\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 143, pp. 1--14.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/143\hfil Existence of positive solutions]
{Existence of positive solutions for \\
nonlinear boundary-value problems in \\
unbounded domains of $\mathbb{R}^{n}$}
\author[F. Toumi,  N. Zeddini\hfil EJDE-2005/143\hfilneg]
{Faten Toumi, Noureddine Zeddini}  % in alphabetical order


\address{D\'{e}partement de Math\'{e}matiques, Facult\'{e} des
Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia}
\email{Faten.Toumi@fsb.rnu.tn}
\email{noureddine.zeddini@ipein.rnu.tn}

\date{}
\thanks{Submitted September 30, 2005. Published December 8, 2005.}
\subjclass[2000]{34B15, 34B27}
\keywords{Green function; nonlinear elliptic equation; positive solution;
\hfill\break\indent Schauder fixed point theorem}

\begin{abstract}
Let $D$ be an unbounded domain in $\mathbb{R}^{n}$ ($n\geq 2$)
with a nonempty compact boundary $\partial D$. We consider the following
nonlinear elliptic problem, in the sense of distributions,
\begin{gather*}
\Delta u=f(.,u),\quad u>0\quad \text{in }D,\\
u\big|_{\partial D}=\alpha \varphi ,\\
\lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\beta \lambda ,
\end{gather*}
where $\alpha ,\beta,\lambda $ are nonnegative constants with
$\alpha +\beta >0$ and $\varphi $ is a nontrivial nonnegative
continuous function on $\partial D$. The function
$f$ is nonnegative and satisfies some appropriate conditions related to
a Kato class of functions, and $h$ is a fixed harmonic
function in $D$, continuous on $\overline{D}$. Our aim is to prove
the existence of positive continuous solutions bounded below by a
harmonic function. For this aim we use the Schauder fixed-point
argument and a potential theory approach.
\end{abstract}

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

\section{Introduction}

 In this paper, we are concerned with the existence and
asymptotic behavior of positive solutions for the following nonlinear
elliptic equation, in the sense of distributions,
\begin{equation}
\Delta u=f(.,u)\quad\mbox{in } D,  \label{E}
\end{equation}
where $D$ is an unbounded domain in $\mathbb{R}^{n}$ ($n\geq 2$)
with a nonempty compact boundary $\partial D$ and $f$ is a nonnegative
measurable function on $D$ that may be singular or sublinear with respect to
the second variable. More precisely we will study the  problem
\begin{equation} \label{Pab}
\begin{gathered}
\Delta u=f(.,u),\quad u>0\quad \text{in }D, \\
u\big|_{\partial D}=\alpha \varphi , \\
\lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\beta \lambda \geq 0,
\end{gathered}
\end{equation}
where $\alpha \geq 0$, $\beta \geq 0$, $\varphi $ is a
nontrivial nonnegative continuous function on $\partial D$, $h$
is the harmonic function in $D$ given by \eqref{e2.2} below
and $f$ satisfies some appropriate conditions related to a Kato class
 (see Definition \ref{def1}) introduced by
Bachar et al in \cite{b1} for $n\geq 3$ and  M\^{a}agli
and Ma\^{a}toug in \cite{m3} for $n=2$.

 In \cite{a2}, Athreya considered \eqref{E} with
a special case of nonlinearity
$f(x,u)=g(u)\leq \max (1,u^{-\alpha })$ for $0<\alpha <1$, on
a simply connected bounded $C^{2}$-domain $\Omega $.
He showed that if $h_{0}$ is a fixed positive
harmonic function in $\Omega $ and $\varphi $ is a nontrivial nonnegative
continuous function on $\partial \Omega $, there exists a constant $c>1$
such that if $\varphi \geq c\,h_{0}$ on $\partial \Omega $, then
\eqref{E} has a positive continuous solution $u$ satisfying $u=\varphi $ on
$\partial \Omega $ and $u\geq h_{0}$ in $\Omega $.

 This result was extended by Bachar et al \cite{b3} on the
half space
$\mathbb{R}_{+}^{n}=\{x=(x_{1},\dots ,x_{n})\in \mathbb{R}^{n}:x_{n}>0\}$
($n\geq 2$). More precisely, they proved that the problem
\begin{gather*}
\Delta u=f(.,u)\quad \text{in }\mathbb{R}_{+}^{n}, \\
u=\varphi \quad \text{in }\partial \mathbb{R}_{+}^{n}, \\
\lim_{x_{n}\to +\infty }\frac{u(x)}{x_{n}}=c\geq 0,
\end{gather*}
has a positive solution $u$ satisfying $u(x)\geq cx_{n}+\rho _{0}(x)$
in $\mathbb{R}_{+}^{n}$, where $\rho _{0}$ is a fixed positive
continuous bounded harmonic function in $\overline{\mathbb{R}_{+}^{n}}$.

 In the sublinear case where $f(x,u)=p(x)u^{\alpha}$, $0<\alpha \leq 1$,
Lair and Wood \cite{l1}  studied the existence
of positive large solutions and bounded ones for the equation
\eqref{E}. In particular they proved the existence of entire bounded
nonnegative solutions in $\mathbb{R}^{n}$ provided that $p$ is locally
 h\"{o}lder continuous and satisfies
$\int_{0}^{\infty }t\max_{|x|=t}(p(x))dt<\infty $.

This result was extended by Bachar and Zeddini in \cite{b2} to more
general function $f(x,u)=q(x)g(u)$. More precisely it is shown
in \cite{b2} that the equation \eqref{E} has at least one
positive continuous bounded solution in $\mathbb{R}^{n}$, provided that the
Green potential of $q$ is continuous bounded in $\mathbb{R}^{n}$ and for all
$\alpha >0$, there exists a constant $k>0$ such that the function
$x\to kx-g(x)$ is nondecreasing on $[\alpha ,\infty )$.

 In this work, we will give two existence results for the problem
\eqref{Pab}. For this aim, we fix a positive harmonic
function $h_{0}$ in $D$, which is continuous and bounded in
$\overline{D}$ such that $\lim_{|x|\to +\infty}h_{0}(x)=0$,
whenever $n\geq 3$. We suppose that the function $f$
satisfies combinations of the following hypotheses:

\begin{enumerate}
\item[(H1)] $f:D\times (0,+\infty )\to [0,+\infty )$ is measurable,
continuous with respect to the second variable.

\item[(H2)] There exists a nonnegative measurable function
$\theta $ on $D\times (0,+\infty )$ such that the function
$t\mapsto \theta (x,t)$ is nonincreasing on $(0,+\infty )$,
 and satisfies
\begin{equation*}
f(x,t)\leq \theta (x,t),\text{ for \ }(
x,t)\in D\times (0,+\infty ).
\end{equation*}

\item[(H3)] The function $\psi $ defined on $D$ by $\psi
(x)=\frac{\theta (x,h_{0}(x))}{
h_{0}(x)}$ belongs to the class $K^{\infty }(D)$.

\item[(H4)] For each $\alpha \geq 0$ and $\beta \geq 0$
with $\alpha +\beta >0$, there exists a nonnegative function
$q_{\alpha,\beta }=q\in K^{\infty }(D)$ such that for each $x\in D$ and
$t\geq s\geq \alpha h_{0}(x)+\beta h(x)$, we have
\begin{gather}
f(x,t)-f(x,s)\leq (t-s)q(x), \label{e1.1}\\
f(x,t)\leq tq(x).
\end{gather}
\end{enumerate}
For the rest of this paper, we denote by $H_{D}\varphi $ the bounded
 continuous solution of the Dirichlet problem
\begin{equation} \label{e1.3}
\begin{gathered}
\Delta w=0\quad \text{in }D,\\
w\big|_{\partial D}=\varphi , \\
\lim_{|x|\to +\infty }\frac{w(x)}{h(x)}=0,
\end{gathered}
\end{equation}
where $\varphi $ is a nonnegative continuous function on $\partial D$
and $h$ is the harmonic function given by \eqref{e2.2}.

 The outline of this paper is as follows. In the second section we
recall and improve some useful results concerning estimates on the Green
function $G_{D}$ of the Laplace operator $\Delta $ in $D$ and some
properties of functions belonging to the Kato class
$K^{\infty }(D)$. In section 3, we will prove a first existence
result for the problem \eqref{Pab}, by using the Schauder fixed-point
theorem. More precisely, we prove the following

\begin{theorem} \label{thm1}
Under the assumptions (H1)--(H3), there exists a constant $c>1$ such
that if $\varphi \geq ch_{0}$ on $\partial D$,
then for each $\lambda \geq 0$, the problem \eqref{Pab} with
$\alpha=\beta=1$ has a positive continuous solution $u$ satisfying
for each $x\in D$,
\begin{equation*}
\lambda h(x)+h_{0}(x)\leq u(x)\leq
\lambda h(x)+H_{D}\varphi (x).
\end{equation*}
\end{theorem}

 In the last section, we use a potential theory approach to prove a
second existence result for the problem \eqref{Pab}.
More precisely, we will prove the following result.

\begin{theorem} \label{thm2}
Under the assumptions (H1) and (H4), if $\alpha \geq 0$ and $\beta \geq 0$
with $\alpha +\beta >0$, then there exists a constant $c_{1}>1$ such that
if $\varphi \geq c_{1}h_{0}$ on $\partial D$
and $\lambda \geq c_{1}$, the problem \eqref{Pab}
has a positive continuous solution $u$ satisfying: For each $x\in D$,
\begin{equation*}
\alpha h_{0}(x)+\beta h(x)\leq u(x)
\leq \alpha H_{D}\varphi (x)+\beta \lambda h(x).
\end{equation*}
\end{theorem}

\subsection*{Notation and preliminaries}
Throughout this paper, we will adopt the following notation.

\begin{enumerate}
\item[i.] $D$ is an unbounded domain in $\mathbb{R}^{n}$
($n\geq 2$) such that the complement of $\overline{D}$ in $\mathbb{R}^{n}$,
 $\overline{D}^{c}=\bigcup_{j=1}^{d}D_{j}$ where
$D_{j}$ is a bounded $C^{1,1}$-domain and
$\overline{D}_{i}\bigcap \overline{D}_{j}=\emptyset$, for $i\neq j$.

\item[ii.] For a metric space $S$, we denote by
$\mathcal{B}(S)$ the set of Borel measurable functions and
$\mathcal{B}_{b}(S)$ the set of bounded ones. $\mathcal{C}(S)$ will
denote the set of continuous functions on $S$. The exponent + means that
only the nonnegative functions are considered.

\item[iii.] $\mathcal{C}_{0}(\overline{D})=\{f\in
\mathcal{C}(\overline{D}):\lim_{|x|\to +\infty }f(x)=0\} $.

\item[iv.] $\mathcal{C}_{b}(D)=\{f\in \mathcal{C}(D):f
\text{ is bounded in }D\} $.
We note that $\mathcal{C}_{0}(\overline{D})$ and $\mathcal{C}_{b}(D)$
 are two Banach spaces endowed with the uniform norm
\begin{equation*}
\|f\|_{\infty }=\sup_{x\in D}|f(
x)|.
\end{equation*}

\item[v.] For $x\in D$, we denote by
$\delta _{D}(x)$ the distance from $x$ to $\partial D$,
\[
\rho _{D}(x)=\frac{\delta _{D}(x)}{\delta _{D}(x)+1},\quad
\lambda _{D}(x)=\delta _{D}(x)(\delta _{D}(x)+1).
\]

\item[vi.] Let $f$ and $g$ be two positive functions on a set $S$.
We denote $f\sim g$, if there exists a constant $c>0$ such that
\begin{equation*}
\frac{1}{c}g(x)\leq f(x)\leq cg(x)\quad \text{for all }x\in S.
\end{equation*}

\item[vii.] For $f\in \mathcal{B}^{+}(D)$, we denote by $Vf$
the Green potential of $f$ defined on $D$ by
\begin{equation*}
Vf(x)=\int_{D}G_{D}(x,y)f(y)dy.
\end{equation*}
Recall that if $f\in L_{\rm loc}^{1}(D)$ and $Vf\in L_{\rm loc}^{1}(D)$,
then we have in the distributional sense (see \cite[p. 52]{c1})
\begin{equation} \label{e1.4}
\Delta (Vf)=-f\quad \text{in  }D.
\end{equation}
Furthermore, we recall that for $f\in \mathcal{B}^{+}(D)$, the
potential $Vf$ is lower semi-continuous in $D$ and if $f=f_{1}+f_{2}$
with $f_{1},f_{2}\in \mathcal{B}^{+}(D)$ and
$Vf\in \mathcal{C}^{+}(D)$, then $Vf_{i}\in \mathcal{C}^{+}(D)$ for
 $i\in \{1,2\} $.

\item[viii.] Let $(X_{t},t>0)$ be the Brownian motion in
$\mathbb{R}^{n}$ and $P^{x}$ be the probability measure on the
 Brownian continuous paths starting at $x$.
For $q\in \mathcal{B}^{+}(D)$, we define the kernel $V_{q}$ by
\begin{equation} \label{e1.5}
V_{q}f(x)=E^{x}\Big(\int_{0}^{\tau_{D}}
e^{-\int_{0}^{t}q(X_{s})ds}f(X_{t})dt\Big),
\end{equation}
where $E^{x}$ is the expectation on $P^{x}$ and
$\tau _{D}=\inf \{t>0:X_{t}\notin D\} $.

If $q\in \mathcal{B}^{+}(D)$ such that $Vq<\infty $,
the kernel $V_{q}$ satisfies the  resolvent equation
(see \cite{c1,m1})
\begin{equation} \label{e1.6}
V=V_{q}+V_{q}(qV)=V_{q}+V(qV_{q}).
\end{equation}
 So for each $u\in \mathcal{B}(D)$ such that
$V(q|u|)<\infty $, we have
\begin{equation} \label{e1.7}
(I-V_{q}(q.))(I+V(q.))u=(I+V(q.))(I-V_{q}(q.))
u=u.
\end{equation}

\item[ix.] We recall that a function $f:[0,\infty )
\to \mathbb{R}$ is called completely monotone if
$(-1)^{n}f^{(n)}\geq 0$, for each $n\in \mathbb{N}$.
Moreover, if $f$ is completely monotone on $[0,\infty )$
then by \cite[Theorem 12a]{w1} there exists a nonnegative
measure $\mu $ on $[0,\infty )$ such that
\begin{equation*}
f(t)=\int_{0}^{\infty }\exp (-tx)d\mu (x).
\end{equation*}
So, using this fact and the Holder inequality we deduce that if $f$ is
completely monotone from $[0,\infty )$ to $(0,\infty)$, then
$\log f$ is a convex function.

\item[x.] Let $f \in \mathcal{B}^{+}(D)$ be such that
$Vf<\infty $. From \eqref{e1.5}, it is easy to see that for
 each $x\in D$, the function $F:\lambda \to V_{\lambda q}f(x)$ is
completely monotone on $[0,\infty )$.

\item[xi.] Let $a\in \mathbb{R}^{n}\backslash \overline{D}$ and $r>0$ such
that $\overline{B(a,r)}\subset \mathbb{R}^{n}\backslash \overline{D}$.
Then we have
\begin{gather*}
G_{D}(x,y)=r^{2-n}G_{\frac{D-a}{r}}(\frac{x-a}{r},\frac{y-a}{r}),
\quad\mbox{for }x,y\in D, \\
\delta _{D}(x)=r\delta _{\frac{D-a}{r}}(\frac{x-a}{r}),\quad
\text{for }x\in D,
\end{gather*}
So without loss of generality, we may suppose  that
$\overline{B(0,1)}\subset \mathbb{R}^{n}\diagdown\overline{D}$.
Moreover, we denote by $D^{\ast }$ the open set
\begin{equation*}
D^{\ast }=\{x^{\ast }\in B(0,1):x\in D\cup \{\infty
\} \} ,
\end{equation*}
where $x^{\ast }=x/|x|^{2}$ is the Kelvin
inversion from $D\cup \{\infty \} $ onto $D^{\ast }$
(see \cite{b1,m3}). Then  for $x,y\in D$,
\begin{equation*}
G_{D}(x,y)=|x|^{2-n}|y|^{2-n}G_{D^{\ast }}(x^{\ast },y^{\ast }).
\end{equation*}
\end{enumerate}

 Also we mention that the letter $C$ will denote a generic
positive constant which may vary from line to line.

\section{Properties of the Green function and the Kato class}

 In this section, we recall and improve some results concerning the
Green function $G_{D}(x,y)$ and the Kato class $K^{\infty }(D)$,
which are stated in \cite{b1} for $n\geq 3$ and in \cite{m3} for $n=2$.


\begin{theorem}[3G-Theorem] \label{3G-thm}
There exists a constant $C_{0}>0$ depending
only on $D$ such that for all $x,y$ and $z$ in $D$
\begin{equation*}
\frac{G_{D}(x,z)G_{D}(z,y)}{G_{D}(x,y)}\leq
C_{0}\Big(\frac{\rho _{D}(z)}{\rho _{D}(x)}G_{D}(x,z)+\frac{\rho
_{D}(z)}{\rho _{D}(y)}G_{D}(y,z)\Big).
\end{equation*}
\end{theorem}

\begin{proposition} \label{prop1}
On $D^{2}$ (that is $x,y\in D$), we have
\begin{equation*}
G_{D}(x,y)\sim \begin{cases}
\frac{1}{|x-y|^{n-2}}\min \Big(1,\frac{\lambda
_{D}(x)\lambda _{D}(y)}{|x-y|^{2}}\Big),& n\geq 3, \\
\log(1+\frac{\lambda _{D}(x)\lambda _{D}(y)}{|x-y|^{2}}),& n=2.
\end{cases}
\end{equation*}
Moreover, for $M>1$ and $r>0$ there exists a constant $C>0$ such that for
each $x\in D$ and $y\in D$ satisfying $|x-y|\geq r$ and
$|y|\leq M$, we have
\begin{equation} \label{e2.1}
G_{D}(x,y)\leq C\frac{\rho _{D}(x)\rho _{D}(y)}{|x-y|^{n-2}}.
\end{equation}
\end{proposition}

\begin{definition} \label{def1} \rm
A Borel measurable function $q$ in $D$ belongs to the
Kato class $K^{\infty}(D)$ if $q$ satisfies
\begin{gather*}
\lim_{\alpha \to 0}\,(\sup_{x\in D}\int_{D\cap
B(x,\alpha )}\frac{\rho _{D}(y)}{\rho _{D}(x)}
G_{D}(x,y)|q(y)|dy)=0,
\\
\lim_{M\to \infty }\,(\sup_{x\in D}\int_{D\cap
(|y|\geq M)}\frac{\rho _{D}(y)}{\rho
_{D}(x)}G_{D}(x,y)|q(y)|dy)=0.
\end{gather*}
\end{definition}

In this  paper,  $h$ denotes the function defined, on $D$, by
\begin{equation} \label{e2.2}
h(x)=c_{n}|x|^{2-n}G_{D^{\ast }}(
x^{\ast },0)=c_{n}\lim_{|y|\to
+\infty }|y|^{n-2}G_{D}(x,y),
\end{equation}
where $c_{n}=\begin{cases}
2\pi & \text{for } n=2, \\
\frac{4\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2}-1)}
&\text{for } n\geq 3.
\end{cases}$.
Then we have the following statement.

\begin{proposition} \label{prop2}
The function $h$ defined by \eqref{e2.2} is harmonic in $D$
and satisfies $\lim_{x\to z\in \partial D}h(x)=0$,
\begin{gather*}
\lim_{|x|\to \infty }\frac{h(x)}{\log|x|}=1\quad \text{ for }n=2, \\
\lim_{|x|\to \infty }h(x)=1\quad \text{for }n\geq 3.
\end{gather*}
Moreover,
\begin{equation} \label{e2.3}
h(x)\sim \begin{cases}
\rho _{D}(x) &\text{for } n\geq 3, \\
\log(1+\rho _{D}(x))&\text{for }n=2.
\end{cases}
\end{equation}
\end{proposition}

The proof of the above proposition  can be found in  \cite[Lemma 4.1]{m3}
 and in \cite{m5}.

\begin{remark}[{\cite[p.427]{d1}}] \label{rmk1} \rm
The function $H_{D}\varphi $ defined in \eqref{e1.3} belongs to
$\mathcal{C}(\overline{D}\cup \{\infty \} )$ and
satisfies
\begin{equation*}
\lim_{|x|\to +\infty }|
x|^{n-2}H_{D}\varphi (x)=C>0.
\end{equation*}
\end{remark}

   In the sequel, we use the notation
\begin{gather}
\|q\|_{D}=\sup_{x\in D}\int_{D}\frac{
\rho _{D}(y)}{\rho _{D}(x)}G_{D}(x,y)|q(y)|dy
\\
\alpha _{q}=\sup_{x,y\in D}\int_{D}\frac{
G_{D}(x,z)G_{D}(z,y)}{G_{D}(x,y)}|q(z)|dz. \label{e2.4}
\end{gather}
It is shown in \cite{b1,m3} that
if $q\in K^{\infty }(D)$, then
\begin{equation}\label{e2.5}
\|q\|_{D}<\infty.
\end{equation}
Now, we remark that from the 3G-Theorem,
\begin{equation*}
\alpha _{q}\leq 2C_{0}\|q\|_{D},
\end{equation*}
where $C_{0}$ is the constant.
Next, we prove that $\alpha _{q}\sim \|q\|_{D}$.

\begin{proposition} \label{prop3}
The following assertions hold
\begin{enumerate}
\item[(i)] For any nonnegative superharmonic function $v$ in
$D$ and any $q$ in $K^{\infty }(D)$,
\begin{equation} \label{e2.6}
\int_{D}G_{D}(x,y)v(y)|q(y)|dy\leq \alpha
_{q}v(x),\quad \forall x\in D.
\end{equation}

\item[(ii)] There exists a constant $C>0$ such that for each
$q\in K^{\infty }(D)$,
\begin{equation*}
C\|q\|_{D}\leq \alpha _{q}.
\end{equation*}
\end{enumerate}
\end{proposition}

\begin{proof}
(i)  Let $v$ be a nonnegative superharmonic function in $D$.
 Then by \cite[Theorem 2.1]{p1} there exists a
 sequence $(f_{k})_{k}$ of nonnegative measurable
functions in $D$  such that the sequence $(v_{k})_{k}$
defined on $D$ by
\begin{equation*}
v_{k}(y):=\int_{D}G_{D}(y,z)f_{k}(z)dz
\end{equation*}
increases to $v$.
Since for each $x\in D$, we have
\begin{equation*}
\int_{D}G_{D}(x,y)v_{k}(y)|q(y)|dy\leq \alpha
_{q}v_{k}(x),
\end{equation*}
the result follows from the monotone convergence theorem.

$(ii)$  We will discuss two cases:
Case 1 ($n\geq 3$).  Using Fatou's Lemma and \eqref{e2.2} we obtain
\[
\int_{D}\frac{h(z)}{h(x)}G_{D}(x,z)|q(z)|dz
\leq \liminf_{|y|\to +\infty}
\int_{D}\frac{G_{D}(x,z)G_{D}(z,y)}{G_{D}(x,y)}|q(z)|dz\leq
\alpha _{q}.
\]
Hence, the result follows from \eqref{e2.3}.

\noindent Case 2 ($n=2$). Let $\varphi _{1}$ be
a positive eigenfunction associated to the first eigenvalue of the Laplacian
in $D^{\ast }$.
 From \cite[Proposition 2.6]{m2}, we have
\begin{equation*}
\varphi _{1}(\xi )\sim \delta _{D^{\ast }}(\xi ),\quad
\forall \xi \in D^{\ast }.
\end{equation*}
Let $v(x)=\varphi _{1}(x^{\ast })$ for $x\in D$.
Then $v$ is superharmonic in $D$ and
\begin{equation*}
v(x)\sim \delta _{D^{\ast }}(x^{\ast })\sim \rho_{D}(x).
\end{equation*}
Applying the assertion (i) to this function $v$ we deduce the result.
\end{proof}

\begin{proposition}[\cite{b1,m3}] \label{prop4}
Let $q$ be a function in $K^{\infty }(D)$ and $v$ be a positive
superharmonic function in $D$.
\begin{enumerate}
\item[(a)] Let $x_{0}\in \overline{D}$. Then
\begin{gather}
\lim_{r\to 0}(\sup_{x\in
D}\int_{B(x_{0},r)\cap D}\frac{v(y)}{
v(x)}G_{D}(x,y)|q(y)|dy)=0, \label{e2.7}
\\
\lim_{M\to +\infty }(\sup_{x\in
D}\int_{(|y|\geq M)\cap D}\frac{
v(y)}{v(x)}G_{D}(x,y)|q(y)|dy)=0. \label{e2.8}
\end{gather}

\item[(b)] The potential $Vq$ is in $\mathcal{C}_{b}(
D)$, $\lim_{x\to z\in \partial D}Vq(x)=0$, and for
$n\geq 3$,
$\lim_{|x| \to +\infty }Vq(x)=0$.

\item[(c)] The function $x\to \frac{\delta_{D}(x)}{|x|^{n-1}}q(x)$
is in $L^{1}(D)$.
\end{enumerate}
\end{proposition}

\begin{example} \label{ex1} \rm
Let $p>n/2$ and $\gamma ,\mu \in \mathbb{R}$ such that
$\gamma <2-\frac{n}{p}<\mu $. Then using the H\"{o}lder inequality and
the same arguments as in \cite[Proposition 3.4]{b1} and
\cite[Proposition 3.6]{m3}, it follows that for each
$f\in L^{p}(D)$, the function defined in $D$ by
$\frac{f(x)}{|x|^{\mu -\gamma }(\delta _{D}(x))^{\gamma }}$ belongs
 to $K^{\infty }(D)$.
Moreover, by taking $p=+\infty $, we find again the results of
\cite{b1,m3}.
\end{example}

\begin{proposition} \label{prop5}
Let $v$ be a nonnegative superharmonic function in $D$ and
$q\in K_{+}^{\infty }(D)$. Then for each $x\in D$ such that
$0<v(x)<\infty $, we have
\begin{equation*}
\exp (-\alpha _{q})v(x)\leq v(x)-V_{q}(qv)(x)\leq v(x).
\end{equation*}
\end{proposition}

\begin{proof}
Let $v$ be a nonnegative superharmonic function in $D$. Then by
\cite[Theorem 2.1]{p1} there exists a sequence $(f_{k})_{k}$ of
nonnegative measurable functions in $D$ such that the sequence
$(v_{k})_{k}$ given in $D$ by
\begin{equation*}
v_{k}(x):=\int_{D}G_{D}(x,y)f_{k}(y)dy
\end{equation*}
increases to $v$.
 Let $x\in D$ such that $0<v(x)<\infty $. Then there
exists $k_{0}\in \mathbb{N}$ such that $0<Vf_{k}(x)<\infty $,
for $k\geq k_{0}$.

 Now, for a fixed $k\geq k_{0}$, we consider the function
$\chi(t)=V_{tq}f_{k}(x)$.
Since the function $\chi $ is completely monotone on
$[0,\infty )$, then $\log \chi $ is convex on
$[0,\infty)$.
Therefore,
\begin{equation*}
\chi (0)\leq \chi (1)\exp \big(-\frac{\chi'(0)}{\chi (0)}\big),
\end{equation*}
which implies
\begin{equation*}
Vf_{k}(x)\leq V_{q}f_{k}(x)\exp \big(\frac{
V(qVf_{k})(x)}{Vf_{k}(x)}\big).
\end{equation*}
Hence, it follows from Proposition \ref{prop3}(i) that
\begin{equation*}
\exp (-\alpha _{q})Vf_{k}(x)\leq V_{q}f_{k}(x).
\end{equation*}
Consequently, from \eqref{e1.6} we obtain
\begin{equation*}
\exp (-\alpha _{q})Vf_{k}(x)\leq Vf_{k}(
x)-V_{q}(qVf_{k}(x))(x)\leq
Vf_{k}(x).
\end{equation*}
By letting $k\to \infty $, we deduce the result.
\end{proof}

\section{Proof of Theorem \ref{thm1}}

Recall that $h_{0}$ is a fixed positive
harmonic function in $D$, which is continuous and bounded in
$\overline{D}$ and $h$ is the function defined by \eqref{e2.2}.
For a fixed nonnegative function $q\in K^{\infty }(D)$, we define
\begin{equation*}
\Gamma _{q}=\{p\in K^{\infty }(D):|p|\leq q\}.
\end{equation*}
To prove Theorem \ref{thm1} we need the following result.

\begin{lemma} \label{lem1}
 Let $q$ be a nonnegative function belonging to $K^{\infty }(D)$.
Then the family of functions
\begin{equation*}
F_{q}=\big\{\int_{D}G_{D}(.,y)h_0(y)p(
y)dy:p\in \Gamma _{q}\big\}
\end{equation*}
is uniformly bounded and equicontinuous in
$\overline{D}\cup \{\infty\} $. Consequently, it is relatively
compact in $\mathcal{C}(\overline{D}\cup \{\infty \} )$.
\end{lemma}

\begin{proof}
Let $q\in K_{+}^{\infty }(D)$ and $L$ the operator defined on
$\Gamma _{q}$ by
\begin{equation*}
Lp(x)=\int_{D}G_{D}(x,y)h_{0}(y) p(y)\, dy.
\end{equation*}
Then by \eqref{e2.6}, we have for each $p\in \Gamma _{q}$ and
$x\in D $,
\begin{equation*}
|Lp(x)|\leq \int_{D}G_{D}(
x,y)h_0(y)q(y)dy\leq \alpha
_{q}h_0(x)\leq \alpha _{q}\|h_0\|_{\infty }.
\end{equation*}
Hence the family $F_{q}:=L(\Gamma _{q})$ is
uniformly bounded.

 Now, let us prove that $L(\Gamma _{q})$ is
equicontinuous on $\overline{D}\cup \{\infty \} $.
Let $x_{0}\in D$ and $r>0$. Let $x\in B(x_{0},r)\cap D $ and
$p\in \Gamma _{q}$.
Since $h_{0}$ is bounded, for $M>0$ we have
\begin{align*}
 \frac{1}{\|h_{0}\|_{\infty}}|Lp(x)-Lp(x_{0})|
&\leq \int_{D}|G_{D}(x,y)-G_{D}(x_{0},y)|q(y)dy\\
&\leq 2\sup_{z\in D}\int_{B(x_{0},2r)\cap D}G_{D}(z,y)q(y)dy \\
&\quad +2\sup_{z\in D}\int_{(|y|\geq M)\cap D}G_{D}(z,y)q(y)dy \\
&\quad +\int_{\Omega }|G_{D}(x,y)-G_{D}(x_{0},y)|q(y)dy,
\end{align*}
where  $\Omega =B^{c}(x_{0},2r)\cap B(0,M)\cap D$.
On the other hand, for every $y\in \Omega $ and $x\in
B(x_{0},r)\cap D$, using \eqref{e2.1}, we obtain
\begin{align*}
|G_{D}(x,y)-G_{D}(x_{0},y)|
&\leq C[\frac{\rho_{D}(x)}{|x-y|^{n-2}}
+\frac{\rho_{D}(x_{0})}{|x_{0}-y|^{n-2}}]\rho_{D}(y)\\
&\leq C\delta _{D}(y)
\leq C\frac{\delta _{D}(y)}{|y|^{n-1}}.
\end{align*}
  Now, since $G_{D}$ is continuous outside the diagonal,
we deduce by the dominated convergence theorem and
Proposition \ref{prop4} (c) that
\begin{equation*}
\int_{\Omega }|
G_{D}(x,y)-G_{D}(x_{0},y)|q(y)dy\to 0\quad
\text{as }|x-x_{0}|\to 0.
\end{equation*}
So, using Proposition \ref{prop4}(a) for $v\equiv 1$, we
deduce that
 $|Lp(x)-Lp(x_{0})|\to 0$ as
$|x-x_{0}|\to 0$, uniformly for all $p\in \Gamma_{q}$.
On the other hand, on $D$, we have
\begin{equation} \label{e3.1}
|Lp(x)|\leq \|h_{0}\|_{\infty
}Vq(x),
\end{equation}
which tends to zero as $x\to \partial D$.
Hence, $L(\Gamma _{q})$ is equicontinuous on
$\overline{D}$.

 Next, we shall prove that $L(\Gamma _{q})$ is
equicontinuous at $\infty $.
 First, we claim that
\begin{equation*}
\lim_{_{|x|\to \infty }}Lp(x)
=\begin{cases}
0 &\text{for }n\geq 3, \\
\int_{D}h_{0}(y)p(y)h(y)dy &\text{for }n=2.
\end{cases}
\end{equation*}
Using \eqref{e3.1} and Proposition \ref{prop4}(b), we obtain $Lp(x)\to 0$ as
$|x|\to \infty $, for $n\geq 3$, uniformly in
$p\in \Gamma _{q}$.

 Finally, we assume that $n=2$ and we put $l=\int_{D}h_0(y)p(y)h(y)dy$.
Since $\lim_{|x|\to +\infty}G_{D}(x,y)=h(y)$, then using Fatou's lemma
and Proposition \ref{prop4}(b), we obtain
\begin{align*}
|l|&\leq \int_{D}h_0(y) q(y)h(y)dy\\
 &\leq \liminf_{|x|\to+\infty }\int_{D}G(x,y)h_0(y)q(y)dy\\
 &\leq \|h_0\|_{\infty }\|Vq\|_{\infty }<+\infty .
\end{align*}
  Now, we shall prove that $\lim_{|x|\to +\infty}Lp(x)=l$.
Let $\varepsilon >0$, then by \eqref{e2.8}, there exists
$M>1$ such that for each $x\in D$ with $|x|\geq 1+M$ we
have
\begin{align*}
|Lp(x)-l|&\leq \int_{D}|G_{D}(x,y)-h(y)|
h_{0}(y)q(y)dy \\
&\leq \varepsilon +\int_{B(0,M)\cap
D}|G_{D}(x,y)-h(y)|h_{0}(y)
q(y)dy.
\end{align*}
On the other hand, using \eqref{e2.1}, for
$y\in B(0,M)\cap D$ and $|x|\geq 1+M$, we have
\[
|G_{D}(x,y)-h(y)|h_0(y)\leq C(\frac{
\delta _{D}(y)}{|y|}+h(y)).
\]
 We deduce from Proposition \ref{prop4}(c) and Lebesgue's
theorem that
$\lim_{|x|\to +\infty}Lp(x)=l$, uniformly in $p\in \Gamma _{q}$.
Thus by Ascoli's theorem $F_{q}$ is relatively compact in
$\mathcal{C}(\overline{D}\cup \{\infty \} )$. This
completes the proof.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1}]
We shall use a fixed-point argument.
 Let $c=1+\alpha _{\psi }$, where $\alpha _{\psi }$ is the constant
defined by \eqref{e2.4} associated to the function $\psi $ given in (H3) and
suppose that
\begin{equation*}
\varphi (x)\geq ch_{0}(x),\quad \forall x\in \partial D.
\end{equation*}
Since $h_{0}$ is a harmonic function in $D$, continuous and bounded in
$\overline{D}$, then the function $w:=H_{D}\varphi -ch_{0}$ is a solution
to the problem
\begin{gather*}
\Delta w=0\quad \text{in }D, \\
w\big|_{\partial D}=\varphi -ch_{0}\geq 0, \\
\lim_{|x|\to +\infty }\frac{w(x)}{h(x)}=0,
\end{gather*}
and by the maximum principle it follows that
\begin{equation} \label{e3.2}
H_{D}\varphi (x)\geq ch_{0}(x),\quad \forall x\in \overline{D}.
\end{equation}
Let $\lambda \geq 0$ and let $\Lambda $ be the non-empty closed bounded
convex set
\begin{equation*}
\Lambda =\{v\in C(\overline{D}\cup \{\infty \}
):h_{0}\leq v\leq H_{D}\varphi \} .
\end{equation*}
Let $S$ be the operator defined on $\Lambda $ by
\begin{equation*}
Sv(x)=H_{D}\varphi (x)-\int_{D}G_{D}(x,y)f(y,v(y)+\lambda h(y))dy.
\end{equation*}
We shall prove that the family $S\Lambda $ is relatively compact
in $C(\overline{D}\cup \{\infty \} )$.
Let $v\in \Lambda $, then by (H2) and (H3) and the fact that $h_{0}$
is positive in $D$, we have for each $y\in D$,
\[
\frac{1}{h_{0}(y)}f(y,v(y)+\lambda
h(y))\leq \frac{\theta (y,h_{0}(y))}{
h_{0}(y)}=\psi (y).
\]
 Hence, we deduce that the function
\begin{equation*}
y\mapsto \frac{1}{h_{0}(y)}f(y,v(y)
+\lambda h(y))\in \Gamma _{\psi }.
\end{equation*}
It follows that the family
\begin{equation*}
\{\int_{D}G_{D}(.,y)f(y,v(y)
+\lambda h(y))dy:v\in \Lambda \} \subseteq F_{\psi }.
\end{equation*}
Thus, from Lemma \ref{lem1}, the family
$\{\int_{D}G_{D}(.,y)f(y,v(y)+\lambda h(y))dy:v\in \Lambda \} $
is relatively compact in $C(\overline{D}\cup \{\infty \} )$.
Since $H_{D}\varphi $\ $\in C(\overline{D}\cup \{\infty \} )$,
we deduce that the family $S(\Lambda)$ is relatively compact in
$C(\overline{D}\cup \{\infty\} )$.

 Next, we shall prove that $S$ maps $\Lambda $ to itself.
It's clear that for all $v\in \Lambda $ we have $Sv(x)
\leq H_{D}\varphi (x),\forall x\in D$.
Moreover, from hypothesis (H2) and \eqref{e2.6}, it follows that
\begin{align*}
\int_{D}G_{D}(x,y)f(y,v(y)+\lambda h(y))dy
&\leq \int_{D}G_{D}(x,y)\theta (y,h_{0}(y))dy\\
&=\int_{D}G_{D}(x,y)\psi (y)h_{0}(y)dy \\
& \leq \alpha _{\psi}h_{0}(x).
\end{align*}
 Hence, using \eqref{e3.2} we obtain
$Sv(x)\geq H_{D}\varphi (x)-\alpha _{\psi }h_{0}(x)\geq h_{0}(x)$,
 which proves that $S(\Lambda
)\subset \Lambda $.

Now, we  prove the continuity of the operator $S$ in $\Lambda $
in the supremum norm. Let $(v_{k})_{k}$ be a sequence in $\Lambda $
which converges uniformly to a function $v$ in $\Lambda $.
 Then, for each $x\in D$, we have
\[
|Sv_{k}(x)-Sv(x)|\leq \int_{D}G_{D}(x,y)|f(
y,v_{k}(y)+\lambda h(y))-f(y,v(y)+\lambda h(y))|dy.
\]
 On the other hand, by hypothesis (H2), we have
\[
|f(y,v_{k}(y)+\lambda h(y))
-f(y,v(y)+\lambda h(y))|\leq
2h_{0}(y)\psi (y)\leq 2\|h_{0}\|_{\infty
}\psi (y).
\]
  Since by Proposition \ref{prop4}(b),
$V\psi $ is bounded, we conclude by the continuity of $f$ with respect
to the second variable and by the dominated convergence theorem that
for all $x\in D$,
\begin{equation*}
Sv_{k}(x)\to  Sv(x) \quad \mbox{as }k\to +\infty .
\end{equation*}
Consequently, as $S(\Lambda )$ is relatively compact
in $C(\overline{D}\cup \{\infty \} )$, we deduce
that the pointwise convergence implies the uniform convergence, namely,
\begin{equation*}
\|Sv_{k}-Sv\|_{\infty }\to 0\quad\mbox{as } k\to +\infty .
\end{equation*}
Therefore, $S$ is a continuous mapping of $\Lambda $ to itself. So
since $S\Lambda $ is relatively compact in
$C(\overline{D}\cup \{\infty \} )$ it follows that $S$ is compact
mapping on $\Lambda$.

 Finally, the Schauder fixed-point theorem implies the
existence of $v\in \Lambda $ such that
\begin{equation*}
v(x)=H_{D}\varphi (x)-\int_{D}G_{D}(
x,y)f(y,v(y)+\lambda h(y))dy.
\end{equation*}
Put $u(x)=v(x)+\lambda h(x)$, for $x\in D$. Then
$u\in C(\overline{D})$ and $u$ satisfies
\begin{equation} \label{e3.3}
u=H_{D}\varphi +\lambda h-\int_{D}G_{D}(.,y)f(
y,u(y))dy.
\end{equation}
Now, we verify that $u$ is a solution of \eqref{Pab} with
$\alpha=\beta=1$. Since $\psi \in K^{\infty }(D)$, it
follows from Proposition \ref{prop4}(c), that
$\psi \in L_{\rm loc}^{1}(D)$. Furthermore, by hypotheses
(H2) and (H3) we have $f(.,u)\leq h_{0}\psi $. This shows
that $f(.,u)\in L_{\rm loc}^{1}(D)$ and $V(f(.,u))\in F_{\psi }$. Then,
from Lemma \ref{lem1}, we have
$V(f(.,u))\in C(\overline{D}\cup \{\infty \} )\subset L_{\rm loc}^{1}(D)$.
Thus, by applying $\Delta $ on both sides of \eqref{e3.3}
and using \eqref{e1.4}, we obtain that $u$ satisfies the elliptic
equation (in the sense of distributions)
\begin{equation*}
\Delta u=f(.,u)\quad \text{in }D\,.
\end{equation*}
Since $H_{D}\varphi =\varphi $ on
$\partial D,\lim_{x\to z\in \partial D}h(x)=0$, and
$\lim_{x\to z\in \partial D}V(f(.,u))(x)=0$, we conclude
that $\lim_{x\to z\in \partial D}u(x)=\varphi (z)$.
On the other hand, since
\begin{equation*}
\lambda h(x)+h_{0}(x)\leq u(x)\leq \lambda h(x)+H_{D}\varphi
(x)
\end{equation*}
and $\lim_{|x|\to +\infty }\frac{
H_{D}\varphi (x)}{h(x)}=\lim_{|x|
\to +\infty }\frac{h_{0}(x)}{h(x)}=0$, we deduce
$\lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\lambda $.
This completes the proof.
\end{proof}

\begin{example} \label{ex2} \rm
Let $D=B^{c}(0,1)$, $p>\frac{n}{2}$, $\sigma >0$ and $\nu >0$.
Let $\varphi$ and $g$  in $\mathcal{C}^{+}(\partial D)$ and put
$h_{0}=H_{D}g$. Then from \cite[p. 258]{a1}, there exists a
constant $c_{0}>0$ such that for each  $x\in D$,
\begin{equation*}
\frac{c_{0}(|x|-1)}{|x|^{n-1}}\leq
h_{0}(x)
\end{equation*}
Moreover, suppose that the function $f$ satisfies (H1) and
\begin{equation*}
f(x,t)\leq t^{-\sigma }\frac{v(x)}{|
x|^{\nu -1+n(\sigma +1)}(|x|-1)
^{1-2\sigma -\frac{n}{p}}},
\end{equation*}
where $v\in L_{+}^{p}(D)$. Then, there exists a constant $c>1$
such that if
$\varphi \geq cg$ on $\partial D$, the problem
\eqref{Pab} with  $\alpha=\beta=1$
 has a positive solution  $u$ in $\mathcal{C}(\overline{D})$
satisfying that for each $x\in D$,
\begin{equation*}
\lambda h(x)+h_{0}(x)\leq u(x)\leq
\lambda h(x)+H_{D}\varphi (x),
\end{equation*}
where $h$ is the function given by \eqref{e2.2}.

 Indeed, (H1) and (H2) are satisfied and by taking
$\gamma =2-\sigma -\frac{n}{p}$and $\mu =2-\frac{n}{p}+\nu $
in Example \ref{ex1}, we deduce that the function
\[
x\mapsto (h_{0}(x))^{-1-\sigma }
\frac{v(x)}{|x|^{\nu -1+n(\sigma +1)
}(|x|-1)^{1-2\sigma -\frac{n}{p}}} \in
K^{\infty }(D),
\]
 which implies that hypothesis (H3) is satisfied.
\end{example}

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

Recall that for a fixed nonnegative function
$q\in K^{\infty}(D)$, we have defined the set
$\Gamma _{q}=\{p\in K^{\infty }(D):|p|\leq q\}$.
Using Propositions \ref{prop3} and \ref{prop4}, with similar
arguments as in \cite[Lemma 4.3]{m3}, we establish the
following lemma.

\begin{lemma} \label{lem2}
Let $q$ be a nonnegative function in $K^{\infty }(D)$
and let $h$ be the function given by \eqref{e2.2}. Then the family of
functions
\begin{equation*}
\mathfrak{F}_{q}(h)=\big\{\frac{1}{h}\int_{D}G(.,y)h(y)p(y)dy:p
\in \Gamma _{q}\big\}
\end{equation*}
is uniformly bounded and equicontinuous in
$\overline{D}\cup \{\infty\} $. Consequently, it is relatively
compact in $\mathcal{C}_{0}(\overline{D})$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm2}]
Let $\alpha \geq 0$, $\beta \geq 0$ with $\alpha +\beta >0$ and
let $q:=q_{\alpha ,\beta }$ be the function in $K^{\infty }(D)$
given by (H4).
Let $c_{1}:=e^{\alpha _{q}}>1$, where $\alpha _{q}$ is
the constant given by \eqref{e2.4}.
Suppose that
\begin{equation*}
\varphi (x)\geq c_{1}h_{0}(x),\quad \forall x\in
\partial D.
\end{equation*}
Then by the maximum principle it follows that
\begin{equation} \label{e4.1}
H_{D}\varphi (x)\geq c_{1}h_{0}(x),\quad \forall x\in \overline{D}.
\end{equation}
Now, let $\lambda \geq c_{1}$ and put
\begin{equation} \label{e4.2}
\begin{gathered}
w(x):=\beta \lambda h(x)+\alpha H_{D}\varphi
(x)\text{, for }x\in D,
\\
v(x):=\alpha h_0 +\beta h(x)\text{, for }x\in D.
\end{gathered}
\end{equation}
Consider the nonempty convex set
\begin{equation*}
\Omega :=\{u\in \mathcal{B}(D):v\leq u\leq w\} .
\end{equation*}
Let $T$ be the operator defined on $\Omega $ by
\begin{equation*}
Tu(x):=w(x)-V_{q}(qw)(x)
+V_{q}(qu-f(.,u))(x).
\end{equation*}
From hypothesis (H4) we have for each $u\in\Omega $
\begin{equation} \label{e4.3}
0\leq f(.,u)\leq uq.
\end{equation}

 Let us prove that the operator $T$ maps $\Omega $ to itself.
By \eqref{e2.6}, it follows that
\begin{equation} \label{e4.4}
\int_{D}G_{D}(x,y)w(y)q(y)dy\leq \alpha _{q}w(x).
\end{equation}
Since $w$ is a harmonic function in $D$ and $Vq<\infty $, by
\eqref{e4.3} and Proposition \ref{prop5}, we have for each $x\in D$,
\[
Tu(x)\geq w(x)-V_{q}(qw)(x)
\geq e^{-\alpha _{q}}w(x)=e^{-\alpha
_{q}}(\beta \lambda h(x)+\alpha H_{D}\varphi (x)).
\]
 Therefore, as $\lambda \geq c_{1}$ and by \eqref{e4.1} we
obtain
\[
Tu(x)\geq \beta h(x)+\alpha h_{0}(x)=v(x).
\]
 On the other hand, we have for each $x\in D$,
\[
Tu(x)\leq w(x)-V_{q}(qw)(x)+V_{q}(qu)(x)
 \leq w(x).
\]
 So $T(\Omega )\subset \Omega $.
Now, let $u_{1},u_{2}\in \Omega $ such that $u_{1}\geq u_{2}$,
then by (H4) we have
\[
Tu_{1}-Tu_{2}=V_{q}(q[u_{1}-u_{2}]-[f(.,u_{1})-f(.,u_{2})])\geq 0.
\]
 Hence, $T$ is a nondecreasing operator on $\Omega $.

 Next, we consider the sequence $(u_{m})_{m\in \mathbb{N}}$ defined by
\begin{equation*}
u_{0}=\beta h+\alpha h_{0}\quad\text{and}\quad
u_{m+1}=Tu_{m}\quad \text{for } m\in \mathbb{N}.
\end{equation*}
Since $\Omega $ is invariant under $T$, we obtain $v=u_{0}\leq u_{1}\leq w$.
Therefore, from the monotonicity of $T$ on $\Omega $, we have
\begin{equation*}
v=u_{0}\leq u_{1}\leq \dots \leq u_{m}\leq u_{m+1}\leq w.
\end{equation*}
Thus, from the monotone convergence theorem and the fact that $f$ is
continuous with respect to the second variable, the sequence
$(u_{m})_{m\in \mathbb{N}}$ converges to a function $u$ satisfying
\begin{equation} \label{e4.5}
u=(I-V_{q}(q.))w+V_{q}(qu-f(.,u)).
\end{equation}
By \eqref{e2.5} and \eqref{e2.6}, we obtain for each $x\in D$,
\begin{equation*}
0\leq V(qu)(x)\leq V(qw)(
x)\leq \alpha _{q}w(x)<\infty .
\end{equation*}
Applying $(I+V(q.))$ on both sides of \eqref{e4.5},
it follows from \eqref{e1.6} and \eqref{e1.7}
that
\begin{equation} \label{e4.6}
u=\beta \lambda h+\alpha H_{D}\varphi -V(f(.,u)).
\end{equation}

Now, let us verify that $u$ is a solution of the problem \eqref{Pab}.
Since $q\in K^{\infty }(D)$ then by Proposition \ref{prop4}, we
obtain $q\in L_{\rm loc}^{1}(D)$.
 By \eqref{e4.3} we have
\begin{equation} \label{e4.7}
f(.,u)\leq qu\leq qw.
\end{equation}
Therefore, since $w$ is continuous in $D$, we obtain that
$f(.,u)\in L_{\rm loc}^{1}(D)$. Using Proposition \ref{prop3} and
\eqref{e4.7},  for each $x\in D$, we have
\[
V(f(.,u))(x)\leq \int_{D}G_{D}(x,y)w(y)q(y)dy\leq \alpha
_{q}w(x).
\]
Then $V(f(.,u))\in L_{\rm loc}^{1}(D)$.
Thus, by applying $\Delta $ on both sides of \eqref{e4.6},
 we deduce that $u$ is a solution of
\begin{equation*}
\Delta u=f(.,u)\quad \text{ in }D
\end{equation*}
(in the sense of distributions).
 Using \eqref{e4.7} we obtain that
\[
f(.,u)\leq \beta \lambda hq+\alpha qH_{D}\varphi \\
\leq \beta \lambda hq+\alpha \|\varphi \|_{\infty }q\,.
\]
Let $g:=\beta \lambda hq+\alpha \|\varphi \|
_{\infty }q$. Since $f(.,u)$ and
$(g-f(.,u))$ are in $\mathcal{B}^{+}(D)$ then
$V(f(.,u))$ and $V(g-f(.,u))$ are two
lower semi-continuous functions.

 On the other hand, by Proposition \ref{prop4}(b) we have
$V(q)\in \mathcal{C}(D)$ and by Lemma \ref{lem2} the function
 $\frac{1}{h}V(hq)\in \mathcal{C}_{0}(\overline{D})$.
 So $Vg$ is a continuous function. This implies that
$V(g-f(.,u))=Vg-V(f(.,u))$ is also an upper semi-continuous
function. Consequently $V(g-f(.,u))$ is in
$\mathcal{C}(D)$. Thus $V(f(.,u))=Vg-V(g-f(.,u))\in \mathcal{C}(D)$.
Therefore $u$ is in $\mathcal{C}(D)$.

 Now using Proposition \ref{prop3}(i) and the fact that
$\lim_{x \to z\in \partial D}h(x)=0$ we deduce that
$\lim_{x \to \partial D}V(hq)(x)=0$. In addition
from Proposition \ref{prop4}(b) we have
$\lim_{x\to \partial D}V(q)(x)=0$. So that
$\lim_{x\to \partial D}V(g)(x)=0$. This in turn implies that
$\lim_{x\to \partial D}V(f(.,u))=0$.
Then by \eqref{e4.6}, we obtain that
$u\big|_{\partial D}=\alpha \varphi $.
On the other hand, we have
\begin{equation*}
\frac{1}{h}V(f(.,u))\leq \beta \lambda \frac{1}{h
}V(hq)+\alpha \|\varphi \|_{\infty }\frac{1
}{h}Vq.
\end{equation*}
Using Propositions \ref{prop2} and \ref{prop4}(b), we obtain that
$\frac{1}{h(x)}V(f(.,u))(x)$ tends to $0$ as
$|x|\to +\infty $ and consequently
$\lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\beta \lambda $.
Hence, $u$ is a positive continuous solution in $D$ of the problem
\eqref{Pab}. This completes the proof.
\end{proof}

\begin{example} \label{ex3} \rm
Let $D=B^{c}(0,1)$ and $0<\gamma \leq 1$. Let $p$ be a nonnegative function
such that the function $q(x)=(\frac{|x|^{n-1}}{|x|-1})^{1-\gamma }p(x)$
is in $K^{\infty}(D)$. Let $\varphi \in \mathcal{C}^{+}(\partial D)$
 and $h_{0}$ be a positive harmonic function in $D$, which belongs to
$\mathcal{C}_{b}(\overline{D})$.
Then, for each $\alpha \geq 0$ and $\beta \geq 0$ with $\alpha +\beta >0$,
there exists a constant $c_{1}>1$
such that if $\varphi \geq c_{1}h_{0}$ on $\partial D$ and $\lambda \geq
c_{1}$, the problem
\begin{gather*}
\Delta u=p(x)u^{\gamma }\quad \text{in }D, \\
u\big|_{\partial D}=\alpha \varphi ,\\
\lim_{|x|\to +\infty }\frac{u(x)}{h(x)}=\beta \lambda \geq 0,
\end{gather*}
has a positive continuous solution on $D$ satisfying that for each
$x\in D$,
\begin{equation*}
\beta h(x)+\alpha h_{0}(x)\leq u(x)\leq \beta \lambda h(x)+\alpha
H_{D}\varphi (x).
\end{equation*}
\end{example}

\subsection*{Acknowledgements}
The authors want to thank Professor Habib Maagli for his valuable
discussions, also to the anonymous referee for his/her valuable
suggestions and comments.


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\end{document}