\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 143, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/143\hfil Picone's identity for a system of PDEs] {Picone's identity for a system of first-order nonlinear partial differential equations} \author[J. Jaro\v{s} \hfil EJDE-2013/143\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 March 31, 2013. Published June 22, 2013.} \subjclass[2000]{35B05} \keywords{Nonlinear differential system; Picone identity; Wirtinger inequality} \begin{abstract} We established a Picone identity for systems of nonlinear partial differential equations of first-order. With the help of this formula, we obtain qualitative results such as an integral inequality of Wirtinger type and the existence of zeros for the first components of solutions in a given bounded domain. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The purpose of this article is to establish a Picone-type identity for the nonlinear differential system \begin{equation} \label{e1.1} \begin{gathered} \nabla u = u A(x) + B(x)\|v\|^{q-2} v,\\ \operatorname{div}v = - C(x)|u|^{p-2}u - D(x)\cdot v, \end{gathered} \end{equation} where $p > 1$ is a constant, $q = p/(p-1)$ is its conjugate, $A(x), D(x) \in C(\overline{\Omega};\mathbb{R}^n)$, $C(x) \in C(\overline{\Omega}, \mathbb{R})$, $B(x) = \operatorname{diag}\{B_1(x),\dots,B_n(x)\}$ is a diagonal matrix with the positive entries defined and continuous in a bounded domain $\Omega \subset \mathbb{R}^n$ with a piecewise smooth boundary $\partial \Omega$ and $u$ and $v$ denote real- and vector-valued functions of $x = (x_1,\dots,x_n)$, respectively, which are continuously differentiable in their domains of definition. Here div and $\nabla$ are the usual divergence and nabla operators, $\|\cdot\|$ is the Euclidean length of a vector in $\mathbb{R}^n$ and the dot is used to denote the scalar product of two vectors in $\mathbb{R}^n$. If the special case $A(x) \equiv 0$ in $\overline{\Omega}$, the system \eqref{e1.1} is equivalent with the second-order half-linear partial differential equation \begin{equation} \operatorname{div} \big( P(x)\|\nabla u\|^{p-2} \nabla u \big) + R(x)\cdot \|\nabla u\|^{p-2} \nabla u + Q(x) |u|^{p-2}u = 0, \label{e1.2} \end{equation} where $$ P(x) = B(x)^{1-p}, \quad R(x) = B(x)^{1-p}D(x), \quad Q(x)= C(x). $$ If the coefficient $P(x)$ is a scalar function, then \eqref{e1.2} reduces to the equation studied in \cite{y1} where the following theorem was proved. \begin{theorem} \label{thmA} Suppose that there exists a nontrivial function $y \in C^1(\overline{\Omega};\mathbb{R})$ such that $y = 0$ on $\partial \Omega$ and \begin{equation} M_\Omega [y] \equiv \int_\Omega \big[ P(x) \big\|\nabla y - \frac{1}{p}\frac{R(x)}{P(x)}y\big\|^p - Q(x)|y|^p\big] dx \leq 0. \label{e1.3} \end{equation} Then every solution $u$ of \eqref{e1.2} must have a zero in $\overline{\Omega}$. \end{theorem} The proof of the above theorem was based on an identity which says that if $u$ is a solution of \eqref{e1.2} satisfying $u(x) \neq 0$ in $\overline{\Omega}$ and $y \in C^1(\overline{\Omega};\mathbb{R})$ is not identically zero in $\Omega$, then \begin{equation} \label{e1.4} \begin{aligned} &\operatorname{div} \Big[|y|^p P(x) \frac{\|\nabla u\|^{p-2}}{|u|^{p-2}u} \nabla u \Big] \\ &= P(x)\big\| \nabla y - \frac{y}{pP(x)}R(x)\big\|^p -Q(x)|y|^p - P(x)\Big\{ \big\| \nabla y - \frac{y}{pP(x)}R(x)\big\|^p \\ &\quad - p\big(\nabla y - \frac{y}{pP(x)}R(x)\big) \cdot \big\|\frac{y}{u}\nabla u \big\|^{p-2} \frac{y}{u}\nabla u +(p-1)\big\|\frac{y}{u}\nabla u\big\|^p\Big\}. \end{aligned} \end{equation} Moreover, if $D(x) \equiv 0$ in $\Omega$, then \eqref{e1.2} reduces to \begin{equation} \operatorname{div} \big[P(x)\|\nabla u\|^{p-2}\nabla u \big] + Q(x)|u|^{p-2} u = 0. \label{e1.5} \end{equation} Identities of Picone type for \eqref{e1.5} (or its special case where $P(x) \equiv 1$ in $\overline{\Omega}$) were established by several authors including Allegretto \cite{a1}, Dunninger \cite{d2}, Kusano et al \cite{k1} and Yoshida \cite{y2} who obtained a variety of qualitative results based on these formulas. For an extension of Picone's identity to the case of pseudo-$p$-Laplacian and anisotropic $p$-Laplacian see Do\v sl\'y \cite{d1} and Fi\v snarov\'a et al \cite{f1}, respectively. As was demonstrated in Ma\v r\'{i}k \cite{m1}, an alternative approach to \eqref{e1.2} and \eqref{e1.5} can be based upon Riccati-type equations and inequalities. While comparison and oscillation theory for equations of the type \eqref{e1.2} and \eqref{e1.5} is well-developed, there appears to be little known for general systems such as \eqref{e1.1}, particularly in the case where $A(x) \neq 0$ or $A(x) \neq D(x)$ in $\Omega$ (for some results concerning the case $p = 2$ see Wong \cite{w1}). The purpose of this article is to generalize Picone's identity for nonlinear partial differential systems of the form \eqref{e1.1} and illustrate its applications by deriving Wirtinger-type inequalities formulated in terms of solutions of the system \eqref{e1.1} and obtaining results about the existence and distribution of zeros of the first component of the solution of \eqref{e1.1}. Our results involve an arbitrary continuous vector-valued function $G(x)$ and particular choices of this function lead to different integral inequalities or criteria for the existence of zeros of first components of solutions of \eqref{e1.1}. They are new even when they are specialized to the case of the damped equation \eqref{e1.2}. This article is organized as follows. In Section 2, the desired generalization of Picone's formula to nonlinear system \eqref{e1.1} is derived and some particular cases of this new identity are discussed. Section 3 contains some applications of the basic formula which include the integral inequalities of the Wirtinger type and theorems about the existence of zeros for components of solutions of system \eqref{e1.1}. \section{Picone's identity} Define $\varphi_p(s) := |s|^{p-2}s,\ s \in \mathbb{R}$, and $\Phi_p(\xi) := \|\xi\|^{p-2}\xi,\ \xi \in \mathbb{R}^n$. Let $\xi, \eta \in \mathbb{R}^n$ and $B$ be a diagonal matrix with positive entries $B_i, \ i = 1, \dots, n$. Define the form $F_B$ by \begin{equation} F_B[\xi,\eta] = \xi \cdot B^{1-p}\Phi_p(\xi) - p\xi \cdot B^{1-p}\Phi_p(\eta) + (p-1)\eta \cdot B^{1-p}\Phi_p(\eta). \label{e2.1} \end{equation} where $B^{1-p} = \operatorname{diag} \{B_1^{1-p},\dots,B_n^{1-p}\}$. The next lemma establishes the generalization of Picone's identity for the nonlinear system \eqref{e1.1}. \begin{lemma} \label{lem2.1} Let $(u,v)$ be a solution of \eqref{e1.1} with $u(x) \neq 0$ in $\overline{\Omega}$. Then, for any $y \in C^1(\overline{\Omega};\mathbb{R})$ and $G \in C(\overline{\Omega},\mathbb{R}^n)$, \begin{equation} \label{e2.2} \begin{aligned} \operatorname{div} \Big[ |y|^p \frac{v}{\varphi_p (u)}\Big] &= \big[\nabla y - yG(x)\big]\cdot B(x)^{1-p} \Phi_p(\nabla y - yG(x)) - C(x)|y|^p \\ &\quad - \big[p\big(A(x)-G(x)\big)+D(x)-A(x)\big]\cdot \frac{|y|^p}{\varphi_p(u)}v \\ &\quad - F_B[\nabla y - y G(x),B(x)y\Phi_q(v)/u]. \end{aligned} \end{equation} \end{lemma} \begin{proof} If $(u,v)$ is a solution of \eqref{e1.1} with $u(x) \neq 0$ and $y \in C^1(\overline{\Omega}, \mathbb{R})$, then a direct computation yields \begin{equation} \operatorname{div} \big[ |y|^p \frac{v}{\varphi_p(u)}\big] = p\frac{\varphi_p(y)}{\varphi_p(u)}\nabla y \cdot v - (p-1) \frac{|y|^p}{|u|^p}\nabla u . v +\frac{|y|^p}{\varphi_p(u)}\operatorname{div}v\,. \label{e2.3} \end{equation} Using \eqref{e1.1}, adding and subtracting the terms $[\nabla y - y G(x)]\cdot B(x)^{1-p}\Phi_p(\nabla y - y G(x))$ and $ pyG(x).B(x)^{1-p}\Phi_p(B(x)\frac{y}{u}\Phi_q(v))$ (=$pyG(x)\cdot \frac{\varphi_p(y)}{\varphi_p(u)}v\big)$ on the right-hand side of \eqref{e2.3}, we obtain \begin{align*} \operatorname{div} \Big[ |y|^p \frac{v}{\varphi_p (u)}\Big] &= \big[\nabla y - yG(x)\big]\cdot B(x)^{1-p}\Phi_p(\nabla y - yG(x))\\ &\quad -C(x)|y|^p - \big[p\big(A(x)-G(x)\big)+D(x)-A(x)\big]\cdot \frac{|y|^p}{\varphi_p(u)}v \\ &\quad -\big\{\big[\nabla y - yG(x)\big]\cdot B(x)^{1-p}\Phi_p\big(\nabla y - y G(x)\big) \\ &\quad - p\big[\nabla y - y G(x)\big]\cdot B(x)^{1-p}\Phi_p\big(B(x)\frac{y}{u}\Phi_q(v)\big) \\ &\quad +(p-1)B(x)\frac{y}{u}\Phi_q(v)\cdot B(x)^{1-p}\Phi_p\big(B(x)\frac{y}{u}\Phi_q(v)\big)\big\}, \end{align*} which is the desired identity \eqref{e2.2}. \end{proof} \begin{remark} \label{rmk2.1} \rm If we put $y(x) \equiv 1$ in \eqref{e2.2} and denote $w = v/\varphi_p(u)$, then \eqref{e2.2} becomes the generalized Riccati equation \begin{equation} \label{e2.4} \begin{aligned} &\operatorname{div} w + \big[pG(x)+(p-1)B(x)\Phi_q(w)\big] \cdot B(x)^{1-p}\Phi_p\big(B(x)\Phi_q(w)\big) \\ &+ \big[ p\big(A(x)-G(x)\big)+D(x)-A(x)\big]\cdot w +C(x) = 0\ . \end{aligned} \end{equation} Moreover, if $G(x) \equiv 0$ and $B(x)$ is a scalar function, then the Riccati-type equation \eqref{e2.4} reduces to \begin{equation} \operatorname{div} w +(p-1)B(x)\|w\|^q + \big[(p-1)A(x)+D(x)\big]\cdot w + C(x) = 0. \label{e2.5} \end{equation} In the particular case where $A(x) \equiv 0$ and $B(x) \equiv 1$ in $\overline{\Omega}$, Equation \eqref{e2.5} has been employed by Ma\v r\'{i}k \cite{m2} as a tool for studying oscillatory properties of damped half-linear PDEs of the form \eqref{e1.2}. \end{remark} \begin{remark} \label{rmk2.2} \rm If $G(x) \equiv 0$ in $\overline{\Omega}$, then \eqref{e2.2} simplifies to \begin{equation} \label{e2.6} \begin{aligned} \operatorname{div}\Big[ |y|^p \frac{v}{\varphi_p(u)}\Big] &= \nabla y \cdot B(x)^{1-p}\Phi_p(\nabla y) - C(x)|y|^p \\ &\quad -\big[ (p-1)A(x)+D(x)\big] \cdot \frac{|y|^p}{\varphi_p(u)}v - F_B[\nabla y, B(x)\frac{y}{u}\Phi_q(v)\big]. \end{aligned} \end{equation} In the particular case $p = 2$, the identity \eqref{e2.6} reduces to the formula used (implicitly) by Wong \cite{w1} in establishing an integral inequality of the Wirtinger type and comparison theorems based on this inequality for the linear system \begin{equation} \nabla u = u A(x) + B(x) v, \quad \operatorname{div}v = - C(x)u - D(x)\cdot v,\label{e2.7} \end{equation} and its Sturmian minorant \begin{equation} \nabla y = y a(x) + b(x) z, \quad \operatorname{div}z = - c(x)y - d(x)\cdot z,\label{e2.8} \end{equation} where the coefficient functions satisfy the same assumptions as above with the only difference that because of the linearity of the problem the matrices $b(x)$ and $B(x)$ are not necessarily diagonal, but are allowed to be any continuous symmetric and positive definite matrices. \end{remark} The choice $G(x) = (1/q)A(x)+(1/p)D(x)$ in \eqref{e2.2} yields \begin{equation} \begin{aligned} &\operatorname{div}\big[ |y|^p \frac{v}{\varphi_p(u)}\big]\\ &= \Big[ \nabla y - y\Big(\frac{A(x)}{q}+\frac{D(x)}{p}\Big)\Big] \cdot B(x)^{1-p}\Phi_p\Big(\nabla y - y\Big(\frac{A(x)}{q}+\frac{D(x)}{p}\Big)\Big) \\ &- C(x)|y|^p - F_B\Big[\nabla y - y\Big(\frac{A(x)}{q} +\frac{D(x)}{p}\Big),B(x)\frac{y}{u}\Phi_q(v)\Big]. \end{aligned}\label{e2.9} \end{equation} Under the further restriction $A(x) \equiv 0$ and $B_1(x) = \dots = B_n(x) =: B(x)$ in $\overline{\Omega}$, the identity \eqref{e2.9} reduces to the following Yoshida's formula for partial differential equations with $p$-gradient terms (see\cite[Theorem 8.3.1]{y1}): \begin{equation} \begin{aligned} &\operatorname{div}\big[ |y|^p \frac{v}{\varphi_p(u)}\big]\\ & = B(x)^{1-p}\big\|\nabla y - \frac{y}{p}D(x)\big\|^p - C(x)|y|^p - F_B\big[ \nabla y - \frac{y}{p}D(x), B(x)\frac{y}{u}\Phi_q(v)\big] \end{aligned} \label{e2.10} \end{equation} which was used in proving Theorem \ref{thmA}. \section{Applications} In what follows, for simplicity we restrict our considerations to the ``isotropic" case where $B_1(x) = \dots = B_n(x) =: B(x)$. In this special case it follows from \cite[Lemma 2.1]{k1} that the form $F_B[\xi,\eta]$ defined by \eqref{e2.1} is positive semi-definite and the equality in $F_B[\xi,\eta] \geq 0$ occurs if and only if $\xi = \eta$. As the first application of the identity \eqref{e2.2} we establish an inequality of the Wirtinger type. \begin{theorem} \label{thm3.1} If there exists a solution $(u,v)$ of \eqref{e1.1} such that $u(x) \neq 0$ in $\overline{\Omega}$ and \begin{equation} \Big[ p \big(A(x)-G(x)\big) + D(x) - A(x) \Big] \cdot \frac{v}{\varphi_p(u)} \geq 0 \label{e3.1} \end{equation} in $\overline{\Omega}$, then the inequality \begin{equation} J_\Omega [y] : = \int_\Omega \big[ B(x)^{1-p} \big\|\nabla y - y G(x) \big\|^p - C(x) |y|^p \big] dx \geq 0 \label{e3.2} \end{equation} holds for any nontrivial function $y \in C^1(\overline{\Omega};\mathbb{R})$ such that $ y = 0$ on $\partial \Omega$. Moreover, if $\big[p(A-G)+D-A\big]\cdot v/\varphi_p(u) \equiv 0$ in $\overline{\Omega}$, then equality in \eqref{e3.2} occurs if and only if $y(x)$ is a solution of \begin{equation} \nabla y = \Big[ G(x)+B(x)\frac{\Phi_q(v)}{u}\Big] y . \label{e3.3} \end{equation} \end{theorem} \begin{proof} Assume that \eqref{e1.1} has a solution $(u,v)$ with $u(x)\neq 0$ in $\overline{\Omega}$ which satisfies \eqref{e3.1}. Let $y(x)$ be a nontrivial continuously differentiable real-valued function such that $y = 0$ on $\partial \Omega$. Integrating \eqref{e2.2} on $\Omega$ and using the divergence theorem we get \begin{align*} 0 &= J_\Omega [y] - \int_\Omega \big[p\big(A(x)-G(x)\big)+D(x)-A(x)\big]\cdot \frac{|y|^p}{\varphi_p(u)}v \,dx \\ &\quad - \int_\Omega F_B[\nabla y - y G(x),B(x)y\Phi_q(v)/u] dx. \end{align*} Since the form $F_B$ is positive semi-definite and the condition \eqref{e3.1} holds, we conclude that $$ 0 \leq J_\Omega [y] $$ as claimed. Clearly, if $\big[p(A-G)+D-A\big]v/\varphi_p(u) \equiv 0$ in $\overline{\Omega}$, then the equality holds in \eqref{e3.2} if and only if $F_B[\nabla y - y G(x),B(x)y\Phi_q(v)/u] \equiv 0$ in $\overline{\Omega}$ which is equivalent with the condition \eqref{e3.3}. \end{proof} As an immediate consequence of the above theorem we have the following result. \begin{corollary} \label{coro3.1} Let $(u,v)$ be a solution of \eqref{e1.1} such that $u(x) \neq 0$ in $\overline{\Omega}$ and \begin{equation} \Big[ p \big(A(x)-G(x)\big) + D(x) - A(x) \Big] \cdot \frac{v}{\varphi_p(u)} \equiv 0 \label{e3.4} \end{equation} in $\overline{\Omega}$. Then, for every nontrivial $y \in C^1(\overline{\Omega};\mathbb{R})$ such that $y = 0$ on $\partial \Omega$, the inequality \eqref{e3.2} is valid. Moreover, the equality holds in \eqref{e3.2} if and only if \begin{equation} \nabla \Big(\frac{y}{u}\Big) = \frac{y}{u}\big(G(x)-A(x)\big) \label{e3.5} \end{equation} in $\Omega$. \end{corollary} \begin{proof} We need to show only that \eqref{e3.5} is equivalent to \eqref{e3.3}. Using the first equation in \eqref{e1.1}, it is easily seen that \begin{align*} \nabla y - \big[ G(x) + B(x)\frac{y}{u}\Phi_q(v)\big] y &= \nabla y - \frac{y}{u}\nabla u + y\big[ A(x)-G(x)\big] \\ &= u \nabla \Big(\frac{y}{u}\Big) + y \big[A(x)-G(x)\big] \\ & = u \Big[ \nabla \Big(\frac{y}{u}\Big) - \frac{y}{u}\big(G(x)-A(x)\big)\Big], \end{align*} from which the assertion follows. \end{proof} In the case where $G(x) \equiv A(x) \equiv D(x)$ in $\overline{\Omega}$, condition \eqref{e3.4} is trivially satisfied and inequality \eqref{e3.2} reduces to $$ \int_\Omega \big[ B(x)^{1-p}\big\|\nabla y - y A(x)\big\|^p - C(x)|y|^p\big] dx \geq 0. $$ Clearly, in this special case the equality in \eqref{e3.2} occurs if and only if $y(x)$ is a constant multiple of $u(x)$. Another choice of $G(x)$ which guarantees the satisfaction of \eqref{e3.4} is $$ G(x) = \frac{(p-1)A(x)+D(x)}{p}. $$ The last result specializes as follows. \begin{corollary} \label{coro3.2} If $(u,v)$ is a solution of \eqref{e1.1} with $u(x) \neq 0$ in $\overline{\Omega}$ and a nontrivial $y \in C^1(\overline{\Omega};\mathbb{R})$ is such that $y = 0$ on $\partial \Omega$, then \begin{equation} J_\Omega[y] = \int_\Omega \Big[ B(x)^{1-p}\big\|\nabla y- y \frac{(p-1) A(x)+D(x)}{p}\big\|^p - C(x)|y|^p\Big] dx \geq 0. \label{e3.6} \end{equation} Furthermore, equality in \eqref{e3.6} occurs if and only if \begin{equation} y(x) = K u(x)\exp \{f(x)\} \quad on \ \overline{\Omega}\label{e3.7} \end{equation} for some constant $K \neq 0$ and some continuous function $f(x)$. \end{corollary} \begin{proof} It suffices to prove \eqref{e3.7}. If \eqref{e3.5} holds, then from \cite[Lemma 2.3]{j1} if follows that there exists a continuous function $f(x)$ such that $y(x)$ is proportional to $u(x)\exp\{f(x)\}$. The proof is complete. \end{proof} The above result can be reformulated as the following theorem which generalizes \cite[Theorem 8.3.2]{y1}. \begin{corollary} \label{coro3.3} If for some nontrivial $C^1$-function $y(x)$ defined on $\overline{\Omega}$ and satisfying $y = 0$ on $\partial \Omega$, the condition \begin{equation} J_\Omega[y] = \int_\Omega \Big[ B(x)^{1-p}\big\|\nabla y- y \frac{(p-1) A(x)+D(x)}{p}\big\|^p - C(x)|y|^p\Big] dx \leq 0 \label{e3.8} \end{equation} holds, then for any solution $(u,v)$ of \eqref{e1.1} the first component $u(x)$ must have a zero in $\overline{\Omega}$. \end{corollary} \begin{thebibliography}{00} \bibitem{a1} W. Allegretto, Y. X. Huang; \emph{A Picone's identity for the p-Laplacian and applications}, Nonlinear Anal. \textbf{32} (1998), 819--830. \bibitem{d1} O. 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