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\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}}



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\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}

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\begin{document}

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\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 $1<p<q<\infty$, which is generated by $A(s)=s^{p-2}+s^{q-2}$,
$s\in\mathbb R^+$.

Let $\gg$ be the potential defined by $\gg'(s)=sA(s)$ for all $s\in\mathbb R^+$, with $\gg(0)=0$. Condition (A2) implies that the mapping $s\mapsto\gg'(s)$ is strictly increasing and continuous
in $\mathbb R^+_0$, so that $\gg$ can be extended by symmetry in~$\mathbb R$ and $\gg$ becomes a symmetric strictly convex function in~$\mathbb R$. In particular, for  $A(s)=s^{p-2}+s^{q-2}$, $s\in\mathbb R^+$,  we have $\gg(s)=s^p/p+s^q/q$, $s\in\mathbb R^+_0$.

In what follows,  a classical solution of problem \eqref{gen} is a function $u\in C^1(
\overline\Omega)$ which satisfies \eqref{gen} in the distributional sense.

By the {\it strong maximum principle} for problem \eqref{gen} we mean the statement that if $u$ is a non-negative classical solution of problem \eqref{gen}, with $u(x_0)=0$ at some point $x_0\in\Omega$, then $u\equiv 0$ in $\Omega$.

In order to describe the Pucci-Serrin strong maximum principle for the inequality \eqref{gen}, we need a further definition. Put $\Phi(s)=sA(s)$ for $s\in\mathbb R^+$ and $\Phi(0)=0$. Then, the function
$$H(s)=s\Phi(s)-\int_0^s\Phi(t)dt\quad\mbox{for all } s\in\mathbb R^+_0$$
is the pre-Legendre transform of $\gg$, since $H(s)=s\gg'(s)-\gg(s)$ for all $s\in\mathbb R^+_0$.

Under hypotheses (A1), (A2), (F1) and (F2), the Pucci-Serrin maximum principle \cite[Theorem 1.1]{pucser2}, see also \cite[Theorem 1.1.1]{pucser}, establishes that the strong maximum principle holds for problem \eqref{gen} if and only if either $f(s)\equiv 0$ for $s\in [0,\mu)$, with $\mu>0$, 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 $1<p<q<\infty$, which plays an important role in mathematical physics. We refer to V.~Benci, P.~D'Avenia, D.~Fortunato and L.~Pisani \cite{ref4} for applications in quantum physics and to L.~Cherfils and Y.~Ilyasov \cite{ref7} for models in plasma physics. As pointed out in the previous section, the weighted $(p,q)$-Laplace operator $\Delta_pu+\Delta_qu$ satisfies the hypotheses of the Pucci-Serrin maximum principle.
This abstract result for the $(p,q)$-Laplacian has been used in several recent works, see e.g. N.~Papageorgiou, V.~R\u{a}dulescu and D.~Repov\v{s} \cite{prramo, prrspringer}.

We assume from now on, without further mentioning, that $p$, $q$ are real numbers, with $1<p<q$, and that $\Omega$ is a bounded domain in~$\RR^N$.

Consider the following nonlinear problem
\bb\label{1}\begin{cases}
\di -\Delta_pu-\Delta_qu+f(u)\geq 0&\quad\mbox{in}\ \Omega,\\
\di u\geq 0&\quad\mbox{on}\ \partial\Omega.\end{cases}
\bbb
The main result of this paper is stated in the following theorem.

\begin{theorem}\label{t1}
Let $f:\mathbb R^+_0\ri\mathbb R^+_0$ be a continuous function such that $f(0)=0$, $f>0$ 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 $r/2\leq s<r$, we get
\begin{equation}\label{5}
s^{N-1}|v_0'(s)|^{p-2}v_0'(s)+s^{N-1}|v_0'(s)|^{q-2}v_0'(s)+\int_{s}^{r}t^{N-1}f(v_0(t))dt=0.
\end{equation}
Taking $s=r/2$ in \eqref{5}, we deduce that
$$v_0'(r/2)<0,$$
since $f$ is positive on $\mathbb R^+$ and $v_0(r/2)=m>0$.

Using this fact in combination with $v_0'(r)\leq 0$, we  claim that  $v_0'(r)< 0$.
Indeed, arguing by contradiction, let
\bb\label{contr} v_0'(r)=0.\bbb
Since $v_0'(r/2)<0$, there exists $a\in (r/2,r]$ such that
$$ v_0'(a)=0\ \mbox{and}\ v_0'(s)<0\ \mbox{for all}\ s\in [r/2,a).$$
Taking $s=a$ in relation \eqref{5} we deduce that $v_0$ vanishes identically in $[a,r]$.

Since $v_0'<0$ in $[r/2,a)$, by Corollary~2.4 of~\cite{AP} the equation in~\eqref{4}
is equivalent in $[r/2,a)$~to
\begin{equation}\label{New}\begin{aligned}
  -(p-1)|v_0'(s)|^{p-2}v_0''(s) -(q-1)|v_0'(s)|^{q-2}&v_0''(s)-
\frac{N-1}{s}|v_0'(s)|^{p-2}v_0'(s)\\
 &-\frac{N-1}{s}|v_0'(s)|^{q-2}v_0'(s)+f(v_0(s))=0.\end{aligned}\end{equation}
Fix $s\in (r/2,a)$. Multiplying  equation \eqref{New} by $v_0'$ and
integrating on $[s,a]$, we get
\begin{equation}\label{6}\begin{aligned}
\di\frac{1}{p'}|v_0'(s)|^{p}+\frac{1}{q'}|v_0'(s)|^{q}&-(N-1)
\int_{s}^{a}\frac{|v_0'(t)|^{p}}{t}dt\\
&-\di (N-1)\int_{s}^{a}\frac{|v_0'(t)|^{q}}{t}dt-F(v_0(s))=0,\end{aligned}
\end{equation}
since $v_0(a)=0$.
On the other hand, since $f\geq 0$, relation \eqref{5} shows that the mapping $$[r/2,r]\ni t\mapsto t^{N-1}\left(|v_0'(t)|^{p-2}+|v_0'(t)|^{q-2}\right)v_0'(t) $$ is negative and non-decreasing. This shows that the mapping $$[r/2,r]\ni t\mapsto t^{N-1}\left(|v_0'(t)|^{p-1}+|v_0'(t)|^{q-1}\right) $$ is decreasing. Since $[r/2,r]\ni t\mapsto t^{N-1}$ is an increasing function, we deduce that
$$[r/2,r]\ni t\mapsto |v_0'(t)|^{p-1}+|v_0'(t)|^{q-1} \quad \mbox{is decreasing.}$$
Now, using the fact that both the real numbers $p-1$ and $q-1$ are positive, we conclude that $|v_0'|$ is decreasing in $[r/2,r]$. Hence,
$$[r/2,r]\ni t\mapsto |v_0'(t)|^{p}+|v_0'(t)|^{q} \quad \mbox{is decreasing.}$$
Now $s\in (r/2,a)$, so that
$$
\di\int_{s}^{a}\frac{|v_0'(t)|^{p}+|v_0'(t)|^{q}}{t}dt\leq
(|v_0'(s)|^{p}+|v_0'(s)|^{q})\int_s^a\frac{dt}{t}=(|v_0'(s)|^{p}+|v_0'(s)|^{q})o(1)$$
as $s\ri a^-$. Therefore
\bb\label{krakow}\lim_{s\rightarrow a^{-}}\frac{\di\int_{s}^{a}\frac{|v_0'(t)|^{p}}{t}dt+
\int_{s}^{a}\frac{|v_0'(t)|^{q}}{t}dt}{|v_0'(s)|^{p}+|v_0'(s)|^{q} }=0.\bbb
Returning now to \eqref{6},  we deduce the following basic estimate
$$
\frac{1}{p'}|v_0'(s)|^{p}+\frac{1}{q'}|v_0'(s)|^{q}=\left(|v_0'(s)|^{p}+|v_0'(s)|^{q}\right) o(1) +F(v_0(s)) \quad  \mbox{as} \ s\rightarrow a^{-}.
$$
Consequently,
\bb\label{se1}\frac{1}{q'}\left( |v_0'(s)|^{p}+|v_0'(s)|^{q}\right)(1+o(1))\leq F(v_0(s)) \quad  \mbox{as} \ s\rightarrow a^{-}.\bbb
Since $v_0'(s)\ri 0$ as $s\ri a^-$ and $1<p<q$, it follows that the left-hand side of \eqref{se1} goes to zero like $|v_0'(s)|^{q}$ as $s\ri a^-$. Therefore
$$\frac{1}{q'}\,|v_0'(s)|^{q}(1+o(1))\leq F(v_0(s)) \quad  \mbox{as} \ s\rightarrow a^{-}.$$
Fix $\epsilon>0$. Then, by \eqref{se1} and for all $s<a$ sufficiently close to $a$, we obtain
$$\left(\frac{1}{q'}\right)^{1/q}\int_{s}^{a}\frac{-v_0'(t)}{F(v_0(t))^{1/q}}dt\leq (1+\epsilon)(a-s).$$
Since $v_0'$ is negative in $(s,a)$, the change of variable $s=v_0(t)$ yields
$$\left(\frac{1}{q'}\right)^{1/q}\int_{0}^{v_0(s)}\frac{ds}{F(s)^{1/q}}\leq (1+\epsilon)(a-s)<\infty ,$$
which contradicts the assumption \eqref{SMP}.
Consequently, \eqref{contr} is false and the claim $v_0'(r)<0$
is completely proved.

\subsection{Conclusion of the proof of Theorem \ref{t1}}
(i) By the construction of $v_0$, we have $u\geq v_0$ in $\rr$. Therefore,
$$-\frac{\partial u}{\partial \nu}(x_{0})=\lim_{t\rightarrow 0^{+}}\frac{u((1-t)x_{0})}{t}\geq\lim_{t\rightarrow 0^{+}}\frac{v_0((1-t)r)}{t} =-v_0'(r)>0,$$
since we supposed, without loss of generality, in the construction above that $B_r$ is centered at the origin.\smallskip

\noindent
(ii) Arguing by contradiction, we assume that $u$ vanishes somewhere in $\Omega$, but $u$ does not vanish identically. Hence,
$$\Omega_+=\{x\in \Omega\,:\, u(x)>0\}\neq \emptyset.$$
Fix a point $z\in \Omega_+$ which is closer to $\partial\Omega_+$ than to $\partial \Omega$ and take the largest ball $B\subset\Omega_+$ centered at $z$. Then, $u(x_{0})=0$ for some $x_{0}\in \partial B$, while $u>0$ in $B$. Clearly, $D u(x_{0})=0$, since $x_{0}$ is an interior minimum point of~$u$ in~$\Omega$.

On the other hand,  (i) applied in $B$ gives
$$\frac{\partial u}{\partial \nu}(x_{0})<0.$$
Hence $D u(x_{0})\neq 0$. This contradicts the fact that $x_{0}$ is an interior minimum point of u. The proof of Theorem \ref{t1} is now complete. \qed

\section*{Perspectives and open problems}
(i) The main result of this paper establishes that the strong maximum principle for the $(p,q)$-Laplace operator holds without any monotonicity assumption on the nonlinearity $f$. Accordingly, the maximum principle holds as soon as the  nonlinear term satisfies a suitable divergent integrability condition near the origin. A related property has been previously established in \cite{dumont}, in the framework of logistic equations with blow-up boundary. In this latter case, no monotonicity hypothesis is necessary and the existence of such singular solutions depends only on a convergent Keller-Osserman integrability condition at infinity. Inspired by \cite[Theorem 1.1]{dumont}, we raise the following

{\it Open problem.} Is condition $\int_{0^+}F(t)^{-1/q}dt=\infty$ used in Theorem \ref{t1} equivalent with the following assumption
$$\limsup_{\alpha\ri 0^+}\lim_{\varepsilon\ri 0}\int_\varepsilon^{\alpha} [F(\alpha)-F(t)]^{-1/q}dt=\infty\ ?$$
We do not have any information concerning the relevance of this growth condition in relationship with the maximum principle.

\smallskip(ii) A very interesting open problem is to establish a version of Theorem \ref{t1} in the case where the $(p,q)$-Laplace operator is replaced by the differential operator ${\rm div}\, \{A(|Du|)Du\}$, when $A$ satisfies assumptions (A1) and (A2).

\smallskip
(iii) We do not know at this stage whether the {\it compact support principle} stated in \cite[Theorem 1.1.2]{pucser} still remains true if the local monotonicity assumption (F2) is removed and only the integrability condition (1.1.7) of~\cite{pucser} is assumed. We raise the same open problem for the {\it dead core principle} stated in \cite[Theorem 8.4.1]{pucser} and we expect that this basic result still remains true without the assumption that the nonlinear term $f$ is non-decreasing on the whole real axis.

\smallskip
(iv) The study of $(p,q)$-Laplace differential operators had a growing interest after the pioneering papers of P.~Marcellini \cite{marce1, marce2} on $(p,q)$-growth conditions. These problems involve integral functionals of the type
$$W^{1,1}(\Omega)\ni u\mapsto\int_\Omega \gg(x,D u)dx,$$
where $\Omega\subseteq\RR^N$ is an open set.
The integrand $\gg:\Omega\times{\mathbb R}^N\rightarrow {\mathbb R}$ satisfied {\it unbalanced} polynomial growth conditions of the type
$$|\xi|^p \lesssim \gg(x,\xi)\lesssim |\xi|^q+1,\quad\mbox{with $1<p<q$,}$$
for every $x\in\Omega$ and $\xi\in{\mathbb R}^N$.

An interesting double phase type operator  considered in the papers of M.~Colombo and G.~Mingione \cite{mingi0,mingi2}, addresses functionals of the type
\begin{equation}\label{eenerg}u\mapsto \int_\Omega (|D u|^p+a(x)|D u|^q)dx,\end{equation}
where $a(x)\geq 0$. The meaning of this functional is also to give a sharper version of the following energy
$$u\mapsto \int_\Omega |D u|^{p(x)}dx,$$
thereby describing sharper phase transitions.

Composite materials with locally different hardening exponents $p$ and $q$ can be described using the energy defined in (\ref{eenerg}). Problems of this type are also motivated by applications to elasticity, homogenization, modelling of strongly anisotropic materials, Lavrentiev phenomenon, etc.

Accordingly, a new double phase model can be given by
\bb\label{comi}\Phi_d(x,|\xi|)=\begin{cases}
 |\xi|^{p}+a(x)|\xi|^{q}&\quad\mbox{if }|\xi|\leq 1,\\
|\xi|^{p_1}+a(x)|\xi|^{q_1}&\quad\mbox{if }|\xi|\geq 1,
\end{cases}\qquad(x,\xi)\in\Omega\times\mathbb R^N,
\bbb
with $a(x)\geq 0$ in $\Omega$.

We consider that a very interesting research direction corresponds to the study of a strong maximum principle for {\it anisotropic} differential operators associated to the functional defined in \eqref{comi}.


% For the acknowledgements section, please do not number the section, please begin it with \section*{Acknowledgements}
\section*{Acknowledgements}
P. Pucci was partly supported
by the Italian MIUR project
{\em Variational methods, with applications to problems in mathematical
physics and geometry} (2015KB9WPT\_009) and is a member of the {\em Gruppo Nazionale per
l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni} (GNAMPA)
of the {\em Istituto Nazionale di Alta Matematica} (INdAM). The
manuscript was realized within the auspices of the INdAM--GNAMPA Project
2018 titled {\em Problemi non lineari alle derivate parziali} (Prot\_U-UFMBAZ-2018-000384),
and of the {\em Fondo Ricerca di Base di Ateneo -- Esercizio 2015} of the University of Perugia,
titled {\em PDEs e Analisi Nonlineare}.

V.D.~R\u adulescu acknowledges the support through a grant  of the Romanian Ministry
of Research and Innovation, CNCS--UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130,
within PNCDI III. He was also supported by the Slovenian Research Agency grants
P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083.


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