\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 $$\label{eq1.1} \begin{gathered} -\mathop{\rm div}(|x|^{-ap}|Du|^{p-2}Du)=\lambda |x|^{-(a+1)p+c}|u|^{p-2}u, \quad \mbox{in } \Omega\\ u= 0, \quad \mbox{on } \partial\Omega, \end{gathered}$$ where $\Omega\subset \mathbb{R}^n$ is an open bounded domain with $C^1$ boundary, $0\in \Omega$, $10$. 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 $10$ such that $$\label{eq1.2} \Big(\int_{\mathbb{R}^n}|x|^{-bq}|u|^{q}\,dx \Big)^{p/q}\leq C_{a,b}\int_{\mathbb{R}^n}|x|^{-ap}|Du|^{p}\,dx,$$ where $$\label{eq1.3} \begin{gathered} -\infty< a<\frac{n-p}p,\quad a\leq b\leq a+1,\\ q=p^*(a,b)=\frac{np}{n-dp},\quad d=1+a-b. \end{gathered}$$ 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 $$\label{eq1.4} \Big(\int_{\Omega}|x|^{-\alpha}|u|^{r}\,dx \Big)^{p/r} \leq C \int_{\Omega}|x|^{-ap}|Du|^{p}\,dx,$$ 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 $0\in \Omega$, $10$ 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 $r0$, it follows that \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} 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 $$\label{eq1.5} \lambda_k:=\inf_{A\in\Gamma_k }\max_{u\in A} \Phi(u)$$ 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 $$\label{eq1.6} \mu_k:= \inf_{h\in\mathcal{O}(S^{k-1}, \mathcal{M}) } \max_{t\in S^{k-1}} \Phi(h(t))$$ 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 $10$, 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 $$\label{eq2.1} h(t)=\begin{cases} \mathop{\rm sign}(t) |t|^q, &\mbox{if } |t|\leq k,\\ \mathop{\rm sign}(t) \{qk^{q-1} |t|+(1-q)k^q \}, & \mbox{if } |t|> k, \end{cases}$$ 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 $$\label{eq2.01} h'(t)=\begin{cases} q |t|^{q-1}, &\mbox{if } |t|\leq k,\\ qk^{q-1}, &\mbox{if } |t|> k \end{cases}$$ and $$\label{eq2.02} H(t)=\begin{cases} \dfrac{q^p}{p(q-1)+1} |t|^{p(q-1)+1}\mathop{\rm sign}(t), & \mbox{if } |t|\leq k,\\[3pt] q^p\big( \dfrac{1}{p(q-1)+1} k^{p(q-1)+1}+k^{p(q-1)}(|t|-k) \big) \mathop{\rm sign}(t), &\mbox{if } |t|> k. \end{cases}$$ It is trivial to verify that $$\label{eq2.2} |H(t)|\leq q |h(t)|(h'(t))^{p-1}, \quad |H(t)||t|^{p-1} \leq q^{p} |h(t)|^p,$$ 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 $$\label{eq2.3} \int_\Omega |x|^{-ap}|Du|^{p-2}Du \cdot D\psi \,dx =\lambda \int_\Omega |x|^{-(a+1)p+c}|u|^{p-2}u \psi \,dx.$$ From the definitions of $h, H, \psi$, (\ref{eq2.2}) implies that \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} By the H\"{o}lder inequality, it follows that \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} where and hereafter $C$ is a universal positive constant independent of $k, q$. Inserting (\ref{eq2.05}) into (\ref{eq2.04}), we see that \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} Equation (\ref{eq2.2}) also implies \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} 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 $$\label{eq2.6} \big(\int_\Omega |x|^{-\alpha} |\eta h(u)|^r\,dx \big)^{p/r} \leq C \int_\Omega |x|^{-ap} |D(\eta h(u))|^p\,dx.$$ Thus, substituting (\ref{eq2.4})--(\ref{eq2.6}) into (\ref{eq2.3}), it is easy to show that \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} 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 \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} Letting $k\to \infty$ in (\ref{eq2.7}), the H\"{o}lder inequality implies \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} 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 $$\label{eq2.10} \big(\int_{\Omega} |x|^{-\alpha} |u|^{qr}\,dx \big)^{p/r} \leq q^p C \big( \int_{\Omega} |x|^{-\alpha} |u|^{pqs}\,dx \big)^{1/s},$$ 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 $10$, $u\geq 0$ 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 $10$, $u\geq 0$ 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}: $$\label{eq3.1} \eta=\frac{(u+\varepsilon)^p-(v+\varepsilon)^p}{(u+\varepsilon)^{p-1}} \quad\mbox{and}\quad \frac{(v+\varepsilon)^p-(u+\varepsilon)^p}{(v+\varepsilon)^{p-1}}$$ in the corresponding equations for $u$ and $v$, respectively, where $\varepsilon$ is a positive parameter. Direct calculation implies $$\label{eq3.2} D \eta=\Big\{1+(p-1) \big( \frac{v+\varepsilon}{u+\varepsilon}\big)^p\Big\} Du -p\big( \frac{v+\varepsilon}{u+\varepsilon}\big)^{p-1}Dv,$$ 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 \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} where the last inequality is a consequence of the following simple calculus inequality (cf. \cite{LP}): $$\label{eq3.4} |w_2|^p> |w_1|^p+p|w_1|^{p-2}w_1\cdot (w_2-w_1)$$ for points in $\mathbb{R}^n, w_1\neq w_2, p>1$. By the Lebesgue's Dominated Convergence Theorem, it follows that $$\label{eq3.5} \lim_{\varepsilon\to 0^+} \int_\Omega |x|^{-(a+1)p+c} \big[\dfrac{u^{p-1}}{u_\varepsilon ^{p-1}}-\dfrac{v^{p-1}}{v_\varepsilon ^{p-1}} \big]( u_\varepsilon ^p- v_\varepsilon ^p)\,dx=0.$$ 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 $10$. 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 $$\label{eq4.1} \int_\Omega |x|^{-(a+1)p+c} \big[\lambda_1 \dfrac{u^{p-1}}{u_\varepsilon ^{p-1}} -\lambda \dfrac{v^{p-1}}{v_\varepsilon ^{p-1}} \big]( u_\varepsilon ^p - v_\varepsilon ^p)\,dx\geq 0,$$ and arguing as before, it follows that $$\label{eq4.2} (\lambda_1 -\lambda)\int_\Omega |x|^{-(a+1)p+c} ( u^p- v^p)\,dx\geq 0.$$ 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 $10$. If $v$ is any eigenfunction corresponding to the eigenvalue $\lambda>\lambda_1>0$ and $\mathcal{N}$ is a nodal domain of $v$, then $$\label{eq4.3} |\mathcal{N}|\geq (C\lambda)^{-1/\sigma}$$ 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 $0 0$ in $\mathcal{N}$, the case $v<0$ being completely analogous. Since $v\in \mathcal{D}_a^{1,p}(\Omega)$, then $v|_\mathcal{N}\in \mathcal{D}_a^{1,p}(\mathcal{N})$. Hence the function $w(x)=v(x)$ if $x\in \mathcal{N}$ and $w(x)=0$ if $\Omega\setminus \mathcal{N}$ belongs to $\mathcal{D}_a^{1,p}(\Omega)$. Using $w$ as a test function in the weak equation satisfied by $v$ yields $$\label{eq4.4} \int_\mathcal{N} |x|^{-ap}|Dv|^p\,dx=\lambda \int_\mathcal{N} |x|^{-(a+1)p+c} |v|^p\,dx.$$ For $r\in (p, \frac{n-p}{np})$, let $\alpha= (1+a)r+n(1-\dfrac rp)$. Then the H\"{o}lder inequality implies that \label{eq4.5} \begin{aligned} \int_\mathcal{N} |x|^{-(a+1)p+c} |v|^p\,dx & \leq |\mathcal{N}|^\sigma \Big( \int_\mathcal{N}|x|^{(-(a+1)p+c +\frac {\alpha p}r)s}\,dx \Big)^{1/s} \Big(\int_\mathcal{N}|x|^{-\alpha}|v|^r \Big)^{p/r}\\ & \leq C |\mathcal{N}|^\sigma\Big(\int_\mathcal{N}|x|^{-\alpha}|v|^r \Big)^{p/r}, \end{aligned} since the choice of $s$ implies that $(-(a+1)p+c+\frac {\alpha p}r)s>-n$. On the other hand, the Caffarelli-Kohn-Nirenberg inequality (\ref{eq1.2}) or (\ref{eq1.4}) implies that $$\label{eq4.6} \Big(\int_\mathcal{N}|x|^{-\alpha}|v|^r \Big)^{p/r} \leq C \int_\mathcal{N} |x|^{-ap}|Dv|^p\,dx,$$ where $C=C(a, \alpha)$. Thus (\ref{eq4.4})--(\ref{eq4.6}) imply (\ref{eq4.3}). \end{proof} \begin{corollary}\label{cor4.3} Each eigenfunction has a finite number of nodal domains. \end{corollary} \begin{proof} Let $\mathcal{N}_j$ be a nodal domain of an eigenfunction associated to some positive eigenvalue $\lambda$. It follows from (\ref{eq4.3}) that $$|\Omega|\geq \sum_{j}|\mathcal{N}_j|\geq (C\lambda)^{-1/\sigma}\sum_j 1$$ and the claim follows. \end{proof} \begin{theorem}\label{thm4.4} Suppose that $10$. $\lambda_1$ 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 $$\label{eq5.1} \int_{\mathcal{N}_i}|Dv_i|^{p}\,dx=\underline{\lambda}_2 \int_{\mathcal{N}_i} |x|^{-(a+1)p+c}|v_i|^{p}\,dx.$$ Then for any $u=\sum_{i=1}^r \alpha_i v_i \in A_r$, it follows that $$\label{eq5.2} \Phi(u)=\sum_{i=1}^r |\alpha_i|^p \Phi(v_i)=\underline{\lambda}_2 \sum_{i=1}^r |\alpha_i|^p J(v_i)=\underline{\lambda}_2 .$$ Thus, it follows that $$\label{eq5.3} \lambda_2:=\inf_{A\in\Gamma_2 }\max_{u\in A} \Phi(u) \leq \max_{u\in A_r} \Phi(u)=\underline{\lambda}_2\,,$$ 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} \begin{thebibliography}{00} \bibitem{AT}A. 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