^p$ space.
\begin{lemma} \label{equivsup}
Let $\Omega$ be a domain in $\mathbb{R}^N$ and $f$ be a measurable
function on $\Omega$. For each $ p>1$, we have
$$
\sup_{t>0}t^{1/p}f^*(t)=\sup_{s>0}s(\alpha_f(s))^{1/p}.
$$
\end{lemma}
\begin{proof}
Let \begin{equation} \label{relation0}
c_1=\sup_{t>0}t^{1/p}f^*(t),\quad
c_2=\sup_{s>0}s(\alpha_f(s))^{1/p}.
\end{equation}
Without loss of generality we may assume that $c_1$ is finite.
Now for $s>0$, take ${t}=(\frac{c_1}{{s}})^p$.
Thus ${t}^{1/p}f^*({t})\leq c_1$. Now by taking
$c=c_1$ in \eqref{relation1}, with $c_1= s t^\frac{1}{p}$,
one can deduce that ${s} (\alpha_f({s}))^{1/p}\leq c_1$, for all
$s>0$. Hence $c_2\leq c_1$. The other way inequality follows
in a similar way.
\end{proof}
The functional $\| \cdot \|_{(p,q)}$ is not a norm on $L(p,q)$.
To obtain a norm, we set
$f^{**}(t):=\frac{1}{t}\int^t_0 f^* (r) dr$
and define
$$
\|f\|^*_{(p,q)} := \|t^{\frac{1}{p}-\frac{1}{q}} f^{**} (t)
\|_{q\,;\,(0,\infty)},\quad \text{for } 1\leq p, q \leq \infty.
$$
For $p>1$, the functional $\| \cdot \|^*_{(p,q)}$ defines a norm
in $L(p,q)$ equivalent to $\|.\|_{(p,q)}$
(see \cite[Lemma 3.4.6]{EdEv}). Endowed with this norm $L(p,q)$
is a Banach space, for $p,q\geq 1$.
In the following proposition we summarize some of the properties
of $L(p,q)$ spaces, see \cite{EdEv,RH} for the proofs.
\begin{proposition}\label{propertiesLorentz}
\begin{itemize}
\item[(i)]
If $p\,>\,0$ and $q_2 \,\geq \, q_1 \, \geq 1$, then
$L(p,q_1) \hookrightarrow L(p, q_2)$
\item[(ii)] If $ p_2\,>p_1\,\geq 1$ and $q_1,\,q_2\geq 1$, then
$L(p_2,q_2)\hookrightarrow L_{\rm loc}(p_1,q_1)$.
\item[(iii)]
H\"{o}lder inequality:
Given $(f,g) \in L(p_1, q_1)\times L(p_2, q_2)$ and $(p,q) \in (1,\infty)\allowbreak\times [1,\infty]$
such that $1/p = 1/p_1 + 1/p_2$, $\, 1/q \leq 1/q_1+ 1/q_2$, then
\begin{equation} \label{Holder}
\|f g\|_{(p,q)} \leq C \|f\|_{(p_1, q_1)} \;\;\|g\|_{(p_2,q_2)},
\end{equation}
where $C$ depends only on $p$.
\item[(iv)]
Let $(p,q) \in (1, \infty) \times (1, \infty)$.
Then the dual space of $L(p,q)$ is isomorphic to $L(p' , q')$
where $ 1/p + 1/p' =1$ and $1/q + 1/q' =1$.
\item[(v)] Let $\gamma>0$. Then
\begin{equation} \label{eq4}
\big\| |f|^\gamma \big\|_{(p,q)}
=\| f \|^{\gamma}_{(\frac{p}{\gamma},\frac{q}{\gamma})}
\end{equation}
\end{itemize}
\end{proposition}
As mentioned before the main interest of
considering the Lorentz spaces is that the usual Sobolev embedding,
the embedding of $\mathcal{D}^{1,p}_0(\Omega)$ in to $L^{p^{*}} (\Omega)$,
can be improved as below (see for example, appendix in \cite{ALT}):
\begin{proposition}[Lorentz-Sobolev embedding]\label{prop:Sobolev}
We have $\mathcal{D}^{1,p}_0(\Omega) \hookrightarrow L(p^*, p)$;\\ i.e.,
there exists $C>0$ such that
$$
\| u\|_{(p^*,\,p)} \leq C \| \nabla u\|_{p},\quad \forall \;u \in \mathcal{D}^{1,p}_0(\Omega).
$$
\end{proposition}
\section{The function space $\mathcal{F}_d$}
For $(d,q) \in[1,\infty)\times [1, \infty)$,
$C_c^{\infty} (\Omega)$ is dense in
the Banach space $L(d, q)$. However, the closure of
$C_c^{\infty} (\Omega)$ in
$L(d, \infty)$ is a closed proper sub space of $L(d, \infty)$
that will henceforth be denoted by
$$
\mathcal{F}_d := \overline{C_c^{\infty} (\Omega)}^{ \|\cdot \|_{(d,\infty)} }
\subset L(d,\infty).
$$
Next we list some of the properties of the space $\mathcal{F}_d$,
see \cite[Proposition 3.1]{ALM} for a proof.
\begin{proposition}\label{prop1}
\begin{itemize}
\item[(i)] For each $d>1$, $L(d,q) \subset \mathcal{F}_d$ when $ 1\leq q < \infty$.
\item[(ii)] For each $a \in \Omega$,
the Hardy potential $x \mapsto |x-a|^{\frac{-N}{d}}$ does not
belong to $\mathcal{F}_{d}$.
\end{itemize}
\end{proposition}
Recall that $L(d,d)=L^d(\Omega)$, hence from (i) it follows that
$L^{N/p}(\Omega)$ is contained in $F_{N/p}$.
Thus Theorem \ref{Exist} readily extends the results in
\cite{AllHua1,FleGosThe}, since $g\in L^{N/p}(\Omega)$ is a part
of their assumptions. Similarly the result in \cite{Huang}
follows as the positive part of weights he considered is bounded
and compactly supported. Note that (ii) shows that $\mathcal{F}_d$ is a
proper subspace of the Lorentz space $L(d,\infty)$.
Now we state a few useful characterizations of the space $\mathcal{F}_d$.
\begin{proposition}\label{equivalent}
The following statements are equivalent
\begin{itemize}
\item[(i)] $f \in\mathcal{F}_d$,
\item[(ii)] $f^*(t)=o(t^{-1/d})$ at 0 and $\infty$; i.e.,
\begin{equation} \label{eq:Tertikas}
\lim_{t \to 0_+} t^{1/d}f^\ast(t)
=0 =
\lim_{t \to \infty} t^{1/d}f^\ast(t).
\end{equation}
\item[(iii)] $\alpha_f(s)=o(s^{-d})$ at 0 and $\infty$; i.e.,
\begin{equation} \label{newchar}
\lim_{s \to 0_+}s (\alpha_f(s))^{1/d}
= 0 = \lim_{s \to \infty}s (\alpha_f(s))^{1/d}.
\end{equation}
\end{itemize}
\end{proposition}
\begin{proof}
(i)$\Rightarrow$(ii): See the first part of \cite[Theorem 3.3]{ALM}.
(ii)$\Rightarrow$(iii): Let (ii) hold. Thus for given $\varepsilon>0$,
there exist $t_1,t_2>0$ such that
\begin{equation}\label{relation2}
t^{1/d} f^\ast(t)<\varepsilon,\quad \forall \,t \in (0, t_1)
\cup (t_2,\infty).
\end{equation}
Let $s_1=\varepsilon\,(t_1)^{-1/d}$ and $s_2=\varepsilon\,(t_2)^{-1/d}$.
Note that
$$
\text{If $s \in (0,s_2)\cup (s_1,\infty)$, then
$t=(\frac{\varepsilon}{s})^d\in (0, t_1) \cup (t_2,\infty)$.}
$$
Now using \eqref{relation2} and \eqref{relation1} with $c=\varepsilon$,
we obtain
\[
s (\alpha_f(s))^{1/d}<\varepsilon, \quad
\forall s \in (0,s_2) \cup (s_1,\infty).
\]
This shows that $\alpha_f(s)=o(s^{-d})$ at 0 and $\infty$.
(iii)$\Rightarrow$ (i): Assume (iii).
Then for a given $\varepsilon>0$, there exist $s_1,s_2$ such that
\begin{equation}\label{eq12}
s(\alpha_f(s))^{1/d}<\varepsilon,\quad
\forall s \in(0,s_1] \cup [s_2,\infty).
\end{equation}
We use \cite[Proposition 3.2]{ALM} to show that $f$ is in $\mathcal{F}_d$.
Let
$$
A_\varepsilon:= \{x: s_1\leq f(x)< s_2\},\quad f_\varepsilon:=f \chi_{A_\varepsilon}.
$$
Note that $|A_\varepsilon|\leq \alpha_f(s_1)<\infty$ and
$f_\varepsilon\in L^\infty(\Omega)$. Let $g=f\chi_{A_\varepsilon^c}$.
Thus it is enough to prove
$$
\|f-f_\varepsilon\|_{(d,\infty)}=\|g\|_{(d,\infty)}<\varepsilon.
$$
Observe that, for $s\in (s_1,s_2)$, $ \alpha_g(s)=\alpha_f(s_2)$
and hence
\begin{equation}\label{eq11}
s(\alpha_g(s))^{1/d} < s_2(\alpha_f(s_2))^{1/d}<\varepsilon, \,\, \forall s\in (s_1,s_2).
\end{equation}
Since $|g|\leq|f|$, we have $\alpha_g(s)\leq \alpha_f(s)$, for all
$s>0$. Now by combining \eqref{eq12} and \eqref{eq11} we obtain
\[
s(\alpha_g(s))^{1/d} < \varepsilon,\quad \forall s>0.
\]
Hence by lemma \ref{equivsup} we obtain $\|g\|_{(d,\infty)}<\varepsilon$.
\end{proof}
Next we give another sufficient condition similar to a condition
of Rozenblum, see \cite[(2.19)]{Rozen},
for a function to be in $\mathcal{F}_d$.
\begin{lemma}\label{IntegrabilityCondn}
Let $h\in L({d}, \infty)$ and $h>0$. If $f$ is such that
$\int_\Omega h^{d-q}|f|^q< \infty$ for some $q\geq d$.
Then $f\in L(d,q)$ and hence in $\mathcal{F}_d$.
\end{lemma}
\begin{proof}
The result is obvious when $q=d$. For $q>d$, let
$g=h^{\frac{d}{q}-1}f$. Then the above integrability condition
yields $g\in L^q(\Omega)$. Using property \eqref{eq4} we obtain
$h^{1-\frac{d}{q}}\in L(\frac{dq}{q-d},\infty)$. Now by H\"{o}lder
inequality \eqref{Holder} we obtain $f\in L(d,q)$ and hence in
$\mathcal{F}_d$ as $L(d,q)\subset \mathcal{F}_d$.
\end{proof}
\begin{remark} \label{rmk3.4} \rm
Let $g\in L^q(\mathbb{R}^N)$ with $q\geq d$ and let
\[
f(x)= |x|^{(\frac{1}{q}-\frac{1}{d} )N}g.
\]
Then using the above lemma one can easily verify that $f\in L(d,q)$.
In general for any $h\in L(d,\infty)$ with $h>0$,
$f=g h^{1-\frac{d}{q}}\in L(d,q)$.
Thus we can obtain Lorentz spaces by interpolating Lebesgue
and weak-Lebesgue spaces suitably.
\end{remark}
Another class of functions contained in $\mathcal{F}_{N/p}$ is provided
by the work of Szulkin and Willem \cite{SWi}.
More specifically they consider the weights $g$ defined by
the conditions:
\begin{equation} \label{eq:SzulinWillem}
\begin{gathered}
g \in L^1_{\rm loc} (\Omega),
\quad
g^+=g_1+g_2\not\equiv 0,
\quad
g_1 \in L^{N/p} (\Omega), \\
\lim_{|x|\to \infty,\, x \in \Omega} |x|^p g_2(x)=0,
\quad
\lim_{x\to a ,\, x\in \overline{\Omega}} | x- a |^p g_2(x)=0
\quad \forall a \in {\overline \Omega}.
\end{gathered}
\end{equation}
The following lemma can be proved using similar arguments as
in \cite[Lemma 4.1]{ALM}.
\begin{lemma}\label{Willem}
Let $g:\Omega\to \mathbb{R}$ be a measurable function such that
\begin{equation}
(i)\lim_{|x|\to \infty,\, x \in \Omega} |x|^p g(x)=0,
\quad
(ii) \lim_{x\to a ,\, x\in \overline{\Omega}} | x- a |^p g(x)=0,
\quad \forall a \in {\overline \Omega}.
\end{equation}
Then there exist finite number of points
$a_1 , \dots, a_m \in \overline{\Omega} $ with the following
property: For every $\varepsilon>0$ there exists $R:= R(\varepsilon) >0$
such that
\begin{gather}
|g(x)| < \frac{\varepsilon}{|x|^p} \quad \text{a.e. }
x \in \Omega \setminus B(0,R) \label{eqn2.2}\\
|g(x)| < \frac{\varepsilon}{|x- a_i|^p} \quad \text{a.e. }
x \in \Omega \cap B(a_i, R^{-1}) \,, \; i=1, \dots, m, \label{eqn2.3}\\
g \in L^{\infty} ( \Omega \setminus A_{\varepsilon} ), \label{eqn2.4}
\end{gather}
where $A_{\varepsilon} := \bigcup_{i=1}^m B (a_i, R^{-1})\cap \Omega$.
\end{lemma}
\begin{theorem} \label{WillemSzulkin}
Let $g:\Omega\to \mathbb{R}$ be as in the previous lemma.
Then $g\in \mathcal{F}_{N/p}$.
\end{theorem}
\begin{proof}
We use Proposition \ref{equivalent}(iii) to show that
$g \in \mathcal{F}_{N/p}$.
For $\varepsilon >0$, let $R$ be given as in the previous lemma.
Let $s_1 :=\varepsilon R^{-p}$. We first show that
$$
s(\alpha_g(s))^{p/N}<\varepsilon,\quad \forall s s \} \big|
\leq \omega_N (\frac{\varepsilon}{s})^{N/p} \,,
$$
where $\omega_N$ is the volume of unit ball in $\mathbb{R}^N$. Thus
\begin{equation}\label{Willemeq1}
s(\alpha_g(s))^{p/N}< C_1\varepsilon,\quad \forall\, s s_2$, using \eqref{eqn2.3} the distribution function
can be estimated as follows:
\begin{align*}
\alpha_{g} (s) &=
\big| \{ x \in \Omega : |g (x)| > s \} \big|
= \big| \{ x \in A_{\varepsilon} : |g (x)| > s \} \big|\\
&\leq \sum_{i=1}^m \big| \{ x \in B (a_i, R^{-1} ) \cap \Omega
: |g (x)| > s \} \big| \\
&\leq \sum_{i=1}^m \big| \{ x \in B (a_i, R^{-1} )
: \varepsilon |x-a_i|^{-p} > s \}\big|\\
&= \sum_{i=1}^m \omega_N ( \frac{\varepsilon}{s} )^{N/p}.
\end{align*}
Therefore,
\begin{equation}\label{Willemeq2}
s (\alpha_g(s))^\frac{p}{N}
\leq C_2 \varepsilon \quad \forall s > s_2,
\end{equation}
where $C_2$ is independent of $\varepsilon$. Now proof follows
using condition (iii) of proposition \ref{equivalent} together with
\eqref{Willemeq1} and \eqref{Willemeq2}.
\end{proof}
As an immediate consequence we have the following remark.
\begin{remark} \label{rmk37} \rm
The positive part of any function satisfying
\eqref{eq:SzulinWillem} belongs to the space $\mathcal{F}_{N/p}$.
In particular Theorem \ref{Exist} summarizes the result by
Willem and Szulkin \cite{SWi}.
\end{remark}
\subsection{Examples}
Now we consider examples of weights that admit a positive
principal eigenvalue for \eqref{eq1} to understand how
the conditions \eqref{eq:SzulinWillem} and the properties
that define the space $\mathcal{F}_{N/p}$ are related to one another.
First, we consider the following functions:
\begin{gather}\label{eq:ModifiedW}
{g}_1(x)
= \frac{1}{\big( \log (2+|x|^2) \big)^{p/N}{(1+|x|^2)^{p/2}}},\\
{ g}_2 (x)
= \frac{1}{|x|^p(1+|x|^2)^{p/2} \big( \log (2+\frac{1}{|x|^2})
\big)^{p/N}}.
\end{gather}
One can verify that ${g}_1,\, { g}_2$ satisfy
\eqref{eq:SzulinWillem} and hence belong to $\mathcal{F}_{N/p}$ and none
of them lies in $L^{N/p}(\mathbb{R}^N)$.
Next we give an example of a weight which is in $\mathcal{F}_{N/p}$
but does not satisfy the condition \eqref{eq:SzulinWillem}.
\begin{example} \label{exa4} \rm
In the cube
$\Omega = \{ (x_1,\dots, x_N) \in \mathbb{R}^N : |x_i| < R \}$
with $0< R <1$
consider the function defined by
\begin{equation}\label{eq:W3}
g_3(x)= \big| x_1 \log (|x_1|) \big|^{-p/N},
\quad x_1 \ne 0.
\end{equation}
Using the condition \eqref{IntegrabilityCondn}, one can verify
that $g_3 \in L (\frac{N}{p}, q)$, for $q>\frac{N}{p}$. But $g_3$
does not satisfy~\eqref{eq:SzulinWillem}. Indeed along the curve
$ x_2 = (x_1)^{\frac{1}{2N}}$, the limit of $|x|^p g_3(x)$ is
infinity as $x$ tends to $0$ and this limit is zero as $x$ tends
to $0$ along the $x_1 $ axis. Thus $g_3$ does not satisfy the
condition~\eqref{eq:SzulinWillem}.
\end{example}
\section{Existence of an eigenvalue and its properties}
\label{Sec:existence}
In this section we prove the existence and the uniqueness of
the positive principal eigenvalue for \eqref{eq1} for $g$
for which $g^+\in \mathcal{F}_{N/p}\setminus \{0\}$.
Moreover we prove a few qualitative properties of that positive
principal eigenvalue.
\subsection{The existence of a minimizer}
We prove the existence using a direct variational principle.
First, we recall the following sets and functional:
\begin{gather*}
\mathcal{D}^+(g)= \{ u\in \mathcal{D}^{1,p}_0(\Omega):\int_{\Omega}g|u|^p>0 \},\quad
M= \{ u\in \mathcal{D}^{1,p}_0(\Omega):\int_{\Omega}g|u|^p=1 \},\\
J(u)=\frac{1}{p}\int_{\Omega}|{\nabla }u|^p,\quad G(u)
=\frac{1}{p}\int_{\Omega}g|u|^p.
\end{gather*}
From the definition of the space $\mathcal{D}^{1,p}_0(\Omega)$, it is obvious
that $J$ is coercive and weakly lower semi-continuous.
Due to the weak assumption on $g^-$, the map $G$ may not
be even continuous. However the map
$$
G^+(u):= \frac{1}{p}\int_{\Omega}g^+|u|^p
$$
is continuous and compact on $\mathcal{D}^{1,p}_0(\Omega)$.
\begin{lemma}\label{compactness}
Let $g^+\in F_{N/p}\setminus\{0\}$. Then $G^+$ is compact.
\end{lemma}
\begin{proof}
Let $\{u_n\}$ converge weakly to $u$ in $X$.
We show that $G^+(u_n)\to G^+(u)$, up to a subsequence.
% pact setapproximate $g^+$ with a $\mathcal{C}_c^\infty(\Omega)$ function in $L(\frac{N}{p}, \infty)$
For $\phi\in\mathcal{C}_c^{\infty}(\Omega)$, we have
\begin{equation}\label{eq5}
p(G^+(u_n)-G^+(u)) = \int_{\Omega}\phi\,(|u_n|^p-|u|^p)
+ \int_{\Omega}(g^+-\phi)\,(|u_n|^p-|u|^p).
\end{equation}
We estimate the second integral using the Lorentz-Sobolev
embedding and the H\"{o}lder inequality as below
\begin{equation}\label{eq7}
\int_{\Omega}|(g^+-\phi)|\,\big|(|u_n|^p-|u|^p)\big |
\leq C\|g^+-\phi\|_{(N/p,\infty)}
\big (\|u_n\|_{(p^*,p)}^p+\|u\|_{(p^*,p)}^p \big )
\end{equation}
where C is a constant which depends only on $N,p$.
Clearly $\{u_n\}$ is a bounded sequence in $L(p^*,p)$.
Let
$$
m:=\sup_{n}\{\|u_n\|_{(p^*,p)}^p+\|u\|_{(p^*,p)}^p\}.
$$
Now using the definition of the space $F_{N/p}$,
for a given $\varepsilon>0$, we choose $g_\varepsilon \in \mathcal{C}_c^{\infty}(\Omega)$
so that
$$
\|g^+-g_{\varepsilon}\|_{(N/p,\infty)}<\frac{p \;\varepsilon}{2 m C}.
$$
Thus by taking $\phi=g_\varepsilon$ in \eqref{eq7} we obtain
\[
\int_{\Omega}|(g^+-g_\varepsilon)|\,\big|(|u_n|^p-|u|^p)\big |
<\frac{p\;\varepsilon}{2}
\]
Since $X\hookrightarrow L^p_{\rm loc}(\Omega)$ compactly,
the first integral in \eqref{eq5} can be made arbitrary small
for large $n$. Thus we choose $n_0\in \mathbb{N}$ so that
\[
\int_{\Omega}g_\varepsilon(|u_n|^p-|u|^p)<\frac{p\varepsilon}{2},\quad
\forall n>n_0.
\]
Hence $|G^+(u_n)-G^+(u)|<\varepsilon$, for $n>n_0$.
\end{proof}
Now we are in a position to prove the existence of a minimizer
for $J$ on $M$.
\begin{theorem}\label{Existence}
Let $\Omega$ be a domain in $\mathbb{R}^N$ with $N>p$.
Let $g\in L^1_{\rm loc}(\Omega)$ and $g^+\in \mathcal{F}_{N/p} \setminus \{0\}$.
Then $J$ admits a minimizer on $M$.
\end{theorem}
\begin{proof}
Since $g \in L^1_{\rm loc} (\Omega)$ and $g^{+} \neq 0$, there
exists $\varphi \in \mathcal{C}_c^{\infty} (\Omega)$ such that
$\int_{\Omega} g|\varphi|^p >0$ (see for example,
\cite[Proposition 4.2]{KPL}) and hence $M\neq \emptyset $. Let
$\{u_n\}$ be a minimizing sequence of $J$ on $M$; i.e.,
$$
\lim_{n \to \infty} J(u_n)=\lambda_1:=\inf_{u\in M}J(u).
$$
By the coercivity of $J,\{u_n\}$ is bounded in $\mathcal{D}^{1,p}_0(\Omega)$ and hence
using the reflexivity of $\mathcal{D}^{1,p}_0(\Omega)$ we obtain a subsequence of
$\{u_n\}$ that converges weakly.
We denote the weak limit by $u$ and the subsequence by $\{u_n\}$
itself. Now using the compactness of $G^+$, we obtain
$$
\lim_{n\to\infty}\int_{\Omega}g^+ |u_n|^p
= \int_{\Omega}g^+ |u|^p.
$$
Now as $u_n\in M$ we write,
$$
\int_{\Omega}g^- |u_n|^p=\int_{\Omega}g^+ |u_n|^p-1
$$
Since the embedding $\mathcal{D}^{1,p}_0(\Omega) \hookrightarrow L^p_{\rm loc}(\Omega)$
is compact, up to a subsequence $u_n\to u$ a.e. in $\Omega$.
Hence by applying Fatou's lemma,
$$
\int_{\Omega}g^- |u|^p\leq\int_{\Omega}g^+ |u|^p-1,
$$
which shows that $\int_{\Omega}g |u|^p\geq 1$. Setting
${\widetilde u} := u/(\int_{\Omega}g |u|^p)^{1/p}$,
the weak lower semi continuity of $J$ yields
\begin{align*}
\lambda_1\leq J(\widetilde u)
=\frac{J(u)}{\int_{\Omega}g |u|^p}\leq J(u)\leq \liminf_{n}J(u_n)
= \lambda_1
\end{align*}
Thus the equality must hold at each step and hence
$\int_{\Omega}g |u|^p = 1$, which shows that $u\in M$ and
$ J (u)= \lambda_1$.
\end{proof}
Note that $R$ is not sufficiently regular to conclude that $u$
is an eigenfunction of \eqref{sol1} corresponding to $\lambda_1$,
using critical point theory.
\begin{proposition}\label{Mineigen}
Let $u$ be a minimizer of $R$ on $\mathcal{D}^+(g)$.
Then $u$ is an eigenfunction of \eqref{eq1}
\end{proposition}
\begin{proof}
For each $\phi \in \mathcal{C}_c^\infty(\Omega)$, using dominated
convergence theorem one can verify that $R$
admits directional derivative along $\phi$.
Now since $u$ is a minimizer of $J$ on $\mathcal{D}^+(g)$ we obtain
$$
\frac{d}{dt}R(u+t\phi)\vert_{t=0}=0.
$$
Therefore,
$$
\int_\Omega |{\nabla }u| ^{p-2} {\nabla }u\cdot{\nabla }\phi
= \lambda_1 \int_\Omega g\, |u|^{p-2} u\, \phi, \quad
\forall \,\phi \in \mathcal{C}_c^\infty(\Omega).
$$
Now we use the density of $\mathcal{C}_c^\infty(\Omega)$ in $\mathcal{D}^{1,p}_0(\Omega)$ to
conclude that
$$
\int_\Omega |{\nabla }u| ^{p-2} {\nabla }u\cdot{\nabla }v
= \lambda_1 \int_\Omega g\, |u|^{p-2} u\, v, \quad \forall \,v \in \mathcal{D}^{1,p}_0(\Omega).
$$
\end{proof}
\subsection{Qualitative properties of $\lambda_1$}
First we prove that the eigenfunctions corresponding to
$\lambda_1$ are of constant sign. Since the eigenfunctions are
not regular enough, the classical strong maximum principle is not
applicable here. In \cite{ALM}, for $p=2$, we use a strong maximum
principle due to Brezis and Ponce \cite{BrezisPonce} to show that
first eigenfunctions are of constant sign. A similar strong
maximum principle is obtained in \cite{KPL}, for quasilinear
operators. From \cite[Proposition 3.2]{KPL} we have the following
lemma.
\begin{lemma}[Strong Maximum principle for $\Delta_p$]\label{StMax}
Let $u\in\mathcal{D}^{1,p}_0(\Omega)$, $V\in L^1_{\rm loc}(\Omega)$ be such that
$u, V\geq 0$ a.e in $\Omega$. If $V|u|^{p-1}\in L^1_{\rm loc}(\Omega)$
and $u$ satisfies the following differential inequality( in the
sense of the distributions)
\[
-\Delta_p(u)+V(x)u^{p-1}\geq 0 \quad \text{in } \Omega,
\]
then either $u\equiv 0$ or $u>0$ a.e.
\end{lemma}
Now using the above lemma we prove the following result.
\begin{lemma}\label{Constantsign}
The eigenfunctions of \eqref{eq1} corresponding to $\lambda_1$
are of constant sign.
\end{lemma}
\begin{proof}
It is clear that the eigenfunctions corresponding to $\lambda_1$
are the minimizers of $R_p$ on $\mathcal{D}_p^+(g)$.
Let $u$ be a minimizer of $R_p$ on $\mathcal{D}_p^+(g)$.
Since $u\neq 0$ either $u^+$ or $u^-$ is non zero.
Without loss of generality we may assume that $u^+\neq 0$.
Now by taking $u^+$ as a test function in \eqref{sol1},
we see that $u^+$ also minimizes $R_p$ on $\mathcal{D}_p^+(g)$.
Thus by Proposition \ref{Mineigen}, $u^+$ also solves \eqref{eq1}
in the weak sense,
\[
-\Delta_p u^{+}-\lambda_1 g {(u^{+})}^{p-1}=0, \quad \text{in } \Omega.
\]
In particular, we have the following differential inequality in
the sense of distributions:
$$
-\Delta_p u^{+}+\lambda_1 g^-{(u^{+})}^{p-1}
= \lambda_1 g^+{(u^{+})}^{p-1}\geq0, \quad \text{in } \Omega.
$$
It is clear that $g^-$ and $u^+$ satisfy all the assumptions of
Lemma \ref{StMax}, provided $g^-(u^+)^{p} \in L^1_{\rm
loc}(\Omega)$. Since $g|u|^p\in L^1(\Omega)$, we have
$(g^-)^{1/q}(u^+)^{p-1} \in L^q(\Omega)$, where $q$ is the
conjugate exponent of $p$. Further,
$(g^-)^{1/p}\in L^p_{\rm loc}(\Omega)$, since
$g\in L^1_{\rm loc}(\Omega)$.
Let us write
$$
g^-{(u^{+})}^{p-1}=(g^-)^{1/p}(g^-)^{1/q}(u^+)^{p-1}.
$$
Now we use H\"{o}lder inequality to conclude that
$g^-{(u^{+})}^{p-1}\in L^1_{\rm loc}(\Omega)$. Now in view of
Lemma \ref{StMax} we obtain $u^+>0$ a.e. and hence $u=u^+$.
Moreover, the zero set of $u$ is of measure zero.
\end{proof}
Indeed, the above lemma shows that $\lambda_1$ is a principal
eigenvalue of \eqref{eq1}.
Next we prove the uniqueness of the positive principal eigenvalue,
using the Picone's identity for the p-Laplacian.
In \cite{AllHua2}, Picone's identity is proved for $\mathcal{C}^1$ functions.
However it is not hard to obtain a similar identity for
less regular functions.
\begin{lemma}[Picone's identity] \label{picone}
Let $u\geq 0, v>0$ a.e. and let $|\nabla v|,|\nabla u|$
exist as measurable functions. Then the following identity holds a.e.
\begin{align*}
&|\nabla u|^p +(p-1)\frac{u^p}{v^p}|\nabla v|^p
-p\frac{u^{p-1}}{v^{p-1}}|\nabla v|^{p-2}\nabla v \\
&= |\nabla u|^p - \nabla (\frac{u^p}{v^{p-1}})\cdot|\nabla v|^{p-2}
\nabla v.
\end{align*}
Further, the left hand side of the above identity is nonnegative.
\end{lemma}
Now we prove the uniqueness of the positive principal eigenvalue.
\begin{lemma}\label{principal}
Let $g\in L(N/p,\infty)$ and let $\lambda>0$ be a positive principal eigenvalue of \eqref{eq1}. Then
\[
\lambda=\lambda_1=\inf\{\int_\Omega|\nabla u |^p: u \in M \}.
\]
\end{lemma}
\begin{proof}
Let $v\in \mathcal{D}^{1,p}_0(\Omega)$ be a positive eigenfunction of \eqref{eq1}
corresponding to $\lambda$. Let $u\in M$ and let $\{\phi_n\}$
in $\mathcal{C}_c^\infty(\Omega)$ be such that
$\|u-\phi_n\|_{\mathcal{D}^{1,p}_0(\Omega)}\to 0$ and $\int_{\Omega}g |u|^p=1$.
Note that $\frac{|\phi_n|^p}{v+\varepsilon}\in \mathcal{D}^{1,p}_0(\Omega)$.
Thus by the Picone's identity (see Lemma \ref{picone}),
we have
\begin{equation}\label{principal:eqn1}
0\leq\int_\Omega|\nabla\phi_n|^p-\int_\Omega|\nabla v|^{p-2}
\nabla v\cdot \nabla \big(\frac{|\phi_n|^p}{(v+\varepsilon)^{p-1}}\big).
\end{equation}
Since $v$ is an eigenfunction of \eqref{eq1} corresponding
to $\lambda$, we have
\begin{equation} \label{principal:eqn2}
\int_\Omega|\nabla v|^{p-2} \nabla v\cdot \nabla
\Big(\frac{\phi_n^p}{(v+\varepsilon)^{p-1}}\Big)
=\lambda\int_\Omega gv^{p-1}\frac{|\phi_n|^p}{(v+\varepsilon)^{p-1}}.
\end{equation}
Now from \eqref{principal:eqn1} and \eqref{principal:eqn2} we
\begin{equation}
0\leq \int_\Omega|\nabla\phi_n|^p
-\lambda\int_\Omega gv^{p-1}\frac{|\phi_n|^p}{(v+\varepsilon)^{p-1}}.
\end{equation}
By letting $\varepsilon \to 0$, the dominated convergence theorem
yields
\[
0\leq \int_\Omega |\nabla\phi_n|^p-\lambda\int_\Omega g |\phi_n|^p.
\]
Now we let $n \to \infty$ to obtain the inequality
\[
0\leq \int_\Omega|\nabla u|^p-\lambda\int_\Omega {g}u^p.
\]
Therefore,
\begin{equation}
\lambda \leq \int_\Omega|\nabla u|^p, \quad \; \forall\, u\in M.
\end{equation}
This completes the proof.
\end{proof}
\begin{remark} \label{remark:principal} \rm
Using Lemma \ref{Constantsign}, we see that $\lambda_1$ is a
positive principal eigenvalue and Lemma \ref{principal} shows
that $\lambda_1$ is the unique positive principal eigenvalue
of \eqref{eq1}. In particular, the eigenfunctions corresponding
to other eigenvalues of \eqref{eq1} must change sign.
\end{remark}
When $\Omega$ is connected, for the simplicity of $\lambda_1$,
we refer to \cite[Theorem 1.3]{KPL}. There, the authors obtained
the simplicity of the first eigenvalue of \eqref{eq1},
if it exists, even for $g$ in $L^1_{\rm loc}(\Omega)$.
\subsection{Radial symmetry of the eigenfunctions}
Now we give sufficient conditions for the radial symmetry
of the eigenfunctions corresponding to the eigenvalue $\lambda_1$
of \eqref{eq1}. Here we assume that the domain $\Omega$
is a ball centered at origin or $\mathbb{R}^N$. Bhattacharya
\cite{Bhattacharya} proved the radial symmetry of the first
eigenfunctions of \eqref{eq1}, when $g\equiv 1$ and $\Omega$ is
ball.
Here we prove that all the positive eigenfunctions corresponding
to $\lambda_1$ are radial and radially decreasing, provided $g$
is nonnegative, radial and radially decreasing. Thus our result
is a two fold generalization of results of Bhattacharya, as we
allow more general weight functions and the domain can be
$\mathbb{R}^N$. Our result uses certain rearrangement
inequalities. We emphasize that here we are not assuming any
conditions on $g$ that ensures $\lambda_1$ is an eigenvalue.
\begin{theorem}
Let $\Omega$ be a ball centered at origin or $\mathbb{R}^N$.
Let $g$ be nonnegative, radial and radially decreasing measurable
function. If $\lambda_1$ is an eigenvalue of \eqref{eq1},
then any positive eigenfunction corresponding to $\lambda_1$
is radial and radially decreasing.
\end{theorem}
\begin{proof}
Let $u$ be a positive eigenfunction of \eqref{eq1} corresponding
to $\lambda_1$. Let $u_*$ and $g_*$ be the symmetric decreasing
rearrangement of $u$ and $g$ respectively. Since $g$ is
nonnegative, radial and radially decreasing, we use property (a)
of Proposition \ref{symmetric} to conclude that $g=g_*$ a.e.
Further, as $u$ is positive by property (c) of
Proposition~\ref{symmetric} we obtain $(u^p)_*=(u_*)^p$ a.e. Now
by the Hardy-Littlewood inequality,
$$
\int_{\Omega}g\, u^p \leq \int_{\Omega}g_* (u^p)_*
= \int_{\Omega}g (u_*)^p .
$$
Also due to Polya-Szego, we have the following inequality:
$$
\int_{\Omega} |{\nabla }u_*|^p\leq \int_{\Omega} |{\nabla }u|^p.
$$
Thus
\begin{equation}\label{eq6}
\frac{1}{\int_{\Omega}g (u_*)^p} \int_{\Omega} |{\nabla }u_*|^p
\leq \frac{1}{\int_{\Omega}g (u)^p}\int_{\Omega} |{\nabla }u|^p.
\end{equation}
Since $u$ is a minimizer of $R_p$ on $\mathcal{D}_p^+(g)$, equality holds
in \eqref{eq6} and hence $u_*$ also minimizes $R_p$ on $\mathcal{D}_p^+(g)$.
Now as $\lambda_1$ is simple, we obtain $u_*=\alpha u$ a.e.
for some $\alpha>0$. This shows that $u$ is radial, radially
decreasing.
\end{proof}
Using the above lemma we see that for
$g(x))=\frac{1}{|x|^p},\, x\in \mathbb{R}^N$ \eqref{eq1} does not
admit a positive principal eigenvalue. A proof for the case
$p=2$ is given in \cite{Solimini}.
\begin{proposition}
Let $g(x)=1/|x|^p$, $x\in \mathbb{R}^N$. Then \eqref{eq1} does
not admit a positive principal eigenvalue.
\end{proposition}
\begin{proof}
From Lemma \ref{principal}, we know that, if $\lambda>\lambda_1$
then $\lambda$ is not a principal eigenvalue of \eqref{eq1}.
Thus, it is enough to show that $\lambda_1$ is not an eigenvalue
of \eqref{eq1}, when $g(x)=\frac{1}{|x|^p}$.
By \cite[Theorem 1.3]{KPL}, if $\lambda_1$ is an eigenvalue
of \eqref{eq1}, then $\lambda_1$ is simple. Further,
if $u$ is an eigenfunction of \eqref{eq1} corresponding
$\lambda_1$, then using the scale invariance of \eqref{eq1},
for each $\alpha\in \mathbb{R}$, one can verify that
$$
v_\alpha(x)=u(\alpha x)
$$
is also an eigenfunction of \eqref{eq1} corresponding
to $\lambda_1$. Now using the simplicity of $\lambda_1$, we obtain
\[
u(x)=|x|^{1-\frac{N}{p}}u(1).
\]
A contradiction as $|x|^{1-\frac{N}{p}}\not\in \mathcal{D}^{1,p}_0(\mathbb{R}^N)$.
\end{proof}
\begin{remark} \label{rmk4.11} \rm
In particular, the above Lemma shows that the best constant
in the Hardy's inequality
\[
\int_{\mathbb{R}^N}|\nabla u|^p \leq C \int_{\mathbb{R}^N}
\frac{1}{|x|^p}|u|^p
\]
is not attained for any $u\in \mathcal{D}^{1,p}_0(\mathbb{R}^N)$.
\end{remark}
\section{An infinite set of eigenvalues}
In this section we discuss the existence of infinitely many
eigenvalues of \eqref{eq1}, using the Ljusternik-Schnirelmann
theory on $\mathcal{C}^1$ manifold due to Szulkin \cite{Szulkin}. Before
stating his result we briefly describe the notion of P.S.
condition and genus.
Let $\mathcal{M}$ be a $\mathcal{C}^1$ manifold and $f\in \mathcal{C}^1(\mathcal{M};\mathbb{R})$. Denote the
differential of $f$ at $u$ by $df(u)$. Then $df(u)$ is an element
of $(T_u \mathcal{M})^*$, the cotangent space of $\mathcal{M}$ at $u$
(see \cite[section 27.4]{Deimling} for definition and properties).
We say that a map $f\in \mathcal{C}^1(\mathcal{M};\mathbb{R})$ satisfies Palais-Smale ( P.S.
for short) condition on $\mathcal{M}$, if a sequence $\{u_n\}\subset \mathcal{M}$ is
such that $f(u_n)\to \lambda$ and $df(u_n)\to 0$ then $\{u_n\}$
possesses a convergent subsequence.
Let $A$ be a closed symmetric (i.e, $-A=A$) subset of $\mathcal{M}$,
the \emph{krasnoselski} genus $\gamma(A)$ is defined to be
the smallest integer $k$ for which there exists a non-vanishing
odd continuous mapping from $A$ to $\mathbb{R}^k$.
If there exists no such map for any $k$, then we define
$\gamma(A)=\infty$ and we set $\gamma(\emptyset)=0$.
For more details and properties of genus we refer to \cite{Rabinowitz}.
From \cite[Corollary 4.1]{Szulkin} one can deduce the following
theorem.
\begin{theorem}\label{Cuesta}
Let $\mathcal{M}$ be a closed symmetric $\mathcal{C}^1$ submanifold of a real
Banach space X and $0\notin \mathcal{M}$. Let $f\in \mathcal{C}^1(\mathcal{M};R)$
be an even function which satisfies P.S. condition on $\mathcal{M}$ and
bounded below. Define
$$
c_j:=\inf_{A\in \varGamma_j}\sup_{x\in A}f(x),
$$
where $\varGamma_j=\{A\subset \mathcal{M}: A$ is compact and symmetric
about origin, $\gamma(A)\geq j\}$. If for a given $j$,
$c_j=c_{j+1}\dots =c_{j+p}\equiv c$, then $\gamma(K_c)\geq p+1$,
where $K_c= \{ x\in M: f(x)=c\,, df(x)=0\}$.
\end{theorem}
Note that the set $M=\{ u \in \mathcal{D}^{1,p}_0(\Omega):
\int_\Omega g |u|^p =1 \}$ may not even possess
a manifold structure from the topology of $\mathcal{D}^{1,p}_0(\Omega)$,
due to the weak assumptions on $g^-$.
However, we show that $M$ admits a $\mathcal{C}^1$ Banach manifold
structure from a subspace contained in $\mathcal{D}^{1,p}_0(\Omega)$.
For $g^-\in L^1_{\rm loc}(\Omega)$, we define
\begin{gather*}
\|u\|_X^p:=\int_\Omega |{\nabla }u|^p + \int_\Omega g^-|u|^p.\\
X:=\{u\in \mathcal{D}^{1,p}_0(\Omega): \|u\|_X<\infty\}.
\end{gather*}
Then one can easily verify the following:
\begin{itemize}
\item $X$ is a Banach space with the norm $\|\cdot\|_X$ and
$X$ is reflexive.
\item Since $g^-$ is locally integrable, $\mathcal{C}_c^\infty(\Omega)$
is contained in $X$.
\item Let $g\in L^1_{\rm loc}(\Omega)$ and $g^+\in \mathcal{F}_{N/p}$.
Then $\mathcal{D}_p^+(g)$ is contained in $X$. This can be seen as
\begin{equation}\label{bound}
\int_{\Omega}g^-|u|^p< \int_{\Omega}g^+|u|^p
\leq C \|g^+\|_{(\frac{N}{p}, \infty)}\|u\|_{\mathcal{D}^{1,p}_0(\Omega)}^p<\infty,
\end{equation}
where $C$ is the constant involving the constants that
are appearing in the Lorentz-Sobolev embedding and the
H\"{o}lder inequality. Note that the first inequality follows as
$\int_{\Omega}g|u|^p>0$, for $u\in \mathcal{D}^+(g)$.
\item $X$ is continuously embedded into $\mathcal{D}^{1,p}_0(\Omega)$.
Thus $X$ embedded continuously into the Lorentz space $L(p^*,p)$
and embedded compactly into $L^p_{\rm loc}(\Omega)$.
\end{itemize}
We denote the dual space of $X$ by $X'$ and the duality action
by $\langle\cdot,\cdot\rangle$.
Using the definition of the norm one can easily see that,
the map $G_p^-$, defined by
\[
G_p^-(u):= \frac{1}{p}\int_{\Omega}g^-|u|^p,
\]
is continuous on $X$. Further, using the dominated convergence
theorem one can verify that $G_p^-$ is continuously differentiable
on $X$ and its derivative is given by
\[
\langle{G_p^-}'(u),v\rangle=\int_\Omega g^-|u|^{p-2}u\,v.
\]
Similarly using the Sobolev embedding and the H\"{o}lder
inequality one can easily verify that $G_p^+$ is $\mathcal{C}^1$ in $\mathcal{D}^{1,p}_0(\Omega)$
and in particular on $X$. The derivative of
$G_p^+$ is given by
\[
\langle{G_p^+}'(u),v\rangle=\int_\Omega g^+|u|^{p-2}u\,v.
\]
Note that for $u\in M$, $\langle G_p'(u),u\rangle=p$ and hence
the map $G_p'(u)\neq0$. Recall that, $c\in \mathbb{R}$
is called a regular value of $G_p$, if $G_p'(u)\neq0$
for all $u$ such that $G_p(u)=c$. Thus we have the following lemma.
\begin{lemma}
Let $\Omega$ be a domain in $\mathbb{R}^N$ with $N>p$.
Let $g\in L_{\rm loc}^1(\Omega)$ be such that
$g^+\in \mathcal{F}_{N/p} \setminus \{0\}$. Then the map $G_p$ is in
$\mathcal{C}^1(X;\mathbb{R})$ and $G_p':X\to X'$ is given by
\[
\langle G_p'(u),v\rangle=\int_\Omega g|u|^{p-2}u\,v.
\]
Further, 1 is a regular value of $G_p$.
\end{lemma}
\begin{remark} \label{rmk5.3} \rm
In view of \cite[Example 27.2]{Deimling}, the above lemma shows
that $M$ is a $\mathcal{C}^1$ Banach submanifold of $X$. Note that $M$ is
symmetric about the origin as the map $G_p$ is even.
\end{remark}
Next we show that $J_p$ satisfies all the conditions to apply
Theorem \ref{Cuesta}.
\begin{lemma}
$J_p$ is a $\mathcal{C}^1$ functional on $M$ and the derivative of $J_p$
is given by
\[
\langle J_p'(u),v\rangle=\int_\Omega|{\nabla }u|^{p-2}
{\nabla }u\cdot{\nabla }v
\]
\end{lemma}
The proof is straight forward and is omitted.
\begin{remark} \label{rmk5.5} \rm
Using \cite[Proposition 6.4.35]{Drabek}, one can deduce that
\begin{equation}\label{representation}
\|dJ_p(u)\|=\min_{\lambda\in\mathbb{R}}\|J_p'(u)- \lambda G_p'(u)\|.
\end{equation}
Thus $dJ_p(u_n)\to 0$ if and only if there exists a sequence
$\{\lambda_n\}$ of real numbers such that
$J_p'(u_n)- \lambda_n G_p'(u_n)\to 0$.
\end{remark}
In the next lemma we prove the compactness of the map $G_p^{+}$,
that we use for showing that the map $J_p$ satisfies P.S.
condition on $M$.
\begin{lemma}\label{weakcontinuity}
The map ${G_p^+}':X\to X'$ is compact.
\end{lemma}
\begin{proof}
Let $u_n\rightharpoonup u$ in $X$ and $v\in X$.
Let $q$ be the conjugate exponent of $p$. Now using
the Lorentz-Sobolev embedding and the H\"{o}lder inequality
available for the Lorentz spaces, one can verify the following:
\begin{gather*}
(|u_n|^{p-2}u_n-|u|^{p-2}u)\in L\Big(\frac{p^*}{p-1},\,\frac{p}{p-1}\Big),\\
(g^+)^{1/q}(|u_n|^{p-2}u_n-|u|^{p-2}u)
\in L(\frac{p}{p-1},\,\frac{p}{p-1})\\
(g^+)^{1/p}|v| \in L(p\,,p)\\
\big \|(g^+)^{1/p}v \big \|_p
\leq C\|g^+\|_{(N/p,\infty)}^{1/p} \|v\|_{(p^*,p)}
\end{gather*}
where $C$ is a constant that depends only on $p,N$.
Now by using the usual H\"{o}lder inequality we obtain
\begin{align*}
& |\langle G_p'(u_n)-G_p'(u), v \rangle|\\
&\leq \int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u |\, |v|\\
&\leq \Big(\int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u )
|^{p/(p-1)}\Big)^{(p-1)/p}
\Big(\int_\Omega g^+|v|^p\Big)^{1/p} \\
&\leq \|g^+\|_{(N/p,\infty)}^{1/p} \|v\|_{(p^*,p)}
\Big(\int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)}
\Big)^{(p-1)/p}
\end{align*}
Thus
\[
\| G_p'(u_n)-G_p'(u)\|\leq \|g^+\|_{(N/p,\infty)}^{1/p}
\Big(\int_\Omega g^+|( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)}
\Big)^{(p-1)/p}
\]
Now it is sufficient to show that
$$
\Big(\int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u )
|^{p/(p-1)}\Big)^{(p-1)/p}\to 0, \quad \text{as } n\to \infty.
$$
Let $\varepsilon>0$ and $g_\varepsilon\in C_c^\infty(\Omega)$ be arbitrary.
\begin{equation}\label{eq10}
\begin{split}
&\int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)} \\
&= \int_\Omega g_\varepsilon |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)}
+ \int_\Omega (g^+-g_\varepsilon ) |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)}
\end{split}
\end{equation}
First we estimate the second integral.
Observe that $\big |( |u_n|^{p-2}u_n-|u|^{p-2}u )\big |^{p/(p-1)}$
is bounded in $L(\frac{p^*}{p},1)$. Let
\begin{gather*}
m= \sup_n \| \big |\big (|u_n|^{p-2}u_n-|u|^{p-2}u \big )
\big |^{p/(p-1)}\|_{\,(\frac{p^*}{p},1)},\\
\int_\Omega |(g^+-g_\varepsilon)| |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)}
\leq C m \big \| \big (g^+-g_\varepsilon \big )
\big \|_{(N/p,\infty)}
\end{gather*}
where the constant C includes all the constants that appear in the
H\"{o}lder inequality and the Lorentz-Sobolev embedding. Now since
$g^+\in F_{N/p}$, from the definition of
$F_{N/p}$, we can choose $g_\varepsilon \in C^\infty_c(\Omega)$
such that
$$
m \| (g^+-g_\varepsilon) \|_{(N/p,\infty)} <\frac{\varepsilon}{2 C}
$$
Thus we can make the second integral in \eqref{eq10} smaller
than $\frac{\varepsilon}{2}$
for a suitable choice of $g_\varepsilon$. Since $X$ is embedded compactly
into $L_{\rm loc}^p(\Omega)$, the first integral converges to zero
up to a subsequence $\{u_{n_k}\}$ of $\{u_n\}$. Hence we obtain
$k_0\in \mathbb{N}$ so that,
\[
\int_\Omega g^+ |( |u_{n_k}|^{p-2}u_n-|u|^{p-2}u )
|^{p/(p-1)}< \varepsilon,\quad \forall\, k>k_0.
\]
Now the uniqueness of limit of subsequence helps us to conclude,
as in Lemma \ref{compactness}, that
$\big(\int_\Omega g^+ |( |u_n|^{p-2}u_n-|u|^{p-2}u )|^{p/(p-1)}
\big)^{(p-1)/p}\to 0$ as $n\to \infty$. Hence the proof.
\end{proof}
\begin{definition} \label{def5.7} \rm
For $\lambda\in \mathbb{R}^+$, we define $A_\lambda:X\to X'$ as
\[
A_\lambda={J_p}'+\lambda\, {G_p^-}'.
\]
\end{definition}
In the next proposition we show that the map $J_p$ indeed satisfies
P.S. condition on the $M$.
\begin{proposition} \label{prop5.8}
$J_p$ satisfies P.S. condition on $M$.
\end{proposition}
\begin{proof}
Let $\{u_n\}$ be a sequence in $M$, such that $J_p(u_n)\to \lambda$
and $dJ_p(u_n)\to 0$. Thus there exists a sequence $\{\lambda_n\}$
such that
\begin{equation}\label{ps}
J_p'(u_n)-\lambda_n
G_p'(u_n)\to 0 \quad \text{as } n\to \infty,
\end{equation}
Since $J_p(u_n)$ is bounded, using the estimate \eqref{bound},
we see that $\{ G_p^{-}(u_n) \}$ is bounded.
Thus the sequence $\{u_n\}$ is bounded in $X$ and hence by the
reflexivity we may assume passing to a subsequence that
$u_n\rightharpoonup u$. Since $G_p^+$ is weakly continuous,
we obtain $G_p^+(u_n)\to G_p^+(u)$.
Now by Fatou's lemma,
\begin{equation}\label{weaklimit}
\int_\Omega g^-|u|^p \leq \liminf \int_\Omega g^+|u_n|^p -1
=\int_\Omega g^+|u|^p -1 .
\end{equation}
Thus $\int_\Omega g|u|^p \geq 1$ and hence $u\neq 0$. Further,
$\lambda_n \to \lambda$ as $n\to \infty$, since
$$
p(J_p(u_n)-\lambda_n)=\langle J_p'(u_n)-\lambda_n
G_p'(u_n),u_n\rangle \to 0.
$$
Now we write \eqref{ps} as
\[
A_{\lambda_n}(u_n)-\lambda_n {G_p^+}{'}(u_n)\to 0.
\]
Since $\lambda_n\to \lambda$, we obtains
$A_{\lambda_n}(u_n)- A_{\lambda}(u_n)\to 0$.
Now the compactness of ${G_p^+}'$ yields
the strong convergence of $A_\lambda(u_n)$ and hence
$\langle A_\lambda(u_n),u_n-u\rangle\to 0$. Since $u_n\rightharpoonup u$,
using \cite[Lemma 4.3]{SWi} one obtain $u_n\to u$.
\end{proof}
We borrow an idea from \cite[Proposition 4.2]{KPL}, for the
proof of the following lemma.
\begin{lemma}
For each $n\in\mathbb{N}$, the set $\Gamma_n\neq \emptyset$.
\end{lemma}
\begin{proof}
The idea is to construct odd continuous maps from $S^{n-1}\to M$,
for each $n\in \mathbb{N}$. Let $\Omega^+= \{x: g^+(x)>0\}$.
Since $|\Omega^+|>0$, using the Lebesgue-Besicovitch differentiation
theorem, one can choose $n$ points $x_1, x_2, \dots x_n$
in $\Omega^+$ such that
$$
\lim_{r\to 0}\frac{1}{|B_r(x_i)|}\int_{B_r(x_i)}g(y)dy=g(x_i)>0.
$$
Thus there exists $R>0$, such that $B_R(x_i)\cap B_R(x_j)=\emptyset$
and
\[
\int_{B_r(x_i)}g(y)dy>0,\,\text{ for } 00
$$
Thus we obtain $v_i= u_i/(\int_{\Omega}g|u_i|^p)^{1/p} \in M$.
Note that the support of $v_i$s are disjoint. Now for
$\alpha=(\alpha_1, \alpha_2, \dots \alpha_n)\in \mathbb{R}^n$
with $\sum|\alpha_i|^p=1$, we have
$\sum \alpha_i v_i \in C_c^\infty(\Omega)$ and
$\int_{\Omega}g|\sum \alpha_i v_i |^p =1$. It is easy to see
that the map $\phi(\alpha)= \sum \alpha_i u_i$ is an odd
continuous map from $S^{n-1}$ into $M$.
Thus $\phi(S^{n-1})$ is compact and symmetric about origin.
Now from the definition of genus it follows that
$\gamma(\phi(S^{n-1}))\geq \gamma(S^{n-1})=n$.
\end{proof}
Now we are in a position to adapt the Ljusternik-Schnirelmann
theorem available for $\mathcal{C}^1$ manifold in our situation and
prove the existence of infinitely many eigenvalues for \eqref{eq1}.
\begin{proof}[Proof of Theorem \ref{infinite}]
Since $J$ and $M$ satisfy all the requirements of
Theorem \ref{Cuesta}, for each $j\in \mathbb{N}$, we have
$\gamma(K_{c_j})\geq 1$. Thus $K_{c_j}\neq\emptyset$
and hence there exist $u_j\in M$ such that $dJ(u_j)=0$
and $J(u_j)=c_j$. Therefore $c_j$ is an eigenvalue
of \eqref{eq1} and $u_j$ is an eigenfunction corresponding to $c_j$.
A proof for the unboundedness of the sequence $\{c_n\}$
is given in \cite{Huang}(see Theorem 2).
For the sake of completeness we adapt their idea in our situation.
Recall that the space $X$ is separable (see \cite[(3.5)]{Adams})
and hence $X$ admits a biorthogonal system $\{e_m,e_m^*\}$,
(see \cite[Proposition 1.f.3]{Linden}) such that
\begin{gather*}
\overline{\{e_m,m:\in \mathbb{N}\}}= X,\quad
e_m^*\in X',\quad \langle e_m^*,e_n\rangle=\delta_{n,m}, \\
\langle e_m^*,x\rangle=0,\quad \forall m \,\Rightarrow x=0.
\end{gather*}
Let
$E_n=\operatorname{span} \{e_1,e_2,\dots,e_n\}$ and let
\[
E_n^{\bot}= \overline{\operatorname{span} \{e_{n+1},e_{n+2},\dots\}}.
\]
Since $E_{n-1}^{\bot}$ is of codimension $n-1$, for any $A\in
\Gamma_{n}$ we have $A\cap E_{n-1}^{\bot}\neq \emptyset$ (see
\cite[Proposition 7.8]{Rabinowitz}). Let
\[
\mu_n=\inf_{A\in \Gamma_{n}}\sup_{A\cap E_{n-1}^{\bot}}J(u),\quad
n=1,2, \dots
\]
Now we show that $\mu_n\to \infty$. If possible let $\{\mu_n\}$
be bounded, then there exists $u_n\in E_{n-1}^{\bot}\cap M$
such that $\mu_n\leq J(u_n)0$.
Since $u_n\in M$, the estimate \eqref{bound} shows that
$u_n$ is indeed bounded in $X$. Thus $u_n\rightharpoonup u$
for some $u\in X$. Now by the choice of biorthogonal system,
for each $m$, $\langle e_m^*,u_n \rangle\to 0$ as $n\to \infty$.
Thus $u_n\rightharpoonup 0$, in $X$ and hence $u=0$,
a contradiction to $\int_{\Omega}g|u|^p\geq 1$
(see the conclusion followed estimate \eqref{weaklimit}).
Therefore, $\mu_n\to \infty$ and hence $c_n\to \infty$ as
$\mu_n\leq c_n$.
\end{proof}
\begin{remark} \label{rmk5.10} \rm
If $g^-\in \mathcal{F}_{N/p} \setminus \{0\}$, then there exists a
sequence $\mu_n$ of negative eigenvalues of \eqref{eq1}
tending to $-\infty$. Further, $\mu_1$ is simple and it is
the unique negative principal eigenvalue of \eqref{eq1}.
\end{remark}
\section{Remarks}
In this section we remark about possible extensions and applications
of weighted eigenvalue problems for the $p$-Laplacian.
One can study the existence of ground states for the $\Delta_p$
operator with a more general subcritical nonlinearities in
the right hand side. More precisely, for given locally
integrable functions $V, g$ on a domain
$\Omega \subset \mathbb{R}^N$ with $V \geq 0$ but $g$
allowed to change sign, we look for the positive solutions
in $\mathcal{D}^{1,p}_0(\Omega)$ for the problem
\begin{equation}\label{extn1}
\Delta_p u + V |u|^{p-2}u = \lambda g |u|^{q-2}u , \quad u \in \mathcal{D}^{1,p}_0(\Omega),
\end{equation}
where $q\in [p,p^*)$ and $10 \}$ and hence a positive
solution of \eqref{extn1}. For the positivity of this minimizer
one can use \cite[Proposition 5.3]{KPL}.
\end{remark}
\begin{remark} \label{rmk6.2} \rm
Let $g$ be as in the above remark. Then the following
generalized Hardy-Sobolev inequality holds
\begin{equation}\label{Hardy-Sobolev}
\Big(\int_\Omega g|u|^q \Big)^{q/p}
\leq \frac{1}{\lambda_1}\int_\Omega \{ |{\nabla }u|^p + V |u|^p \} ,
\quad \forall \, u \in \mathcal{D}^{1,p}_0(\Omega) ,\, \int_{\Omega}g|u|^q>0
\end{equation}
where $\lambda_1$ is the minimum of
${\int_{\Omega}\{|{\nabla }u|^p+V|u|^p\}}$ on $M_q$.
Further the best constant is attained. This extends
the results of Visciglia \cite{Visciglia} for $p\neq 2$.
\end{remark}
\begin{remark} \label{rmk6.3} \rm
The existence of a simple eigenvalue for \eqref{eq1} can be applied
to study the bifurcation phenomena of the solutions for
the semilinear problem of the type
\begin{equation} \label{OurProblem}
-\Delta_p u= \lambda \big( a(x) u + b(x) r(u) \big), \quad u \in \mathcal{D}^{1,p}_0(\Omega)
\end{equation}
for a real parameter $\lambda$ when $a, b$ are in certain sub class of weak Lebesgue space with a suitable growth condition on $r$. Such a result is available for $p=2$ see in \cite{ALM}. We deal with this question in a subsequent work.
\end{remark}
\subsection*{Acknowledgments}
The author wants to thank Prof. Mythily Ramaswamy and
Prof. S. Kesavan for their useful discussions and for providing
several critical comments that greatly improved this manuscript.
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