\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 122, pp. 1--6.\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 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/122\hfil Picone's identity] {Picone's identity for the p-biharmonic operator with applications} \author[J. Jaro\v s\hfil EJDE-2011/122\hfilneg] {Jaroslav Jaro\v s} % in alphabetical order \address{Jaroslav Jaro\v s \newline Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia} \email{jaros@fmph.uniba.sk} \thanks{Submitted July 17, 2011. Published September 27, 2011.} \thanks{Supported by grant 1/0481/08 from the Slovak agency VEGA} \subjclass[2000]{35B05, 35J70} \keywords{$p$-biharmonic operator; Picone's identity} \begin{abstract} In this article, a Picone-type identity for the weighted $p$-bihar\-monic operator is established and comparison results for a class of half-linear partial differential equations of fourth order based on this identity are derived. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} The purpose of this article is to present a Picone-type identity for the weighted $p$-biharmonic operator extending the known formula for a pair of ordinary iterated Laplacians with positive weights $a$ and $A$ which says that if $u, v, a\Delta u$ and $A\Delta v$ are twice continuously differentiable functions with $v(x) \neq 0$, then $$\begin{split} &\operatorname{div} \big[ u\nabla (a \Delta u) - a \Delta u \nabla u - {u^2 \over v} \nabla (A\Delta v) + A\Delta v \nabla \big({u^2 \over v}\big) \big] \\ &= - {u^2 \over v}\Delta (A\Delta v) + u\Delta (a\Delta u) + (A-a)(\Delta u)^2 \\ &\quad - A\big(\Delta u - u {\Delta v \over v}\big)^2 + 2 A {\Delta v \over v} | \nabla u - {u \over v} \nabla v|^2 \end{split} \label{e1.1}$$ (see \cite{d3}). Here $\operatorname{div}, \nabla, \Delta$ are the usual divergence, nabla and Laplace operators and $|\cdot|$ denotes the Euclidean length of a vector in $\mathbb{R}^n$. In \cite{d3}, the integrated form of \eqref{e1.1} was used to obtain a variety of qualitative results (including Sturmian comparison theorems, integral inequalities of the Wirtinger type and lower bounds for eigenvalues) for a pair of linear elliptic partial differential equations of the form \begin{gather} \Delta (a(x)\Delta u) - c(x) u = 0, \label{e1.2}\\ \Delta (A(x) \Delta v) - C(x) v = 0 \label{e1.3} \end{gather} (or for the inequalities $u[\Delta(a(x)\Delta u) - c(x)u] \leq 0$ and $\Delta(A(x)\Delta v) - C(x) v \geq 0$) considered in a bounded domain $G \subset \mathbb{R}^n$ with a piecewise smooth boundary $\partial G$. We extend the formula \eqref{e1.1} to the case where $\Delta (a \Delta u)$ and $\Delta (A\Delta v)$ are replaced by the more general weighted $p$-biharmonic operators $\Delta (a |\Delta u|^{p-2}\Delta u)$ and $\Delta (A |\Delta v|^{p-2}\Delta v), p > 1,$ respectively, and show that some of results in \cite{d3} remain valid for half-linear partial differential equations \begin{gather} \Delta (a(x)|\Delta u|^{p-2}\Delta u) - c(x)|u|^{p-2} u = 0, \label{e1.4} \\ \Delta (A(x)|\Delta v|^{p-2}\Delta v) - C(x)|v|^{p-2} v = 0 \label{e1.5} \end{gather} which reduce to \eqref{e1.2} and \eqref{e1.3} when $p = 2$. This article is organized as follows. In Section 2, we establish several forms of the desired generalization of Picone-Dunninger formula. Next, in Section 3, we illustrate applications of the basic identities by deriving Sturmian comparison theorems and other qualitative results concerning differential equations and inequalities involving the weigthed $p$-bilaplacian. For related results in the particular case $n = 1$ see \cite{j1} (general $p > 1$) and \cite{t1} ($p = 2$). Picone identities for various kinds of half-linear partial differential equations of the second order and their applications can be found in the monographs \cite{d1,y1}. \section{Picone's Identity} Let $G$ be a bounded domain in $\mathbb{R}^n$ with a piecewise smooth boundary $\partial G$ and let $a \in \mathrm{C}^2(\bar{G},\mathbb{R}_+ )$, $A \in \mathrm{C}^2(\bar{G},\mathbb{R}_+)$, $c \in \mathrm{C}(\bar{G},\mathbb{R})$ and $C \in \mathrm{C}(\bar{G}, \mathbb{R})$ where $\mathbb{R}_+ = (0, \infty)$. For a fixed $p > 1$ define the function $\varphi_p: \mathbb{R} \to \mathbb{R}$ by $\varphi_p(s) = |s|^{p-2}s$ for $s \neq 0$ and $\varphi_p(0) = 0$, and consider partial differential operators of the form \begin{gather*} l[u] = \Delta (a(x) \varphi_p(\Delta u)) - c(x)\varphi_p(u),\\ L[v] = \Delta (A(x) \varphi_p(\Delta v)) - C(x)\varphi_p(v) \end{gather*} with the domains $\mathcal{D}_l(G)$ (resp. $\mathcal{D}_L(G)$) defined to be the sets of all functions $u$ (resp. $v$) of class $\mathrm{C}^2(\bar{G},\mathbb{R})$ such that $a(x)\varphi_p(u)$ (resp. $A(x)\varphi_p(v)$) are in $\mathrm{C}^2(G,\mathbb{R}) \cap \mathrm{C}(\bar{G},\mathbb{R})$. Also, denote by $\Phi_p$ the form defined for $X, Y \in \mathbb{R}$ and $p > 1$ by $$\Phi_p(X,Y) := X\varphi_p(X) + (p-1)Y\varphi_p(Y) - p X\varphi_p(Y).$$ From the Young inequality it follows that $\Phi_p(X,Y) \geq 0$ for all $X, Y \in \mathbb{R}$ and the equality holds if and only if $X = Y$. We begin with the following lemma which can be verified by a routine computation. We call it a \emph{weaker form of Picone's identity} because of the relative weak hypothesis that $u$ is an arbitrary twice continuously differentiable function which does not need to satisfy any differential equation or inequality nor even to be in the domain of the operator $l$. \begin{lemma}\label{lem2.1} If $u \in \mathrm{C}^2(\bar{G},\mathbb{R}), v \in \mathcal{D}_L(G)$ and $v$ does not vanish in $G$, then $$\begin{split} &\operatorname{div} \big[-{|u|^p \over \varphi_p(v)}\nabla (A\varphi_p(\Delta v)) + A \varphi_p(\Delta v)\nabla \big({|u|^p \over \varphi_p(v)}\big)\big] \\ &= -{|u|^p \over \varphi_p(v)}L[v] + A|\Delta u|^p - C|u|^p - A\Phi_p\big(\Delta u, u {\Delta v \over v}\big) \\ &\quad + p(p-1)A|u|^{p-2}\varphi_p\big({\Delta v \over v}\big)\big|\nabla u - {u \over v}\nabla v \big|^2. \end{split} \label{e2.1}$$ \end{lemma} An integration of \eqref{e2.1} with the use of the divergence theorem gives the Picone identity in the integral form $$\begin{split} &- \int_{\partial G} {|u|^p \over \varphi_p(v)} {\partial (A\varphi_p(\Delta v)) \over \partial \nu}ds + \int_{\partial G} (p-1)A\varphi_p\big({\Delta v \over v}\big) \big[{\varphi_p \over v}\big(v {\partial u \over \partial \nu} -u{\partial v \over \partial \nu}\big)\big] ds\\ &+ \int_{\partial G} A \varphi_p\big({\Delta v \over v}\big) \varphi_p(u) {\partial u \over \partial \nu}ds \\ &= - \int_G {|u|^p \over \varphi_p(v)} L[v] dx + \int_G \big[A|\Delta u|^p - C|u|^p\big] dx\\ &\quad + \int_G \big[ p(p-1)A|u|^{p-2}\varphi_p\big({\Delta v \over v}\big) \big|\nabla u - {u \over v}\nabla v\big|^2 - A \Phi_p\big(\Delta u, u{\Delta v \over v}\big)\big] dx, \end{split} \label{e2.2}$$ where $\partial/\partial \nu$ denotes the exterior normal derivative, which extends the formula in \cite[Theorem 2.1]{d3}. Adding to \eqref{e2.1} the obvious identity $$\operatorname{div}\big[ u \nabla (a \varphi_p(\Delta u)) - a \varphi_p(\Delta u)\nabla u\big] = ul[u] - a |\Delta u|^p +c |u|^p,$$ which holds for any $u \in \mathcal{D}_l(G)$, yields the following stronger form of Picone's formula. \begin{lemma} \label{lem2.2} If $u \in \mathcal{D}_l(G), v \in \mathcal{D}_L(G)$ and $v(x) \neq 0$ in $G$, then $$\begin{split} &\operatorname{div} \big[ u\nabla (a \varphi_p(\Delta u)) - a \varphi_p(\Delta u) \nabla u - {|u|^p \over \varphi_p(v)} \nabla (A\varphi_p(\Delta v))\\ &\quad + A \varphi_p(\Delta v)\nabla \big({|u|^p \over \varphi_p(v)}\big)\big]\\ &= - {|u|^p \over \varphi_p(v)}L[v] + u l[u] + (A-a)|\Delta u|^p + (c-C)|u|^p - A\Phi_p\big(\Delta u, u {\Delta v \over v}\big) \\ &\quad + p(p-1)A|u|^{p-2}\varphi_p\big({\Delta v \over v}\big) \big|\nabla u - {u \over v}\nabla v\big|^2. \end{split} \label{e2.3}$$ \end{lemma} Again, integrating \eqref{e2.3} and using the divergence theorem we easily obtain the following integral version of the formula which generalizes the result from Dunninger \cite[Theorem 2.2]{d3}: $$\begin{split} &\int_{\partial G} {u \over \varphi_p(v)} \big[ \varphi_p(v){\partial (a\varphi_p(\Delta u)) \over \partial \nu} - \varphi_p(u){\partial(A\varphi_p(\Delta v)) \over \partial \nu} \big] ds \\ &\quad + \int_{\partial G} (p-1)A\varphi_p\big({\Delta v \over v}\big) \big[{\varphi_p(u) \over v} \big(v {\partial u \over \partial \nu} -u{\partial v \over \partial \nu}\big)\big]ds\\ &\quad + \int_{\partial G} {1 \over \varphi_p(v)}{\partial u \over \partial \nu} \big[ A\varphi_p(u)\varphi_p(\Delta v) - a \varphi_p(v)\varphi_p(\Delta u)\big] ds\\ &= \int_G {u \over \varphi_p(v)} \{\varphi_p(v)l[u] - \varphi_p(u)L[v]\} dx\\ &\quad + \int_G \big[(A-a)|\Delta u|^p + (c-C)|u|^p\big] dx\\ &\quad + \int_G \big[ p(p-1)A|u|^{p-2}\varphi_p\big({\Delta v \over v}\big) \big|\nabla u - {u \over v}\nabla v\big|^2 - A\Phi_p\big(\Delta u, u{\Delta v \over v}\big)\big] dx. \end{split} \label{e2.4}$$ \section{Applications} As a first application of identity \eqref{e2.2} we prove the following result. \begin{theorem} \label{thm3.1} If there exists a nontrivial function $u \in \mathrm{C}^2(\bar{G},\mathbb{R})$ such that \begin{gather} u = 0 \quad\text{on } \partial G , \label{e3.1}\\ M_p[u] \equiv \int_G \big[A(x)|\Delta u|^p - C(x)|u|^p\big] dx \leq 0, \label{e3.2} \end{gather} then there does not exist a $v \in \mathcal{D}_L(G)$ which satisfies \begin{gather} L[v] \geq 0 \quad\text{in } G, \label{e3.3}\\ v > 0 \quad\text{on }\partial G, \label{e3.4}\\ \Delta v < 0 \quad\text{in } G . \label{e3.5} \end{gather} \end{theorem} \begin{proof} Suppose to the contrary that there exists a $v \in \mathcal{D}_L(G)$ satisfying \eqref{e3.3}-\eqref{e3.5}. Since $v > 0$ on $\partial G$ and $\Delta u < 0$ in $G$, the maximum principle implies that $v > 0$ on $\bar{G}$. Thus, the integral identity \eqref{e2.2} is valid and it implies, in view of the hypotheses \eqref{e3.1}-\eqref{e3.5}, that \begin{align*} 0 &\geq M_p[u] - \int_G {|u|^p \over \varphi_p(v)}L[v]dx \\ & = - \int_G \big[p(p-1)A|u|^{p-2}\varphi_p\big({\Delta v \over v}\big) \big|\nabla u - {u \over v}\nabla v\big|^2-A\Phi_p\big(\Delta u, u {\Delta v \over v}\big)\big] dx \\ &\geq - \int_G p(p-1)A|u|^{p-2}\varphi_p\big({\Delta v\over v}\big) \big|\nabla u - {u \over v} \nabla v \big|^2 dx \geq 0. \end{align*} It follows that $\nabla u - {u \over v} \nabla v = 0$ in $G$ and therefore $u/v = k$ in $\bar{G}$ for some nonzero constant $k$. Since $u = 0$ on $\partial G$ and $v > 0$ on $\partial G$, we have a contradiction. Hence no $v$ satisfying \eqref{e3.3}-\eqref{e3.5} can exist. \end{proof} \begin{theorem} \label{thm3.2} If there exists a nontrivial $u \in \mathrm{C}^2(\bar{G},\mathbb{R})$ which satisfies \eqref{e3.1} and \eqref{e3.2}, then every solution $v \in \mathcal{D}_L(G)$ of the inequality \eqref{e3.3} satisfying \eqref{e3.5} and $$v(x) > 0 \quad\text{for some } x \in G \label{e3.6}$$ has a zero in $\bar{G}$. \end{theorem} \begin{proof} If the function $v$ satisfies \eqref{e3.3}, \eqref{e3.5} and \eqref{e3.6}, then either $v(x) < 0$ for some $x \in \partial G$, and so $v$ must vanish somewhere in $G$, or $v \geq 0$ on $\partial G$. In the latter case, however, Theorem \ref{thm3.1} implies that $v(x) = 0$ for some $x \in \partial G$, and the proof is complete. \end{proof} As an immediate consequence of Theorem \ref{thm3.2} we obtain the following integral inequality of the Wirtinger type. \begin{corollary} \label{coro3.1} If there exists a $v \in \mathcal{D}_L(G)$ such that $L[v] = 0$, $v > 0$ and $\Delta v < 0$ in $G$, then for any nontrivial function $u \in \mathrm{C}^2(\bar{G},\mathbb{R})$ satisfying $u = 0$ on $\partial G$, we have $$\int_G A(x)|\Delta u|^p dx \geq \int_G C(x) |u|^p dx\,.$$ \end{corollary} As a further application of Picone's identities established in Section 2 we derive the Sturmian comparison theorem. It belongs to weak comparison results in the sense that the conclusion with respect to $v$ applies (similarly as in Theorem \ref{thm3.2}) to $\bar{G}$ rather than $G$. \begin{theorem} \label{thm3.3} If there exists a nontrivial $u \in \mathcal{D}_l(G)$ such that \begin{gather} \int_G ul[u] dx \leq 0, \label{e3.7} \\ u = {\partial u \over \partial \nu} = 0 \quad\text{on } \partial G, \label{e3.8}\\ V_p[u] \equiv \int_G \big[(a-A)|\Delta u|^p + (C-c)|u|^p \big] dx \geq 0, \label{e3.9} \end{gather} then every $v \in \mathcal{D}_L(G)$ which satisfies \eqref{e3.3}, \eqref{e3.5}, \eqref{e3.6} has a zero in $\bar{G}$. \end{theorem} \begin{proof} Suppose that $v(x) \neq 0$ in $\bar{G}$. Then, condition \eqref{e3.6} implies that $v(x) >0$ for all $x \in \bar{G}$ and from the integral Picone's identity \eqref{e2.4} we obtain, in view of \eqref{e3.3},\eqref{e3.5} and \eqref{e3.7}-\eqref{e3.9}, that \begin{align*} 0 &= V_p[u] +\int_G ul[u]dx - \int_G {|u|^p \over v^{p-1}} L[v]dx \\ &\quad - \int_G \big[ p(p-1)A|u|^{p-2}{|\Delta v|^{p-1} \over v^{p-1}} \big|\nabla u - {u \over v}\nabla v\big|^2 - A\Phi_p\big(\Delta u, u{\Delta v \over v}\big)\big] dx \\ &\leq - \int_G p(p-1)A|u|^{p-2}{|\Delta v|^{p-1} \over v^{p-1}}\big|\nabla u - {u \over v}\nabla v\big|^2 dx \leq 0 . \end{align*} Consequently, $\nabla(u/v) = 0$ in $G$; that is, $u/v = k$ in $G$, and hence on $\bar{G}$ by continuity, for some nonzero constant $k$. However, this cannot happen since $u = 0$ on $\partial G$ whereas $v > 0$ on $\partial G$. This contradiction shows that $v$ must vanish somewhere in $\bar{G}$. \end{proof} As a final application of the Picone identity \eqref{e2.4} we obtain a lower bound for the first eigenvalue of the nonlinear eigenvalue problem \begin{gather} \Delta(|\Delta u|^{p-2} \Delta u) = \lambda |u|^{p-2} u \quad \text{in } G, \label{e3.10}\\ u = \Delta u = 0 \quad \text{on }\partial G \label{e3.11} \end{gather} investigated by Dr\'abek and \^Otani \cite{d2}. They proved that for any $p > 1$ the Navier eigenvalue problem \eqref{e3.10}-\eqref{e3.11} considered on a bounded domain $G \in \mathbb{R}^n$ with a smooth boundary $\partial G$, has a principal eigenvalue $\lambda_1$ which is simple and isolated and that there exists strictly positive eigenfunction $u_1$ in $G$ associated with $\lambda_1$ and satisfying $\partial u_1/\partial \nu < 0$ on $\partial G$. Actually, our technique based on the identity \eqref{e2.4} allows to consider more general nonlinear eigenvalue problem \begin{gather} l[u] = \lambda |u|^{p-2} u \quad \text{in } G, \label{e3.12}\\ u = 0, \quad \Delta u + \sigma {\partial u \over \partial \nu} = 0 \quad \text{on }\ \partial G, \label{e3.13} \end{gather} where $0 \leq \sigma \leq +\infty$ (the case $\sigma = +\infty$ corresponds to the boundary condition $\partial u/ \partial \nu = 0$) and $l{u} \equiv \Delta (a \varphi_p (\Delta u)) - c \varphi_p(u)$ as before. \begin{theorem} \label{thm3.4} Let $\lambda_1$ be the first eigenvalue of \eqref{e3.12}-\eqref{e3.13} and $u_1 \in \mathcal{D}_l(G)$ be the corresponding eigenfunction. If there exists a function $v \in \mathcal{D}_L(G)$ such that \begin{gather} v > 0 \quad \text{in } \bar{G}, \label{e3.14}\\ \Delta v \leq 0 \quad \text{in } G \label{e3.15} \end{gather} and if $V_p[u_1] \geq 0$, then $\lambda_1 \geq \inf_{x \in G} \big[{L[v] \over v^{p-1}}\big] .$ \end{theorem} \begin{proof} The identity \eqref{e2.4}, in view of the above hypotheses, implies that \begin{align*} &\lambda_1\int_G|u_1|^p dx - \int_G |u_1|^p {L[v] \over v^{p-1}} dx \\ &= V_p[u_1] + \int_G \big[p(p-1)A|u_1|^{p-2}{|\Delta v|^{p-1} \over v^{p-1}} \big|\nabla u_1 - {u_1 \over v} \nabla v\big|^2 + A\Phi_p(\Delta u_1, u_1\Delta v/v)\big] dx\\ &\quad + \int_{\partial G} \sigma^{p-1}a |{\partial u_1 \over \partial \nu}|^p ds \geq 0, \end{align*} from which the conclusion readily follows. \end{proof} \begin{thebibliography}{0} \bibitem{d1} O. 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