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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 113, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/113\hfil Existence of positive bounded solutions]
{Existence of positive bounded solutions for some nonlinear
polyharmonic elliptic systems}

\author[S. Gontara, Z. Z. El Abidine \hfil EJDE-2010/113\hfilneg]
{Sabrine Gontara, Zagharide Zine El Abidine}

\address{Sabrine Gontara \newline
D\'epartement de math\'ematiques,
Faculte des sciences de Tunis,
campus universitaire, 2092 Tunis, Tunisia}
\email{sabrine-28@hotmail.fr}

\address{Zagharide Zine El Abidine \newline
D\'epartement de math\'ematiques,
Faculte des sciences de Tunis, campus universitaire,
2092 Tunis, Tunisia}
\email{Zagharide.Zinelabidine@ipeib.rnu.tn}

\thanks{Submitted June 6, 2010. Published August 16, 2010.}
\subjclass[2000]{34B27, 35J40}
\keywords{Green function; Kato class; positive bounded solution;
\hfill\break\indent
 Shauder fixed point theorem; polyharmonic elliptic system}

\begin{abstract}
 We prove existence results for positive bounded continuous
 solutions of a nonlinear polyharmonic system by using a potential
 theory approach and properties of a large functional class
 $K_{m,n}$ called Kato class.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

The goal is to study the existence of positive continuous bounded
solutions for the nonlinear elliptic higher order system
\begin{equation}
\begin{gathered}
(-\Delta )^{m}u+\lambda qg(v)=0\quad\text{in }B, \\
(-\Delta )^{m}v+\mu pf(u)=0\quad\text{in }B, \\
\lim_{x\to \xi \in \partial B}\frac{u(x)}{(1-|x| ^{2})^{m-1}}
=\varphi (\xi ), \\
\lim_{x\to \xi \in \partial B}\frac{v(x)}{(1-| x|^{2})^{m-1}}=\psi
(\xi ),
\end{gathered}  \label{1.1}
\end{equation}
where $m$ is a positive integer,
$B=\{ x\in \mathbb{R} ^n:| x|<1\} $ is the unit ball of
$\mathbb{R}^n$ $(n\geq 2)$,
$\partial B=\{ x\in\mathbb{R} ^n:| x| =1\} $ is the boundary
of $B$, $\lambda $, $\mu $, are
nonnegative constants and $\varphi $, $\psi $ are two nontrivial
nonnegative continuous functions on $\partial B$.

 For the case $m=1$, the existence of solutions for nonlinear
elliptic systems has been extensively studied for both bounded
and unbounded $C^{1,1}$domain $D$ in $\mathbb{R}^n$
$(n\geq 3)$ (see [8, 9, 11-13]).

 The polyharmonic operator $(-\Delta )^{m}$,
$m\in\mathbb{N}^{\ast }$, has been studied several years later.
Indeed, Boggio \cite{Boggio} showed that the Green function
$G_{m,n}$ of the operator $(-\Delta)^{m}$ on $B$ with Dirichlet
boundary conditions
$u=\frac{\partial }{\partial \nu }u=\dots
 =\frac{\partial ^{m-1}}{\partial \nu ^{m-1}}u=0$ on $\partial B$,
is given by:
\begin{equation}
G_{m,n}(x,y)=k_{m,n}| x-y| ^{2m-n}\int_{1}^{\frac{[
x,y] }{| x-y| }}\frac{(\nu ^{2}-1)^{m-1}}{\nu ^{n-1} }d\nu ,
\label{1.2}
\end{equation}
where $k_{m,n}$ is a positive constant,
$\frac{\partial }{\partial \nu }$ is the outward normal derivative
and for $x$, $y$ in $B$,
$[x,y] ^{2}=|x-y|^{2}+( 1-|x|^{2}) (1-|y|^{2})$.

 From its expression, it is clear that $G_{m,n}$ is
nonnegative in $ B^{2}$. This does not hold for the Green function
of $(-\Delta )^{m}$ in an arbitrary bounded domain (see for
example \cite{Gar}). It is well known that for $m=1$, we do not
have this restriction. In \cite{IHSM}, the properties of the Green
function $G_{m,n}$ of $(-\Delta )^{m}$ on $B$ allowed the authors
to introduce a large functional class called Kato class denoted by
$ K_{m,n}$ (see Definition \ref{def1} below). This class played a key role
in the study of some nonlinear polyharmonic equation (see
\cite{IHSM,Sonia,MTZ}). For the case $m=1$, the Kato class has
been introduced and studied for general domain possibly unbounded
in \cite{BM,Bachar,M} for $n\geq 3$ and \cite {zeddini} for $n=2$.

\begin{definition}[\cite{IHSM}] \label{def1} \rm
A borel measurable function $q$ on $B$ belongs to the
Kato class $K_{m,n}$ if $q$ satisfies the  condition
\[
\lim_{\alpha \to 0}\Big( \sup_{x\in B}
 \int_{B\cap B(x,\alpha )}\big( \frac{\delta (y)}{\delta
(x)}\big) ^{m}G_{m,n}(x,y)| q(y)| dy\Big) =0.
\]
Here and always $\delta (x)=1-| x| $, is the Euclidian distance
between $x$ and $\partial B$.
\end{definition}

As typical example of functions belonging to the class $K_{m,n}$,
we have

\begin{example}[\cite{Sonia}] \label{exa1} \rm
 The function $q$ defined in $B$ by
\[
q(x)=\frac{1}{(\delta (x))^{\lambda }(\log\frac{2}{\delta (x)})^{\mu }},
\]
is in $K_{m,n}$ if and only if $\lambda <2m$ and
$\mu \in\mathbb{R}$ or $\lambda =2m$ and $\mu >1$.
\end{example}

 Before presenting our main results, we lay out a number of potential
theory tools and some notations which will be used throughout the paper.
We are mainly concerned with the bounded continuous solution
$H\varphi $ of the Dirichlet problem
\[
\begin{gathered}
\Delta u=0\quad \text{in }B \\
u\big|_{\partial B}=\varphi ,
\end{gathered}
\]
where $\varphi $ is a nonnegative continuous function on
$\partial B$. We remark that the function defined on $B$ and
denoted by $H^{m}\varphi :x\to (1-| x| ^{2})^{m-1}H\varphi (x)$
is a bounded continuous solution of the problem
\begin{equation}
\begin{gathered}
(-\Delta )^{m}u=0\quad \text{in }B \\
\lim_{x\to \xi \in \partial B}\frac{u(x)}{(1-| x|
^{2})^{m-1}}=\varphi (\xi ).
\end{gathered}  \label{1.4}
\end{equation}
For simplicity, we denote
\[
C_0(B)=\{w\text{ continuous on }B\text{ and }\lim_{x\to \xi \in
\partial B}w(x)=0\}
\]
and
\[
C(\overline{B})=\{w\text{ continuous on }\overline{B}\}.
\]
 We also refer to $V_{m,n}f$ the $m$-potential of a
nonnegative measurable function $f$ on $B$ by
\[
V_{m,n}f(x)=\int_{B}G_{m,n}(x,y)\,f(y)dy,\quad \text{for }x\in B.
\]
Recall that for each nonnegative measurable function $f$ on $B$
such that $f$ and $V_{m,n}f$ are in $L_{\rm loc}^{1}(B)$, we have
\[
(-\Delta )^{m}(V_{m,n}f)=f,
\]
in the distributional sense.


 The outline of this paper is as follows. In section $2$, we
collect some preliminary results about the Green function and the
Kato class $ K_{m,n} $. In section $3$, a careful analysis about
continuity is performed. In particular, we prove the following
result.

\begin{theorem} \label{thm1}
Let $m-1\leq \beta \leq m$, $q\in K_{m,n}$, then the function
$v$ defined on $B$ by
\[
v(x)=\int_{B}\Big(\frac{\delta (y)}{\delta (x)}\Big) ^{\beta
}G_{m,n}(x,y)\,| q(y)| dy
\]
is in $C(\overline{B})$ and if $m-1\leq \beta <m$, we have
$\underset{ x\to \xi \in \partial B}{\lim }v(x)=0$.
\end{theorem}

 Based on these properties of the Green's function $G_{m,n}$
and Kato class $K_{m,n}$, we establish in section 4 the first existence
result stated in Theorem \ref{thm2} below. The following conditions are
considered

\begin{itemize}
\item[(H1)] The functions $f$, $g:(0,\infty)\to [ 0,\infty )$
are nondecreasing and continuous.

\item[(H2)] The functions $p$ and $q$
are measurable nonnegative in $B$ such that the functions
\[
x\mapsto \frac{p(x)}{( \delta (x)) ^{m-1}}\quad\text{and}\quad
x\mapsto \frac{q(x)}{( \delta (x)) ^{m-1}}
\]
belong to the Kato class $K_{m,n}$.

\item[(H3)] We suppose that
\begin{gather*}
\lambda _0=\inf_{x\in B}\frac{H^{m}\varphi (x)}
{V_{m,n}(qg(H^{m}\psi )) (x)}>0, \\
\mu _0=\inf_{x\in B}\frac{H^{m}\psi (x)}
{V_{m,n}(pf(H^{m}\varphi )) (x)}>0.
\end{gather*}
\end{itemize}

\begin{theorem} \label{thm2}
Assume {\rm (H1)--(H3)}. Then for each
$\lambda \in [0,\lambda _0)$ and each $\mu \in [ 0,\mu _0)$,
the problem \eqref{1.1} has a positive continuous solution
$(u,v)$ satisfying for each $x\in B$,
\begin{equation}
\begin{gathered}
( 1-\frac{\lambda }{\lambda _0}) H^{m}\varphi (x)\leq u(
x) \leq H^{m}\varphi (x), \\
( 1-\frac{\mu }{\mu _0}) H^{m}\psi (x)\leq v( x)
\leq H^{m}\psi (x).
\end{gathered}  \label{1.5}
\end{equation}
\end{theorem}

 In section 5, we study the system \eqref{1.1} when the
functions $f$ and $g$ are non-increasing and $\lambda =\mu =1$. More
precisely, we fix a nontrivial nonnegative continuous function
$\Phi $ on $ \partial B$ and we suppose the following hypotheses

\begin{itemize}
\item[(H4)] The functions $f$, $g:(0,\infty )\to
[ 0,\infty )$ are non-increasing and continuous.

\item[(H5)] The functions $p$ and $q$ are measurable
nonnegative in $B$ such that the functions
\[
\widetilde{p}:x\mapsto p(x)\frac{f(H^{m}\Phi (x))}{( \delta (x))
^{m-1}H\Phi (x)},\quad
\widetilde{q}:x\mapsto q\,(x)\frac{g(H^{m}\Phi
(x))}{( \delta (x)) ^{m-1}H\Phi (x)}
\]
belong to the Kato class $K_{m,n}$.
\end{itemize}

 Using a fixed point argument, we prove in section 5 the following
second existence result.

\begin{theorem} \label{thm3}
Assume that $\lambda =\mu =1$ and that {\rm (H4)--(H5)}
are satisfied. Suppose that there exists $\gamma>1$ such
that  $\varphi \geq \gamma\Phi $
and $\psi \geq \gamma\Phi $ on $\partial B$.
Then \eqref{1.1} has a
positive continuous solution satisfying for each $x\in B$
\begin{equation}
\begin{gathered}
H^{m}\Phi (x)\leq u(x)\leq H^{m}\varphi (x), \\
H^{m}\Phi (x)\leq v(x)\leq H^{m}\psi (x).
\end{gathered} \label{gha}
\end{equation}
\end{theorem}

 Note that for $m=1$ we find again the result of \cite{Sameh} which
was our original motivation for deriving our study. The last section is
reserved to examples.
We conclude this section by giving some notation.

$(i)$ Let $f$ and $g$ be nonnegative functions on a set $S$. We
write $ f(x)\approx g(x)$ for $x\in S$ if there is $c>0$ not
depending on $x$ such that
\[
\frac{1}{c}g(x)\leq f(x)\leq cg(x),\quad \forall x\in S.
\]

$(ii)$ For $s$, $t\in \mathbb{R}$, we denote
$s\wedge t=\min (s,t)$ and $s\vee t=\max (s,t)$.

$(iii)$ For any measurable function $f$ on $B$, we use the notation
\[
\alpha _{f}:=\underset{x,y\in B}{\sup
}\int_{B}\frac{G_{m,n}(x,z)G_{m,n}(z,y) }{G_{m,n}(x,y)}| f(z)| dz.
\]
Finally, we mention that the letter $c$ will be a positive
generic constant which may vary from line to line.

\section{Properties of the Green function $G_{m,n}$ and
class $K_{m,n}$}

 To make the paper self contained, this section is devoted to recall
some results established in \cite{IHSM,Ben} that will be
useful in our study.

\begin{proposition}[3G-Theorem] \label{prop1}
There exists $C_{m,n}>0$ such that for each $x$, $y$, $z\in B$
\begin{equation}
\frac{G_{m,n}(x,z)G_{m,n}(z,y)}{G_{m,n}(x,y)}
\leq C_{m,n}\Big[
\Big( \frac{ \delta (z)}{\delta (x)}\Big)
^{m}G_{m,n}(x,z)+\Big( \frac{\delta (z)}{ \delta (y)}\Big)
^{m}G_{m,n}(y,z)\big] .  \label{1.3}
\end{equation}
\end{proposition}

\begin{proposition} \label{prop2}
On $B^{2}$, the following estimates hold

(i) For $2m<n$,
\begin{equation}
G_{m,n}(x,y)\approx | x-y| ^{2m-n}\Big( 1\wedge \frac{ (\delta
(x)\delta (y))^{m}}{| x-y| ^{2m}}\Big) . \label{2.1}
\end{equation}

(ii) For $2m=n$,
\begin{equation}
G_{m,n}(x,y)\approx \log\Big(1+\frac{(\delta (x)\delta
(y))^{m}}{| x-y| ^{2m}}\Big) .  \label{2.2}
\end{equation}

(iii) For $2m>n$,
\begin{equation}
G_{m,n}(x,y)\approx (\delta (x)\delta (y))^{m-\frac{n}{2}}
(1\wedge \frac{(\delta (x)\delta (y))^{n/2}}{| x-y| ^n}
) .  \label{2.3}
\end{equation}
\end{proposition}

\begin{proposition} \label{prop3}
On $B^{2}$ there exists $c>0$ such that
\begin{equation}
c(\delta (x)\delta (y))^{m}\leq G_{m,n}(x,y).  \label{ll}
\end{equation}
Moreover if $| x-y| \geq r$, we have
\begin{equation}
G_{m,n}(x,y)\leq c\frac{(\delta (x)\delta (y))^{m}}{r^n}.  \label{yy}
\end{equation}
\end{proposition}

\begin{proposition} \label{prop4}
Let $q$ be a function in $K_{m,n}$, then

(i) The constant $\alpha _{q}$ is finite.

(ii) The function $x\mapsto (\delta (x))^{2m-1}q(x)$ is in
$ L^{1}(B)$.
\end{proposition}

\begin{proposition} \label{prop5}
For each nonnegative function $q\in K_{m,n}$ and $h$ a nonnegative
harmonic in $B$ we have for $x\in B$
\begin{equation}
\int_{B}G_{m,n}(x,y)(1-| y| ^{2})^{m-1}h(y)q(y)dy\leq \alpha
_{q}(1-| x| ^{2})^{m-1}h(x).  \label{rr}
\end{equation}
In particular,
\begin{equation}
\sup_{x\in B}\int_{B}\Big(\frac{\delta (y)}{\delta (x)}\Big)
^{m-1}G_{m,n}(x,y)q(y)dy\leq 2^{m-1}\alpha _{q}.  \label{aa}
\end{equation}
\end{proposition}

\section{Modulus of Continuity}

 The objective of this section is to prove Theorem \ref{thm1}.
Let $q$ be the function defined in $B$ by
\[
q(x)=\frac{1}{(\delta (x))^{\lambda }}.
\]
It is shown in \cite{IHSM} that the function $q\in K_{m,n}$ if
and only if $\lambda <2m$ and $V_{m,n}q$ is bounded if and only if
$\lambda <m+1$. More precisely, we give in the following sharp
estimates, on the $m$-potential $V_{m,n}q$, which improve the
inequalities given in \cite[Proposition 3.10]{IHSM}.

\begin{proposition} \label{prop6}
On $B$, the following estimates hold:

(i) $V_{m,n}q(x)\approx (\delta (x))^{m}$ if $\lambda <m$,

(ii) $V_{m,n}q(x)\approx (\delta (x))^{m}\log (\frac{2}{\delta (x)})$
if $\lambda =m$,

(iii) $V_{m,n}q(x)\approx (\delta (x))^{2m-\lambda }$ if $m<\lambda
<m+1$.
\end{proposition}

 To prove Proposition \ref{prop6}, we need the next two lemmas.
In what follows, for $x\in B$, we denote
\begin{gather*}
D_{1}=\{ y\in B,\text{ }| x-y| ^{2}\leq \delta (x)\delta
(y)\} , \\
D_{2}=\{ y\in B,\text{ }| x-y| ^{2}\geq \delta (x)\delta
(y)\} .
\end{gather*}

\begin{lemma}[\cite{Ben}] \label{lem1}
Let $x\in B$.

(1) If $y\in D_{1}$, then
\[
\frac{3-\sqrt{5}}{2}\delta (x)\leq \delta (y)\leq
\frac{3+\sqrt{5}}{2}\delta (x)\quad\text{and}\quad
| x-y| \leq \frac{1+\sqrt{5}}{2}( \delta (x)\wedge \delta (y)) .
\]

(2) If $y\in D_{2}$, then
\[
\delta (x)\vee \delta (y)\leq \frac{\sqrt{5}+1}{2}| x-y| .
\]
In particular, we have
\[
B(x,\frac{\sqrt{5}-1}{2}\delta (x))\subset D_{1}\subset
B(x,\frac{\sqrt{5}+1 }{2}\delta (x)).
\]
\end{lemma}

\begin{lemma} \label{lem2}
 For each $x\in B$,
\[
\log (\frac{2}{\delta (x)})\approx (1+\int_{D_{2}}\frac{1}{| x-y|
^n}dy).
\]
\end{lemma}

\begin{proof}
In \cite[Example 6]{So}, the authors showed that
\[
\int_{B}\frac{G_{1,n}(x,y)}{\delta (y)}dy\underset{\delta
(x)\to 0}{\sim }c\delta (x)\log (\frac{2}{\delta (x)}).
\]
Then, since the functions
$x\mapsto \int_{B}\frac{G_{1,n}(x,y)}{\delta (y)} dy $ and
$x\mapsto \delta (x)\log (\frac{2}{\delta (x)})$ are positive
continuous in $B$ we deduce that
\begin{equation}
\int_{B}\frac{G_{1,n}(x,y)}{\delta (y)}dy\approx \delta
(x)\log (\frac{2}{ \delta (x)})\text{ for all }x\in B.  \label{n}
\end{equation}
Now for $x\in B$, we write
\[
\int_{B}\frac{G_{1,n}(x,y)}{\delta
(y)}dy=\int_{D_{1}}\frac{G_{1,n}(x,y)}{ \delta
(y)}dy+\int_{D_{2}}\frac{G_{1,n}(x,y)}{\delta (y)}dy.
\]
So to prove the result, it is sufficient by \eqref{n} to show
\begin{equation}
\int_{D_{1}}\frac{G_{1,n}(x,y)}{\delta (y)}dy\approx
 \delta (x)  \label{x}
\end{equation}
and
\begin{equation}
\int_{D_{2}}\frac{G_{1,n}(x,y)}{\delta (y)}dy\approx \delta
(x)\int_{D_{2}} \frac{1}{| x-y| ^n}dy. \label{u}
\end{equation}
To this end, we distinguish two cases.

\noindent\textbf{Case 1:}  $n\geq 3$. Let $x\in B$.
 By using \eqref{2.1}, we have
\begin{equation}
\int_{D_{1}}\frac{G_{1,n}(x,y)}{\delta (y)}dy\approx
\frac{1}{\delta (x)} \int_{D_{1}}\frac{1}{| x-y| ^{n-2}}dy.
\label{p}
\end{equation}
On the other hand, by Lemma \ref{lem1},
\[
\int_{B( x,\frac{\sqrt{5}-1}{2}\delta (x)) }\frac{1}{|
x-y| ^{n-2}}dy\leq \int_{D_{1}}\frac{1}{| x-y| ^{n-2}}dy\leq
\int_{B( x,\frac{\sqrt{5}+1}{2}\delta (x)) }\frac{1 }{|
x-y| ^{n-2}}dy,
\]
which implies
\[
\int_0^{\frac{\sqrt{5}-1}{2}\delta (x)}r\,dr\leq
\int_{D_{1}}\frac{1 }{| x-y| ^{n-2}}dy\leq
\int_0^{\frac{\sqrt{5}+1 }{2}\delta (x)}r\,dr.
\]
Hence, we deduce that
\begin{equation}
\int_{D_{1}}\frac{1}{| x-y| ^{n-2}}dy\approx ( \delta
(x)) ^{2}.  \label{h}
\end{equation}
By \eqref{p} and \eqref{h} we deduce \eqref{x}.
Furthermore, by \eqref{2.1} and the definition of $D_{2}$, we have
for $x\in B$ and $y\in D_{2}$
\[
G_{1,n}(x,y)\approx \frac{\delta (x)\delta (y)}{| x-y| ^n}.
\]
So we have clearly \eqref{u}.

\noindent\textbf{Case 2:} $n=2$.
Let $y\in D_{1}$ and $x\in B$, then using that
$\log (1+t)\leq ct^{1/2}$ for $t\geq 0$, we obtain
\[
\log 2\leq \log ( 1+\frac{\delta (x)\delta (y)}{| x-y|
^{2}}) \leq c( \frac{\delta (x)\delta (y)}{| x-y|
^{2}}) ^{1/2},
\]
this together with \eqref{2.2} and Lemma \ref{lem1} imply
\[
\frac{1}{c\delta (x)}\int_{B( x,\frac{\sqrt{5}-1}{2}\delta
(x)) }dy
\leq \int_{D_{1}}\frac{G_{1,n}(x,y)}{\delta(y)}dy
\leq c\int_{B( x, \frac{\sqrt{5}+1}{2}\delta (x))}\frac{1}{| x-y| } dy.
\]
So, we obtain
\[
\frac{1}{c\delta (x)}\int_0^{\frac{\sqrt{5}-1}{2}\delta
(x)}r\,dr\leq \int_{D_{1}}\frac{G_{1,n}(x,y)}{\delta (y)}dy\leq
c\int_0^{\frac{\sqrt{5}+1}{2}\delta (x)}dr.
\]
Hence, we obtain the claim \eqref{x}.
On the other hand, since
$\frac{\delta (x)\delta (y)}{|x-y| ^{2}}\in [ 0,1] $ for $x\in B$
 and $y\in D_{2}$ and using the fact that $\log (1+t)\approx t$
for $t\in [0,1] $, we obtain
\[
\int_{D_{2}}\frac{G_{1,n}(x,y)}{\delta (y)}dy\approx \delta
(x)\int_{D_{2}} \frac{1}{| x-y| ^{2}}dy,
\]
which gives \eqref{u} for $n=2$.
This completes the proof.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop6}]
In \cite{IHSM}, the authors proved the result
(i) and the upper estimates of $V_{m,n}q$ if $\lambda \in [ m,m+1) $.
Let us prove the lower estimates. First we need to show that
\begin{equation}
\int_{D_{1}}\frac{G_{m,n}(x,y)}{(\delta (y))^{\lambda }}dy\geq c(\delta
(x))^{2m-\lambda }\quad \text{for }x\in B.  \label{2}
\end{equation}
For this, we remark by Proposition \ref{prop2} and the definition of $D_{1}$
that for each $n$, $m\in\mathbb{N}^{\ast }$
\[
G_{m,n}(x,y)\geq c| x-y| ^{2m-n},\quad x\in B,\; y\in
D_{1}.
\]
It follows from Lemma \ref{lem1}, that
\begin{align*}
\int_{D_{1}}\frac{G_{m,n}(x,y)}{(\delta (y))^{\lambda }}dy
&\geq \frac{c}{
(\delta (x))^{\lambda }}\int_{D_{1}}| x-y| ^{2m-n}dy \\
&\geq \frac{c}{(\delta (x))^{\lambda }}\int_{B( x,\frac{\sqrt{5}-1}{2}
\delta (x)) }| x-y| ^{2m-n}dy \\
&\geq \frac{c}{(\delta (x))^{\lambda }}\int_0^{\frac{\sqrt{5}-1}{2}\delta
(x)}r^{2m-n}r^{n-1}dr \\
&\geq c(\delta (x))^{2m-\lambda }.
\end{align*}
Then \eqref{2} is proved for each $m$ and $n$ and so (iii)
holds.

It remains to prove the lower estimate in (ii); i.e., for
$\lambda =m$.
Since $\frac{\delta (x)\delta (y)}{| x-y| ^{2}} \in
[ 0,1] $, for $y\in D_{2}$, $x\in B$ and using the fact
that $ \log (1+t)\approx t$ for $t\in [ 0,1] $, we obtain
immediately by Proposition \ref{prop2},
\begin{equation}
G_{m,n}(x,y)\approx \frac{(\delta (x)\delta (y))^{m}}{| x-y|
^n},\quad \text{for }y\in D_{2}, \; x\in B. \label{hh}
\end{equation}
Now let $x\in B$, by writing
\[
V_{m,n}q(x)=\int_{D_{1}}\frac{G_{m,n}(x,y)}{(\delta
(y))^{m}}dy+\int_{D_{2}} \frac{G_{m,n}(x,y)}{(\delta (y))^{m}}dy,
\]
it follows from \eqref{2} and \eqref{hh} that
\[
V_{m,n}q(x)\geq c(\delta (x))^{m}\Big( 1+\int_{D_{2}}\frac{1}{|
x-y| ^n}dy\Big) .
\]
Now, using Lemma \ref{lem2}, we deduce that
\[
V_{m,n}q(x)\geq c(\delta (x))^{m}\log ( \frac{2}{\delta (x)}) .
\]
This completes the proof.
\end{proof}

\begin{proposition} \label{prop7}
Let $x_0\in \overline{B}$ and $q\in K_{m,n}$. Then we have
\[
\lim_{\alpha \to 0}\Big( \sup_{x\in B} \int_{B\cap B(x_0,\alpha
)}\frac{G_{m,n}(x,y)G_{m,n}(y,z)}{G_{m,n}(x,z)} | q(y)| dy\Big)
=0
\]
uniformly in $z\in B$.
\end{proposition}

\begin{proof}
Let $\varepsilon >0$, then by the definition of $K_{m,n}$, there
is $r>0$
such that
\[
\sup_{x\in B}\int_{B\cap B(x,r)}\Big( \frac{\delta
(y)}{\delta (x)}\Big) ^{m}G_{m,n}(x,y)| q(y)| dy\leq \varepsilon.
\]
Now, let $x_0\in \overline{B}$, $x, z\in B$ and $\alpha >0$
then by \eqref{1.3}
\begin{align*}
&\int_{B\cap B(x_0,\alpha )}\frac{G_{m,n}(x,y)G_{m,n}(y,z)}{G_{m,n}(x,z)}
| q(y)| dy \\
&\leq 2C_{m,n}\sup_{\xi \in B}\int_{B\cap
B(x_0,\alpha )}\Big(\frac{\delta (y)}{\delta (\xi )}\Big)
^{m}G_{m,n}(\xi ,y)| q(y)| dy.
\end{align*}
Furthermore, from \eqref{yy}, for each $x\in B$, we have
\begin{align*}
&\int_{B\cap B(x_0,\alpha )}\Big(\frac{\delta (y)}{\delta (x)}\Big) ^{m}G_{m,n}(x,y)| q(y)| dy \\
&\leq \int_{B\cap B(x_0,\alpha )\cap (| x-y| <r)}
\Big(\frac{\delta (y)}{\delta (x)}\Big) ^{m}G_{m,n}(x,y)|
q(y)| dy \\
&\quad +\int_{B\cap B(x_0,\alpha )\cap (| x-y| \geq r)}
\Big(\frac{\delta (y)}{\delta (x)}\Big) ^{m}G_{m,n}(x,y)|
q(y)| dy \\
&\leq \varepsilon +\frac{c}{r^n}\int_{B\cap B(x_0,\alpha )}(\delta
(y))^{2m}| q(y)| dy \\
&\leq \varepsilon +\frac{c}{r^n}\int_{B\cap B(x_0,\alpha
)}(\delta (y))^{2m-1}| q(y)| dy.
\end{align*}
Using Proposition \ref{prop4} (ii), we deduce the result by letting
$\alpha \to 0$.
\end{proof}

\begin{corollary} \label{coro1}
Let $m-1\leq \beta \leq m$, $x_0\in \overline{B}$, then for
each $q\in K_{m,n}$,
\[
\lim_{\alpha \to 0}\Big( \sup_{x\in B} \int_{B\cap B(x_0,\alpha )}\Big(\frac{\delta (y)}{\delta (x)}\Big) ^{\beta }G_{m,n}(x,y)| q(y)| dy\Big) =0.
\]
\end{corollary}

\begin{proof}
For $\beta =m-1$, the result was proved in \cite{MTZ}.
For $\beta \in ( m-1,m] $, we deduce from Proposition \ref{prop6}, that
\begin{equation}
h(x):=\int_{B}G_{m,n}(x,y)\frac{1}{( \delta (y))
^{\lambda }} dy\approx ( \delta ( x) )
^{\beta },\text{ }x\in B, \label{3}
\end{equation}
where $\lambda =2m-\beta $ if $\beta \in ( m-1,m) $ and
$\lambda <m$ if $\beta =m$.
Let $\varepsilon >0$, then by Proposition \ref{prop7} there exists
 $\alpha >0$ such
that for each $z\in B$ we have
\[
\sup_{x\in B}\int_{B\cap B(x_0,\alpha )}\frac{
G_{m,n}(x,y)G_{m,n}(y,z)}{G_{m,n}(x,z)}| q(y)| dy\leq \varepsilon .
\]
By Fubini's theorem, we have
\begin{align*}
&\int_{B\cap B(x_0,\alpha )}h(y)G_{m,n}(x,y)| q(y)| dy
\\
&=\int_{B}( \int_{B\cap B(x_0,\alpha
)}\frac{G_{m,n}(x,y)G_{m,n}(y,z) }{G_{m,n}(x,z)}| q(y)| dy)
\frac{G_{m,n}(x,z)}{
( \delta (z)) ^{\lambda }}dz \\
&\leq \varepsilon h(x).
\end{align*}
Which together with \eqref{3} imply
\begin{align*}
&\sup_{x\in B}\int_{B\cap B(x_0,\alpha )}\Big(\frac{\delta (y)}{\delta (x)}\Big) ^{\beta }G_{m,n}(x,y)| q(y)| dy \\
&\leq c\text{ }\sup_{x\in B}\int_{B\cap B(x_0,\alpha )}\frac{
h(y)}{h(x)}G_{m,n}(x,y)| q(y)| dy
\leq c\varepsilon .
\end{align*}
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
 Let $\beta \in [ m-1,m] $, $x_0\in \overline{B}$ and
$\varepsilon >0$. By Corollary \ref{coro1}, there exists $\alpha >0$
such that
\begin{equation}
\sup_{\xi \in B}\int_{B\cap B(x_0,2\alpha )}\Big(\frac{\delta (y)}{\delta (\xi )}\Big) ^{\beta }G_{m,n}(\xi ,y)|
q(y)| dy\leq \varepsilon .  \label{m}
\end{equation}
We distinguish following two cases.

\noindent \textbf{Case 1:} $\beta \in [ m-1,m)$.
First we  prove that $v$ is continuous on $B$.
For this aim we fix $x_0\in B$ and $x$, $z\in B\cap B(x_0,\alpha )$.
So we have
\begin{align*}
| v(x)-v(z)|
&\leq \int_{B}| \frac{G_{m,n}(x,y) }{(\delta
(x))^{\beta }}-\frac{G_{m,n}(z,y)}{(\delta (z))^{\beta }}
| (\delta (y))^{\beta }| q(y)| dy \\
&\leq \int_{B\cap B(x_0,2\alpha )}| \frac{G_{m,n}(x,y)}{(\delta
(x))^{\beta }}-\frac{G_{m,n}(z,y)}{(\delta (z))^{\beta }}| (\delta
(y))^{\beta }| q(y)| dy \\
&\quad +\int_{B\cap B^{c}(x_0,2\alpha )}| \frac{G_{m,n}(x,y)}{(\delta
(x))^{\beta }}-\frac{G_{m,n}(z,y)}{(\delta (z))^{\beta }}| (\delta
(y))^{\beta }| q(y)| dy \\
&\leq 2\sup_{\xi \in B}\int_{B\cap B(x_0,2\alpha
)}\Big(\frac{\delta (y)}{\delta (\xi )}\Big) ^{\beta
}G_{m,n}(,y)|
q(y)| dy \\
&\quad +\int_{B\cap B^{c}(x_0,2\alpha )}| \frac{G_{m,n}(x,y)}{(\delta
(x))^{\beta }}-\frac{G_{m,n}(z,y)}{(\delta (z))^{\beta }}| (\delta
(y))^{\beta }| q(y)| dy \\
&= I_{1}+I_{2}.
\end{align*}
If $| y-x_0| $ $\geq 2\alpha $ then $| y-x| \geq \alpha $ and
$|y-z| \geq \alpha $.

So applying \eqref{yy}, for all $x\in B\cap B(x_0,\alpha )$ and $y\in
B\cap B^{c}(x_0,2\alpha )$, we have
\[
\Big(\frac{\delta (y)}{\delta (x)}\Big) ^{\beta }G_{m,n}(x,y)\leq
c(\delta (y))^{\beta +m}\leq c(\delta (y))^{2m-1}.
\]
On the other hand, for $y\in B\cap B^{c}(x_0,2\alpha )$,
$x\mapsto \frac{ G_{m,n}(x,y)}{(\delta (x))^{\beta }}$ is
continuous in $B\cap B(x_0,\alpha )$. Hence since $x\mapsto
(\delta (x))^{2m-1}q(x)$ is in $L^{1}(B)$ then by
the dominated convergence theorem, we obtain
\[
I_{2}=\int_{B\cap B^{c}(x_0,2\alpha )}\big| \frac{G_{m,n}(x,y)}{
(\delta (x))^{\beta }}-\frac{G_{m,n}(z,y)}{(\delta (z))^{\beta }}\big|
(\delta (y))^{\beta }| q(y)| dy\to 0
\]
as $| x-z| \to 0$.
This together with \eqref{m} imply that $v$ is
continuous on $B$.

 Next, we  show that
\begin{equation}
v(x)\to 0\quad \text{as }\delta (x)\to 0.  \label{mmm}
\end{equation}
For this we consider $x_0\in \partial B$ and
$x\in B(x_0,\alpha )\cap B$,
then
\begin{align*}
v(x) &= \int_{B\cap B(x_0,2\alpha )}\Big( \frac{\delta (y)}{\delta (x)}
\Big) ^{\beta }G_{m,n}(x,y)| q(y)| dy \\
&\quad +\int_{B\cap B^{c}(x_0,2\alpha )}\Big( \frac{\delta (y)}{\delta (x)}
\Big) ^{\beta }G_{m,n}(x,y)| q(y)| dy \\
&\leq \sup_{\xi \in B}\int_{B\cap B(x_0,2\alpha
)}\Big( \frac{ \delta (y)}{\delta (\xi )}\Big) ^{\beta
}G_{m,n}(\xi ,y)|
q(y)| dy \\
&\quad +\int_{B\cap B^{c}(x_0,2\alpha )}\Big( \frac{\delta (y)}{\delta (x)}
\Big) ^{\beta }G_{m,n}(x,y)| q(y)| dy \\
&= J_{1}+J_{2}.
\end{align*}
For $y\in B\cap B^{c}(x_0,2\alpha )$ we have $| y-x| \geq \alpha
$. So from \eqref{yy} we obtain
\[
\Big( \frac{\delta (y)}{\delta (x)}\Big) ^{\beta }G_{m,n}(x,y)\leq
c(\delta (x))^{m-\beta }\to 0\text{ as }\delta (x)\to 0.
\]
Then by the same arguments as above, we deduce that
$J_{2}\to 0$ as $ \delta (x)\to 0$. This together
with \eqref{m} gives \eqref{mmm}.

\noindent \textbf{Case 2:} $\beta =m$.
We point out that for $y\in B$, the function
$x\mapsto\frac{G_{m,n}(x,y)}{ (\delta (x))^{m}}$ is continuous in
$\overline{B}$ outside the diagonal. So using similar arguments
as in the case 1 we prove that $v\in C(\overline{B })$.
This completes the proof.
\end{proof}

\begin{proposition} \label{prop8}
Let $m-1\leq \beta <m$ and $q$ be a nonnegative function in $K_{m,n}$.
Then the family of functions
\[
\Big\{ \int_{B}\Big(\frac{\delta (y)}{\delta (x)}\Big)
^{\beta }G_{m,n}(x,y)f(y)dy,\text{ }| f| \leq q\Big\}
\]
is relatively compact in $C_0(B)$.
\end{proposition}

The proof of the above proposition is similar to the one of
Theorem \ref{thm1}.  So we omit it.

\section{Proof of Theorem \ref{thm2}}

Assume that the hypotheses {\rm (H1)--(H3)} are satisfied.
Then  for $x\in B$ we have
\begin{gather}
\lambda _0V_{m,n}( qg( H^{m}\psi ) ) (x)\leq
H^{m}\varphi (x),  \label{4.1} \\
\mu _0V_{m,n}( pf( H^{m}\varphi ) ) (x)\leq
H^{m}\psi (x).  \label{4.2}
\end{gather}
 Let $\lambda \in [ 0,\lambda _0)$ and
$\mu \in [ 0,\mu _0)$. We define the sequences
$( u_{k})_{k\geq 0}$ and $ ( v_{k}) _{k\geq 0}$ by
\begin{gather*}
v_0=H^{m}\psi \\
u_{k}=H^{m}\varphi -\lambda V_{m,n}( qg(v_{k})) \\
v_{k+1}=H^{m}\psi -\mu V_{m,n}(pf(u_{k})).
\end{gather*}
We will prove that for all $k\in\mathbb{N}$,
\begin{gather}
0<( 1-\frac{\lambda }{\lambda _0}) H^{m}\varphi \leq u_{k}\leq
u_{k+1}\leq H^{m}\varphi , \label{4.3} \\
0<( 1-\frac{\mu }{\mu _0}) H^{m}\psi \leq v_{k+1}\leq v_{k}\leq
H^{m}\psi .  \label{4.4}
\end{gather}
 From \eqref{4.1} we have that for each $x\in B$,
\begin{align*}
u_0(x)
&= H^{m}\varphi (x)-\lambda V_{m,n}(qg(v_0))( x) \\
&\geq H^{m}\varphi (x)-\frac{\lambda }{\lambda _0}H^{m}\varphi (x) \\
&= ( 1-\frac{\lambda }{\lambda _0}) H^{m}\varphi (x)>0.
\end{align*}
So
\[
v_{1}( x) -v_0( x) =-\mu V_{m,n}(pf(u_0))(x) \leq 0.
\]
On the other hand, since $g$ is nondecreasing we have
\[
u_{1}( x) -u_0( x)
 =\lambda V_{m,n}[ q( g(v_0)-g(v_{1})) ] ( x) \geq 0.
\]
Since $f$ is nondecreasing and using that
\begin{equation}
u_0( x) \leq H^{m}\varphi (x),  \label{4}
\end{equation}
we deduce from \eqref{4.2} that
\[
v_{1}(x)
= H^{m}\psi (x)-\mu V_{m,n}( pf(u_0)) (x)
\geq ( 1-\frac{\mu }{\mu _0}) H^{m}\psi (x)>0.
\]
This implies that
\[
u_{1}( x) \leq H^{m}\varphi (x).
\]
Finally, we obtain
\begin{gather*}
0<( 1-\frac{\lambda }{\lambda _0}) H^{m}\varphi \leq u_0\leq
u_{1}\leq H^{m}\varphi , \\
0<( 1-\frac{\mu }{\mu _0}) H^{m}\psi \leq v_{1}\leq v_0\leq
H^{m}\psi .
\end{gather*}
This implies that \eqref{4.3} and \eqref{4.4} hold for $k=0$ and we
conclude for any $k\in \mathbb{N}$ by induction.

 Therefore, the sequences $(u_{k}) _{k\geq 0}$ and
$ ( v_{k}) _{k\geq 0}$ converge respectively to
two functions $u$ and $v$ satisfying
\begin{equation}
\begin{gathered}
0<( 1-\frac{\lambda }{\lambda _0}) H^{m}\varphi \leq u\leq
H^{m}\varphi , \\
0<( 1-\frac{\mu }{\mu _0}) H^{m}\psi \leq v\leq H^{m}\psi .
\end{gathered}  \label{4.9}
\end{equation}
Now, since $g$ is nondecreasing continuous, we obtain by
\eqref{4.4} that
for each $(x,y)\in B^{2}$
\[
0\leq G_{m,n}(x,y)q(y)g(v_{k})\leq ||g(H^{m}\psi )||_{\infty
}G_{m,n}(x,y)q(y).
\]
Moreover, since $x\mapsto \frac{q(x)}{( \delta (x))
^{m-1}}\in K_{m,n}$ then by \eqref{aa}, we have for each $x\in B$,
\[
y\mapsto G_{m,n}(x,y)q(y)\in L^{1}(B).
\]
So using the continuity of $g$ and the dominated convergence
theorem we deduce that
\[
\lim_{k\to \infty }V_{m,n}(qg(v_{k}))=V_{m,n}(qg(v)),
\]
and so we have that for each $x\in B$,
\begin{equation}
u( x) =H^{m}\varphi (x)-\lambda V_{m,n}(qg(v))( x) .
\label{4.5}
\end{equation}
Similarly we prove that for each $x\in B$,
\begin{equation}
v( x) =H^{m}\psi (x)-\mu V_{m,n}(pf(u))(x).  \label{4.6}
\end{equation}

Next, we claim that $( u,v) $ satisfies
\begin{gather*}
( -\Delta ) ^{m}u=-\lambda qg(v), \\
( -\Delta ) ^{m}v=-\mu pf(u).
\end{gather*}
Indeed, since $g(v)$ is bounded and
$x\mapsto \frac{q(x)}{(\delta(x)) ^{m-1}}\in K_{m,n}$,
we deduce by Proposition \ref{prop4} that
\[
qg(v)\in L_{\rm loc}^{1}(B).
\]
On the other hand by Theorem \ref{thm1}, we have
\[
x\mapsto \frac{1}{( \delta (x))
^{m-1}}\int_{B}G_{m,n}(x,y)q(y)dy \in C_0(B).
\]
Therefore, using that $g(v)$ is bounded we get
\begin{equation}
V_{m,n}(qg(v))\in C_0(B),  \label{yyy}
\end{equation}
which implies
\[
V_{m,n}(qg(v))\in L_{\rm loc}^{1}(B).
\]
So we have in the distributional sense
\[
( -\Delta ) ^{m}V_{m,n}(qg(v))=qg(v)\quad\text{in }B.
\]
Similarly,
\[
( -\Delta ) ^{m}V_{m,n}(pf(u))=pf(u)\quad\text{in }B.
\]
Now, applying the operator $( -\Delta ) ^{m}$ in
\eqref{4.5} and \eqref{4.6}, it follows by \eqref{4.9} that
$(u,v)$ is a
positive bounded solution of
\begin{gather*}
(-\Delta )^{m}u+\lambda qg(v)=0\quad\text{in }B, \\
(-\Delta )^{m}v+\mu pf(u)=0\quad\text{in }B.
\end{gather*}
 From \eqref{4.5} and \eqref{yyy}, we deduce that $u$ is continuous
in $B$. Similarly $v$ is continuous.

 Finally, by \eqref{1.4}, \eqref{4.5} and Theorem \ref{thm1}, we obtain
\[
\lim_{x\to \xi \in \partial B}\frac{u(x)}{( 1-|x|^{2})
^{m-1}}=\varphi (\xi ).
\]
Similarly,
\[
\lim_{x\to \xi \in \partial B}\frac{v(x)}{( 1-|x|^{2})
^{m-1}}=\psi ( \xi ) .
\]
This completes the proof.

\section{Proof of Theorem \ref{thm3}}

 Assume that $\lambda =\mu =1$ and the hypotheses
(H4)  and (H5) are satisfied. Let
${\widetilde{p}}$ and ${ \widetilde{q}}$ be the functions in
$K_{m,n}$ given by hypothesis (H5).
 Put $\gamma =1+\alpha _{\widetilde{p}}+\alpha _{\widetilde{ q}}$,
where $\alpha _{\widetilde{p}}$ and $\alpha _{\widetilde{q}}$ are
the constants associated respectively to the functions
${\widetilde{p}}$ and ${ \widetilde{q}}$.

Let us consider two nonnegative continuous functions $\varphi $
and $\psi $ on $\partial B$ such that $\varphi \geq \gamma \Phi $
and $\psi \geq \gamma \Phi $. It follows that for each $x\in B$,
\begin{equation}
H^{m}\varphi (x)\geq \gamma H^{m}\Phi (x),\quad
H^{m}\psi (x)\geq \gamma H^{m}\Phi (x).  \label{5.1}
\end{equation}
Let $S$ be the non-empty closed convex set given by
\[
S=\{w\in C_0(B):H^{m}\Phi \leq w\leq H^{m}\psi \}.
\]
We define the operator $T$ on $S$ by
\[
Tw=H^{m}\psi -V_{m,n}( pf[ H^{m}\varphi -V_{m,n}(qg(w))]) .
\]
We aim to prove that $T$ has a fixed point in $S$. First, we shall
prove that $TS$ is relatively compact in $C_0(B)$. Let $w\in S$,
then since $w\geq H^{m}\Phi $ we deduce from hypothesis
(H4)  that
\[
V_{m,n}(qg(w)) \leq V_{m,n}(qg(H^{m}\Phi ))
= V_{m,n}((\delta (.))^{m-1}\widetilde{q}H\Phi ).
\]
Which implies by (H5) and \eqref{rr} that
\begin{equation}
V_{m,n}(qg(w))\leq \alpha _{\widetilde{q}}H^{m}\Phi .  \label{5.2}
\end{equation}
This together with \eqref{5.1} imply
\begin{align*}
H^{m}\varphi -V_{m,n}(qg(w))
&\geq \gamma H^{m}\Phi -\alpha _{\widetilde{q}
}H^{m}\Phi \\
&= (1+\alpha _{\widetilde{p}})H^{m}\Phi \\
&\geq H^{m}\Phi .
\end{align*}
Hence, using (H4), we have
\begin{equation}
pf[ H^{m}\varphi -V_{m,n}(qg(w))] \leq pf(H^{m}\Phi )=(\delta
(.))^{m-1}\widetilde{p}H\Phi .  \label{z"}
\end{equation}
This yields
\begin{equation}
pf[ H^{m}\varphi -V_{m,n}(qg(w))] \leq \| H\Phi
\| _{\infty }(\delta (.))^{m-1}\widetilde{p}.  \label{za}
\end{equation}
Then using Proposition \ref{prop8} with $\beta =m-1$, we deduce that the
family of functions
\[
\{ V_{m,n}(pf[ H^{m}\varphi -V_{m,n}(qg(w))]):w\in S\}
\]
is relatively compact in $C_0(B)$. So since $H^{m}\psi \in
C_0(B)$, we conclude that the family $TS$ is relatively compact
in $C_0(B)$.

Next, we shall prove that $T(S)\subset S$. For all $w\in S$, we have
obviously
\[
Tw(x)\leq H^{m}\psi (x),\quad \forall x\in B.
\]
On the other hand, by \eqref{z"}, we have
\begin{align*}
V_{m,n}(pf[ H^{m}\varphi -V_{m,n}(qg(w))]
&\leq V_{m,n}((\delta(.))^{m-1}\widetilde{p}H\Phi ) \\
&\leq V_{m,n}(\widetilde{p}H^{m}\Phi ).
\end{align*}
Then, by (H5) and \eqref{rr} we have
\begin{equation}
V_{m,n}(pf[ H^{m}\varphi -V_{m,n}(qg(w))] \leq \alpha
_{ \widetilde{p}}H^{m}\Phi .  \label{e}
\end{equation}
Which implies by \eqref{5.1}, that for each $x\in B$
\begin{align*}
Tw(x) &\geq H^{m}\psi (x)-\alpha _{\widetilde{p}}H^{m}\Phi (x) \\
&\geq (\gamma -\alpha _{\widetilde{p}})H^{m}\Phi (x) \\
&\geq (1+\alpha _{\widetilde{q}})H^{m}\Phi (x) \\
&\geq H^{m}\Phi (x),
\end{align*}
which proves that $T(S)\subset S$.

 Now, we prove the continuity of the operator $T$ in $S$ for the
supremum norm. Let $(w_{k})_{k\in \mathbb{N}}$ be a sequence in
$S$ which converges uniformly to a function $w$ in $S$.
 Since $g$ is nonincreasing we deduce by $(H_{5})$ that
\[
qg(w_{k})\leq qg(H^{m}\Phi )\leq \| H\Phi \| _{\infty
}(\delta (.))^{m-1}\widetilde{q}.
\]
Now, it follows from (H5) and \eqref{aa}, that for each $x\in B$,
\[
y\mapsto ( \delta (y)) ^{m-1}G_{m,n}(x,y)\widetilde{q}(y)\in
L^{1}(B).
\]
We conclude by the dominated convergence theorem that for all
$x\,\in \,B$,
\begin{equation}
\lim_{k\to \infty }V_{m,n}(qg(w_{k}))( x)
=V_{m,n}(qg(w))( x)  \label{5.4}
\end{equation}
and so from the continuity of $f$, we have
\[
\lim_{k\to \infty }p( x) f[H^{m}\varphi
(x)-V_{m,n}(qg(w_{k}))(x)]=p( x) f[H^{m}\varphi
(x)-V_{m,n}(qg(w))(x)].
\]
By \eqref{za}, for each $x,y$ in $B$,
\[
G_{m,n}(x,y)p(y)f[H^{m}\varphi (y)-V_{m,n}(qg(w_{k}))(y)]
\leq c(\delta (y))^{m-1}\widetilde{p}(y)G_{m,n}(x,y).
\]
Then since $\widetilde{p}\in K_{m,n}$, we get by \eqref{aa} and
the dominated convergence theorem that for each $x\in B$,
\[
Tw_{k}(x)\to Tw(x)\quad\text{as }k\to +\infty .
\]
Consequently, since $T(S)$ is relatively compact in $C_0(B)$, we
deduce that the pointwise convergence implies the uniform
convergence, namely,
\[
{\Vert Tw_{k}-Tw\Vert }_{\infty }\to 0\quad \text{as }
k\to +\infty .
\]
Therefore, $T$ is a continuous mapping of $S$ to itself. So, since
$T(S)$ is relatively compact in $C_0(B)$, it follows that $T$ is
a compact mapping on $S$. Finally, the Schauder
fixed-point theorem implies the existence of a
function $w\in S$ such that $w=Tw$. We put for $x\in B$
\begin{equation}
u(x)=H^{m}\varphi (x)-V_{m,n}(qg(w))(x)  \label{y}
\end{equation}
and
$v(x)=w(x)$.
Then
\[
v(x)=H^{m}\psi (x)-V_{m,n}(pf(u))(x).
\]
It is clear that $(u,v)$ satisfies \eqref{gha} and it remains
to prove that $ ( u,v) $ satisfies \eqref{1.1} with $\lambda =\mu=1$.

Since $0\leq qg(v)\leq c(\delta (.))^{m-1}\widetilde{q}$ then by
Proposition \ref{prop4}, it follows that $qg(v)\in L_{\rm loc}^{1}(B)$ and from
\eqref{5.2}, we have $ V_{m,n}(qg(v))\in L_{\rm loc}^{1}(B)$.
Hence $u$ satisfies (in the distributional sense)
\[
(-\Delta )^{m}u=-(-\Delta )^{m}V_{m,n}(qg(w))=-qg(v).
\]
On the other hand,
\[
(-\Delta )^{m}v=-(-\Delta )^{m}V_{m,n}(pf[ H^{m}\varphi
-V_{m,n}(qg(v)) ] ).
\]
Using \eqref{za} and Proposition \ref{prop4} we deduce that
$pf[H^{m}\varphi -V_{m,n}(qg(v))] \in L_{\rm loc}^{1}(B)$.
Moreover, by \eqref{e} we get
\[
V_{m,n}(pf(u))=V_{m,n}(pf[ H^{m}\varphi -V_{m,n}(qg(v))] )\in
L_{\rm loc}^{1}(B).
\]
Hence, we have in the distibutional sense
\[
(-\Delta )^{m}v=-pf(u).
\]
Finally, let $\xi \in \partial B$, then since
$qg(v)\leq c(\delta (.))^{m-1}\widetilde{q}$, we deduce by
Theorem \ref{thm1} for $\beta =m-1$, that
\[
\underset{x\to \xi }{\lim }\frac{V_{m,n}(qg(v))(x)}{(1-|
x^{2}| )^{m-1}}=0.
\]
Hence by \eqref{1.4} and \eqref{y} we have
\[
\lim_{x\to \xi } \frac{u(x)}{(1-| x|^{2})^{m-1}}
=\varphi (\xi )-\lim_{x\to \xi }
\frac{ V_{m,n}(qg(v))(x)}{(1-| x^{2}| )^{m-1}}=\varphi (\xi ).
\]
Similarly,
\[
\lim_{x\to \xi } \frac{v(x)}{(1-| x| ^{2})^{m-1}}
=\psi (\xi )-\lim_{x\to \xi }\frac{
V_{m,n}(pf(u))}{(1-| x| ^{2})^{m-1}}=\psi (\xi ).
\]
This completes the proof.

\section{Examples}

In this section, we give  examples that illustrate the existence
results for \eqref{1.1}. In the following two
examples (H3) is satisfied.

\begin{example} \label{exa2} \rm
Let $\varphi $ be a continuous function on $\partial B$ such that
there exists $c_0>0$ satisfying $\varphi (x)\geq c_0$ for all
$x\in \partial B$. Let $p$ be a nonnegative function on $B$ such
that $p_0=\frac{p}{ (\delta (.))^{m-1}}$ is in $K_{m,n}$ and
$q$ be a nonnegative measurable function satisfying for each $x\in
B$, $q(x)\leq \frac{c}{( \delta (x)) ^{\lambda }}$ with
$\lambda <m$. We consider $f$,
$g:(0,\infty )\to [0,\infty )$ nondecreasing and continuous functions.
Then (H3) is satisfied. Indeed, let $x\in B$, by \eqref{aa}, we have
\[
V_{m,n}(p)(x)\leq 2^{m-1}\alpha _{p_0}(\delta (x))^{m-1}.
\]
So
\begin{align*}
\frac{H^{m}\varphi (x)}{V_{m,n}(pf(H^{m}\psi ))(x)}
 &\geq \frac{(1-| x| ^{2})^{m-1}c_0}{2^{m-1}\alpha
_{p_0}\| f(H\psi )\| _{\infty }(\delta (x))^{m-1}}\\
&\geq \frac{c_0}{2^{m-1}\alpha _{p_0}\| f(H\psi )\|
_{\infty }}>0,
\end{align*}
which implies that $\lambda _0>0$.

 Now since $\psi $ is a nonnegative continuous function, then there
exists $c>0$ such that for all $x\in B$, $H\psi (x)\geq c\delta (x)$.
So we have
\[
\frac{H^{m}\psi (x)}{V_{m,n}(qg(H^{m}\varphi ))(x)}\geq
\frac{c\delta (x)(1-| x| ^{2})^{m-1}}{\| g(H\varphi
)\| _{\infty }V_{m,n}q(x)}.
\]
Since $q(x)\leq \frac{c}{( \delta (x)) ^{\lambda }}$,
$\lambda <m, $ we have by Proposition \ref{prop6} that
\[
V_{m,n}(q)(x)\approx (\delta (x))^{m}.
\]
So
\[
\frac{(1-| x| ^{2})^{m-1}H\psi (x)}{V_{m,n}(qg(H^{m} \varphi))(x)}
\geq \frac{c\delta (x)(1-| x| ^{2})^{m-1} }{\|
g(H\varphi )\| _{\infty }( \delta (x)) ^{m}}
\geq \frac{c}{\| g(H\varphi )\| _{\infty }}>0.
\]
This proves that $\mu _0>0$.
\end{example}

\begin{example} \label{exa3} \rm
Let $\varphi $ and $\psi $ two nonnegative continuous functions
on $\partial B$.
 We consider $f$, $g:(0,\infty )\to [ 0,\infty )$
nondecreasing and continuous functions. Since the functions
$H^{m}\varphi $
and $H^{m}\psi $ are nonnegative bounded, then there exist
$a_{1}\geq 0$, $a_{2}\geq 0$ such that $a_{1}+a_{2}>0$ and for
each $x\in B$,
\[
f(H^{m}\varphi (x))\leq a_{1}H^{m}\varphi (x)+a_{2},\quad
g(H^{m}\psi (x))\leq a_{1}H^{m}\psi (x)+a_{2}.
\]
We assume
\begin{itemize}
\item[(A1)] $a_{1}\varphi \approx a_{1}\psi$;

\item[(A2)] $a_{2}p\leq a_{2}\frac{c}{(\delta (x))^{\sigma }}$
$a_{2}q\leq a_{2} \frac{c}{(\delta (x))^{\sigma }}$
 with $\sigma <m$.

\end{itemize}
Then (H3) is satisfied. Indeed for each $x\in B$, we have
\[
V_{m,n}(qg(H^{m}\psi )(x)\leq a_{1}V_{m,n}(qH^{m}\psi
)(x)+a_{2}V_{m,n}(q)(x).
\]
By \eqref{rr}, we have
\[
V_{m,n}(qH^{m}\psi )(x)\leq \alpha _{q}H^{m}\psi (x),
\]
and by Proposition \ref{prop6},
\[
V_{m,n}(q)(x)\leq c( \delta (x)) ^{m}.
\]
Then
\begin{align*}
V_{m,n}(qg(H^{m}\psi ))(x)
&\leq a_{1}\alpha _{q}H^{m}\psi (x)+a_{2}c(
\delta (x)) ^{m} \\
&\leq c( \delta (x)) ^{m-1}(a_{1}H\psi (x)+a_{2}\delta (x)).
\end{align*}
So using that there exists $c>0$ such that for all $x\in B$,
$H\varphi
(x)\geq c\delta (x)$, we obtain
\begin{align*}
\frac{H^{m}\varphi (x)}{V_{m,n}( qg(H^{m}\psi ) )(x)}
&\geq c\frac{(a_{1}+a_{2})H\varphi (x)}{a_{1}H\psi (x)
 +a_{2}\delta (x)} \\
&\geq c\frac{a_{1}H\psi (x)+a_{2}\delta (x)}{a_{1}H\psi (x)
 +a_{2}\delta (x)}
= c>0.
\end{align*}
Hence $\lambda _0>0$. Similarly we have $\mu _0>0$.
 Note that if $a_{1}=0$ then  hypothesis (A1) is
satisfied for each $\varphi $ and $\psi $ and if $a_{2}=0$ then
the hypothesis (A2) is satisfied for each $p$ and $q$.
\end{example}

 Now, as an application of Theorem \ref{thm2}, we give the following example.

\begin{example} \label{exa4} \rm
Let $\lambda $, $\mu $ be nonnegative constants, and $\varphi $,
$\psi $ be two nontrivial nonnegative continuous functions on
$\partial B$. Let $ f(t)=t^{\alpha }$ and $g(t)=t^{\beta }$, where
$\alpha $, $\beta >0$.
Now, let $\sigma <m$. We take $p$ and $q$ two nonnegative
measurable functions satisfying for each $x\in B$,
\[
p(x)\leq \frac{c}{( \delta (x)) ^{\sigma }},\quad
q(x)\leq \frac{c}{( \delta (x)) ^{\sigma }}.
\]
Using similar arguments as above in Example \ref{exa2}, we show that
(H3) is satisfied.
Then for each $\lambda \in [0,\lambda _0)$ and each
$\mu \in [ 0,\mu _0)$, the problem
\begin{gather*}
(-\Delta )^{m}u+\lambda qv^{\alpha }=0\quad\text{in }B, \\
(-\Delta )^{m}v+\mu pu^{\beta }=0\quad\text{in }B, \\
\lim_{x\to \xi \in \partial B}\frac{u(x)}{(1-|
x| ^{2})^{m-1}}=\varphi (\xi ), \\
\lim_{x\to \xi \in \partial B}\frac{v(x)}{(1-| x|
^{2})^{m-1}}=\psi (\xi ),
\end{gather*}
has positive continuous solution $(u,v)$ satisfying \eqref{1.5}.
\end{example}

We end this section by giving an example as application
of Theorem \ref{thm3}.

\begin{example} \label{exa5} \rm
Let $\alpha >0$, $\beta >0$, $f(t)=t^{-\alpha }$ and
$g(t)=t^{-\beta }$.
 Let $p$ and $q$ two nonnegative measurable functions such that
\[
p(x)\leq \frac{c}{(\delta (x))^{\lambda }}\quad
\text{with }\lambda <m(1-\alpha ),
\]
and
\[
q(x)\leq \frac{c}{(\delta (x))^{\mu }}\quad \text{with }\mu <m(1-\beta ).
\]
Let $\varphi $, $\psi $ and $\Phi $ nontrivial nonnegative
continuous functions on $\partial B$. Then there exists a constant
$\gamma >1$ such that if $\varphi \geq \gamma \Phi $ and
$\psi \geq \gamma \Phi $ on $\partial B$, the problem
\begin{gather*}
(-\Delta )^{m}u+qv^{-\alpha }=0\quad\text{in }B, \\
(-\Delta )^{m}v+pu^{-\beta }=0\quad\text{in }B, \\
\lim_{x\to \xi \in \partial B}\frac{u(x)}{(1-|
x| ^{2})^{m-1}}=\varphi (\xi ), \\
\lim_{x\to \xi \in \partial B}\frac{v(x)}{(1-| x|
^{2})^{m-1}}=\psi (\xi ),
\end{gather*}
has a positive continuous solution satisfying \eqref{gha}.
\end{example}

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\end{document}