\documentclass{article} \let\rmdefault\sfdefault \def\modra#1{{\color{blue}\bm{#1}}} \def\cervena#1{{\color{red}\bm{#1}}} \def\separuj{\par\smallskip\hrule\kern 0.5pt\hrule \smallskip} \def\separujB{\par\hrule\kern 0.5pt\hrule} \newenvironment{block}{}{} \usepackage{amsfonts,amsmath,amsthm,url,bm} \usepackage{fancybox} \usepackage{mathpazo} \usepackage[latin2]{inputenc} \usepackage[IL2]{fontenc} \newtheorem{theorem}{Theorem} \newtheorem{corollary}{Corollary} \newtheorem{lemma}{Lemma} \newtheorem{Theorem}{Theorem} \def\theTheorem{\Alph{Theorem}} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem{remark}{Remark} \sloppy \everymath{\displaystyle} \usepackage[pdftex,nodirectory]{web} \def\titlepageTrailer{} \margins{.15in}{.15in}{12pt}{.15in} % left,right,top, bottom \screensize{4.5in}{6in} % web.sty dimensions \parindent 0 pt \usepackage{mdwlist} \usepackage{eso-pic} \definecolor{mygreen}{RGB}{120,190,20} \definecolor{mygreen}{RGB}{10,80,40} \definecolor{webgreen}{RGB}{10,80,40} \definecolor{seda}{gray}{0.31} \definecolor{webgreen}{RGB}{120,190,20} \AddToShipoutPicture{\hbox to 0 pt{\hbox to \paperwidth{\color{mygreen}\vrule width 0.5em height\paperheight\color{black}%\hskip -0.5 em \hskip 0 pt plus 1 fill \raise 1 pt\hbox {\normalfont\tiny \color{gray}\textbf{CDDEA 2010, Rajecké Teplice} (\thepage/12)} \hskip 0 pt plus 1 fill }}}% \def\qed{} \def\lambdamin{\lambda_{\text{\rm{min}}}} \def\lambdamax{\lambda_{\text{\rm{max}}}} \makeatletter\let\over\@@over\makeatother \def\theenumi{\roman{enumi}} \def\labelenumi{\textrm{\upshape{(\theenumi)}}} \def\konst{\textrm{const}} \def\div{\mathop{\hbox{\rm div}}} \def\meas{\mathop{\hbox{\rm meas}}} \def\sgn{\mathop{\hbox{\rm sgn}}} \def\laplac{\Delta} \def\R{\mathbb{R}} \def\N{\mathbb{N}} \def\dxi{\,\mathrm{d}\xi\,} \def\dx{\,\mathrm{d}x\,} \def\dS{\,\mathrm{d}\sigma\,} \def\dt{\,\mathrm{d}t\,} \def\dT{\,\mathrm{d}T\,} \def\du{\,\mathrm{d}u\,} \def\ds{\,\mathrm{d}s\,} 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\def\ss#1#2{\left\langle#1,#2\right\rangle} \makeatletter \renewcommand\maketitle {% \thispagestyle{empty}% \null\bigskip\bigskip \ifeqforpaper\vspace*{2\baselineskip}% \else \vbox to\titleauthorproportion\textheight\bgroup% \fi \noindent\makebox[\linewidth]{\parbox{\linewidth}% {\bfseries\color{\webuniversity@color}\ifeqforpaper\large\fi \centering\webuniversity}}\par\ifeqforpaper\else\minimumskip\fi \vspace{\stretch{1}}% \noindent\makebox[\linewidth]{% \parbox{\hproportionwebtitle\linewidth}% {\bfseries\color{\webtitle@color}\ifeqforpaper\Large\else \large\fi\centering\webtitle}}\par\ifeqforpaper \vspace{2\baselineskip}\else\minimumskip\vspace{\stretch{1}}\fi \noindent\makebox[\linewidth]{% \parbox{\hproportionwebauthor\linewidth}% {\bfseries\color{\webauthor@color}\ifeqforpaper \large\fi\centering\webauthor}} \ifeqforpaper\else \egroup % end of \vbox for title and author \fi\bigskip \optionalpagematter \par\vspace{\stretch{1}} \ifx\web@directory@option y\webdirectory\fi \par\ifeqforpaper\else\minimumskip\fi\vspace{\stretch{1}} \vfill\noindent\begingroup \trailerFontSize\titlepageTrailer\par\endgroup \newpage } \makeatother \pagestyle{empty}% %\usepackage[inactive]{fancytooltips} \begin{document} \def\TooltipRefmark{\hbox{\ \ }} \def\TooltipExtratext{\hbox{\ \ }} \title{Conjugacy criteria for half-linear ODE \\in theory of PDE\\ with generalized $p$-Laplacian\\ and mixed powers\\[15pt]} \author{Robert Ma\v r\'\i k\\[6mm]Dpt. of Mathematics\\ Mendel University\\Brno, CZ } \date{} \maketitle \begin{equation} \begin{aligned} \div\left(A(x)\norm{\nabla y}^{p-2}\nabla y\right)&+ \ss{\vec b(x)}{\norm{\nabla y}^{p-2}\nabla y}\\&+ c(x)|y|^{p-2}y+\sum_{i=1}^m c_i(x)|y|^{p_i-2}y=e(x), \end{aligned} \tag{E} \end{equation} \begin{itemize} \item $x=(x_1,\ldots,x_n)_{i=1}^n\in\R^n$, $p>1$, $p_i>1$, \item $A(x)$ is elliptic $n\times n$ matrix with differentiable components, $c(x)$ and $c_i(x)$ are H\"older continuous functions, $\vec b(x)=\bigl(b_1(x),\ldots,b_n(x)\bigr)$ is continuous $n$-vector function, \item $\nabla=\left({\partial \over\partial x_1},\ldots,{\partial \over\partial x_n}\right)_{i=1}^n$ and $\div={\partial \over\partial x_1}+\cdots+{\partial \over\partial x_n}$ is are the usual nabla and divergence operators, \item $q$ is a conjugate number to the number $p$, i.e., $q=\frac p{p-1}$, \item $\ss{\cdot}{\cdot}$ is the usual scalar product in $\R^n$, $\Vert{\cdot}\Vert$ is the usual norm in $\R^n$, $\Vert A\Vert =\sup\left\{\Vert Ax\Vert: x\in \R^n \text{ with } \Vert x\Vert =1\right\}=\lambdamax$ is the spectral norm \item \textbf{solution} of \eqref{eq:E} in $\Omega\subseteq \R^n$ is a differentiable function $u(x)$ such that $A(x)\Vert\nabla u(x)\Vert^{p-2}\nabla u(x)$ is also differentiable and $u$ satisfies \eqref{eq:E} in $\Omega$ \item $ S(a)=\{x\in\R^n: \Vert x\Vert =a\}$, \\$ \Omega(a)=\{x\in\R^n:a\leq \Vert x\Vert \}$, \\$ \Omega(a,b)=\{x\in\R^n:a\leq\Vert x\Vert \leq b\}$ \end{itemize} \newpage % \begin{equation} % {\shadowbox{$\div\Bigl(A(x)\Vert\nabla u\Vert^{p-2}\nabla u\Bigr) + \ss{\vec b(x)}{\Vert\nabla u\Vert^{p-2}\nabla u}+c(x)|u|^{p-2}u=0$}} \tag{E} % \end{equation} \nadpis {Concept of oscillation for ODE} \begin{equation} u''+c(x)u=0 \label{eq1} \end{equation} \begin{itemize} \item Equation \eqref{eq1} is oscillatory if each solution has infinitely many zeros in $[x_0,\infty)$. \item Equation \eqref{eq1} is oscillatory if each solution has a zero $[a,\infty)$ for each $a$. \item Equation \eqref{eq1} is oscillatory if each solution has conjugate points on the interval $[a,\infty)$ for each $a$. \item All definition are equivalent (no accumulation of zeros and Sturm separation theorem). \item Equation is oscillatory if $c(x)$ is large enough. Many oscillation criteria are expressed in terms of the integral $\int^\infty c(x)\dx$ (Hille and Nehari type) \item There are oscillation criteria which can detect oscillation even if $\int^\infty c(x)\dx$ is extremly small. These criteria are in fact series of conjugacy criteria. \end{itemize} \newpage \nadpis Equation with mixed powers \begin{equation} \label{eq:Sun} (p(t)u')'+c(t)u+\sum_{i=1}^m c_i(t)|u|^{\alpha _i}\sgn u=e(t) \end{equation} where $\alpha_1>\cdots >\alpha_m>1>\alpha_{m+1}>\cdots>\alpha_n>0$. \begin{Theorem}[Sun,Wong (2007)] \label{theorem:sun_wong} If for any $T\geq 0$ there exists $a_1$, $b_1$, $a_2$, $b_2$ such that $T\leq a_1p$ for every $i$). % S(r)}\cervena{\lambdamax(x)}}$ plays a crucial role in the linear % case and $\boxed{\rho(r)\geq \max_{x\in S(r)}\cervena{\frac{\Vert % {A(x)}\Vert ^p_F}{\lambdamin^{p-1}(x)}}}$ plays similar role % if $p>1$. This phenomenon can be observed also in other % oscillation criteria than Theorems B and C. We know that % $\rho(r)\geq \lambda(r)$. Why such a discrepancy appears? \end{itemize*} \newpage \begin{equation} \begin{aligned} \div\left(A(x)\norm{\nabla y}^{p-2}\nabla y\right)&+ \ss{\vec b(x)}{\norm{\nabla y}^{p-2}\nabla y}\\&+ \modra{c(x)|y|^{p-2}y}+\cervena{\sum_{i=1}^m c_i(x)|y|^{p_i-2}y}=\cervena{e(x)}, \end{aligned} \tag{E} \label{eq:E} \end{equation} \nadpis Modus operandi \begin{itemize} \item Get rid of terms $\sum_{i=1}^m c_i(x)|y|^{p_i-2}y$ and $e(x)$ (join with $c(x)|y|^{p-2}y$) and convert the problem into \begin{equation*} \div\left(A(x)\norm{\nabla y}^{p-2}\nabla y\right)+ \ss{\vec b(x)}{\norm{\nabla y}^{p-2}\nabla y}+\modra{C(x)|y|^{p-2}y}=0. \end{equation*} \item Derive Riccati type inequality in $n$ variables. \item Derive Riccati type inequality in $1$ variable. \item Use this inequality as a tool which transforms results from ODE to PDE. \end{itemize} \newpage Using generalized AG inequality $\sum \alpha _i\geq \prod\left(\frac{\alpha_i}{\eta_i}\right)^{\eta_i}$, if $\alpha_i\geq 0$, $\eta_i>0$ and $\sum \eta_i=1$ we eliminate the right-hand side and terms with mixed powers. \begin{lemma}\label{lemma:est1} Let either $y>0$ and $e(x)\leq 0$ or $y<0$ and $e(x)\geq 0$. Let $\eta_i>0$ be numbers satisfying $\sum_{i=0}^m{\eta_i}=1$ and $\eta_0+\sum_{i=1}^m p_i\eta_i=p$ and let $c_i(x)\geq 0$ for every $i$. Then \begin{equation*}%\label{eq:est1} \frac{1}{|y|^{p-2}y}\left(-e(x)+\sum_{i=1}^m c_i(x)|y|^{p_i-2} y\right)\geq C_1(x), \end{equation*} where \begin{equation} \label{eq:C1} C_1(x):=\left|\frac{e(x)}{\eta_0}\right|^{\eta_0} \prod_{i=1}^m\left(\frac{c_i(x)}{\eta_i}\right)^{\eta_i}. \end{equation} \end{lemma} %\begin{remark} \textbf{Remark:} The numbers $\eta_i$ from Lemma \ref{lemma:est1} exist, if $p_i>p$ for some $i$. %\end{remark} % The following lemma is a modification of Lemma \ref{lemma:est1} in % the case $e(x)\equiv 0$. \begin{lemma}\label{lemma:est10} Suppose $c_i(x)\geq 0$. Let $\eta_i>0$ be numbers satisfying $\sum_{i=1}^m{\eta_i}=1$ and $\sum_{i=1}^m p_i\eta_i=p$. Then \begin{equation*}%\label{eq:est10} \frac{1}{|y|^{p-2}y}\sum_{i=1}^m c_i(x)|y|^{p_i-2}y\geq C_2(x), \end{equation*} where \begin{equation} \label{eq:C2} C_2(x):=\prod_{i=1}^m\left(\frac{c_i(x)}{\eta_i}\right)^{\eta_i} \end{equation} \end{lemma} % \begin{remark} \textbf{Remark:} The numbers $\eta_i$ from Lemma \ref{lemma:est10} exist iff $p_i>p$ for some $i$ and $p_j0$. % \begin{enumerate} % \item If $\alpha<\beta$ and $b>0$, then $b-ax^\alpha\geq -x^\beta \left(\frac{a(\beta-\alpha)}{b\beta}\right)^{\frac\beta\alpha} \frac{b\alpha}{\beta-\alpha}$ % \label{pa} % \item If $\alpha>\beta$ and $b\geq0$, then $ax^\alpha+b\geq x^\beta \left(\frac{a(\alpha-\beta)}{b\beta}\right)^{\frac\beta\alpha} \frac{b\alpha}{\alpha-\beta}$ % \label{pb} % \end{enumerate} % \end{lemma} % Another possibility how to remove the right hand side and terms with % mixed powers is available if we rewrite % \begin{equation*} % \frac{1}{|y|^{p-2}y}\left(-e(x)+\sum_{i=1}^m c_i(x)|y|^{p_i-2}y\right) % \end{equation*} % into the form % \begin{equation*} % \sum_{i=1}^m \left(c_i(x)|y|^{p_i-p}-\frac{\epsilon_i e(x)}{|y|^{p-2}y} \right), \quad \epsilon_i>0, \quad \sum_{i=1}^m\epsilon_i=1 % \end{equation*} % study the family of min/max problems % for terms in this sum. % \bigskip % \begin{lemma}\label{lemma:estimate2} % Let $e(x)<0$ and $y>0$. Then % \begin{equation*}%\label{eq:estimate2} % \sum_{i=1}^m c_i(x)|y|^{p_i-p}-\frac{e(x)}{|y|^{p-2}y} % \geq C_3(x), % \end{equation*} % where % \begin{multline} % \label{eq:C3} % C_3(x):=\sum_{i\in I_1} % \left(\left[\frac{[c_i(x)]_+(p_i-p)}{\epsilon_i|e(x)|(p-1)}\right]^{(p-1)/(p_i-1)}\frac{\epsilon_i|e(x)|(p_i-1)}{p_i-p}\right)\\ % - \sum_{i\in I_2}\left(\left[\frac{[-c_i(x)]_+(p-p_i)}{\epsilon_i|e(x)|(p-1)}\right]^{(p-1)/(p_i-1)}\frac{\epsilon_i|e(x)|(p_i-1)}{p-p_i}\right), % \end{multline} % $I_1=\{i\in[1,m]\cap \N:p_i>p\}$ and $I_2=\{i\in[1,m]\cap \N:p_i0$, $\sum_{i=1}^m\epsilon_i=1$. Moreover, if % $I_2=\{\}$, then the inequality $e(x)<0$ can be relaxed to % $e(x)\leq 0$. % \end{lemma} % \newpage \begin{lemma}\label{lemma:cC} Let $y$ be a solution of \eqref{eq:E} which does not have zero on $\Omega$. Suppose that there exists a function $C(x)$ such that \begin{equation*} C(x)\leq c(x)+\sum_{i=1}^m c_i(x)|y|^{p_i-p}-\frac{e(x)}{|y|^{p-2}y} % \label{ineq:C} \end{equation*} Denote $\vec w(x)=A(x)\frac{\norm{\nabla y}^{p-2}\nabla y}{|y|^{p-2}y}$. The function $\vec w(x)$ is well defined on $\Omega$ and satisfies the inequality \begin{equation} \label{eq:RIC} \div \vec w+(p-1)\Lambda(x) \norm{\vec w}^q+\ss{\vec w}{A^{-1}(x)\vec b(x)}+C(x)\leq 0 \end{equation} where \begin{equation}\label{eq:Lambda} \Lambda(x)= \begin{cases} \lambda_{{\max}}^{1-q}(x)& % \text{ for } 12. \end{cases} \end{equation} \end{lemma} \begin{lemma}\label{lemma:alpha} Let \eqref{eq:RIC} hold. Let $l>1$, $l^*=\frac{l}{l-1}$ be two mutually conjugate numbers and $\alpha \in C^1(\Omega,\R^+)$ be a smooth function positive on $\Omega$. Then \begin{multline*} % \label{eq:RIC2} \div (\alpha(x)\vec w)+ (p-1)\frac {\Lambda(x)\alpha^{1-q}(x)}{l^*} \norm{\alpha(x)\vec w}^q\\ -\frac{l^{p-1}\alpha(x)}{ p^p \Lambda^{p-1}(x)}\norm{A^{-1}(x)\vec b(x)-\frac{\nabla \alpha(x)}{\alpha(x)}}^p +\alpha(x)C(x)\leq 0 \end{multline*} holds on $\Omega$. If $\norm{A^{-1}\vec b-\frac{\nabla \alpha}\alpha}\equiv 0$ on $\Omega$, then this inequality holds with $l^*=1$. \end{lemma} \newpage \begin{theorem}\label{lemma:radialODE} Let the $n$-vector function $\vec w$ satisfy inequality \begin{equation*} \div \vec w+C_0(x)+(p-1)\Lambda_0(x)\norm{\vec w}^q\leq 0 \end{equation*} on $\Omega(a,b)$. Denote $\tilde C(r)=\int_{S(r)}C_0(x)\dS$ and $\tilde R(r)=\int_{S(r)}\Lambda_0^{1-p}\dS$. Then the half-linear ordinary differential equation \begin{equation*}%\label{eq:radialODE} \left(\tilde R(r) |u'|^{p-2}u\right)'+\tilde C(r) |u|^{p-2}u=0, \qquad {}'=\frac{\mathrm{d}}{\dr} \end{equation*} is disconjugate on $[a,b]$ and it possesses solution which has no zero on $[a,b]$. \end{theorem} \begin{theorem}\label{th1} Let $l>1$. Let $l^*={1}$ if $\norm{\vec b}\equiv 0$ and $l^*=\frac{l}{l-1}$ otherwise. Further, let $c_i(x)\geq 0$ for every $i$. Denote \begin{equation*}%\label{eq:tildeR} \tilde R(r)=(l^*)^{p-1}\int_{S(r)}\Lambda^{1-p}(x)\dS \end{equation*} and \begin{equation*} \tilde C(r)=\int_{S(r)}c(x)+C_1(x)-\frac{l^{p-1}}{ p^p \Lambda^{p-1}(x)}\norm{A^{-1}(x)\vec b(x)}^p\dS, \end{equation*} where $\Lambda(x)$ is defined by \eqref{eq:Lambda} and $C_1(x)$ is defined by \eqref{eq:C1}. Suppose that the equation \begin{equation*}%\label{eq:th1} \left(\tilde R(r)|u'|^{p-2}u'\right)'+\tilde C(r) |u|^{p-2}u=0 \end{equation*} has conjugate points on $[a,b]$. If $e(x)\leq 0$ on $\Omega(a,b)$, then equation \eqref{eq:E} has no positive solution on $\Omega(a,b)$. If $e(x)\geq 0$ on $\Omega(a,b)$, then equation \eqref{eq:E} has no negative solution on $\Omega(a,b)$. \end{theorem} \begin{theorem}[non-radial variant of Theorem \ref{th1}]\label{th1a} Let $l>1$ and let $\Omega\subset\Omega(a,b)$ be an open domain with piecewise smooth boundary such that $\meas(\Omega \cap S(r))\neq 0$ for every $r\in[a,b]$. Let $c_i(x)\geq 0$ on $\Omega$ for every $i$ and let $\alpha(x)$ be a function which is positive and continuously differentiable on $\Omega$ and vanishes on the boundary and outside $\Omega$. Let $l^*=1$ if $\norm{A^{-1}\vec b-\frac{\nabla \alpha}{\alpha}}\equiv 0$ on $\Omega$ and $l^*=\frac{l}{l-1}$ otherwise. In the former case suppose also that the integral \begin{equation*} \int_{S(r)}\frac{\alpha(x)}{ \Lambda^{p-1}(x)}\norm{A^{-1}(x)\vec b(x)-\frac{\nabla\alpha(x)}{\alpha(x)}}^p\dS \end{equation*} which may have singularity on $\partial \Omega$ if $\Omega\neq\Omega(a,b)$ is convergent for every $r\in[a,b]$. Denote \begin{equation*} \tilde R(r)=(l^*)^{p-1}\int_{S(r)}\alpha(x)\Lambda^{1-p}(x)\dS \end{equation*} and \begin{equation*} \tilde C(r)=\int_{S(r)}{\cervena{\alpha(x)}}\left(c(x)+C_1(x)-\frac{l^{p-1}}{ p^p \Lambda^{p-1}(x)}\norm{A^{-1}(x)\vec b(x)-\frac{\nabla\alpha(x)}{\alpha(x)}}^p\right)\dS, \end{equation*} where $\Lambda(x)$ is defined by \eqref{eq:Lambda} and $C_1(x)$ is defined by \eqref{eq:C1} and suppose that equation \begin{equation*} \left(\tilde R(r)|u'|^{p-2}u'\right)'+\tilde C(r) |u|^{p-2}u=0 \end{equation*} has conjugate points on $[a,b]$. If $e(x)\leq 0$ on $\Omega(a,b)$, then equation \eqref{eq:E} has no positive solution on $\Omega(a,b)$. If $e(x)\geq 0$ on $\Omega(a,b)$, then equation \eqref{eq:E} has no negative solution on $\Omega(a,b)$. \end{theorem} \newpage \begin{theorem}\label{th2} Let $l$, $\Omega$, $\alpha(x)$, $\Lambda(x)$ and $\tilde R(r)$ be defined as in Theorem \ref{th1a} and let $c_i(x)\geq 0$ and \cervena{$e(x)\equiv 0$} on $\Omega(a,b)$. Denote \begin{equation*} \tilde C(r)=\int_{S(r)}\alpha(x)\left(c(x)+C_2(x)-\frac{l^{p-1}}{ p^p \Lambda^{p-1}(x)}\norm{A^{-1}(x)\vec b(x)-\frac{\nabla \alpha(x)}{\alpha(x)}}^p\right)\dS, \end{equation*} where $C_2(x)$ is defined by \eqref{eq:C2}. If the equation %\eqref{eq:th1} \begin{equation*} \left(\tilde R(r)|u'|^{p-2}u'\right)'+\tilde C(r) |u|^{p-2}u=0 \end{equation*} has conjugate points on $[a,b]$, then every solution of equation \eqref{eq:E} has zero on $\Omega(a,b)$. \end{theorem} \bigskip\bigskip\bigskip {\rightskip 2cm \leftskip 2cm Similar theorems can be derived also for estimates of terms with mixed powers based on different methods than AG inequality % (for example % \eqref{eq:C3}) (see R. M., Nonlinear Analysis TMA 73 (2010)). } \end{document}