\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 16, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/16\hfil The eigenvalue problem] {The eigenvalue problem for a singular quasilinear elliptic equation} \author[Benjin Xuan\hfil EJDE-2004/16\hfilneg] {Benjin Xuan} \address{Benjin Xuan \hfill\break Department of Mathematics, University of Science and Technology of China \hfill\break Department of mathematics, Universidad Nacional, Bogota Colombia} \email{wenyuanxbj@yahoo.com} \date{} \thanks{Submitted August 15, 2003. Published February 6, 2004.} \thanks{Supported by grants 10101024 and 10371116 from the National Natural Science \hfill\break\indent Foundation of China.} \subjclass[2000]{35J60} \keywords{Singular quasilinear elliptic equation, eigenvalue problem,\hfill\break\indent Caffarelli-Kohn-Nirenberg inequality} \begin{abstract} We show that many results about the eigenvalues and eigenfunctions of a quasilinear elliptic equation in the non-singular case can be extended to the singular case. Among these results, we have the first eigenvalue is associated to a $C^{1,\alpha}(\Omega)$ eigenfunction which is positive and unique (up to a multiplicative constant), that is, the first eigenvalue is simple. Moreover the first eigenvalue is isolated and is the unique positive eigenvalue associated to a non-negative eigenfunction. We also prove some variational properties of the second eigenvalue. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \section{Introduction}\label{intro} In this paper, we shall study the eigenvalue problem of the singular quasilinear elliptic equation \begin{equation} \label{eq1.1} \begin{gathered} -\mathop{\rm div}(|x|^{-ap}|Du|^{p-2}Du)=\lambda |x|^{-(a+1)p+c}|u|^{p-2}u, \quad \mbox{in } \Omega\\ u= 0, \quad \mbox{on } \partial\Omega, \end{gathered} \end{equation} where $\Omega\subset \mathbb{R}^n$ is an open bounded domain with $C^1$ boundary, $0\in \Omega$, $1
0$. For $a=0, c=p$, there are many results about the eigenvalues and eigenfunctions of problem (\ref{eq1.1}), such as $\lambda_1$ is associated to a $C^{1,\alpha}(\Omega)$ eigenfunction which is positive in $\Omega$ and unique (up to a multiplicative constant), that is, $\lambda_1$ is simple. Moreover $\lambda_1$ is isolated, and is the unique positive eigenvalue associated to a non-negative eigenfunction (cf. \cite{LP, AT, CM} and references therein). In this paper, we will show that many results about the eigenvalues and eigenfunctions in the case where $a=0, c=p$ can be extended to the more general case where $0\leq a<(n-p)/p$, $c>0$. The starting point of the variational approach to these problems is the following weighted Sobolev-Hardy inequality due to Caffarelli, Kohn and Nirenberg \cite{CKN}, which is called the Caffarelli-Kohn-Nirenberg inequality. Let $1
0$ such that \begin{equation} \label{eq1.2} \Big(\int_{\mathbb{R}^n}|x|^{-bq}|u|^{q}\,dx \Big)^{p/q}\leq C_{a,b}\int_{\mathbb{R}^n}|x|^{-ap}|Du|^{p}\,dx, \end{equation} where \begin{equation}\label{eq1.3} \begin{gathered} -\infty< a<\frac{n-p}p,\quad a\leq b\leq a+1,\\ q=p^*(a,b)=\frac{np}{n-dp},\quad d=1+a-b. \end{gathered} \end{equation} Let $\Omega\subset \mathbb{R}^n$ be an open bounded domain with $C^1$ boundary and $0\in \Omega$, and let $\mathcal{D}_a^{1,p}(\Omega)$ be the completion of $C_0^\infty(\mathbb{R}^n)$, with respect to the norm $\|\cdot\|$ defined by $$ \|u\|=\Big(\int_{\Omega}|x|^{-ap}|Du|^{p}\,dx \Big)^{1/p}. $$ From the boundedness of $\Omega$ and the standard approximation argument, it is easy to see that (\ref{eq1.2}) holds for any $u\in \mathcal{D}_a^{1,p}(\Omega)$ in the sense \begin{equation} \label{eq1.4} \Big(\int_{\Omega}|x|^{-\alpha}|u|^{r}\,dx \Big)^{p/r} \leq C \int_{\Omega}|x|^{-ap}|Du|^{p}\,dx, \end{equation} for $1\leq r\leq \frac{np}{n-p},\ \alpha \leq (1+a)r+n(1-\frac rp)$; that is, the imbedding $\mathcal{D}_a^{1,p}(\Omega) \hookrightarrow L^r(\Omega, |x|^{-\alpha})$ is continuous, where $L^r(\Omega, |x|^{-\alpha})$ is the weighted $L^r$ space with norm $$ \|u\|_{r, \alpha}:=\|u\|_{L^r(\Omega, |x|^{-\alpha})} =\Big( \int_{\Omega}|x|^{-\alpha}|u|^{r}\,dx\Big)^{1/r}. $$ In fact, we have the following compact imbedding result which is an extension of the classical Rellich-Kondrachov compactness theorem (cf. \cite{CC} for $p=2$ and \cite{XB} for the general case). For the convenience of the reader, we include its proof here. \begin{theorem}[Compact imbedding theorem] \label{thm1.1} Let $\Omega\subset \mathbb{R}^n$ be an open and bounded domain with $C^1$ boundary and $0\in \Omega$, $1
0$
with $B_\rho(0)\subset \Omega$ is a ball centered at the origin
with radius $\rho$, it follows that $\{u_m\}\subset
W^{1,p}(\Omega\setminus B_\rho(0))$. Then the classical
Rellich-Kondrachov compactness theorem guarantees the existence of
a convergent subsequence of $\{u_m\}$ in $L^r(\Omega\setminus
B_\rho(0))$. By taking a diagonal sequence, we can assume without
loss of generality that $\{u_m\}$ converges in
$L^r(\Omega\setminus B_\rho(0))$ for any $\rho>0$.
On the other hand, for any $1\leq r< \frac{np}{n-p}$, there exists a $b\in (a, a+1]$
such that $r 0$, and
$u\in \mathcal{D}_a^{1,p}(\Omega)$ is a solution of (\ref{eq1.1}).
Then $u\in L^\infty(\Omega, |x|^{-\alpha})$ and
$u\in C^{1, \alpha}(\Omega\setminus \{0\})$ for some $\alpha\geq 0$.
\end{theorem}
\begin{proof} By the standard elliptic regularity theory (e.g. \cite{TP}),
it suffices to show the $L^\infty$ boundedness of $u$. To do this, we apply
the Moser's iteration as in \cite{GV} and \cite{CG}.
For $k>0, q\geq1$, we define two $C^1$ functions on $\mathbb{R}$, $h$ and $H$ by
\begin{equation} \label{eq2.1}
h(t)=\begin{cases}
\mathop{\rm sign}(t) |t|^q, &\mbox{if } |t|\leq k,\\
\mathop{\rm sign}(t) \{qk^{q-1} |t|+(1-q)k^q \}, & \mbox{if } |t|> k,
\end{cases}
\end{equation}
and $H(t)= \int_0^t (h'(s))^p\,ds$. Thus, it is easy to see that
$h'(t)\geq 0$ for all $t\in \mathbb{R}$ and
$H(u(x)) \in \mathcal{D}_a^{1,p}(\Omega)$ if $u\in \mathcal{D}_a^{1,p}(\Omega)$.
In fact, a simple calculation shows that
\begin{equation} \label{eq2.01}
h'(t)=\begin{cases} q |t|^{q-1}, &\mbox{if } |t|\leq k,\\
qk^{q-1}, &\mbox{if } |t|> k
\end{cases}
\end{equation}
and
\begin{equation}\label{eq2.02}
H(t)=\begin{cases}
\dfrac{q^p}{p(q-1)+1} |t|^{p(q-1)+1}\mathop{\rm sign}(t),
& \mbox{if } |t|\leq k,\\[3pt]
q^p\big( \dfrac{1}{p(q-1)+1} k^{p(q-1)+1}+k^{p(q-1)}(|t|-k) \big)
\mathop{\rm sign}(t), &\mbox{if } |t|> k.
\end{cases}
\end{equation}
It is trivial to verify that
\begin{equation} \label{eq2.2}
|H(t)|\leq q |h(t)|(h'(t))^{p-1}, \quad
|H(t)||t|^{p-1} \leq q^{p} |h(t)|^p,
\end{equation}
for all $t\in \mathbb{R}$. In fact, for all $|t|\leq k,\ q\geq 1$, we see that
$$
|H(t)|=\frac{q^p}{p(q-1)+1} |t|^{p(q-1)+1}
\leq q |h(t)|(h'(t))^{p-1}=q^p |t|^{p(q-1)+1}
$$
and
$$
|H(t)||t|^{p-1}=\frac{q^p}{p(q-1)+1}|t|^{pq}\leq q^{p} |h(t)|^p=q^p |t|^{pq}.
$$
For $|t|> k$, $q\geq 1$, a direct calculation shows that
\begin{align*}
&|H(t)|- q |h(t)|(h'(t))^{p-1}\\
&=q^p\big( (\frac{1}{p(q-1)+1}-1) k^{p(q-1)+1}+(1-q)k^{p(q-1)}(|t|-k) \big)
\leq 0
\end{align*}
and
\begin{align*}
|H(t)||t|^{p-1}- q^{p} |h(t)|^p
&=q^p\big( (\frac{1}{p(q-1)+1}-1) k^{p(q-1)+1}|t|^{p-1}\\
&\quad -q^pk^{p(q-1)}(|t|-k)^p+k^{p(q-1)}(|t|^p-k^p) \big) \leq 0
\end{align*}
Let $\psi(x)=\eta^p H(u(x))$ be a test function defined in $\Omega$,
where $\eta$ is a non-negative smooth function in $\Omega$ to be specified later.
Then from (\ref{eq1.1}), it follows that
\begin{equation} \label{eq2.3}
\int_\Omega |x|^{-ap}|Du|^{p-2}Du \cdot D\psi \,dx
=\lambda \int_\Omega |x|^{-(a+1)p+c}|u|^{p-2}u \psi \,dx.
\end{equation}
From the definitions of $h, H, \psi$, (\ref{eq2.2}) implies that
\begin{equation} \label{eq2.04}
\begin{aligned}
& \int_\Omega |x|^{-ap}|Du|^{p-2}Du \cdot D\psi \,dx\\
&= \int_\Omega |x|^{-ap}\eta^p |Du|^{p-2}Du \cdot DH(u) \,dx
+ p \int_\Omega |x|^{-ap}\eta^{p-1}H(u) |Du|^{p-2}Du \cdot D\eta \,dx \\
&\geq \int_\Omega |x|^{-ap}\eta^p |Dh(u)|^p\,dx
- p q \int_\Omega |x|^{-ap}\eta^{p-1} |Du|^{p-1} |D\eta| |h(u)|(h'(u))^{p-1} \,dx.
\end{aligned}
\end{equation}
By the H\"{o}lder inequality, it follows that
\begin{equation}\label{eq2.05}
\begin{aligned}
&p q \int_\Omega |x|^{-ap}\eta^{p-1} |Du|^{p-1} |D\eta| |h(u)|(h'(u))^{p-1} \,dx\\
&\leq \dfrac12 \int_\Omega |x|^{-ap}\eta^p |Dh(u)|^p\,dx
+ C q^p \int_\Omega |x|^{-ap} |h(u)|^p |D\eta|^{p} \,dx,
\end{aligned}
\end{equation}
where and hereafter $C$ is a universal positive constant independent of $k, q$.
Inserting (\ref{eq2.05}) into (\ref{eq2.04}), we see that
\begin{equation} \label{eq2.4}
\begin{aligned}
& \int_\Omega |x|^{-ap}|Du|^{p-2}Du \cdot D\psi \,dx\\
& \geq \dfrac12 \int_\Omega |x|^{-ap}\eta^p |Dh(u)|^p\,dx
- C q^p \int_\Omega |x|^{-ap} |h(u)|^p |D\eta|^{p} \,dx\,.
\end{aligned}
\end{equation}
Equation (\ref{eq2.2}) also implies
\begin{equation}\label{eq2.5}
\begin{aligned}
\lambda \int_\Omega |x|^{-(a+1)p+c}|u|^{p-2}u \psi \,dx
& =\lambda \int_\Omega |x|^{-(a+1)p+c}|u|^{p-2}u \eta^p H(u) \,dx\\
& \leq \lambda q^p \int_\Omega |x|^{-(a+1)p+c} \eta^p |h(u)|^p\,dx.
\end{aligned}
\end{equation}
For any $r\in (p, \dfrac{np}{n-p})$, let $\alpha=n+(a+1)r-\dfrac{nr}p
\in (ar, (a+1)r)$, from the Caffarelli-Kohn-Nirenberg inequality (\ref{eq1.4}),
it follows that
\begin{equation} \label{eq2.6}
\big(\int_\Omega |x|^{-\alpha} |\eta h(u)|^r\,dx \big)^{p/r}
\leq C \int_\Omega |x|^{-ap} |D(\eta h(u))|^p\,dx.
\end{equation}
Thus, substituting (\ref{eq2.4})--(\ref{eq2.6}) into (\ref{eq2.3}),
it is easy to show that
\begin{equation} \label{eq2.7}
\begin{aligned}
&\big(\int_\Omega |x|^{-\alpha} |\eta h(u)|^r\,dx \big)^{p/r}\\
& \leq q^p C \big\{\int_\Omega |x|^{-ap} |h(u)|^p |D\eta|^{p} \,dx
+ \int_\Omega |x|^{-(a+1)p+c} \eta^p |h(u)|^p\,dx\big\}.
\end{aligned}
\end{equation}
For each $x_0\in \bar \Omega$, let $\eta\in C_0^\infty(B_{2R}(x_0)), R<1$,
such that
$$
0\leq \eta \leq 1, \eta\equiv 1 \mbox{ in }B_{R}(x_0),\ \ |D\eta|<2/R.
$$
Then (\ref{eq2.7}) implies that
\begin{equation} \label{eq2.8}
\begin{aligned}
&\big(\int_{B_{R}(x_0)} |x|^{-\alpha} |h(u)|^r\,dx \big)^{p/r}\\
&\leq q^p C \big\{ \int_{B_{2R}(x_0)} \dfrac{|x|^{-ap}}{ R^p} |h(u)|^p \,dx
+\int_{B_{2R}(x_0)} |x|^{-(a+1)p+c} |h(u)|^p\,dx\big\}.
\end{aligned}
\end{equation}
Letting $k\to \infty$ in (\ref{eq2.7}), the H\"{o}lder inequality implies
\begin{equation}\label{eq2.9}
\begin{aligned}
&\big(\int_{B_{R}(x_0)} |x|^{-\alpha} |u|^{qr}\,dx \big)^{p/r}\\
& \leq q^p C \big\{
\int_{B_{2R}(x_0)} \dfrac{|x|^{-ap}}{ R^p} |u|^{pq} \,dx
+ \int_{B_{2R}(x_0)} |x|^{-(a+1)p+c} |u|^{pq}\,dx\big\}\\
& \leq q^p C \big(\int_{B_{2R}(x_0)} |x|^{-\alpha} |u|^{pqs}\,dx \big)^{1/s},
\end{aligned}
\end{equation}
where
$$
s\in \big(\max\big\{1, \frac{n-\alpha}{n-(a+1)p+c},\frac{n-\alpha}{n-ap}\big\},
\frac rp\big).
$$
A simple covering argument yields that
\begin{equation}\label{eq2.10}
\big(\int_{\Omega} |x|^{-\alpha} |u|^{qr}\,dx \big)^{p/r}
\leq q^p C \big( \int_{\Omega} |x|^{-\alpha} |u|^{pqs}\,dx \big)^{1/s},
\end{equation}
that is,
$$
\|u\|_{L^{qr}(\Omega, |x|^{-\alpha})}
\leq (Cq)^{1/q} \|u\|_{L^{pqs}(\Omega, |x|^{-\alpha})},
$$
which is a reversed H\"{o}lder inequality and implies that
$u\in L^{qr}(\Omega, |x|^{-\alpha})$ for all $q> 1$.
Then letting $q=\chi^m, m=0,1,2,\cdots, \chi=\frac r{ps}>1$,
the Moser's iteration technique (cf. \cite{MJ}) implies
\begin{align*}
\|u\|_{L^{ps\chi^N}(\Omega, |x|^{-\alpha})}
&\leq \prod_{m=0}^{N-1} (C\chi^m)^{\chi^{-m}} \|u\|_{L^{ps}(\Omega, |x|^{-\alpha})}\\
&\leq C^\sigma \chi^\tau \|u\|_{L^{ps}(\Omega, |x|^{-\alpha})}\\
&\leq C\|u\|_{L^{ps}(\Omega, |x|^{-\alpha})},
\end{align*}
where $\sigma=\sum_{m=0}^{N-1}\chi^{-m},\ \tau=\sum_{m=0}^{N-1}m\chi^{-m}$.
Letting $N\to \infty$, we therefore obtain
$\|u\|_{L^{\infty}(\Omega, |x|^{-\alpha})} <\infty$.
\end{proof}
Based on the above regularity result, the strong maximum principle due to
Vazquez \cite{VJ} implies the following positivity of nonnegative eigenfunction.
\begin{corollary} \label{cor2.2}
Suppose that $1 0$, $u\geq 0$ is an
eigenfunction corresponding to $\lambda>0$. Then $u>0$ in $\Omega\setminus \{0\}$.
\end{corollary}
\section{Simplicity of $\lambda_1$}
In this section, we prove the simplicity of $\lambda_1$, that is, any two
eigenfunctions both corresponding to $\lambda_1$ are proportional.
\begin{theorem} \label{thm3.1}
Suppose that $1 0$, $u\geq 0$ and
$v\geq 0$ are eigenfunctions both corresponding to $\lambda_1$.
Then $u$ and $v$ are proportional.
\end{theorem}
\begin{proof}
From Theorem \ref{thm2.1}, $u, v$ are bounded. We use the modified test-functions
as in \cite{LP}:
\begin{equation}\label{eq3.1}
\eta=\frac{(u+\varepsilon)^p-(v+\varepsilon)^p}{(u+\varepsilon)^{p-1}}
\quad\mbox{and}\quad
\frac{(v+\varepsilon)^p-(u+\varepsilon)^p}{(v+\varepsilon)^{p-1}}
\end{equation}
in the corresponding equations for $u$ and $v$, respectively, where $\varepsilon$
is a positive parameter. Direct calculation implies
\begin{equation} \label{eq3.2}
D \eta=\Big\{1+(p-1) \big( \frac{v+\varepsilon}{u+\varepsilon}\big)^p\Big\} Du
-p\big( \frac{v+\varepsilon}{u+\varepsilon}\big)^{p-1}Dv,
\end{equation}
and, by symmetry, the gradient of the test-function in the corresponding equations for $v$ has a similar expression with $u$ and $v$ interchanged. Set
$$
u_\varepsilon=u+\varepsilon,\quad v_\varepsilon=v+\varepsilon.
$$
Inserting the chosen test-functions into their respective equations and
adding these, it follows that
\begin{equation}\label{eq3.3}
\begin{aligned}
& \lambda_1 \int_\Omega |x|^{-(a+1)p+c} \big[\dfrac{u^{p-1}}{u_\varepsilon ^{p-1}}
-\dfrac{v^{p-1}}{v_\varepsilon ^{p-1}} \big]( u_\varepsilon ^p- v_\varepsilon ^p)
\,dx \\
&= \int_\Omega |x|^{-ap} \Big\{1+(p-1)\big(\frac{v_\varepsilon}{u_\varepsilon}\big)^p
\Big\}|Du_\varepsilon|^p\,dx\\
& \quad + \int_\Omega |x|^{-ap} \Big\{1+(p-1) \big( \frac{u_\varepsilon}
{v_\varepsilon}\big)^p\Big\}|Dv_\varepsilon|^p \,dx\\
& \quad - \int_\Omega |x|^{-ap} p \big( \frac{v_\varepsilon}{u_\varepsilon}
\big)^{p-1}|Du_\varepsilon|^{p-2} Du_\varepsilon \cdot Dv_\varepsilon \,dx\\
& \quad - \int_\Omega |x|^{-ap} p \big( \frac{u_\varepsilon}{v_\varepsilon}
\big)^{p-1}|Dv_\varepsilon|^{p-2} Dv_\varepsilon \cdot Du_\varepsilon \,dx\\
& = \int_\Omega |x|^{-ap} ( u_\varepsilon ^p- v_\varepsilon ^p)
( |D\log u_\varepsilon| ^p-|D\log v_\varepsilon| ^p)\,dx \\
& \quad - \int_\Omega |x|^{-ap} p v_\varepsilon ^p |D\log u_\varepsilon|^{p-2}
D\log u_\varepsilon \cdot (D\log v_\varepsilon- D\log u_\varepsilon)\,dx \\
& \quad - \int_\Omega |x|^{-ap} p u_\varepsilon ^p |D\log v_\varepsilon|^{p-2}
D\log v_\varepsilon \cdot (D\log u_\varepsilon- D\log v_\varepsilon)\,dx \\
&
\geq 0,
\end{aligned}
\end{equation}
where the last inequality is a consequence of the following simple
calculus inequality (cf. \cite{LP}):
\begin{equation}\label{eq3.4}
|w_2|^p> |w_1|^p+p|w_1|^{p-2}w_1\cdot (w_2-w_1)
\end{equation}
for points in $\mathbb{R}^n, w_1\neq w_2, p>1$. By the Lebesgue's Dominated
Convergence Theorem, it follows that
\begin{equation}\label{eq3.5}
\lim_{\varepsilon\to 0^+} \int_\Omega |x|^{-(a+1)p+c}
\big[\dfrac{u^{p-1}}{u_\varepsilon ^{p-1}}-\dfrac{v^{p-1}}{v_\varepsilon ^{p-1}}
\big]( u_\varepsilon ^p- v_\varepsilon ^p)\,dx=0.
\end{equation}
The same argument as in \cite{LP} implies that $vDu=uDv$ a.e. in
$\Omega$, which implies that $u$ and $v$ are proportional.
\end{proof}
\section{Isolation of $\lambda_1$}
In this section, we prove the isolation of $\lambda_1$. First, we show that
only the first eigenfunctions are non-negative.
\begin{theorem}\label{thm4.1}
Suppose that $1 0$. If $v\geq 0$ is any
eigenfunction corresponding to the eigenvalue $\lambda$, then $\lambda=\lambda_1$.
\end{theorem}
\begin{proof}
Let $u\geq 0$ denote a first eigenfunction, then the same procedure as in Section 3
yields
\begin{equation}\label{eq4.1}
\int_\Omega |x|^{-(a+1)p+c} \big[\lambda_1 \dfrac{u^{p-1}}{u_\varepsilon ^{p-1}}
-\lambda \dfrac{v^{p-1}}{v_\varepsilon ^{p-1}} \big]( u_\varepsilon ^p
- v_\varepsilon ^p)\,dx\geq 0,
\end{equation}
and arguing as before, it follows that
\begin{equation}
\label{eq4.2}
(\lambda_1 -\lambda)\int_\Omega |x|^{-(a+1)p+c} ( u^p- v^p)\,dx\geq 0.
\end{equation}
This leads to a contradiction, if $\lambda>\lambda_1$, since $u$ can be
replaced by any of the functions $2u, 3u, 4u,\cdots$. Thus $\lambda=\lambda_1$.
\end{proof}
From Theorem \ref{thm4.1}, for any eigenvalue $\lambda>\lambda_1$, the
corresponding eigenfunction $v$ must change sign.
Next, we need an estimate of the measure of the nodal domains of
an eigenfunction $v$. We recall that a nodal domain of $v$ is a
connected component of $\Omega \setminus \{x\in \Omega: u=0\}$.
\begin{theorem} \label{thm4.2}
Suppose that $1 0$. If $v$ is any eigenfunction
corresponding to the eigenvalue $\lambda>\lambda_1>0$ and $\mathcal{N}$ is a nodal
domain of $v$, then
\begin{equation}\label{eq4.3}
|\mathcal{N}|\geq (C\lambda)^{-1/\sigma}
\end{equation}
for some positive constant $C>0$, where $\sigma =1-\frac pr-\frac1s$,
$r\in (p,\frac{n-p}{np})$, $s>\frac r{r-p}$ if $c\geq n-\frac{np}r$,
$s\in (\frac r{r-p}$, $\frac{nr}{nr-cr-np})$ if $0 0$. $\lambda_1$ is isolated.
\end{theorem}
\begin{proof} Suppose, on the contrary, there exists a sequence of
eigenvalues $\{\nu_m\}$ such that $\nu_m\neq \lambda_1$ and $\nu_m\to \lambda_1$
as $m\to \infty$. Let $u_m$ be an eigenfunction associated to $\nu_m$ such that
$\|u_m\|_{\mathcal{D}_a^{1,p}(\Omega)}=1$. Thus, up to a subsequence, $\{u_m\}$
converge weakly in $\mathcal{D}_a^{1,p}(\Omega)$ and strongly in
$L^p(\Omega, |x|^{-(a+1)p+c})$ to a function $u\in \mathcal{D}_a^{1,p}(\Omega)$.
Furthermore, the limit function $u$ is an eigenfunction associated to the first
eigenvalue $\lambda_1$. Without loss of generality, assume that $u\geq 0$.
Then for any $\delta>0$, by the Egorov theorem, $u_m$ converges uniformly to $u$
on a subset $\Omega_\delta\subset \Omega$, with
$|\Omega\setminus \Omega_\delta|<\delta$. Let $\mathcal{N}_m$ be a nodal domain of
$u_m$ such that $u_m<0$ in $\mathcal{N}_m$, then $|\mathcal{N}_m|\to 0$ as
$m\to \infty$, which contradicts (\ref{eq4.3}).
\end{proof}
\section{Variational property of the second eigenvalue}
Since $\lambda_1$ is isolated in the spectrum and there exist
eigenvalues different from $\lambda_1$, it makes sense to define
the second eigenvalue of (\ref{eq1.1}) as
$$
\underline{\lambda}_2:=\inf\{\lambda\in \mathbb{R}: \lambda
\mbox{ is eigenvalue and } \lambda>\lambda_1 \}> \lambda_1.
$$
It follows from the closure of the set of eigenvalues of (\ref{eq1.1}) that
$\underline{\lambda}_2$ is a different eigenvalue of (\ref{eq1.1}) from $\lambda_1$.
\begin{theorem}\label{thm5.1}
$\underline{\lambda}_2 =\lambda_2$, where $\lambda_2$ is defined by (\ref{eq1.5}).
\end{theorem}
\begin{proof}
It is trivial that $\underline{\lambda}_2 \leq\lambda_2$. It suffices to show
that $\lambda_2\leq \underline{\lambda}_2$. Suppose that $v$ is the eigenfunction
associated to $\underline{\lambda}_2$, then from Theorem \ref{thm4.1} and
Corollary \ref{cor4.3}, let $\mathcal{N}_1, \cdots, \mathcal{N}_r, r\geq 2$ denote
the nodal domains of $v$. For $i=1,\cdots, r$, set
$$
v_i(x)=\begin{cases}
\dfrac {v(x)}{\big[\int_{\mathcal{N}_i}|x|^{-(a+1)p+c} |v|^p\,dx\big]^{1/p}},
&\mbox{ if } x\in \mathcal{N}_i \\
0, &\mbox{ if } x\in \Omega\setminus \mathcal{N}_i.
\end{cases}
$$
It is easy to see that $v_i \in \mathcal{D}_a^{1,p}(\Omega)$. Let $\mathcal{F}_r$
denote the subspace of $\mathcal{D}_a^{1,p}(\Omega)$ spanned by
$\{v_1, \cdots, v_r\}$ and $A_r=\{u\in \mathcal{F}_r\,:\, J(u)=1 \}$.
For each $u\in \mathcal{F}_r,\ u=\sum_{i=1}^r \alpha_i v_i$, it follows that
$$
J(u)=\sum_{i=1}^r |\alpha_i|^p J(v_i)=\sum_{i=1}^r |\alpha_i|^p.
$$
Thus the set $A_r$ can also be represented as
$$
A_r=\big\{\sum_{i=1}^r \alpha_i v_i : \sum_{i=1}^r |\alpha_i|^p=1 \big\}.
$$
It is easy to see that $A_r$ is compact, symmetric and $\gamma(A_r)=r\geq 2$,
that is, $A_r\in \Gamma_2$.
On the other hand, inserting $v_i$ into the corresponding equation of $v$ yields
\begin{equation}\label{eq5.1}
\int_{\mathcal{N}_i}|Dv_i|^{p}\,dx=\underline{\lambda}_2 \int_{\mathcal{N}_i}
|x|^{-(a+1)p+c}|v_i|^{p}\,dx.
\end{equation}
Then for any $u=\sum_{i=1}^r \alpha_i v_i \in A_r$, it follows that
\begin{equation} \label{eq5.2}
\Phi(u)=\sum_{i=1}^r |\alpha_i|^p \Phi(v_i)=\underline{\lambda}_2
\sum_{i=1}^r |\alpha_i|^p J(v_i)=\underline{\lambda}_2 .
\end{equation}
Thus, it follows that
\begin{equation} \label{eq5.3}
\lambda_2:=\inf_{A\in\Gamma_2 }\max_{u\in A} \Phi(u)
\leq \max_{u\in A_r} \Phi(u)=\underline{\lambda}_2\,,
\end{equation}
which implies the conclusion.
\end{proof}
The above argument implies the following further variation characterization
of the second eigenvalue (cf. \cite{DR, CDG, ACCG} for $a=0, c=p$).
\begin{theorem}\label{thm5.2}
$\underline{\lambda}_2 =\lambda_2=\mu_2= \inf_{h\in \,athcal {F}} \max_{u\in h([-1,1])}
\Phi(u)$, where $\mathcal{F}:=\{h\in C([-1,1], \mathcal{M}): h(\pm 1)=\pm e_1\}$
and $e_1\in \mathcal{M}$ is the positive eigenfunction associated to $\lambda_1$.
\end{theorem}
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Electron. J. Diff. Eqns., Vol. {\bf 2001}(2001), No. 33, 1-9.
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Proceedings of 2000 Seminar in Differential Equations, Kvilda (Czech Republic),
to appear.
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\emph{The beginning of the Fu\v{c}ik spectrum for the p-Laplacian},
J. of Diff. Eqns., Vol. {\bf 159}(1999), 212-238.
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Addendum in Proc. Amer. Math. Soc., Vol. {\bf 116}(1992), 583-584.
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\end{document}
0$, it follows that
\begin{equation} \label{eq1.04}
\begin{aligned}
& \int_{|x|<\delta}|x|^{-\alpha}|u_m-u_j|^{r}\,dx \\
& \leq \Big( \int_{|x|<\delta}|x|^{-(\alpha-br)\frac q{q-r}}\,dx\Big)^{1-\frac rq}
\Big( \int_{\Omega}|x|^{-br}|u_m-u_j|^{r}\,dx\Big)^{r/q}\\
& \leq C \Big( \int_0^\delta r^{n-1-(\alpha-br)\frac q{q-r}}\,dr\Big)^{1-\frac rq}\\
& =C \delta ^{n-(\alpha-br)\frac q{q-r}},
\end{aligned}
\end{equation}
where $C>0$ is a constant independent of $m$. Since
$\alpha< (1+a)r+n(1-\frac rp)$, it follows that $n-(\alpha-br)\frac q{q-r}>0$.
Therefore, for a given $\varepsilon>0$, we first fix $\delta>0$ such that
$$
\int_{|x|<\delta}|x|^{-\alpha}|u_m-u_j|^{r}\,dx \leq \frac{\varepsilon}2, \quad
\forall\ m, j\in \mathbb{N}.
$$
Then we choose $N\in \mathbb{N}$ such that
$$
\int_{\Omega\setminus B_\delta(0)}|x|^{-\alpha}|u_m-u_j|^{r}\,dx
\leq C_\alpha \int_{\Omega\setminus B_\delta(0)} |u_m-u_j|^{r}\,dx
\leq \frac{\varepsilon}2, \quad \forall\ m, j\geq N,
$$
where $C_\alpha=\delta ^{-\alpha}$ if $\alpha\geq 0$ and $C_\alpha=
(\mathop{\rm diam}(\Omega) )^{-\alpha}$ if $\alpha< 0$. Thus
$$
\int_{\Omega}|x|^{-\alpha}|u_m-u_j|^{r}\,dx \leq \varepsilon , \quad
\forall\ m, j\geq N,
$$
that is, $\{u_m\}$ is a Cauchy sequence in $L^q(\Omega, |x|^{-bq})$.
\end{proof}
For studying the eigenvalue problem (\ref{eq1.1}), we introduce the
following two functionals on $\mathcal{D}_a^{1,p}(\Omega)$:
$$
\Phi(u):= \int_{\Omega} |x|^{-ap}|Du|^{p}\,dx,\quad
J(u):=\int_{\Omega} |x|^{-(a+1)p+c}|u|^p\,dx.
$$
For $c>0$, $J$ is well-defined. Furthermore,
$\Phi, J\in C^1(\mathcal{D}_a^{1,p}(\Omega),\mathbb{R})$, and a real value
$\lambda$ is an eigenvalue of problem (\ref{eq1.1}) if and only if there
exists $u\in \mathcal{D}_a^{1,p}(\Omega)\setminus\{0\}$ such that
$\Phi'(u)=\lambda J'(u)$. At this point, let us introduce the set
$$
\mathcal{M}:=\{u\in \mathcal{D}_a^{1,p}(\Omega): J(u)=1 \}.
$$
Then $\mathcal{M}\neq \emptyset$ and $\mathcal{M}$ is a $C^1$ manifold in
$\mathcal{D}_a^{1,p}(\Omega)$. It follows from the standard Lagrange multiplier
argument that eigenvalues of (\ref{eq1.1}) correspond to critical values of
$\Phi|_{\mathcal{M}}$. From Theorem \ref{thm1.1}, $\Phi$ satisfies the (PS)
condition on $\mathcal{M}$. Thus a sequence of critical values of
$\Phi|_{\mathcal{M}}$ comes from the Ljusternik-Schnirelman critical point
theory on $C^1$ manifolds. Let $\gamma(A)$ denote the Krasnoselski genus on
$\mathcal{D}_a^{1,p}(\Omega)$ and for any $k\in \mathbb{N}$, set
$$
\Gamma_k:=\{A\subset \mathcal{M}: A \mbox{ is compact, symmetric and }
\gamma(A)\geq k\}.
$$
Then the values
\begin{equation} \label{eq1.5}
\lambda_k:=\inf_{A\in\Gamma_k }\max_{u\in A} \Phi(u)
\end{equation}
are critical values and hence are eigenvalues of problem (\ref{eq1.1}).
Moreover, $\lambda_1\leq \lambda_2\leq \cdots \leq \lambda_k\leq \cdots \to+\infty$.
One can also define another sequence of critical values minimaxing $\Phi$ along
a smaller family of symmetric subsets of $\mathcal{M}$. Let us denote by $S^k$
the unit sphere of $\mathbb{R}^{k+1}$ and
$$
\mathcal{O}(S^k, \mathcal{M}):= \{h\in C(S^k, \mathcal{M}): h \mbox{ is odd}\}.
$$
Then for any $k\in \mathbb{N}$, the value
\begin{equation}\label{eq1.6}
\mu_k:= \inf_{h\in\mathcal{O}(S^{k-1}, \mathcal{M}) }
\max_{t\in S^{k-1}} \Phi(h(t))
\end{equation}
is an eigenvalue of (\ref{eq1.1}). Moreover $\lambda_k\leq \mu_k$. This new
sequence of eigenvalues was first introduced by \cite{DR} and later used
in \cite{CM1, CM} for $a=0, c=p$.
From the Caffarelli-Kohn-Nirenberg inequality (\ref{eq1.2}) or (\ref{eq1.4}),
it is easy to see that
$$
\lambda_1=\mu_1=\inf\{\Phi(u): u\in \mathcal{D}_a^{1,p}(\Omega), J(u)=1\}>0,
$$
and the corresponding eigenfunction $e_1\geq 0$.
To obtain the properties of the eigenvalues of problem (\ref{eq1.1}),
first we need some boundedness and regularity results of the eigenfunctions
of problem (\ref{eq1.1}). In section 2, based on the Moser's iteration technique,
we shall deduce the $L^\infty$ boundedness and
$C^{1,\alpha}(\Omega\setminus \{0\})$ regularity results. In section 3,
we shall obtain the simplicity of the first eigenvalue $\lambda_1$.
In section 4, we shall prove that the first eigenvalue $\lambda_1$ is isolated.
Section 5 is concerned with the properties of the second eigenvalue $\lambda_2$.
\section{Regularity results}
In this section, we will prove the $L^\infty$ boundedness and
$C^{1,\alpha}(\Omega\setminus \{0\})$ regularity results of the weak solution
to problem (\ref{eq1.1}) (cf. \cite{CC, CG} for the case $p=2$).
\begin{theorem} \label{thm2.1}
Assume that $1