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If you miss one of these packages it is recommended to copy all files from its subfolder to your folder. \usepackage{authblk} %obligatory package \usepackage[usenames]{xcolor} %obligatory package \usepackage{lastpage} %obligatory package \usepackage{enumerate} %recommended package %\usepackage{mathtools} %recommended package %\usepackage{bm} %recommended package %\usepackage[noadjust]{cite} %recommended package %\usepackage[width=136mm]{caption} %Recommended if the paper containes figures or tables %\usepackage[dvipdfm]{hyperref} %Use this option in dvi->pdf mode (e.g. in case you prefer eps figures) %obligatory package %\usepackage[dvips]{hyperref} %Use this option in dvi->ps->pdf mode %obligatory package \usepackage{hyperref} %Use this in pdflatex mode %obligatory package \definecolor{ForestGreen}{rgb}{0.15,0.416,0.18} \definecolor{EgyptBlue}{rgb}{0.063,0.2,0.65} \hypersetup{ colorlinks=true, linkcolor=EgyptBlue, citecolor=EgyptBlue, urlcolor=ForestGreen } \linespread{1.05} \hoffset -1in \voffset -1in \oddsidemargin 25mm \textwidth 160mm \topmargin 10mm \headheight 10mm \headsep 10mm \textheight 237mm \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{definition} \newtheorem{remark}[theorem]{Remark} \theoremstyle{definition} \newtheorem{example}[theorem]{Example} \numberwithin{equation}{section} \numberwithin{table}{section} \numberwithin{figure}{section} \title{The maximum principle with lack of monotonicity} %Please write the title of your paper here. Please use capital letters only for the first letter of the title and for proper names %Please write the full names of all authors here, using the order first name - middle names (optionally initials only) - last (family) name. \author[1]{\textbf{Patrizia Pucci}\footnote{Corresponding author. Email: {\tt patrizia.pucci@unipg.it}}} \author[2, 3]{\textbf{Vicen\c{t}iu D. R\u{a}dulescu}} %Please enter the affiliations and addresses of the authors here. \affil[1]{Dipartimento di Matematica e Informatica, Universit\`a di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy} \affil[2]{Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia} \affil[3]{Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Krak\'ow, Poland} \affil[4]{Department of Mathematics, University of Craiova, 200585 Craiova, Romania} %In case you have no co-authors, use the form below for your name and affiliation(s) instead the above one. %\author{\textbf{First Author}\footnote{Email: firstauthor@firstuniversity.com}} %\affil{First University, 1 University Street, City Name, H--1234, Country Name\\ %Second University, 2 University Square, City Name H--9876, Country Name} %Please write here the short title of your paper not exceeding 65 characters including spaces \newcommand\shorttitle{The maximum principle with lack of monotonicity} %Please fill here the names of all authors, first name and middle name should be abbreviated to single letter \newcommand\authorsshort{P. Pucci and V.D. R\u{a}dulescu} %Please rewrite .png to .eps if you use dvi mode \newcommand{\ejqtdelogo}{tink.png} %\newcommand{\ejqtdelogo}{tink.eps} % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Please do not change this paragraph.% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % \makeatletter \renewcommand{\maketitle}{\bgroup\setlength{\parindent}{0pt} \begin{picture}(20,20) \put(-50,-5){\includegraphics[width=2.9truecm]{\ejqtdelogo}} \end{picture} \vspace{1truecm} \begin{center}{\vbox{\titlefont\@title}}\end{center} \vspace{0.5truecm} \begin{center}{\@author} \end{center} \egroup } \renewcommand{\thefootnote}{\fnsymbol{footnote}} \renewcommand{\@fnsymbol}[1]{% \ifcase#1 \or {\,\Letter\!} \or\textasteriskcentered\or \textasteriskcentered\textasteriskcentered \else\@ctrerr\fi} \makeatother \makeatletter \newcommand{\hbibitem}[4]{\bibitem{#1}{#2} \def\@tempa{#3}% \def\@tempb{#4}% \ifx\@tempa\@empty\ifx\@tempb\@empty{}{}\else{}{\url{https://doi.org/#4}}\fi\else {\href{http://www.ams.org/mathscinet-getitem?mr=#3}{MR#3}}\ifx\@tempb\@empty{}\else{; \url{https://doi.org/#4}}\fi\fi} \makeatother \renewcommand{\Affilfont}{\small\normalfont} \renewcommand{\Authfont}{\Large} \renewcommand\Authands{ and } \newcommand*{\titlefont}{\fontsize{18}{21.6}\selectfont\textbf} \makeatletter \renewcommand\@author{\ifx\AB@affillist\AB@empty\AB@author\else \ifnum\value{affil}>\value{Maxaffil}\def\rlap##1{##1}% \AB@authlist\\[\affilsep]\vbox{\AB@affillist} \else \AB@authors\fi\fi} \makeatother \makeatletter \def\Year{2017} \def\IssueNumber{XX} \def\ps@plain{ \def\@oddhead{\ifnum\thepage=1\hss\baselineskip8pt \vtop to 0 pt{\vskip-0.4truecm\noindent\hbox{\hspace{1.3truecm}\Large Electronic Journal of Qualitative Theory of Differential Equations\hss\linebreak}% \vskip 0.1truecm \noindent\hbox{\hspace{1.3truecm}\footnotesize \Year, No.\ {\bf \IssueNumber}, {1--\begin{NoHyper}\pageref{LastPage}\end{NoHyper}};\enspace \href{https://doi.org/10.14232/ejqtde.\Year.1.\IssueNumber}{https:/\!/doi.org/10.14232/ejqtde.\Year.1.\IssueNumber}} \hfill \hbox{\footnotesize \href{https://www.math.u-szeged.hu/ejqtde/}{www.math.u-szeged.hu/ejqtde/}} \vss} \else \hss\textit{\shorttitle}\hss\hbox to 0pt{\hss\thepage}\fi}\def\@oddfoot{} \def\@evenhead{\hbox to 0pt{\thepage\hss} \hss\textit{\authorsshort}\hss} \def\@evenfoot{} } \makeatother % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %You may define your own macros here e.g \def\R{\mathbb{R}} \newcommand\dom{\operatorname{dom}} \newcommand{\eps}[1]{{#1}_{\varepsilon}} \newcommand{\RR}{\mathbb R} \newcommand{\NN}{\mathbb N} \newcommand{\PP}{\mathbb P} \newcommand{\ZZ}{\mathbb Z} \newcommand{\ri}{\rightarrow} \newcommand{\di}{\displaystyle} \newcommand{\ep}{\varepsilon} \newcommand{\bb}{\begin{equation}} \newcommand{\bbb}{\end{equation}} \def\gg{\mathcal G} \def\rr{\mathcal R} \def\ee{\mathcal E} \renewcommand{\le}{\leqslant} \renewcommand{\leq}{\leqslant} \renewcommand{\ge}{\geqslant} \renewcommand{\geq}{\geqslant} \begin{document} \maketitle \pagestyle{plain} \begin{center} \noindent \begin{minipage}{0.85\textwidth}\parindent=15.5pt \medskip \begin{center} \noindent Received XX XXXXXX 201X, appeared XX XXXXXX 20XX % Please do not change this line, it will be edited by the technical editors. \smallskip \noindent Communicated by Handling Editor \medskip \centerline{This paper is dedicated with esteem to Professor L\'aszl\'o Hatvani} \centerline{on the occasion of his 75th anniversary} \end{center} \bigskip {\small{ \noindent {\bf Abstract.} We establish a maximum principle for the weighted $(p,q)$-Laplacian, which extends the general Pucci-Serrin strong maximum principle to this quasilinear abstract setting. The feature of our main result is that it does not require any monotonicity assumption on the nonlinearity. The proof combines a local analysis with techniques on nonlinear differential equations.} \smallskip % Please enter at most 6 keywords here with lowercase letters separated by commas. \noindent {\bf{Keywords:}} generalized maximum principle, $(p,q)$-operator, nonlinear differential inequality, normal derivative, positive solution. \smallskip % Please enter at most 5 Mathematics Subject Classification codes here. Please use 2010 classification codes, which can be found on the following link: http://www.ams.org/msc//msc2010.html. \noindent{\bf{2010 Mathematics Subject Classification:}} 35J60, 35B50, 35B51, 35R45,.} %35A16} \end{minipage} \end{center} %\bigskip %\begin{center}{\footnote{This paper is dedicated with esteem to Professor L\'aszl\'o Hatvani on the %occasion of his 70th anniversary}}\end{center} \section{Introduction} The maximum principle is a basic tool in the mathematical analysis of partial differential equations. This is an extremely useful instrument when studying the qualitative behavior of solutions of differential equations and inequalities. The roots of the maximum principle go back to C.F.~Gauss, who already knew the maximum principle for harmonic functions in 1839, in close relationship with the mean value formula. Let us first recall some of the major steps related to the understanding of the maximum principle. Let $\Omega$ be a bounded domain in $\RR^N$ such that $\partial\Omega$ has the interior sphere property at any point. The maximum principle asserts that if $u:\overline\Omega\ri\RR$ is a smooth function such that \bb\label{supharm}\begin{cases} \di -\Delta u\geq 0&\quad\mbox{in }\ \Omega,\\ \di u= 0&\quad\mbox{on }\ \partial\Omega, \end{cases} \bbb then $u\geq 0$ in $\Omega$. A stronger version of the maximum principle has been deduced by E.~Hopf \cite{hopf, hopf1}. The Hopf lemma asserts that if $u$ satisfies \eqref{supharm}, then the following alternative holds: {\it either} $u$ vanishes identically in $\Omega$ {\it or} $u$ is positive in $\Omega$ and its exterior normal derivative $\partial u/\partial\nu<0$ on $\partial\Omega$. G.~Stampacchia \cite{stampa} showed that the strong maximum principle continues to remain true in the case of certain {\it linear perturbations} of the Laplace operator. More precisely, let $a\in L^\infty(\Omega)$ be such that, for some $\alpha>0$, $$\int_\Omega (|D u|^2+a(x)u^2)dx\geq\alpha \|u\|^2_{H^1_0(\Omega)}\quad\mbox{for all}\ u\in H^1_0(\Omega).$$ Stampacchia's maximum principle asserts that if $$\begin{cases} \di -\Delta u+a(x)u\geq 0&\quad\mbox{in }\ \Omega,\\ \di u= 0&\quad\mbox{on }\ \partial\Omega, \end{cases} $$ then either $u\equiv 0$ in $\Omega$ or $u>0$ in $\Omega$ and $\partial u/\partial\nu<0$ on $\partial\Omega$. J.-L. V\'azquez \cite{vazquez} observed that the maximum principle remains true for suitable {\it nonlinear perturbations} of the Laplace operator, subject to {\it monotonicity assumptions} on the nonlinear term. More precisely, let $f:\mathbb R^+_0\ri\RR$, $\mathbb R^+_0=[0,\infty)$, be a continuous non-decreasing function such that $f(0)=0$ and $$\int_{0^+}F(t)^{-1/2}dt=\infty, \quad \mbox{where }\ F(t)=\int_0^tf(s)ds.$$ Under these assumptions, V\'azquez proved that if $u\in C^2(\Omega)\cap C(\overline\Omega)$ satisfies $$\begin{cases} \di -\Delta u+f(u)\geq 0&\quad\mbox{in }\ \Omega,\\ \di u\geq 0&\quad\mbox{on }\ \partial\Omega, \end{cases} $$ then either $u\equiv 0$ in $\Omega$ or $u>0$ in $\Omega$. We point out that the Keller-Osserman type growth assumption \bb\label{koss}\int_{0^+}F(t)^{-1/2}dt=\infty\bbb holds true for ``superlinear" nonlinearities. For instance, $f(t)=t^q$, with $t\in\mathbb R^+_0$ and $q\geq 1$, satisfies the hypotheses of the V\'azquez maximum principle. Condition \eqref{koss} is also satisfied by some nonlinearities for which $f(t)/t$ is not bounded at the origin, for instance $f(t)=t(\log t)^2$, $t\in\mathbb R^+$, $\mathbb R^+=(0,\infty)$. The necessity of \eqref{koss} is due to P.~Benilan, H.~Br\'ezis and M.~Crandall \cite{benilan}, while for the $p$-Laplacian it is due to J.-L. V\'azquez \cite{vazquez}. In this latter case, relation \eqref{koss} becomes $$\int_{0^+}F(t)^{-1/p}dt=\infty.$$ For other classes of differential operators, necessity is due to J.I.~Diaz \cite[Theorem 1.4]{diaz} and P.~Pucci, J.~Serrin and H.~Zou \cite[Corollary 1]{psz}. In a series of papers, P.~Pucci and J.~Serrin \cite{pucser1, pucser2, pucser3} extended the maximum principle into several directions and under very general assumptions. For instance, P.~Pucci and J.~Serrin considered the following canonical divergence structure inequality \bb\label{gen} -{\rm div}\, \{A(|Du|)Du\}+f(u)\geq 0\quad\mbox{in}\ \Omega,\bbb where the function $A=A(s)$ and the nonlinearity $f$ satisfy the following conditions: \smallskip \noindent(A1) $A\in C(\RR^+)$; \smallskip \noindent(A2) the mapping $s\mapsto sA(s)$ is strictly increasing in $\RR^+$ and $sA(s)\ri 0$ as $s\ri 0$; \smallskip \noindent(F1) $f\in C(\RR^+_0)$; \smallskip \noindent(F2) $f(0)=0$ and $f$ is non-decreasing on some interval $(0,\delta)$, $\delta>0$. \smallskip Condition (A2) is a minimal requirement for ellipticity of \eqref{gen}, allowing moreover singular and degenerate behavior of the operator $A$ at $s=0$, that is, at critical points $x\in\Omega$ of $u$, such that $(Du)(x)=0$. The differential operator ${\rm div}\, \{A(|Du|)Du\}$ is called the {\it $A$-Laplace operator}. An important example of $A$-Laplace operator that fulfills hypotheses (A1) and (A2) is the $(p,q)$-Laplace operator $\Delta_pu+\Delta_qu$, with $1
0$, or $f(s)>0$ for $s\in(0,\delta)$ and $$\int_0^\delta\frac{ds}{H^{-1}(F(s))}=\infty.$$ For further details on the maximum principle we refer to the monographs by L.E.~Fraenkel \cite{frank}, D.~Gilbarg and N.S.~Trudinger \cite{gilbarg}, and M.H.~Protter and H.F.~Weinberger \cite{prot}. \section{Strong maximum principle for the $(p,q)$-Laplacian} The {\it global} monotonicity assumption on the nonlinearity $f$ plays a central role in the statement of the V\'azquez maximum principle. This hypothesis is replaced with the {\it local} monotonicity condition (F2) in the strong maximum principle of Pucci and Serrin, namely $f$ is assumed to be non-decreasing on some interval $(0,\delta)$. Our purpose in this paper is to prove that the monotonicity constraint on $f$ can be removed and that only the growth of the nonlinearity near zero guarantees the maximum principle. This will be done for the $(p,q)$-Laplace operator $\Delta_pu+\Delta_qu$, with $1
0$ in $\mathbb R^+$ and
\begin{equation}\label{SMP}\int_{0^+}F(t)^{-1/q}dt=\infty,\end{equation}
where $F(t)=\int_0^tf(s)ds$.
(i) Let $u\in C^1(\overline\Omega)$ be a positive solution of problem \eqref{1} and assume that $u(x_0)=0$ for some $x_0\in\partial\Omega$. If $\partial\Omega$ satisfies the interior sphere condition at $x_0$, then the normal derivative of $u$ at $x_0$ is negative.
(ii) Let $u\in C^1(\overline\Omega)$ be a non-negative solution of problem \eqref{1}. Then the following alternative holds: either $u$ vanishes identically in $\Omega$ or $u$ is positive in $\Omega$.
\end{theorem}
The proof is based on some {\it local estimates} and uses some ideas found in the papers by S.~Dumont, L.~Dupaigne, O.~Goubet and V.~R\u{a}dulescu \cite{dumont} and L.~Dupaigne \cite{dupaigne}. A central role in our arguments is played by the comparison of $u$ with the minimal solution of a suitable nonlinear second order differential equation
in a small ring.
Theorem \ref{t1} establishes that the maximum principle associated to problem \eqref{1} holds even for nonlinearities which are not monotone in {\it any} interval $(0,\delta)$. A class of functions of this type is given by $f(t)=t^a(1+\cos t^{-1})$ for all $t\in\mathbb R^+$, where $a>q-1$.
The interest for the study of non-negative solutions in problem \eqref{1} is due to reaction-diffusion models. In these prototypes $u$ is viewed as the density of a reactant and the region where $u=0$ is called the {\it dead core}, that is where no reaction takes place. We refer to P. Pucci and J. Serrin \cite{pucserdc} for a thorough analysis of dead core phenomena in the setting of quasilinear elliptic equations.
\subsection{An associated $(p,q)$-Dirichlet problem on a small ring} Let $u\in C^1(\overline\Omega)$ be a positive solution of problem \eqref{1}. Assume that there exists $x_0\in\partial\Omega$ such that $u(x_0)=0$. Since $\partial\Omega$ has the interior sphere property at $x_0$, there exists small $r>0$ and a ball $B_r$ of radius $r$ such that $B_r\subset\Omega$ and $\partial B_r\cap\partial\Omega =\{x_0\}$. Passing eventually to a translation, we can assume that $B_r$ is centered at the origin.
Let $\rr=B_r\setminus B_{r/2}$ and put
$$m=\min\{u(x)\, :\, x\in \partial B_{r/2}\}.$$
Since $u$ is positive, it follows that $m>0$.
Consider the following nonlinear boundary value problem
\bb\label{ring}\begin{cases}
\di -\Delta _{p}v-\Delta_qv+f(v)= 0& \quad \mbox{in} \quad \rr,\\
\di v= 0& \quad \mbox{on} \quad \partial B_{r},\\
\di v= m& \quad \mbox{on} \quad \partial B_{r/2}.
\end{cases}
\end{equation}
The energy functional $\ee:W^{1,q}(\rr)\ri\RR$ associated to problem \eqref{ring} is
$$\ee(v)=\frac{1}{p}\int _{\rr}|D v|^{p}dx+\frac{1}{q}\int _{\rr}|D v|^{q}dx+\int_{\rr}F(v)dx.$$
The manifold
$$M=\{ v\in W^{1,q}(\rr)\,:\, v\geq 0\ \mbox{in}\ \rr,\ v=0\ \mbox{on}\ \partial B_r,\ v=m\ \mbox{on}\ \partial B_{r/2}\},$$
and the minimization problem
$$\inf\{\ee(v)\,:\, v\in M\}$$
associated to \eqref{ring}, are well defined.
Since $\ee$ is coercive, it follows that any minimizing sequence $(v_n)_n\subset M$ of $\ee$ is bounded. By reflexivity, up to a subsequence, not relabelled, we deduce that there exists $v_0\in M$ such that
$$v_n\rightharpoonup v_0 \quad\mbox{in } \ W^{1,q}(\rr).$$
Moreover, $\ee(v_0)\leq \liminf_{n\ri\infty}\ee(v_n)$ by the weakly lower semicontinuity of $\ee$. Hence $v_0$ minimizes $\ee$ over $M$. Consequently,
$$-\Delta _{p}v_0-\Delta_qv_0+f(v_0)= 0 \quad \mbox{in} \quad \rr,$$
$v_0=0$ on $\partial B_r$ and $v_0=m$ on $\partial B_{r/2}$. These arguments also show that $v_0$ is a {\it minimal solution} of problem \eqref{ring}.
The same conclusion can be obtained after observing that the functions $0$ (resp., $u$) are subsolution (resp., supersolution) of problem \eqref{ring} and then using the same approach as in the proof of Proposition 2.1 and Corollary 2.2 in \cite{dumont}. We point out that the minimality principle stated in \cite[Corollary 2.2]{dumont} holds true with no monotonicity assumption on the nonlinear term $f$. Details on the method of lower and upper solutions for the $(p,q)$-Laplace operator can be found in A.~Araya and A. Mohammed \cite[Lemma 2.3]{araya}, see also \cite[Example 1.1 (ii)]{araya}.
\smallskip
In view of the invariance of $\rr$ and of the $(p,q)$-Laplace operator, the function $v_0\circ R $ is still a non-negative solution of problem \eqref{ring}, for any rotation $R$ of the Euclidean space. Moreover, the minimality of $v_0$ implies that
$$v_0(x)\leq v_0(R(x)) \quad \mbox{for all } x\in \rr.$$
Applying this inequality at $y=R^{-1}(x)$, we deduce that $v_0$ is a radial function. Therefore, \eqref{ring} along $v_0$ can be written in the equivalent form as
\begin{equation}\label{4}
\begin{cases}
\di\big(s^{N-1}|v_0'|^{p-2}v_0'\big)'+\big(s^{N-1}|v_0'|^{q-2}v_0'\big)+f(v_0(s))=0\quad \mbox{for all } s\in\left({r}/{2},r\right),\\
\di v_0(r)=0, \qquad \di v_0(r/2)=m.
\end{cases}
\end{equation}
\subsection{Boundary behavior of the comparison function $v_0$} In what follows we shall prove that the derivative of $v_0$ at both $r/2$ and $r$ is negative.
First note that
$$v_0'(r)\leq 0$$
since $v_0$ is non-negative in $(r/2,r)$ and $v_0(r)=0$.
Our aim is to show that
$$v_0'(r/2)<0\quad \mbox{and}\quad v_0'(r)<0.$$
Multiplying by $s^{N-1}$ the equation \eqref{4} and integrating on $[s,r]$, where $r/2\leq s