\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{On the spectrum of the p-biharmonic operator} { Abdelouahed El Khalil, Siham Kellati \& Abdelfattah Touzani} \begin{document} \setcounter{page}{161} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2002-Fez conference on Partial Differential Equations,\newline Electronic Journal of Differential Equations, Conference 09, 2002, pp 161--170. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % On the spectrum of the p-biharmonic operator % \thanks{ {\em Mathematics Subject Classifications:} 35P30, 34C23. \hfil\break\indent {\em Key words:} p-biharmonic operator, Duality mapping, Palais-Smale condition, \hfil\break\indent unbounded weight. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published December 28, 2002.} } \date{} \author{Abdelouahed El Khalil, Siham Kellati \& Abdelfattah Touzani} \maketitle \begin{abstract} This work is devoted to the study of the spectrum for p-biharmonic operator with an indefinite weight in a bounded domain. \end{abstract} \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{rem}[thm]{Remark} \newtheorem{cor}[thm]{Corrolary} \section{Introduction} Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$, $N\geq 1$, not necessary regular; $1\frac{N}{2p}& \mbox{for }\frac{N}{p}\geq 2\\ =1 &\mbox{for }\frac{N}{p}<2. \end{cases} $$We assume that |\Omega^{+}_{\rho}|\not=0, where \Omega^{+}_{\rho}=\{x\in \Omega; \rho(x)>0\} and \lambda\in \mathbb{R}. We consider the eigenvalue problem$$\begin{gathered} \Delta_p^{2}u=\lambda\rho(x)|u|^{p-2}u \quad\mbox{in } \Omega\\ u\in W_{0}^{2,p}(\Omega). \end{gathered} \eqno{(1.1)} $$Here \Delta_p^{2}:=\Delta(|\Delta u|^{p-2}\Delta u), the operator of fourth order called the p-biharmonic operator. For p=2, the linear operator \Delta_{2}^{2}=\Delta^{2}=\Delta.\Delta is the iterated Laplacian that multiplied with positive constant appears often in Navier-Stokes equations as being a viscosity coefficient. Its reciprocal operator denoted (\Delta^{2})^{-1} is the celebrated Green's operator \cite{Lio}. It is important to indicate that here we don't suppose any boundary conditions on the high order partial derivatives \frac{\partial^{2}u}{\partial x_{i}\partial x_{j}} on the boundary set \partial\Omega of the domain \Omega. The particular case \rho\equiv 1 and u=\Delta u=0 on \partial\Omega was considered recently by Dr\'abek and \^Otani \cite{Dra}. There the authors proved the existence, the simplicity, and the isolation of the first eigenvalue of (1.1) by using a transformation of a problem to a known Poisson's problem, and using the well-known advanced regularity of Agmon-Douglis-Niremberg \cite{Gi-Tr}. Note that this transformation processus is not applicable to our situation because the quantity \Delta u does not necessary vanished on \partial \Omega and the eventual regularity is not required in any bounded domain. The main objective of this work is to show that problem (1.1) has at least one non-decreasing sequence of positive eigenvalues (\lambda_{k})_{k\geq 1}, by using the Ljusternich-schnirelmann theory on C^{1} manifolds, see e.g. \cite{Szu}. Our approach is based only on some properties of the considered operator. So that we give a direct characterization of \lambda_{k} involving a minimax argument over sets of genus greater than k. We set$$\lambda_{1}=\inf\big\{\|\Delta v\|_p^{p}, v\in W_{0}^{2,p}(\Omega); \int_{\Omega}\rho(x)|v|^{p}dx=1\big\}, $$where \|.\|_p denotes the L^{p}(\Omega)-norm. It is not difficult to show that \|\Delta u\|_p defines a norm in W_{0}^{2,p}(\Omega) and W_{0}^{2,p}(\Omega) equipped with this norm is a uniformly convex Banach space for 12 and r>\frac{N}{2p}. Let u,v \in W_{0}^{2,p}(\Omega). By H\"older's inequality, we have$$ \big|\int_{\Omega} \rho(x) |u(x)|^{p-2}u(x)v(x)dx\big|\leq \| \rho\|_{r}\|u\|_{s}^{p-1}\| v\|_{p_{2}} $$where \frac{1}{p_{2}}=\frac{1}{p}-\frac{2}{N} and s is given by$$ \frac{p-1}{s}+\frac{1}{p_{2}}+\frac{1}{r}=1.\eqno{(3.2)} $$Therefore,$$\frac{p-1}{s}=1-\frac{1}{r}-\frac{1}{p_{2}}> 1-\frac{2p}{N}-\frac{1}{p_{2}}=\frac{p-1}{p_{2}}. $$Then it suffices to take$$\max(1,p-1)\frac{N}{2p}$. In this case $W_{0}^{2,p}(\Omega)\hookrightarrow L^{q}(\Omega)$, for any $q\in[p,+\infty[$. There is $q\geq p$ such that $\frac{1}{q}+\frac{1}{r}+\frac{p-1}{p}=\frac{1}{q}+\frac{1}{r}+\frac{1}{p'}=1$.\\ We obtain that $$\frac{1}{q}=\frac{1}{p}-\frac{1}{r}\leq \frac{1}{p}.\eqno{(3.4)}$$ By H\"older's inequality, we arrive at $$\big|\int_{\Omega} \rho(x) |u(x)|^{p-2}u(x)v(x)dx\big| \leq \|\rho\|_{r}\|u\|_p^{p-1}\| v\|_{q},$$ for any $u,v\in W_{0}^{2,p}(\Omega)$. Then $B'$ is also well defined.\\ Third case: $\frac{N}{p}<2$ and $r=1$. In this case $W_{0}^{2,p}(\Omega)\hookrightarrow C(\overline{\Omega})\cap L^{\infty}(\Omega)$. Then for any $u,v \in W_{0}^{2,p}(\Omega)$, we have $$\big|\int_{\Omega} \rho(x) |u(x)|^{p-2}u(x)v(x)dx\big|<\infty,$$ with $\rho\in L^{1}(\Omega)$, and $B'$ is well defined. Step 2. $B'$ is completely continuous. Let $(u_{n}) \subset W_{0}^{2,p}(\Omega)$ be a sequence such that $u_{n}\to u$ weakly in $W_{0}^{2,p}(\Omega)$. We have to show that $B'(u_{n})\to B'(u)$ strongly in $W_{0}^{2,p}(\Omega)$, i.e.,\\ $$\sup_{v\in W_{0}^{2,p}(\Omega)\,\, \|\Delta v\|_p\leq 1} \Big|\int_{\Omega}\rho[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u]v\, dx\big|\to 0,\quad \mbox{as } n\to +\infty.$$ For this end, we distinguish three cases as in step 1 above. For $\frac{N}{p}> 2$, and $r>\frac{N}{2p}$. Let $s$ be as in (3.3). Then \begin{align*} &\sup_{v\in W_{0}^{2,p}(\Omega),\, \|\Delta v\|_p\leq 1} \Big|\int_{\Omega}\rho\big[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u\big]v dx\Big|\\ &\leq \sup_{v\in W_{0}^{2,p}(\Omega),\, \|\Delta v\|_p\leq 1} \big[\|\rho\|_{r} \big\| |u_{n}|^{p-2}u_{n}-|u|^{p-2}u\big\|_{\frac{s}{p-1}} \|v\|_{p_{2}}\big]\\ &\leq c\|\rho\|_{r} \big\| |u_{n}|^{p-2}u_{n}-|u|^{p-2}u\big\|_{\frac{s}{p-1}}, \end{align*} where $c$ is the constant of Sobolev's embedding \cite{Ada}. On the other hand, the Nemytskii's operator $u\mapsto |u|^{p-2}u$ is continuous from $L^{s}(\Omega)$ into $L^{\frac{s}{p-1}}(\Omega)$, and $u_{n}\to u$ weakly in $W_{0}^{2,p}(\Omega)$. So, we deduce that $u_{n}\to u$ strongly in $L^{s}(\Omega)$ because $s0$ such that $$\|v\|_{q}\leq c\|\Delta v\|_p,\quad \forall v\in W_{0}^{2,p}(\Omega).$$ Thus $$\sup_{\stackrel{v\in W_{0}^{2,p}(\Omega)}{\|\Delta v\|_p\leq 1}} \big|\int_{\Omega}\rho[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u]v\,dx\big| \leq c\|\rho\|_{r}\big\| |u_{n}|^{p-2}u_{n}-|u|^{p-2}u \big\|_p^{p-1}.$$ From the continuity of $u\mapsto|u|^{p-1}u$ from $L^{p}(\Omega)$ into $L^{p'}(\Omega)$, and from the compact embedding of $W_{0}^{2,p}(\Omega)$ in $L^{p}(\Omega)$, we have the desired result.\\ If $\frac{N}{p}<2$ and $r=1,$ $W_{0}^{2,p}(\Omega)\subset C(\overline{\Omega}),$ then we obtain $$\sup_{\stackrel{v\in W_{0}^{2,p}(\Omega)}{\|\Delta v\|_p\leq 1}} \big|\int_{\Omega}\rho\big[|u_{n}|^{p-2}u_{n}-|u|^{p-2}u\big]v\,dx\big| \leq c\|\rho\|_{1}\sup_{\overline{\Omega}} \big| |u_{n}|^{p-2}u_{n}-|u|^{p-2}u\big|,$$ where $c$ is the constant given by embedding of $W_{0}^{2,p}(\Omega)$ in $C(\overline{\Omega})\cap L^{\infty}(\Omega)$. It is clear that $$\sup_{\overline{\Omega}}\big| |u_{n}|^{p-2}u_{n}-|u|^{p-2}u\big|\to 0,\quad \mbox{as } n\to +\infty.$$ Hence $B'$ is completely continuous, also in this case.\\ (ii) $\{u_{n}\}$ is bounded in $W_{0}^{2,p}(\Omega)$. Hence without loss of generality, we can assume that $u_{n}$ converges weakly in $W_{0}^{2,p}(\Omega)$ for some function $u\in W_{0}^{2,p}(\Omega)$ and $\|\Delta u_{n}\|_p\to c$. For the rest we distinct two cases: \\ If $c=0$ then $u_{n}$ converges strongly to $0$ in $W_{0}^{2,p}(\Omega)$. If $c\not =0$, then we argue as follows: $$\langle \Delta_p^{2}u_{n},u_{n}-u\rangle =\|\Delta u_{n}\|_p^{p}-\langle \Delta_p^{2}(u_{n}),u\rangle .$$ Applying $\epsilon_{n}$ of (3.1) to $u$, we deduce that $$\Theta_{n}:=\langle \Delta_p^{2}(u_{n}),u\rangle -\|\Delta u\|_p^{p}\langle B'(u_{n}),u\rangle \to 0\quad \mbox{as } n\to +\infty.\eqno{(3.5)}$$ Thus $$\langle\Delta_p^{2}u_{n},u_{n}-u\rangle =\|\Delta u_{n}\|_p^{p}-\Theta_{n}-\|\Delta u_{n}\|_p^{p} \langle B'(u_{n}),u\rangle .$$ That is, $$\langle\Delta_p^{2}u_{n},u_{n}-u\rangle =\|\Delta u_{n}\|_p^{p}(1-\langle B'(u_{n}),u\rangle )-\Theta_{n}.$$ Hence, $$\limsup_{n\to +\infty} \langle \Delta_p^{2}u_{n},u_{n}-u \rangle \leq c^{p}{\limsup_{n\to +\infty}}(1-\langle B'(u_{n}),u\rangle ).$$ On the other hand, from (i) $B'(u_{n})\to B'(u)$ in $W^{-2,p'}(\Omega)$ and $pB(u)=1$, because $pB(u_{n})=1$ for all $n\in\mathbb{N}^*$. So $pB(u)= \langle B'(u),u\rangle =1$. This yields that \begin{align*} 1-\langle B'(u_{n}),u \rangle =& \langle B'(u),u>-0$such that $$c\|\Delta u\|_p\leq\|v\|\leq \frac{1}{c}\|\Delta u\|_p.$$ This implies that the set $$V=F_{k}\cap\{v\in W_{0}^{2,p}(\Omega):B(v)\leq \frac{1}{p} \}$$ is bounded. Thus$V$is a symmetric bounded neighbourhood of$0 \in F_{k}$. By (f) in \cite[Prop. 2.3]{Szu},$\gamma(F_{k}\cap\cal{M})= k. $Because$F_{k}\cap\cal{M}$is compact and$\Gamma_{k}\not=\emptyset$. Now, we claim that$\lambda_{k}\to +\infty$, as$k\to +\infty$. Indeed let be$(e_{n},e_{j}^*)_{n,j}$a bi-orthogonal system such that$e_{n}\in W_{0}^{2,p}(\Omega)$and$e_{j}^*\in W^{-2,p'}(\Omega)$, the$e_{n}$are linearly dense in$W_{0}^{2,p}(\Omega)$; and the$e_{j}^*$are total for$W^{-2,p'}(\Omega)$, see e.g. \cite{Szu}. For$k\in\mathbb{N}^*$, set $$F_{k}=\mathop{\rm span}\{e_{1},\dots,e_{k}\},\quad F_{k}^{\bot}=\mathop{\rm span}\{e_{k+1,e_{k+2},\dots}\}.$$ By (g) of Proposition 2.3 in \cite{Szu}, we have for any A$\in \Gamma_{k},A\cap F_{k-1}^{\bot}\not=\emptyset$. Thus $$t_{k}:=\inf_{A\in\Gamma_{k}}\sup_{u\in A\cap F_{k-1}^{\bot}}pA(u)\to +\infty.$$ Indeed, if not, for$k$is large, there exists$u_{k}\in F_{k-1}^{\bot}$with$\|u_{k}\|_p=1$such that $$t_{k}\leq pA(u_{k})\leq M,$$ for some$M>0$independent of$k$. Thus$\|\Delta u_{k}\|_p\leq M$. This implies that$(u_{k})_{k}$is bounded in$W_{0}^{2,p}(\Omega)$. For a subsequence of$\{u_{k}\}$if necessary, we can assume that$\{u_{k}\}$converge weakly in$W_{0}^{2,p}(\Omega)$and strongly in$L^{p}(\Omega)$. By our choice of$F_{k-1}^{\bot}$, we have$u_{k}\hookrightarrow 0$weakly in$W_{0}^{2,p}(\Omega)$. Because$\langle e_{n}^*,e_{k}\rangle=0$,$\forall k\geq n$. This contradicts the fact that$\|u_{k}\|_p=1\, \forall k$. Since$\lambda_{k}\geq t_{k}$, the claim is proved. This completes the proof. \hfill$\square$\begin{cor} \label{cor3.1} (i)$\lambda_{1}=\inf\{\|\Delta v\|_p^{p}, v\in W_{0}^{2,p}(\Omega); \int_{\Omega}\rho(x)|v|^{p}dx=1\}$.\\ (ii)$0<\lambda_{1}\leq\lambda_{2}\leq\dots\leq\lambda_{n}\to +\infty$. \end{cor} \paragraph{Proof} (i) For$u\in \cal{M}$, we put$K_{1}=\{u,-u\}$. It is clear that$\gamma (K_{1})=1 $, that$A$is even and that $$pA(u)=\max_{K_{1}}pA\geq\inf_{K\in \Gamma_{1}}\max_{K}pA.$$ Hence $$\inf_{u\in\cal{M}}pA(u)\geq \inf_{K\in \Gamma_{1}}\max_{K}pA=\lambda_{1}.$$ On the other hand,$\forall K\in\Gamma_{1},\ \forall u\in K$, $$\sup_{K}pA\geq pA(u)\geq \inf_{u\in\cal{M}}pA(u).$$ So $$\inf_{K\in \Gamma_{1}}\max_{K}pA =\lambda_{1}\geq\inf_{u\in\cal{M}}pA(u).$$ Thus $$\lambda_{1}=\inf_{u\in\cal{M}}pA(u)=\inf\{\|\Delta v\|_p^{p}, v\in W_{0}^{2,p}(\Omega):\int_{\Omega}\rho(x)|v|^{p}dx=1\}.$$ (ii) For all$i\geq j$,$\Gamma_{i}\subset\Gamma_{j}$. From the definition of$\lambda_{i},i\in\mathbb{N}^*$, we have$\lambda_{i}\geq\lambda_{j}$.$\lambda_{n}\to +\infty$is already proved in Theorem \ref{thm3.1}. Which completes the proof. \hfill$\square$\begin{cor}\label{coro3.2} Assume that$|\Omega^{-}_{\rho}|\not=0$with$\Omega^{-}_{\rho}=\{x\in\Omega: \rho(x)<0\}$. Then$\Delta_p^{2}$has a decreasing sequence of the negative eigenvalues$(\lambda_{-n})(\rho)_{n\geq 0}$, such that$\lim_{n\to +\infty}\lambda_{-n}=-\infty$. \end{cor} \paragraph{Proof} First, remark that$\Omega^{-}_{\rho}=\Omega^{+}_{(-\rho)},$so$|\Omega^{+}_{(-\rho)}|=|\Omega^{-}_{\rho}|\not=0$. From Theorem \ref{thm3.1},$\Delta_p^{2}$has an increasing sequence of the positive eigenvalues$\lambda_{n}(-\rho)$, such that$\lim_{n\to+\infty}\lambda_{n}(-\rho)=+\infty$. Note that$\lambda_{n}(-\rho)$satisfies $$\Delta_p^{2}u=\lambda_{n}(-\rho)(-\rho)|u|^{p-2}u =-\lambda_{n}(-\rho)\rho|u|^{p-2}u,$$ for$u\in W_{0}^{2,p}(\Omega)$. Put$ \lambda_{-n}(\rho):=-\lambda_{n}(-\rho)$then$\lambda_{n}(-\rho)_{n\geq 0}$is an increasing positive sequence so$(\lambda_{-n})(\rho)_{n\geq 0}$is a negative decreasing sequence. On the other hand,$\lim_{n\to +\infty}\lambda_{n}(-\rho)=+\infty$. So $$\lim_{n\to +\infty}\lambda_{-n}(\rho)=-\infty.$$ \begin{thebibliography}{00} \frenchspacing \bibitem{Ada} R. A{\sc dams}, {\em Sobolev spaces},{ Academic Press, New-York (1975).} \bibitem{Dra} P . D{\sc r\'abek} and M. \^O{\sc tani}, {\em Global bifurcation result for the p-biharmonic operator}, Electronic Journal of Differential Equations, Vol. 2001(2001), No. 48, 1-19. \bibitem{Gi-Tr} D. G{\sc ilbarg} and N{\sc eil} S. T{\sc rudinger}, {\em Elliptic Partial Differential Equations of second order}, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo {(1983).} \bibitem{Lin} P. L{\sc indqvist}, {\em On the equation$\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u=0$}, Proc. Amer. Math. Soc., 109 (1990), 157-164. \bibitem{Lio} J. L. L{\sc ions}, {\em Quelques m\'ethodes de r\'esolution des probl\`emes aux limites non lin\'eaires,} {Dunod, Paris} {(1969).} \bibitem{Szu}A. S{\sc zulkin}, {\em Ljusternick-Schnirelmann theory on$C^{1}\$-manifolds}, Ann. Inst. Henri Poincar\'e, Anal. Non., 5 (1988), 119-139. \end{thebibliography} \noindent\textsc{Abdelouahed El Khalil} (e-mail: lkhlil@hotmail.com) \\ \textsc{Siham Kellati} (e-mail: siham360@caramail.com)\\ \textsc{Abdelfattah Touzani} (e-mail: atouzani@iam.net.ma)\\[2pt] Department of Mathematics and Informatic,\\ Faculty of Sciences Dhar-Mahraz, \\ P.O. Box 1796 Atlas-Fez, Morocco \end{document}