#\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 12, pp. 1--9.\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--2004/12\hfil On the second eigenvalue] {On the second eigenvalue of a Hardy-Sobolev operator} \author[K. Sreenadh\hfil EJDE--2004/12\hfilneg] {K. Sreenadh} \address{Konijeti Sreenadh \hfill\break T.I.F.R. Centre, Post Box No.1234, Bangalore-560012, India} \email{srinadh@math.tifrbng.res.in} \date{} \thanks{Submitted August 12, 2003. Published Janaury 22, 2004.} \subjclass[2000]{35J20, 35J70, 35P05, 35P30} \keywords{$p$-Laplacian, Hardy-Sobolev operator, unbounded domain} \begin{abstract} In this note, we study the variational characterization and some properties of the second smallest eigenvalue of the Hardy-Sobolev operator $L_{\mu}:=-\Delta_{p}-\frac{\mu}{|x|^p}$ with respect to an indefinite weight $V(x)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Propositioin} \newtheorem{corollary}[theorem]{Corollary} \section{ Introduction} Let $\Omega$ be a domain in $\mathbb{R}^N$ containing $0$. We recall the classical Hardy-Sobolev inequality which states that, for $1N/p$ and a closed subset $S$ of measure zero in $\mathbb{R}^N$ such that $\Omega\backslash S$ is connected and $V\in L^{r}_{\rm loc}(\Omega\backslash S)$. \end{itemize} Here we note that there is no global integrability condition assumed on $V^{-}$. This work is motivated by the work in \cite{SW}. The eigenvalue problem with indefinite weights has been studied for the case $\mu=0$ by Szulkin-Willem \cite{SW}. However, some important properties, of the smallest eigenvalue $\lambda_1$, such as simplicity and being isolated were shown only for $p=2$. Recently the author in \cite{Sr2} proved the simplicity of $\lambda_1$ and sign changing nature of eigenfunctions corresponding to other eigenvalues when $\Omega$ is bounded. Infact in \cite{Sr2} the author studied these properties for $L_\mu$. Following the same arguments, one can prove these results in the present case. However, showing that $\lambda_1$ is isolated and characterization of the second smallest eigenvalue, were open questions. To prove these properties, we follow the ideas in \cite{Sr1} and in \cite{CDG}. Here we should mention that our results are new even for the case $\mu=0$. We use the following results in later sections. \begin{proposition}[Boccardo-Murat \cite{BM}] \label{prop1.1} Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ and let $u_{n} \in W^{1,p}(\Omega)$ satisfy $-\Delta_{p} u_{n}=f_{n}+g_{n} \quad \mbox{in } \mathcal{D}'(\Omega)$ and \begin{itemize} \item[(i)] $u_{n} \to u$ weakly in $W^{1,p}(\Omega)$ \item[(ii)] $u_{n} \to u$ in $L^p(\Omega)$ \item[(iii)] $f_{n} \to f$ in $W^{-1,p^{'}}$ \item[(iv)] $g_{n}$ is a bounded sequence of Radon measures. \end{itemize} Then there exists a subsequence $\{u_{n}\}$ of $\{u_{n}\}$ such that $\nabla u_{n} \to \nabla u$ a.e. in $\Omega$. \end{proposition} \begin{proposition}[Brezis-Lieb \cite{BL}] \label{prop1.2} Let $f_{n}\to f$ a.e in $\Omega$ as $n\to \infty$ and $f_{n}$ be bounded in $L^p(\Omega)$, for some $p>1$. Then $\lim_{n\to \infty} \{\|f_{n}\|_{p}-\|f_{n}-f\|_{p}\}=\|f\|_{p}.$ \end{proposition} Let $X$ be a Banach space and let $M=\{u\in X\;\vline \; g(u)=0\}$ with $g\in C^{1}$. Also let $f:X \to \mathbb{R}$ be a $C^{1}$ functional and let $\tilde{f}$ be the restriction of $f$ to $M$. Then we have the following form of the Mountain pass Theorem \cite{St}. \begin{proposition} \label{prop1.3} Let $u,v\in M$ with $u\not\equiv v$ and suppose that $c:=\inf_{h\in \Gamma}\max_{w\in h(t)} f(w)>\max\{f(u), f(v)\}$ where $\Gamma:=\{h\in C([-1,+1],M)\;\vline h(-1)=u \quad \text{and}\quad h(1)=v\}\ne \emptyset$ Also suppose that $\tilde{f}$ satisfies Pailse-Smale (PS) condition on $M$. Then $c$ is a critical value of $\tilde{f}$. \end{proposition} We define the norm $\|\tilde{f}'\|_*=\inf \{\|f'(u)-tg'(u)\|_{X^{*}}: t\in \mathbb{R}\}.$ The variational characterization of the smallest eigenvalue is given by $\lambda_1=\inf_{0\not\equiv u\in W^{1,p}_0(\Omega)} \frac{\int_{\Omega} |\nabla u|^pdx -\int_{\Omega}\frac{|u|^p}{|x|^p}dx}{\int_{\Omega}|u|^p V(x) dx}$ and the corresponding eigenfunction is denoted by $\phi_1$, which is unique under the condition $\int_{\Omega}|\phi|^pV(x)dx=1$ (see \cite{Sr2}). We will prove the following property. \begin{theorem} \label{thmI} The eigenvalue $\lambda_1$ is isolated in the spectrum of $L_{\mu}$. \end{theorem} We will establish the following variational characterisation of the second smallest eigenvalue: $\lambda_2=\inf_{\gamma\in \Gamma} \sup_{u\in \gamma} \frac{\int_{\Omega} |\nabla u|^pdx -\int_{\Omega}\frac{|u|^p}{|x|^p}dx}{\int_{\Omega}|u|^p V(x) dx},$ where $\Gamma=\left\{\gamma\in C\left([-1,1]:M\right)\;\vline \gamma(-1)=-\phi_1, \gamma(1) =\phi_1\right\}$ and $M$ is defined as in the next section. We show also the following property of $\lambda_2$. \begin{theorem} \label{thmII} If $V_a \le V_b$, then $\lambda_2(V_a)\ge \lambda_2(V_b)$. \end{theorem} \section{Proofs of results} In this section we show that $\lambda_1$ is siolated and give a variational characterization for second smallest eigenvalue of $L_{\mu}$. \begin{lemma} \label{lm2.1} The mapping $u\longmapsto \int_{\Omega}V^{+}|u|^pdx$ is weakly continuous. \end{lemma} The proof of this lemma follows from (\ref{eq:new1}) and (H1). We refer the reader to \cite{SW} for more details.\\ Now, we consider the set $M=\Big\{ u \in D^{1,p}_0(\Omega)\;\vline \;\int_{\Omega}|u|^pV(x)=1\Big\}.$ Since $M$ is not a manifold in $D^{1,p}_0(\Omega)$, we define $X=\{u\in D^{1,p}_0(\Omega)\;\vline\; \|u\|_X <\infty \}$, where $\|u\|_{X}^p:=\int_{\Omega}|\nabla u|^p dx + \int_{\Omega} |u|^p V^{-} dx.$ Then $M$ is a $C^{1}$-manifold as a subset of the space $X$. On this space, we define the functional $J_{\mu}(u)=\frac{\int_{\Omega}|\nabla u|^p\,dx -\int_{\Omega}\frac{\mu}{|x|^p}|u|^p\,dx}{\int_{\Omega}|u|^p V\,dx}.$ Let $\tilde{J_\mu}$ denote the restriction of $J_\mu$ to $M$. and let $\|u\|_{L^p(V)}^p=\int_{\Omega}|u|^pV(x) dx$. \begin{lemma} \label{lm2.2} The functional $\tilde{J_{\mu}}$ satisfies the Palais-Smale condition at any positive level. \end{lemma} \begin{proof} Let $\{u_{n}\}$ be a sequence in $M$ such that $J_{\mu}(u_{n})\to \lambda>0$ and $$\label{eq:b1} \langle J_{\mu}(u_{n}),\phi\rangle -J_{\mu}(u_{n})\int_{\Omega}|u_{n}|^{p-2}u_{n} \phi Vdx =o(1).$$ Using Hardy-Sobolev inequality and $u_n\in M$, it follows that $u_{n}$ is bounded in $X$ which gives the existence of a subsequence $\{u_{n}\}$ of $\{u_{n}\}$ and $u$ such that $u_{n}\to u$ weakly in $D^{1,p}_0(\Omega)$. Since $\lambda>0$ we may assume that $J_{\mu}(u_{n})\ge 0$. Using Lemma \ref{lm2.1} and (\ref{eq:b1}), we get $\langle J_{\mu}' (u_{n})-J_{\mu}'(u), u_{n}-u\rangle +J_{\mu}(u_{n}) \int_{\Omega}\left[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u\right](u_{n}-u) V^{-}dx =o(1).$ By Fatou's Lemma, \begin{align*} 0&=\int_{\Omega} \lim_{n\to \infty}\left[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u\right] [u_{n}-u]V^{-}\\ &\le \liminf_{n\to \infty}\int_{\Omega}\left[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u\right] [u_{n}-u]V^{-}dx. \end{align*} Also, $u_{n}$ satisfies $-\Delta_{p}u_{n}-\frac{\mu}{|x|^p}|u_{n}|^{p-2}u_{n}-J_{\mu}(u_{n})|u_{n}| ^{p-2}u_{n} V(x) =o(1) \quad \text{in} \quad \mathcal{D}'(\Omega_{m}),$ where $\Omega_{m}$ is a bounded domain such that $\Omega=\cup_{m=1}^{\infty} \Omega_{m}$. By Proposition \ref{prop1.1}, noting that $\frac{\mu}{|x|^p} |u_{n}|^{p-2} u_{n}+ J_{\mu}(u_n) |u_n|^{p-2}u_n V^{-}$ is a bounded sequence of Radon measures, there exists a subsequence $\{u_{n}^{m}\}$ of $\{u_n\}$ an $u$ such that $\nabla u_{n}^{m}\to \nabla u$ a.e., in $\Omega_{m}$. By the process of diagonalization we can choose a subsequence $\{u_{n}\}$ such that $\nabla u_{n}\to \nabla u$ a.e. in $\Omega$. By Proposition \ref{prop1.2}, we have \begin{gather} \label{eq:2new} \|u_{n}-u\|_{1,p}^p=\|u_{n}\|_{1,p}^p-\|u\|_{1,p}^p+o(1)\\ \label{eq:2new2} \|\frac{u_{n}-u}{|x|}\|_{L^p(1)}^p=\|\frac{u_{n}}{|x|}\|_{L^p(1)}^p-\|\frac{u}{|x|}\|_{L^p(1)}^p+o(1). \end{gather} We also have, by Fatau's lemma, \begin{align*} &\int_{\Omega} V^{-} ( |u_{n}|^p+|u|^p-|u_{n}|^{p-2} u_{n} u -|u|^{p-2} u u_{n} )dx \\ & \ge \int_{\Omega} V^{-} \left( |u_n|^p+|u|^p\right) -\Big( \int_{\Omega} V^{-} |u_n|^p \Big)^{(p-1)/p} \Big(\int_{\Omega} V^{-}|u|^p\Big)^{1/p}\\ &\quad -\Big(\int_{\Omega} V^{-} |u|^p\Big)^{(p-1)/p} \Big(\int_{\Omega} V^{-}|u_n|^p\Big)^{1/p}\\ &=\Big[\Big(\int_{\Omega}V^- |u_n|^p\Big)^{(p-1)/p} - \Big( \int_{\Omega}V^- |u|^p \Big)^{(p-1)/p}\Big]\\ &\quad\times \Big[ \Big(\int_{\Omega} V^- |u_n|^p\Big)^{1/p} -\Big(\int_{\Omega} V^- |u|^p \Big) ^{\frac{1}{p}}\Big] \ge 0\,. \end{align*} Now using (\ref{eq:2new}) and (\ref{eq:2new2}), \begin{align*} o(1)=& \langle J_{\mu}(u_{n})-J_{\mu}(u), (u_{n}-u)\rangle +J_{\mu}(u_{n})\int_{\Omega}[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u](u_{n}-u) V^{-}dx\\ \ge& \int_{\Omega}|\nabla u_{n}-\nabla u|^p-\int_{\Omega}\frac{\mu}{|x|^p} |u_{n}-u|^p +o(1)\\ \ge& \big(1-\frac{\mu}{\lambda_{N}}\big) \|u_{n}-u\|_{1,p}+o(1). \end{align*} i.e., $u_{n} \to u$ in $D^{1,p}_0(\Omega)$. Notice that \begin{align*} o(1) = & \langle J_{\mu}(u_n)-J_{\mu}(u), u_{n}-u\rangle \\ =& \int_{\Omega} V^{-} \left( |u_n|^{p-2}u_n - |u|^{p-2}u\right) (u_n-u) dx+o(1) \ge 0\,. \end{align*} Therefore, $\int_{\Omega} V^{-} |u_{n}|^p dx \to \int_{\Omega} V^{-} |u|^pdx$ and hence $\|u_n\|_{X}\to \|u\|_{X}$. \end{proof} Observe that $\tilde{J_\mu}(u)\ge \lambda_1$ and $\tilde{J_\mu}(\pm \phi_1)=\lambda_1$. So $+ \phi_1$ and $-\phi_1$ are two global minima of $\tilde{J_\mu}$. Now consider $\Gamma=\{ \gamma\in C([-1,1];M) \;\vline \;\gamma(-1)=-\phi_1, \gamma(1)=\phi_1\}.$ By Proposition \ref{prop1.3}, there exists $u\in X$ such that $\tilde{J}_{\mu}'(u) =0$ and $J_{\mu}(u)={\mathcal{C}}$, where $$\label{Cvalue} \mathcal{C} =\inf_{\gamma\in\Gamma} \sup_{u\in\gamma} \tilde{J}_{\mu}(u).$$ \begin{lemma} \label{lm2.3} \begin{itemize} \item[(i)] $M$ is locally arc wise connected \item[(ii)] Any connected open subset $B$ of $M$ is arcwise connected \item[(iii)] if $B'$ is a component of an open set $A$, then $\partial B' \cap B$ is empty. \end{itemize} \end{lemma} The proof of this lemma follows from the fact that $M$ is a Banach Manifold. For a proof we refer the reader to \cite{CDG}. Define $\mathcal{O}=\{u\in M \;\vline \; \tilde{J_{\mu}}(u)< r \}$ \begin{lemma} \label{lm2.4} Each component of $\mathcal{O}$ contains a critical point of $\tilde{J_{\mu}}$. \end{lemma} \begin{proof} Let $\mathcal{O}_1$ be a component of $\mathcal{O}$ and let $d=\inf \{\tilde{J_\mu}(u), u\in \mathcal{O}_1\}$, where $\overline{\mathcal{O}_1}$ is $X$-closure of $\mathcal{O}$. Suppose this infimum is achieved by $v \in \overline{\mathcal{O}_1}$. Then by Lemma \ref{lm2.3} this cannot be in $\partial \mathcal{O}_1$ and hence $v$ is in $\mathcal{O}_1$ and is a critical point of $\tilde{J_\mu}$. Now we show that $d$ is achieved. Let $u_n\in \mathcal{O}_1$ be a minimizing sequence with $\tilde{J_\mu}(u_n)\le d+\frac{1}{n^2}$. By Ekeland Variational Principle, we get $v_n \in \mathcal{O}_1$ such that \begin{gather} \label{eq:Lem2.41} \tilde{J_\mu}(v_n)\le \tilde{J_\mu}(u_n), \\ \|v_n -u_n \|_X \le \frac{1}{n}, \\ \tilde{J_\mu}(v_n)\le \tilde{J_\mu}(v)+\frac{1}{n}\|v_n -v\|_X, \quad \forall v\in \mathcal{O}_1\,. \label{eq:Lem2.42} \end{gather} From (\ref{eq:Lem2.41}) it follows that $\tilde{J_\mu}(v_n)$ is bounded. Now we claim that $\|\tilde{J_\mu}'(v_n)\|_{*}\to 0$. We fix $n$ and choose $w\in X$ tangent to $M$ at $v_n$, i.e., $\int_\Omega |v_n|^{p-2}v_n w V=0$. Now we consider the path $u_t =\frac{v_n +t w}{\|v_n+tw\|_{L^p(V)}}.$ Since $\tilde{J_\mu}(v_n)\le d+\frac{1}{n}0\}$. Then $u \big|_{\mathcal{O}} \in D^{1,p}_0(\mathcal{O})$ \end{lemma} \begin{proof} Let $u_{n} \in C_{c}(\Omega)\cap D^{1,p}_0(\Omega)$ such that $u_{n}\to u$ in $D^{1,p}_0(\Omega)$. Then $u_{n}^{+}\to u^{+}$ in $D^{1,p}_0(\Omega)$. Let $v_{n}=\min(u_{n},u)$ and let $\phi:\mathbb{R}\to \mathbb{R}$ be a $C^{1}$ function such that $$\phi(t)= \begin{cases} 0&\text{for } t\le 1/2\\ 1&\text{for } t\ge 1 \end{cases}$$ and $|\phi'|\le 1$. Let $\psi_{r}(x)=\phi\big(d(x,S)/r\big)$ where $d(x,S)=\mathop{\rm dist}(x,S)$. Then $$\psi_{r}(x)\begin{cases} 0 &\text{for } d(x,S)\le r/2 \\ 1 &\text{for } d(x,S)\ge r \end{cases}$$ and $|\nabla \psi_{r}(x)| \le C/r$ for some constant $C$. Now we define $w_{n,r}(x)=\psi_{r}v_{n}(x)\big|_\mathcal{O}$. Since $\psi_{r}v_{n}\in C(\overline{\Omega})$, we have $w_{n,r} \in C(\overline{\mathcal{O}})$ and vanishes on the boundary $\partial \mathcal{O}$. Indeed for $x\in \partial \mathcal{O}\cap S$ then $\psi_{r}(x)=0$ and so $w_{n,r}(x)=0$. If $x\in \partial \mathcal{O} \cap \Omega$ and $x\notin S$ then $u(x)=0$(since $u$ is continuous except at 0) and so $v_{n}(x)=0$ . If $x\in \partial \Omega$ then $u_{n}(x)=0$ and hence $v_{n}(x)=0$. So in all the cases $w_{n,r}(x)=0$ for $x \in \partial \mathcal{O}$. Therefore, $w_{n,r}\in D^{1,p}_0(\mathcal{O})$. \begin{align*} \int_{\Omega}|\nabla(w_{n,r})-\nabla(\psi_{r}u)|^p &=\int_{\mathcal{O}}|(\nabla \psi_{r}) v_{n}+\psi_{r} \nabla v_{n} -(\nabla \psi_{r}) u -\psi_{r} \nabla u|^p\, dx\\ &\le \|\nabla \psi_{r}v_{n}-\nabla \psi_{r}u\|_{L^p(\mathcal{O})}^p +\|\psi_{r}\nabla v_{n}- \psi_{r} \nabla u\|_{L^p(\mathcal{O})}^p \end{align*} which goes to $0$ as $n\to \infty$. i.e., $w_{n,r}\to \psi_{r}u\big|_{\mathcal{O}}$ in $D^{1,p}_0(\mathcal{O})$. Now $\int_{\mathcal{O}}|\nabla \psi_{r}u +\psi_{r} \nabla u -u|^p \le \int_{\mathcal{O}}|\psi_{r} \nabla u -\nabla u|^p +\int_{\mathcal{O}\cap \{r/2<|x| \lambda_1, it has to change sign in \Omega (see \cite{Sr2}). Let O_1 and O_2 be positive and negative nodal domains of u_a respectively such that \[ \int_{O_1}V_{a}\; (u_{a}^{+})^p\,dx<\int_{O_1}V_{b}\; (u_{a}^{+})^p\,dx \quad \text{and } \int_{O_2}V_{a}\;(u_{a}^{-})^p\,dx \le \int_{O_2}V_{b}\; (u_{a}^{-})^p\,dx.$ By Lemma \ref{lm2.7}, $u_a \big|_{O_1}\in D^{1,p}_0(O_1)$ and also in $L^p(O_1, V^-)$. We have $\lambda_1(O_1,V_{b})\le\frac{\int_{O_1} |\nabla u_a|^p -\frac{\mu}{|x|^p}|u_a|^p}{\int_{O_1}|u_a|^pV_{b}}<\lambda_2(V_{a}).$ Therefore, $\lambda_1(O_1,V_{b})<\lambda_2(V_{a})$. Simillarily $\lambda_1(O_2,V_{b})\le \lambda_2(V_{a})$. Now we modify $O_1$ and $O_2$ to get $\tilde{O_1}$ and $\tilde{O_2}$ with empty intersection and $\lambda_1(\tilde{O_1},V_b)<\lambda_2(V_{a})$ and $\lambda_1(\tilde{O_2},V_b)<\lambda_2$. For $\eta>0$, let $O_1(\eta) = \{x\in O_1 \;\vline \;dist(x,O_1^{c})>\eta\}$. Then $\lambda_1(O_1(\eta),V_b)\ge \lambda_1(O_1,V_b)$ and $\lambda_1(O_1(\eta),V_b)\to \lambda_1(O_1,V_b)$ as $\eta \to 0$. Therefore, there exists $\eta_0 >0$ such that $\lambda_1(O_1(\eta),V_b)<\lambda_2(V_{a})$ for $0<\eta<\eta_0$. Let $x\in \partial O_2\cap \Omega$ and $0<\eta <\min\{\eta_0, \mathop{\rm dist}(x_0, \Omega^{c})\}$. Now define $\tilde{O_2}=O_2\cup B(x_0, \eta/2)$. Then $\tilde{O_2}\cap O_1(\eta) =\emptyset$, $\lambda_1 (\tilde{O_2},V_b)<\lambda_1(O_2, V_b)<\lambda_2(V_{a})$. Now we consider the function $v=v_1-v_2$, where $v_i$ are the extensions by zero outside $\tilde{O_i}$ of the eigenfunctions associated to $\lambda_1 (\tilde{O_i}, V_b)$. Then $v$ satisfies (\ref{eq:b2}). \end{proof} \subsection*{Acknowledgements} The Author was supported by grant 40/1/2002-R\&D-II/165 from the National Board for Higher Mathematics(NBHM), DAE, Govt. of India. \begin{thebibliography}{00} \bibitem{BM} L. Boccardo and F. Murat, \textit{Almost everywhere convergence of gradients of solutions to elliptic and parabolic equations}, Nonlinear Analysis TMA Vol. 19, no.6, 581-597, 1992. \bibitem{BL} H. 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