\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 243, 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} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/243\hfil Generalized Picone's identity] {Generalized Picone's identity and its applications} \author[K. Bal \hfil EJDE-2013/243\hfilneg] {Kaushik Bal} \address{Kaushik Bal \newline School of Mathematical Sciences\\ National Institute for Science Education and Research\\ Institute of Physics Campus\\ Bhubaneshwar-751005, Odisha, India} \email{kausbal@gmail.com} \thanks{Submitted July 24, 2013. Published November 8, 2013.} \subjclass[2000]{35J20, 35J65, 35J70} \keywords{Quasilinear elliptic equation; Picone's identity; comparison theorem} \begin{abstract} In this article we give a generalized version of Picone's identity in a nonlinear setting for the $p$-Laplace operator. As applications we give a Sturmian Comparison principle and a Liouville type theorem. We also study a related singular elliptic system. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The classical Picone's identity states that, for differentiable functions $v>0$ and $u\geq 0$, we have \begin{equation}\label{pic2} |\nabla u|^2+\frac{u^2}{v^2}|\nabla v|^2-2\frac{u}{v}\nabla u\nabla v =|\nabla u|^2-\nabla(\frac{u^2}{v})\nabla v\geq0 \end{equation} Later Allegreto-Huang \cite{AlHu} presented a Picone's identity for the $p$-Laplacian, which is an extension of \eqref{pic2}. As an immediate consequence, they obtained a wide array of applications including the simplicity of the eigenvalues, Sturmian comparison principles, oscillation theorems and Hardy inequalities to name a few. This work motivated a lot of generalization of the Picone's identity in different cases see \cite{BoDo, TaJaYo, Ty} and the reference therein. In a recent paper Tyagi \cite{Ty} proved a generalized version of Picone's identity in the nonlinear framework, asking the question about the Picone's identity which can deal with problems of the type: \begin{gather*} -\Delta u=a(x)f(u)\quad\text{in }\Omega,\\ u=0\quad\text{on }\partial\Omega. \end{gather*} where $\Omega$ is a open, bounded subset of $\mathbb{R}^n$. They proved that for differentiable functions $v>0$ and $u\geq 0$ we have \begin{equation}\label{n-pic2} |\nabla u|^2+\frac{|\nabla u|^2}{f'(v)} +(\frac{u\sqrt{f'(v)}\nabla v}{f(v)}-\frac{\nabla u}{\sqrt{f'(v)}})^2 =|\nabla u|^2-\nabla(\frac{u^2}{f(v)})\cdot\nabla v\geq0 \end{equation} where $f(y)\neq 0$ and $f'(y)\geq1$ for all $y\neq0$; $f(0)=0$. Moreover $|\nabla u|^2-\nabla(u^2/f(v))\cdot\nabla v=0$ holds if and only if $u=c v$ for an arbitrary constant $c$. In this article, we generalize the main result of Tyagi \cite{Ty} for the $p$-laplacian operator; i.e, we will give a nonlinear analogue of the Picone's identity for the $p$-Laplacian operator. In this work, we assume the following hypothesis: \begin{itemize} \item $\Omega$ denotes any domain in $\mathbb{R}^n$. \item $10$ and $u\geq0$ be two non-constant differentiable functions in $\Omega$. Also assume that $f'(y)\geq (p-1)[f(y)^{\frac{p-2}{p-1}}]$ for all $y$. Define \begin{gather*} L(u,v)=|\nabla u|^p-\frac{p u^{p-1}\nabla u|\nabla v|^{p-2}\nabla v}{f(v)} +\frac{u^pf'(v)|\nabla v|^p}{[f(v)]^2}.\\ R(u,v)=|\nabla u|^p-\nabla(\frac{u^p}{f(v)})|\nabla v|^{p-2}\nabla v. \end{gather*} Then $L(u,v)=R(u,v)\geq0$. Moreover $L(u,v)=0$ a.e. in $\Omega$ if and only if $\nabla (\frac{u}{v})=0$ a.e. in $\Omega$. \end{theorem} \begin{remark} \rm When $p=2$ and $f(y)=y$ we get the Classical Picone's Identity \eqref{pic2} for Laplacian and when $p=2$ we get back its nonlinear version \eqref{n-pic2}. \end{remark} \begin{proof}[Proof of Theorem \ref{PICN}] Expanding $R(u,v)$ by direct calculation we get $L(u,v)$. To show $L(u,v)\geq0$ we proceed as follows, \begin{align*} L(u,v)&=|\nabla u|^p-\frac{p u^{p-1}\nabla u|\nabla v|^{p-2}\nabla v}{f(v)}+\frac{u^pf'(v)|\nabla v|^p}{[f(v)]^2}\\ &=|\nabla u|^p+\frac{u^pf'(v)|\nabla v|^p}{[f(v)]^2}-\frac{pu^{p-1}|\nabla u||\nabla v|^{p-1}}{f(v)}\\ &\quad+\frac{pu^{p-1}|\nabla v|^{p-2}}{f(v)}\{|\nabla u||\nabla v|-\nabla u\nabla v\}\\ &=p\Bigl(\frac{|\nabla u|^p}{p}+\frac{(u|\nabla v|)^{(p-1)q}}{q[f(v)]^q}\Bigr)-\frac{p}{q}\frac{(u|\nabla v|)^{(p-1)q}}{[f(v)]^q} -\frac{pu^{p-1}|\nabla u||\nabla v|^{p-1}}{f(v)}\\ &\quad+\frac{u^pf'(v)|\nabla v|^p}{[f(v)]^2}+\frac{pu^{p-1}|\nabla v|^{p-2}}{f(v)}\{|\nabla u||\nabla v|-\nabla u.\nabla v\} \end{align*} Recall from Young's inequality, for non-negative $a$ and $b$, we have \begin{equation} ab\leq \frac{a^p}{p}+\frac{b^q}{q} \end{equation} where $\frac{1}{p}+\frac{1}{q}=1$. Equality holds if $a^p=b^q$.\\ So using Young's Inequality we have, \begin{equation}\label{yi} p\Bigl(\frac{|\nabla u|^p}{p}+\frac{(u|\nabla v|)^{(p-1)q}}{q[f(v)]^q}\Bigr) \geq \frac{pu^{p-1}|\nabla u||\nabla v|^{p-1}}{f(v)} \end{equation} Which is possible since both $u$ and $f$ are non negative. Equality holds when \begin{equation}\label{yie} |\nabla u|=\frac{u}{[f(v)]^{\frac{q}{p}}}|\nabla v| \end{equation} Again using the fact that, $f'(y)\geq (p-1)[f(y)^{\frac{p-2}{p-1}}]$ we have \begin{equation}\label{cf} \frac{u^pf'(v)|\nabla v|^p}{[f(v)]^2} \geq \frac{p}{q}\frac{(u|\nabla v|)^{(p-1)q}}{[f(v)]^q} \end{equation} Equality holds when \begin{equation}\label{cfe} f'(y)= (p-1)[f(y)^{\frac{p-2}{p-1}}]\,. \end{equation} Combining \eqref{yi} and \eqref{cf} we obtain $L(u,v)\geq0$. Equality holds when \eqref{yie} and \eqref{cfe} together with $|\nabla u||\nabla v|=\nabla u.\nabla v$ holds simultaneously. Solving for \eqref{cfe} one obtains $f(v)=v^{p-1}$. So when, $L(u,v)(x_0)=0$ and $u(x_0)\neq0$, then \eqref{yi} together with $f(v)=v^{p-1}$ and $|\nabla u||\nabla v|=\nabla u.\nabla v$ yields, \[ \nabla \big(\frac{u}{v}\big)(x_0)=0. \] If $u(x_0)=0$, then $\nabla u=0$ a.e. on $\{u(x)=0\}$ and $\nabla \big(\frac{u}{v}\big)(x_0)=0$. \end{proof} \section{Applications} We begin this section with the application of the above Picone's identity in the nonlinear framework. As is well understood today that Picone's identity plays a significant role in the proof of Sturmian comparison theorems, Hardy-Sobolev inequalities, eigenvalue problems, determining Morse index etc. In this section, following the spirit of \cite{AlHu}, we will give some applications of the nonlinear Picone's identity. \subsection*{Hardy type result} We start this part with a theorem which can be applied to prove Hardy type inequality following the same method as in \cite{AlHu}. \begin{theorem} Assume that there is a $v\in C^1$ satisfying \[ -\Delta_p v\geq\lambda g f(v)\, \quad v>0\quad\text{in }\Omega. \] for some $\lambda>0$ and nonnegative continuous function $g$. Then for any $u\in C_c^{\infty}(\Omega)$; $u\geq0$ it holds that \begin{equation}\label{eer} \int_{\Omega}|\nabla u|^p\geq\lambda\int_{\Omega}g|u|^p \end{equation} where, $f$ satisfies $f'(y)\geq (p-1)[f(y)^{\frac{p-2}{p-1}}]$. \end{theorem} \begin{proof} Let $\Omega_0\subset\Omega$, $\Omega_0$ be compact. Take $\phi\in C^{\infty}_0(\Omega)$, $\phi>0$. By Theorem \ref{PICN}, we have \begin{align*} 0&\leq\int_{\Omega_0} L(\phi,v)\leq\int_{\Omega} L(\phi,v)\\ &=\int_{\Omega} R(\phi,v) =\int_{\Omega}|\nabla \phi|^p-\nabla(\frac{{\phi}^p}{f(v)}) |\nabla v|^{p-2}\nabla v\\ &=\int_{\Omega}|\nabla \phi|^p+\nabla(\frac{{\phi}^p}{f(v)})\Delta_p v\\\ &\leq \int_{\Omega}|\nabla \phi|^p-\lambda\int_{\Omega}g{\phi}^p. \end{align*} Letting $\phi\to u$, we have \eqref{eer}. \end{proof} \subsection*{Sturmium Comparison Principle} Comparison principles always played an important role in the qualitative study of partial differential equation. We present here a nonlinear version of the Sturmium comparison principle. \begin{theorem} Let $f_1$ and $f_2$ are the two weight functions such that $f_1< f_2$ and $f$ satisfies $f'(y)\geq (p-1)[f(y)^{\frac{p-2}{p-1}}]$. If there is a positive solution $u$ satisfying \begin{equation}\nonumber -\Delta_p u=f_1(x)|u|^{p-2}u\;\mbox{for}\;x\in\Omega,\;\;u=0\quad\text{on }\partial\Omega. \end{equation} Then any nontrivial solution $v$ of \begin{equation}\label{nti} \begin{gathered} -\Delta_p v=f_2(x)f(v)\quad\text{for }x\in\Omega,\\ u=0\quad\text{on }\partial\Omega, \end{gathered} \end{equation} must change sign. \end{theorem} \begin{proof} Let us assume that there exists a solution $v>0$ of \eqref{nti} in $\Omega$. Then by Picone's identity we have \begin{align*} 0&\leq\int_{\Omega} L(u,v) =\int_{\Omega} R(u,v)\\ &=\int_{\Omega}|\nabla u|^p-\nabla(\frac{u^p}{f(v)})|\nabla v|^{p-2}\nabla v\\ &=\int_{\Omega}f_1(x)u^p-f_2(x)u^p\\ &=\int_{\Omega}(f_1-f_2)u^p<0, \end{align*} which is a contradiction. Hence, $v$ changes sign in $\Omega$. \end{proof} \subsection*{Liouville type result} In this section we present a Liouville type result for $p$-Laplacian. Existence of solution for some equation having non-variational structure is generally obtained using the bifurcation method and by obtaining a priori estimates. With this in mind we give a proof of Liouville type result motivated by \cite{ItLoSa}. \begin{theorem} Let $c_0>0$, $p>1$ and $f$ satisfy $f'(y)\geq (p-1)[f(y)^{\frac{p-2}{p-1}}]$. Then the inequality \begin{equation}\label{lio} -\Delta_p v\geq c_0 f(v) \end{equation} has no positive solution in $W^{1,p}_{\rm loc}(\mathbb{R}^n)$. \end{theorem} \begin{proof} We start by assuming that $v$ is a positive solution of \eqref{lio}. Choose $R>0$ and let $\phi_1$ be the first eigenfunction corresponding to the first eigenvalue $\lambda_1(B_R(y))$ such that $\lambda_1(B_R(y))0,\quad v>0\quad\text{in }\Omega\\ u=0,\quad v=0\quad\text{on }\partial\Omega. \end{gathered} \end{equation} where $f$ satisfies $f'(y)\geq (p-1)[f(y)^{\frac{p-2}{p-1}}]$. We have the following result. \begin{theorem} Let $(u,v)$ be a weak solution of \eqref{kio} and $f$ satisfy $f'(y)\geq (p-1)[f(y)^{\frac{p-2}{p-1}}]$. Then $u=c_1 v$ where $c_1$ is a constant. \end{theorem} \begin{proof} Let $(u,v)$ be the weak solution of \eqref{kio}. Now for any $\phi_1$ and $\phi_2$ in $W^{1,p}_0(\Omega)$, we have \begin{gather}\label{jhio} \int_{\Omega}|\nabla u|^{p-2}|\nabla u|\nabla {{\phi}_1}dx =\int_{\Omega} f(v)\phi_1 dx,\\ \label{jhik} \int_{\Omega}|\nabla u|^{p-2}|\nabla u|\nabla {{\phi}_2}dx=\int_{\Omega} \frac{[f(v)]^2}{u^{p-1}} \phi_2 dx. \end{gather} Choosing $\phi_1=u$ and $\phi_2=u^p/f(v)$ in \eqref{jhio} and \eqref{jhik} we obtain \[ \int_{\Omega}|\nabla u|^pdx=\int_{\Omega}uf(v)dx =\int_{\Omega} \nabla(\frac{u^p}{f(v)})|\nabla v|^{p-2}\nabla v dx. \] Hence we have \[ \int_{\Omega} R(u,v)dx=\int_{\Omega}\bigl(|\nabla u|^p-\nabla(\frac{u^p}{f(v)})|\nabla v|^{p-2}\nabla v\bigr) dx=0. \] By the positivity of $R(u,v)$ we have that $R(u,v)=0$ and hence \begin{center} $\nabla(\frac{u}{v})=0$ \end{center} which gives $u=c_1 v$ where $c_1$ is a constant. \end{proof} \subsection*{Acknowledgements} The author would like to thank the anonymous referee for his/her useful comments and suggestions. \begin{thebibliography}{0} \bibitem{AlHu} Walter Allegretto, Yin Xi Huang. \newblock A {P}icone's identity for the {$p$}-{L}aplacian and applications. \newblock {\em Nonlinear Anal.}, 32(7):819--830, 1998. \bibitem{BaBaGi} Mehdi Badra, Kaushik Bal, Jacques Giacomoni; \newblock A singular parabolic equation: Existence, stabilization. \newblock {\em J. 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