\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 26, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/26\hfil Sturmian comparison results]
{Sturmian comparison results for \\
quasilinear elliptic equations
in $\mathbb{R}^n$}

\author[Tadie\hfil EJDE-2007/26\hfilneg]
{Tadie}

\address{Tadie \newline
Mathematics Institut \\
Universitetsparken 5 \\
 2100  Copenhagen, Denmark.\newline
Department of Mathematics,  Makerere University,
P.O. Box 7062, Kampala, Uganda}
\email{tad@math.ku.dk}

\thanks{Submitted October 19, 2006. Published February 12, 2007.}
\subjclass[2000]{35B05, 35B50, 35J70}
\keywords{$p$-Laplacian; Picone's identity; comparison methods;
\hfill\break\indent maximum principle}

\begin{abstract}
 We obtain Sturmian comparison results for the nonnegative solutions
 to Dirichlet problems associated with  $p$-Laplacian operators.
 From  Picone-type identities  \cite{k2,t2},
 we obtain results comparing solutions of two types of equations.
 We also present results related to those operators using
 Picone-type identities.
\end{abstract}

\maketitle \numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

 In this work $\Omega$ denotes an open and bounded subset of $\mathbb{R}^n$,
 $n\geq 2 $ with $\partial \Omega \in  C^\ell$, $\ell\geq 1$. Also
$a \in C^1(\overline{\Omega};  0,\infty ))$,
$c \in C( \overline{\Omega};\mathbb{R})$ and  functions
$f,g   \in C^1(\overline{\Omega}; \mathbb{R}) $.
Define in $\Omega $ the operators
\begin{equation}
\begin{gathered}
 pu:= \nabla.\{ a(x) \Phi(\nabla u) \} \\
 Pu := \nabla.\{ a(x) \Phi(\nabla u) \}   + c(x) \phi(u).
\end{gathered} \label{e0.1}
\end{equation}
Associated with the functions $f$ and $g$ define
\begin{equation}
Fu :=  Pu   +   f(x,   u ),     Gu:= Pu + g(x,   u) \label{e0.2}
\end{equation}
where for $(\zeta, t) \in \mathbb{R}^n \times \mathbb{R}$,
$ \Phi(\zeta)= |\zeta|^{\alpha -1}\zeta ,\phi(t)=|t|^{\alpha -1} t $ and
 $\alpha > 0 $. Solutions of  \eqref{e0.1} or \eqref{e0.2}
  with regular boundary  data \newline
(e.g. $u|_{\partial \Omega} = g \in C(\overline{\partial \Omega} ) $)
  will be  supposed to belong to the space
\begin{equation}
 D_p(\Omega) := \{ w \in C^1(   \overline{\Omega} ;   \mathbb{R} ) :
 a(x) \Phi(\nabla w) \in C^1( \Omega;   \mathbb{R} ) \cap
 C(  \overline{\Omega} ;   \mathbb{R}   )   \}\,. \label{e0.3}
\end{equation}
 For any other  similar domain $E$, $D_P(E) $  is defined similarly.

\subsection{Picone-type formulae}
Similar to \cite[Theorem 1.1]{k1},
 let  $E$ be  a bounded domain in $\mathbb{R}^n$ ($n\geq 2$)  with  a
regular boundary (e.g.  $\partial E \in C^\ell$,   $\ell\geq 1$),
  and  define for $\alpha>0 $ and $f, g \in
C(\overline{E}\times \mathbb{R};  \mathbb{R}) $  the operators
\begin{equation}
\begin{gathered}
  Fu:= \nabla.\{ a\Phi(\nabla u)\} + c \phi(u)+ f(x,u)\\
  Gv:= \nabla.\{ A\Phi(\nabla v)\} + C \phi(v)+ g(x,v)
\end{gathered} \label{e0.4}
\end{equation}
where $ a,A \in C^1(\overline{E};  \mathbb{R}_+)$,
    $c, C \in C(\overline{E};\mathbb{R})$.

\begin{lemma} \label{lem0.1}
If $u , v \in D_P(E)$ with $v\neq 0$ in $E$, then from
\[
 \nabla .\big\{ \frac u{\phi(v)} [\phi(v) a \Phi(\nabla u)] \big\}
= a|\nabla u|^{\alpha +1} + uFu -c|u|^{\alpha+1} - uf(x,u),
\]
and
\begin{align*}
\nabla . \big\{u\phi(u)   \frac{A\Phi(\nabla v)}{\phi(v)}\big \}
&= (\alpha +1) A \phi(u/v)   \nabla u .\Phi(\nabla v)
 - \alpha A |\frac uv \nabla v|^{\alpha+1}\\
&\quad  + \frac u{\phi(v)} \phi(u) Gv
  - C|u|^{\alpha+1} - \frac u{\phi(v)} \phi(u) g(x,v),
\end{align*}
we obtain
\begin{equation} \label{e0.5}
\begin{aligned}
&\nabla .\big\{  \frac u{\phi(v)}[ \phi(v) a \Phi(\nabla u) -
\phi(u)A \Phi(\nabla v)]   \big\}\\
&=(a-A)|\nabla u|^{\alpha+1} + (C-c)|u|^{\alpha+1}\\
&\quad +A \big\{   |\nabla u|^{\alpha+1} - (\alpha+1)|\frac uv
\nabla v|^{\alpha-1}
  \nabla u. (\frac uv \nabla v) + \alpha |\frac uv \nabla v|^{\alpha+1} \big\}
\\
&\quad +  \frac u{\phi(v)}\big\{  [\phi(v)Fu - \phi(u)Gv] + [\phi(u)g(x,v) -
  \phi(v) f(x,u)]  \big\} .
\end{aligned}
\end{equation}
\end{lemma}

 The following important inequality is also from  \cite[Lemma 2.1]{k1}:
For all $\alpha>0$ and all $\xi,   \eta \in \mathbb{R}^n $,
\begin{equation} \label{e0.6}
Y(\xi,   \eta) :=   |\xi|^{\alpha+1} + \alpha |\eta|^{\alpha+1} -
(\alpha+1)|\eta|^{\alpha-1} \xi.\eta \geq 0\,.
\end{equation}
The equality holds if and only if  $\xi = \eta$.
For $u,   v\in C^1 $ define
\[
 Z(u,v) := Y( \nabla u , \nabla v) .
\]

\subsection*{Some identities}
 If $a=A$,   $c=C$,   $Fu=Gv=0$ in $E$ then \eqref{e0.5} becomes
\begin{equation} \label{e0.7}
\begin{aligned}
&\nabla .\big\{ \frac u{\phi(v)}   a[ \phi(v)  \Phi(\nabla u) -
\phi(u) \Phi(\nabla v)] \big\} \\
&=  a \{   |\nabla u|^{\alpha+1} - (\alpha+1)|\frac uv \nabla v|^{\alpha-1}
  \nabla u. (\frac uv \nabla v) + \alpha |\frac uv \nabla v|^{\alpha+1} \}
\\\
&\quad +  u \phi(u) \Big[\frac{g(x,v)}{\phi(v)} -  \frac{f(x,u)}{\phi(u)}\Big]\\
&:= aZ(u,v)+ u \phi(u) \Big[\frac{g(x,v)}{\phi(v)}
-  \frac{f(x,u)}{\phi(u)}\Big].
\end{aligned}
\end{equation}
Define
\[
\chi(x,t):= \frac{f(x,t)}{\phi(t)} .
\]
 For the functions $u$ and $v$ above, if $ \Omega \subset E $
    is open , non empty  and
 $f(x,t)\equiv g(x,t) $, then after
     integrating  \eqref{e0.6} over $\Omega$ we get for positive $u$ and $v$
\begin{equation} \label{e0.8a}
\begin{aligned}
  &\int_{\partial \Omega} au \big\{
  |\nabla u|^{\alpha-1}\frac{\partial u}{\partial \nu_\Omega} - \phi( \frac vu)
  |\nabla v|^{\alpha-1} \frac{\partial v}{\partial \nu_\Omega} \big\} ds \\
 &=\int_\Omega \left[ aZ(u,v) + |u|^{\alpha+1}
  \{ \chi(x,v) - \chi(x,u)\}\right] dx\,.
\end{aligned}
\end{equation}
After interchanging $u$ and $v$,
\begin{equation} \label{e0.8b}
\begin{aligned}
  &\int_{\partial \Omega} av \big\{
  |\nabla v|^{\alpha-1} \frac{\partial v}{\partial \nu_\Omega}- \phi(\frac uv)
  |\nabla u|^{\alpha-1} \frac{\partial u}{\partial \nu_\Omega}   \big\} ds \\
 &=\int_\Omega [a Z(v,u) + |v|^{\alpha+1}
  \{ \chi(x,u) - \chi(x,v)\}] dx
 \end{aligned}
\end{equation}
where $\nu_\Omega$ denotes the  outward normal unit vector to
$\partial  \Omega$.

For the operators $F$ and $G$ in \eqref{e0.1}-\eqref{e0.2},
if $u$ and $v$ satisfy respectively
$ Fu=Gv=0$ in $\Omega $, Equation \eqref{e0.6} leads to
\begin{equation} \label{e0.9}
\begin{aligned}
&\nabla .\{   \frac u{\phi(v)}   a[ \phi(v)  \Phi(\nabla u) -
\phi(u) \Phi(\nabla v)] \}  \\
&:= aZ(u,v)+ u \phi(u)   \chi(x,v)  \quad \text{if } v>0 \text{ in } \Omega, \\
% \intertext{and}
& \nabla .\{   \frac v{\phi(u)}    a[ \phi(u)  \Phi(\nabla v) -
\phi(v) \Phi(\nabla u)] \}  \\
&:= aZ(v,u)- v \phi(v)   \chi(x,v)  \quad \text{if }   u>0 \text{ in } \Omega .
\end{aligned}
\end{equation}


\begin{remark} \label{rmk1} \rm
 It is a classical result that if $u $ and $v $  are
continuous and piecewise-$C^1$ in $\overline{\Omega} $ and
 for $pw:= \nabla . \{a(x)\Phi(\nabla w) \} $ satisfies weakly
\begin{gather*}
 G_1u:=pu + g(x,u) \geq 0 \geq pv +  g(x,v)  \quad \text{in } \Omega   ; \\
 u \leq v  \quad  \text{in }   \overline{\Omega}\, ,
\end{gather*}
then if $g \in C(\Omega \times \mathbb{R}) $ is non decreasing in
its second argument , the existence of such $u$ and $v$ leads to
the existence of a solution $w\in D_P(\Omega) $  of
$ pw + g(x,w) =0$ in $\Omega ; w|_{\partial \Omega } = w_0 $
for any continuous $w_0 $ satisfying $u\leq w_0 \leq v $ on $\partial \Omega$.
\end{remark}

\begin{remark}\label{rmk2} \rm
 Let $\Omega $ be bounded, $\Omega' $ be an open subset of
$\Omega   ,   c \in C(\overline{\Omega}) $ and
$h \in C(   \overline{\Omega} \times \mathbb{R}   ) $.
It is known  (e.g. \cite{d1,s1}) that if $u, v \in D_p(\Omega) $
satisfy  (weakly) for
$H(w) := \nabla . \{a(x)\Phi(\nabla w) \} + c(x) \phi(w) + h(x,w)$,
\begin{equation}
Hu  \geq Hv  \quad \text{in } \Omega   ; \quad
  (u-v)|_{\partial \Omega'}\leq 0
\label{e0.10}
\end{equation}
then $(u-v)\leq 0 $ in $\Omega' $   provided that
 $\forall x\in \Omega $,
$ c(x) \phi(w) + h(x,w) $
is non increasing  in $w $ for
$  |w| \leq \max\{   |u|_{L^\infty(\Omega)} , |v|_{L^\infty(\Omega)} \} $.
\end{remark}

\section{Main Results}

Let $a, c ,  \dots  $ be as defined in the Introduction.
Define  in $\Omega $ the  equations:
\begin{gather}
Pu:=\nabla.\{ a(x) \Phi(\nabla u) \}   + c(x) \phi(u)=0  ,   \label{eP} \\
Fv:=\nabla.\{ a(x) \Phi(\nabla v) \}   + c(x) \phi(v)   +   f(x,   v )=0,
  \label{eF} \\
G_1w := \nabla.\{ a(x) \Phi(\nabla w) \}  + g(x,w)=0   . \label{eG}
\end{gather}
Following the Remarks  \ref{rmk1}-\ref{rmk2}, we have the following result for the problem
\begin{equation}
G_1w := pw + g(x,w) =0  \quad \text{in } \Omega ; \quad
  w|_{\partial \Omega} =0 \label{eGo}
\end{equation}

\begin{theorem} \label{thm1.1}
(1)   Assume that for all $x$ in $\Omega$,   $g$ is increasing in the
second argument  and that $a(x)>0 $ is constant in $\Omega$.
Then if there is a strictly  positive $v\in D_P(\Omega) $ which satisfies
$G_1v \leq 0 $ in $\Omega$ and $v|_{\partial \Omega} \geq 0 $,
then \eqref{eGo} has a solution $u\in D_P(\Omega) $ which satisfies
$0\leq u \leq v $ in $\Omega$.
\\
(2)   If for all $x$ in $\Omega$,   $g $ is non increasing  in the
second argument  then \eqref{eGo} has at most one solution in $D_P(\Omega) $.
\end{theorem}


\begin{theorem} \label{thm1.2}
 Assume that $\Omega $ is bounded and  connected  and
$c \in C(\overline{\Omega}) $ is non positive.
\begin{enumerate}
\item   Let   $u\in D_p(\Omega) $  be   a solution of
\begin{gather*}
 Pu:=\nabla.\{   a(x) \Phi(\nabla u)   \}   + c(x) \phi(u)=0 \quad\text{in }
  \Omega \\
  u|_{\partial \Omega} =0 .
\end{gather*}
 Then $u>0$   in   $\Omega $  if
$\mathop{\rm meas}\{ x\in \Omega :  u(x)>0 \} >0 $.

\item   For  the solutions $w\in D_p(\Omega) $ of
 \begin{gather*}
 Fu:=\nabla.\{ a(x) \Phi(\nabla w)  \} + c(x) \phi(w) + f(x,w)=0
\quad\text{in }  \Omega \\
  w|_{\partial \Omega} =0
\end{gather*}
the same conclusion  holds  provided that in $\overline{\Omega}$,
  $f(x, t) \leq 0 $  for $t \geq 0 $.
\end{enumerate}
\end{theorem}


\begin{theorem} \label{thm1.3} \quad
\begin{enumerate}
\item  Assume that  for all $x \in \Omega$,   $f(x, t) \geq 0$ for
    $t\geq 0 $ . Then if  \eqref{eP} has a  strictly positive solution $u$ which satisfies
$u|_{\partial \Omega}=0 $ ,
\eqref{eF} cannot have a solution strictly positive in $\Omega$ .
Consequently if  \eqref{eP} has a positive solution $u$  with the
boundary condition
 $u|_{\partial \Omega}= 0    $ then  any  non negative solution $v$  of
\eqref{eF}   has a zero inside $\Omega$.

\item  If \eqref{eP} has a solution strictly positive in $\Omega $ then if
for all $x \in \Omega $,  $f(x, t) \leq 0$    for    $t\geq 0 $,
\eqref{eF} has no nontrivial and nonnegative solution $v$ satisfying
$v|_{\partial \Omega}=0 $.
\end{enumerate}
\end{theorem}

\begin{theorem} \label{thm1.4}
Let  $f \in C(\overline{\Omega}\times \mathbb{R}; \mathbb{R})$ and
let $u,   v \in D_p(\Omega) $ be two  solutions of
$$
Fw:=\nabla.\{   a\Phi(\nabla w)   \} + c \phi(w) + f(x,w) =0   ; \quad 
 w>0 \quad \text{in }  \Omega; \quad  w|_{\partial \Omega}=0   .
 $$
(1)   If for all $x$ in $\Omega$,   $t \mapsto  \chi(x,t)= f(x,t)/\phi(t) $
is  strictly  increasing  and positive  in $t>0 $ then
\begin{itemize}
\item[(i)]   the two solutions  intersect  in $\Omega $ ;

\item[(ii)]   if for some open $D\subset \Omega$,    $v \geq u$
 in  $ D $ then
\begin{equation} \label{a1}
 \int_{\partial D} au \big\{
  |\nabla u|^{\alpha-1} \frac{\partial u}{\partial \nu_D} - \phi(\frac uv)
  |\nabla v|^{\alpha-1} \frac{\partial v}{\partial \nu_D} \big\} ds \geq 0
\end{equation}
 and  if in addition   $u= v$   on   $\partial D$,
  then
\begin{equation} \label{a2}
\begin{gathered}
\int_D \big\{ a Z(v,u) +  |v|^{\alpha+1} X(x, u:v) \big\} dx \leq 0 \quad and \\
\int_D \big\{   a Z(u,v) + |u|^{\alpha+1}X(x,  v:u) \big\} dx \geq 0,
\end{gathered}
\end{equation} %\label{e1.1}
where $X(x, w:z):= \chi(x,w) - \chi(x,z)$.
\end{itemize}
(2)  If for all $x$ in $\Omega $
\begin{itemize}
\item[(i)] $  t \mapsto  \chi(x,t)= f(x,t)/\phi(t) $ is
 positive and  strictly  decreasing in $t>0 $    or
\item[(ii)]   if  $f$ is  positive and decreasing in $t>0 $
  then the two solutions coincide.
\end{itemize}
(3)   For connected $\Omega$, the problem
$$
Pw= \nabla.\{ a\Phi(\nabla w)\} + c \phi(w)= 0 \quad\text{in }  \Omega ;
\quad   w|_{\partial \Omega} =0
$$
has at most one non negative solution in $D_P(\Omega) $.
\\
This problem has at most one strictly positive solution even if $\Omega$
is not connected.
\end{theorem}


\section{Proofs of the main results}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
(1) Taking in account  remark \ref{rmk1}, we just need to
 build a subsolution $w \in D_P(\Omega) $, such that
\[
 G_1w\geq 0 \geq G_1v \quad\text{and}\quad
 0\leq w \leq v  \quad\text{in }\Omega.
\]
Because $v>0 $ in $\Omega $ we consider any nonnegative
$U \in C(\overline{\Omega})
 $ which is piecewise afine; i.e.,  there exists
$\mathcal{N}:=\{\eta_i ;  i=1,2,\dots ,M \}$ and some
 finite number (pairwise disjoint) of subsets $B_i$,
$1\leq i \leq N $ of $\Omega$ such that  with
$x=(x_1, x_2, \dots ,x_n) \in \Omega$
\begin{itemize}
\item[(i)]  $    B:= \bigcup_{i=1}^N  B_i \subset \Omega $;

\item[(ii)]  $    \forall i,    U(x) = \sum_{i=1}^n \eta_i x_i  < v(x)$
  for  $x\in B_i $;

\item[(iii)]  $  U|_{\partial B}=0 $  and is extended by $0$ outside
$B $ in $\Omega$.
\end{itemize}
Thus  as $a(x) $ is positive and constant in $\Omega $,
\[
 GU = g(x,U)\geq 0 \geq Gv \quad\text{and}\quad
 0\leq U \leq v \quad\text{in }\Omega .
\]
The solution $u$ of $pu+g(x,U)=0 $ in $\Omega ;u|_{\partial \Omega}=0 $
is in  $D_P(\Omega) $ and satisfies
 $  G_1u = pu + g(x,u)\geq 0 \geq G_1v $ and
$ 0\leq u \leq v $ in $\Omega $ . Thus  from Remark \ref{rmk1},
this leads to the existence of such a  required solution.

\noindent (2)   Let $g$ be decreasing in the second argument.
Suppose that  there are two distinct solutions
 $ u $ and $v \in D_P(\Omega) $ such that for some subset $B$
of $\Omega $ whose measure is strictly positive
$ v>u $ in $B $ and $(u-v)|_{\partial B}=0 $. In that case,
as $g$ is decreasing,
\[
 pu - pv = g(x,v) - g(x,u)\leq 0 \quad\text{in } B \quad
\text{and}\quad (u-v)|_{\partial B}\geq 0.
\]
 This  leads  to $u \geq v $ in $B$,  conflicting with the assumption.
Therefore any such two solutions  have to coincide in $\Omega$.
\end{proof}

The proof of Theorem \ref{thm1.2}, follows from the lemma below.

\begin{lemma} \label{lem2.1}
(1)   Let $u \in D_p(\Omega) $ be a solution of
\begin{equation} \label{ePo}
\begin{gathered}
pu:=\nabla.\{a(x) \Phi(\nabla u) \}   = 0  \quad\text{in  }  \Omega ; \\
 u|_{\partial \Omega} =0   ; \quad    \mathop{\rm meas} \{\Omega^+ \}  > 0
\end{gathered}
\end{equation}
where $\Omega^+:=\{ x\in \Omega : u(x)> 0 \} $ and
$\Omega^-:=\{ x\in \Omega : u(x)> 0 \}$.
Then $u \geq 0 $ a.e. in $\Omega $.
Moreover if in addition $\Omega $ is connected then $u>0$   in
   $\Omega $.

\noindent(2)   The same conclusions hold  for the problems
\begin{equation} \label{eP1}
\begin{gathered}
 Pu:=\nabla.\{ a(x) \Phi(\nabla u) \}  + c(x) \phi(u)   = 0
  \quad\text{in }  \Omega   ; \\
u|_{\partial \Omega} =0  ;\quad \mathop{\rm  meas} \{\Omega^+ \}  > 0
\end{gathered}
\end{equation}
 where $c \in C( \overline{\Omega};\mathbb{R}) $  remains non positive
 in $\Omega $.

The same conclusion holds for the operator $F$ if in
$\overline {\Omega} \times \mathbb{R}_+ $  the function $f$ is non positive.
\end{lemma}

\begin{proof}
 (1)   Let $k:= \max_{\Omega^-}    |u(x)| $ and  the function
 $v(x) := u(x)_+   +   k $.

As $(\nabla u - \nabla v )|_{\Omega^+}\equiv 0$,   $Z(u,v)=0 $ and weakly
in $ \Omega^+ $,
$$
\nabla .\big\{ \frac u{\phi(v)}   a[ \phi(v)  \Phi(\nabla u) -
\phi(u) \Phi(\nabla v)] \big\}= \frac u{\phi(v)} \{   \phi(v) - \phi(u)   \}
 \nabla [a(x)\Phi(\nabla u) ]= 0
$$
by \eqref{e0.5} and \eqref{ePo}. So, as $v$ is constant in $\Omega^-$,
$$
\nabla .\big\{ \frac u{\phi(v)}   a[ \phi(v)  \Phi(\nabla u) -
\phi(u) \Phi(\nabla v)] \big\}
 = \begin{cases}
 aZ(u,k)   &\text{in } \Omega^- ,\\
 0   &\text{otherwise}.
\end{cases}
$$
This implies  after integration over $\Omega $ that
\[
 0 = \int_{\Omega^-} a (x)Z(u,k)dx
 = \int_{\Omega^-} a(x)|\nabla u|^{\alpha +1} dx > 0
\]
 which is absurd unless $\mathop{\rm meas}\{ \Omega^- \}=0$.
The fact that $a \in C^1(\overline{\Omega}; (0, \infty ))$ makes
 the operator $p$ here  satisfy the conditions required for the case
 of the following  maximum principle.

\begin{quote} \cite[Theorem 2.2]{d1}
  If the bounded domain $\Omega $ is connected , $p\in ( 1 ,\infty) $
and $u \in W_{\rm loc}^{1,p}(\Omega)\bigcap C^0(\Omega) $ satisfies
$ - \mathop{\rm div} A(x, Du)   +   \Lambda |u|^{p-2} u \geq 0$,
$u \geq 0$ in   $\Omega $
for a constant $\Lambda \in \mathbb{R} $ then either $u \equiv 0 $ or
 $u>0 $ in $ \Omega$.
\end{quote}

(2)   If $c \leq  0 $ in $\Omega $ and  $\mathop{\rm meas}\{\Omega^- \} >0 $
 proceeding as above with $v$ defined as before,
\begin{equation} \label{eP2}
\begin{aligned}
 &\nabla .\big\{ \frac u{\phi(v)}   a[ \phi(v)  \Phi(\nabla u) -
\phi(u) \Phi(\nabla v)] \big\} \\
&= \begin{cases}
 u \{   pu +c\phi(u)   \} - u \phi(\frac uv)\{   pu + c\phi(v) \}
  &\text{in }  \Omega^+ \\
 aZ(u,k) + u \{   pu + c\phi(u)   \} - u \phi(\frac uv) c \phi(v)
  &\text{in } \Omega^- .
\end{cases} \\
&= \begin{cases}
 u   pu \{ 1 - \phi(\frac uv)\}  & \text{in }   \Omega^+ \\
 aZ(u,k) + upu  & \text{in }   \Omega^-  .
\end{cases}
\end{aligned}
\end{equation}
 From \eqref{eP1}, $upu = -c \phi(u) \geq 0 $ in $\Omega $ provided that $c$
 is non positive  there.

For the operator $F$ ,  \eqref{eP2} reads
\begin{equation}
\begin{aligned}
 &\nabla .\big\{ \frac u{\phi(v)}   a[ \phi(v)  \Phi(\nabla u) -
\phi(u) \Phi(\nabla v)] \big\} \\
 &= \begin{cases}
-c(x) u \phi(\frac uv){\phi(v)}\{\phi(v) - \phi(u)\} - u \phi( \frac uv)
\{ f(x,v)-f(x,u) \} \\
+ u \phi( \frac uv) f(x,v) -uf(x,u)  & \text{in } \Omega^+ \\[4pt]
 aZ(u,k) + \frac u{\phi(k)} \{   -\phi(u)[ c(x) \phi(k) + f(x,k) ] \\
 + \phi(u)f(x,k) - \phi(k) f(x,u)  \}    &\text{in } \Omega^-
\end{cases} \\
&= \begin{cases}
 -c(x) u\phi(u)\{ 1- \phi(\frac uv)\} + uf(x,u) \{ \phi(\frac uv) -1\}
&  \text{in }  \Omega^+ \\
aZ(u,k)  - c (x) u\phi(u) - uf(x,u)  & \text{in } \Omega^-  .
\end{cases}
\end{aligned} \label{eP3}
\end{equation}
Integrating of  both sides of \eqref{eP2}  and \eqref{eP3}  over $\Omega $
 provides an absurdity as the left would be zero while the right
would be strictly positive, unless $\Omega^- $ has measure zero.
This completes the proof.
\end{proof}


\begin{proof}[Proof of  Theorem \ref{thm1.3}]
(1) If $v $ and $u$ are respectively    solutions of
\begin{equation} \label{e2.2}
\begin{gathered}
Fv = 0   ; \quad   v>0  \quad  \text{in } \Omega  \quad \text{and } \\
Pu = 0   ; \quad   u\geq 0  \quad  \text{in } \Omega;
\quad u|_{\partial \Omega} =0
\end{gathered}
\end{equation}
with $f \in C(   \overline{\Omega} \times \mathbb{R} ; [ 0 , \infty))$ .
As in \eqref{e0.5}  we have
$$
 \nabla .\big\{  \frac u{\phi(v)}[ \phi(v) a \Phi(\nabla u) -
\phi(u)a \Phi(\nabla v)] \big\}
= aZ(u,v) + u \phi(   \frac uv   )f(x,v) > 0\,.
$$
Then integrating  both sides of the equation leads to a contradiction.

\noindent(2) Similarly if in \eqref{e2.2},  $u>0 $ in $\Omega $
and $v|_{\partial \Omega}=0 $  after interchanging $u$ and $v$
in \eqref{e0.5} we get to
$$
 \nabla .\big\{ \frac v{\phi(u)}    a[ \phi(u)  \Phi(\nabla v) -
\phi(v) \Phi(\nabla u)]\big\} =aZ(v,u) - vf(x,v) > 0.
$$
Then we complete as above.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.4}]
The statement \eqref{a1} follows from \eqref{e0.8a}.
Adding \eqref{e0.8a} and \eqref{e0.8b}, we get
\begin{align*}
&\int_{\partial D} a(u-v)\{   \Phi(\nabla u) - \Phi(\nabla v)   \}.
{\nu_D }ds  \\
&= \int_D \big\{ a Z(u,v) + a Z(v,u) + [|u|^{\alpha+1}
- |v|^{\alpha+1}](   \chi(x,v) - \chi(x,u)   ) \big\} dx
\end{align*}
leading to  \eqref{a2}.
For the two solutions, \eqref{e0.6} (and  interchanging $u$ and $v$)
leads (after integration over $\Omega$) to
\begin{equation}
\begin{aligned}
0 &\leq \int_\Omega a Z(u,v)  dx\\
& = - \int_\Omega  u\phi(u) \Big\{\frac{f(x,v)}{\phi(v)} -
\frac{ f(x,u)}{\phi(u)} \Big\}dx  \\
&= - \int_\Omega |u|^{\alpha+1}\{   \chi(x,v) - \chi(x,u)   \}dx      .
  \end{aligned}  \label{e1.2a}
\end{equation}
  and
\begin{equation}
\begin{aligned}
0&\leq \int_\Omega a Z(v,u)dx \\
 &= - \int_\Omega  v\phi(v) \Big\{\frac{f(x,u)}{\phi(u)} -
  \frac{f(x,v)}{\phi(v)} \Big\}dx \\
&= \int_\Omega|v|^{\alpha+1}\{   \chi(x,v) - \chi(x,u)   \}dx .
    \end{aligned}  \label{e1.2b}
\end{equation}
  Assume that $ \chi(x,t) $ is increasing:
  If we suppose that $v > u$  in  $\Omega$  then \eqref{e1.2a}
    provides a contradiction and if we suppose that $u>v$,
\eqref{e1.2b} would lead to a contradiction.
  Assume that $\chi(x,t)$ is decreasing and define
$\Omega_+  :=\{ x\in \Omega  :   X(x):=\chi(x,v) - \chi(x,u) >0 \}$ and \quad 
$\Omega_- :=\{ x\in \Omega  :   X(x):=\chi(x,v) - \chi(x,u) <0 \}$.
Then (without loss of generality)
$0< v < u $ in $\Omega_+$ and $ v>u>0$ in $\Omega_-$  whence
\begin{equation}
 \begin{gathered}
    \int_{\Omega_+} |v|^{\alpha+1}X(x) dx
\leq \int_{\Omega_+}   |u|^{\alpha+1}X(x)dx ,\\
   \int_{\Omega_-} |v|^{\alpha+1}X(x) dx
\leq \int_{\Omega_-} |u|^{\alpha+1}X(x)dx.
\end{gathered} \label{e1.2c}
\end{equation}
    This  implies from \eqref{e1.2a} and \eqref{e1.2b} that
    $$
0 \leq \int_\Omega |v|^{\alpha+1} X(x)dx
  \leq \int_\Omega |u|^{\alpha+1} X(x)dx \leq 0
$$
 whence
$\int_\Omega Z(u,v)dx=0$,  leading to $v \equiv u $ in
$\Omega$ by \eqref{e0.6}.
If $f$ is nonnegative and decreasing in $t$, $\chi $ is decreasing in  $t$
and the same conclusion is reached.

\noindent(3)   The statement follows immediately from \eqref{e0.8a}
or \eqref{e0.8b} as we would get for any such two solutions
  $0 = \int_\Omega a(x) Z(u,v) dx  $ the right hand side being
strictly positive unless $u \equiv v$ in $\Omega$.
\end{proof}


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\end{document}