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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 92, pp. 1--11.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}


\begin{document}
\title[\hfilneg EJDE-2006/92\hfil Existence results]
{Existence results for nonlinear elliptic equations in 
 bounded domains of $\mathbb{R}^n$}

\author[M. Zribi\hfil EJDE-2006/92\hfilneg]{Malek Zribi}
\address{Malek Zribi \\
d\'{e}partement de math\'{e}matiques, facult\'{e} des sciences de tunis,
campus universitaire, 1060 Tunis, Tunisia}
\email{malek.zribi@insat.rnu.tn}

\date{}
\thanks{Submitted May 4, 2006. Published August 15, 2006.}
\subjclass[2000]{34B27, 34J65}
\keywords{Green function; elliptic equation; positive solutions}

\begin{abstract}
 We establish existence results for the boundary-value problem  
 $\Delta u+f(.,u)=0$ in a smooth bounded domain in $\mathbb{R}^n$ 
 $(n\geq 2)$, where  $f$ satisfies some appropriate conditions 
 related to a Kato class. The proofs are based on various 
 techniques related to potential theory.
\end{abstract}

\maketitle

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

\section{Introduction}

Let $\Omega $ be a $C^{1,1}$ bounded domain in $\mathbb{R}^n$ $(n\geq 2)$.
In this paper we study the existence and the asymptotic behaviour of bounded
solutions to the nonlinear elliptic boundary-value problem 
\begin{equation}  \label{e1.1}
\begin{gathered} \Delta u+f(.,u)=0\quad \text{in }\Omega \\ u>0,\quad
\text{in }\Omega \\ u=g \quad\text{on } \partial \Omega, \end{gathered}
\end{equation}
where $g$ is a nonnegative continuous function on $\partial \Omega$ and $f$
satisfies some convenient conditions. The question of existence of solutions
of \eqref{e1.1} has been studied by several authors in both bounded and
unbounded domains with various nonlinearities; see for example \cite%
{b1,b2,d1,e1,k1,l1,l2,l3,m2,m3,m4,m5,s1,t1,y1,z1,z2} and references therein.
Note that solutions of these problems are understood in distributional sense.

Our tools are based essentially on some inequalities satisfied by the Green
function $G(x,y)$ of $(-\Delta )$ in $\Omega $ which allow to some
properties of functions belonging to the Kato class $K(\Omega )$ which
contains properly the classical one; see \cite{a1,c1}. The class $K( \Omega )
$ has been introduced in \cite{m5}, for $n\geq 3$ and \cite{m2,z1} for $n=2$
as follows.

We denote by $\delta (x)$ the Euclidian distance between $x$ and $\partial
\Omega $.

\begin{definition} \label{def1} \rm
A Borel measurable function $q$ in $\Omega $ belongs to the Kato
class $K(\Omega )$ if $q$ satisfies
\begin{equation} \label{e1.2}
\lim_{\alpha \to 0}\big(\sup_{x\in \Omega
}\int_{\Omega \cap B(x,\alpha )}\frac{\delta (y)}{\delta (x)}
G(x,y)| q(y)| dy\big)=0.
\end{equation}
\end{definition}

For the sake of simplicity we set $Hg$ the bounded continuous solution of
the Dirichlet problem 
\begin{gather*}
\Delta u=0\quad\text{in }\Omega \\
u=g\quad \text{on } \partial \Omega,
\end{gather*}
where $g$ is a nonnegative continuous function on $\partial \Omega$. We also
refer to $Vf$ the potential of a measurable nonnegative function $f$,
defined on $\Omega $ by 
\begin{equation*}
Vf(x)=\int_{\Omega }G(x,y)f(y)dy.
\end{equation*}

Our plan in this paper is as follows. The section 2 is devoted to collect
some preliminary results about the Green function $G(x,y)$ and the
properties of the Kato class $K(\Omega )$.

In section 3, we establish an existence result for \eqref{e1.1} where the
combined effects of a singular and a sublinear term in the nonlinearity $f$
are considered. Our motivation in this section comes from paper \cite{s1},
where Shi and Yao investigated the existence of nonnegative solutions for
the elliptic problem 
\begin{gather*}
\Delta u+K(x)u^{-\gamma }+\lambda u^{\alpha }=0\quad \text{in }\Omega  \\
u(x)>0\quad \text{in }\Omega  \\
,u=0\quad \text{on }\partial \Omega ,
\end{gather*}%
where $\gamma $ and $\alpha $ in $(0,1)$ are two constants, $\lambda $ is a
real parameter and $K$ is in $C^{0,\beta }(\overline{\Omega })$. Using this
result. Sun and Li \cite{y1} gave a similar result in $\mathbb{R}^{n}$ ($%
n\geq 2$). In fact they proved an existence result for the problem 
\begin{gather*}
\Delta u+p(x)u^{-\gamma }+q(x)u^{\alpha }=0\quad \text{in }\mathbb{R}^{n} \\
u(x)>0,\quad x\in \mathbb{R}^{n} \\
u(x)\rightarrow 0,\quad \text{as }|x|\rightarrow \infty ,
\end{gather*}%
where $\gamma $ and $\alpha $ in $(0,1)$ are two constants and $p,q$ are two
nonnegative functions in $C_{\mathrm{loc}}^{\beta }(\mathbb{R}^{n})$ such
that $p+q\neq 0$.

The pure singular elliptic equation 
\begin{equation}  \label{e1.3}
\Delta u+p(x)u^{-\gamma }=0,\quad \gamma >0,\; x\in D\subseteq \mathbb{R}^n
\end{equation}
has been extensively studied for both bounded and unbounded domains $D$ in $%
\mathbb{R}^n(n\geq 2)$. We refer to \cite{d1,e1,k1,l2,l3} and references
therein) for various existence and uniqueness results related to solutions
for equation \eqref{e1.3}.

For more general situations M\^{a}agli and Zribi showed in \cite{m4} that
the problem 
\begin{gather*}
\Delta u+\varphi (.,u)=0,\quad x\in D \\
u=0 \quad\text{on } \partial D \\
\lim_{| x| \to \infty }u(x)=0,\quad \text{if $D$ is unbounded}
\end{gather*}
admits a unique positive solution if $\varphi $ is a nonnegative measurable
function on $(0,\infty )$, which is nonincreasing and continuous with
respect to the second variable and satisfies

\begin{itemize}
\item[(H0)] For all $c>0$, $\varphi (.,c)$ is in $K_{n}^{\infty }(D)$, 
where $K_{n}^{\infty }(D)$ is the classical Kato class; see \cite{z2}.
\end{itemize}

On the other hand, the problem \eqref{e1.1} with a sublinear term $f(.,u)$
have been studied in $\mathbb{R}^n$ by Brezis and Kamin in $[3]$. Indeed,
the authors proved the existence and the uniqueness of a positive solution
for the problem 
\begin{gather*}
\Delta u+\rho (x)u^{\alpha }=0\quad \text{in }\mathbb{R}^n, \\
\liminf_{| x| \to \infty }u(x)=0,
\end{gather*}
with $0<\alpha <1$ and $\rho $ is a nonnegative measurable function
satisfying some appropriate conditions.

Thus a natural question to ask is for more general singular and sublinear
terms combined in the nonlinearity, whether or not \eqref{e1.1} has a
solution which we aim to study in this section. In fact we are interested in
solving the following problem (in the sense of distributions) 
\begin{equation}  \label{e1.4}
\begin{gathered} \Delta u+\varphi (.,u)+\psi (.,u)=0,\quad \text{in }\Omega
\\ u>0,\quad \text{in }\Omega \\ u=0 \quad \text{on } \partial \Omega\,.
\end{gathered}
\end{equation}

Here $\varphi $ and $\psi $ are required to satisfy the following hypotheses:

\begin{itemize}
\item[(H1)] $\varphi $ is a nonnegative Borel measurable function on $\Omega
\times (0,\infty )$, continuous and nonincreasing with respect to the second
variable.

\item[(H2)] For all $c>0$, $x\to \varphi ( x,c\delta (x))$ is in $K(\Omega )$%
.

\item[(H3)] $\psi $ is a nonnegative Borel measurable function on $\Omega
\times (0,\infty )$, continuous with respect to the second variable such
that there exist a nontrivial nonnegative function $p$ and a nonnegative
function $q\in K(\Omega )$ satisfying for $x\in \Omega $ and $t>0$, 
\begin{equation}  \label{e1.5}
p(x)h(t)\leq \psi (x,t)\leq q(x)f(t),
\end{equation}
where $h$ is a measurable nondecreasing function on $[0,\infty )$ satisfying 
\begin{equation}  \label{e1.6}
\lim_{t\to 0^{+}}\frac{h(t)}{t}=+\infty
\end{equation}
and $f$ is a nonnegative measurable function locally bounded on [ $0,\infty )
$ satisfying 
\begin{equation}  \label{e1.7}
\limsup_{t\to \infty }\frac{f(t) }{t}<\| Vq\| _{\infty }.
\end{equation}
\end{itemize}

Using a fixed point argument, we shall prove the following existence result.

\begin{theorem} \label{thm1}
Assume (H1)--(H3). Then the problem \eqref{e1.4} has a
positive solution $u\in C_{b}(\Omega )$ such that for each
$x\in \Omega $,
\begin{equation*}
a\delta (x)\leq u(x)\leq V(\varphi (.,a\delta ))(x)+bVq(x),
\end{equation*}
where $a,b$ are positive constants.
\end{theorem}

Typical examples of nonlinearities satisfying (H1)-(H3) are: 
\begin{gather*}
\varphi (x,t)=p(x)(\delta (x))^{\gamma }t^{-\gamma }; \quad \gamma \geq 0, \\
\psi (x,t)=q(x)t^{\alpha }\log(1+t^{\beta }), \quad \alpha ,\beta \geq 0
\end{gather*}
such that $\alpha +\beta <1$, where $p$ and $q$ are two nonnegative
functions in $K(\Omega )$.

In this section, using different techniques from those used by Shi and Yao 
\cite{s1}, we improve their results in the sense of distributional solutions.

In section 4, we consider the nonlinearity $f(x,t)=-\varphi (x,t)$ and we
suppose that $g$ is nontrivial, then using a potential theory approach we
investigate an existence result and an uniqueness result for the problem 
\begin{equation}  \label{e1.8}
\begin{gathered} 
\Delta u-\varphi (.,u)=0\quad \text{in }\Omega \\ 
u>0\quad \text{in }\Omega \\ u=g \quad\text{on } \partial \Omega, 
\end{gathered}
\end{equation}
where $\varphi $ is required to satisfy the following three conditions:

\begin{itemize}
\item[(H4)] $\varphi $ is a nonnegative measurable function on $\Omega
\times [ 0,\infty )$, continuous and nondecreasing with respect to the
second variable.

\item[(H5)] $\varphi (.,0)=0 $.

\item[(H6)] For all $c>0$, $\varphi (.,c)$ is in $K( \Omega )$.
\end{itemize}

Our main result is the following.

\begin{theorem} \label{thm2}
Assume (H4)-(H6). Then the problem \eqref{e1.8} has a unique
positive solution $u$ such that
$0<u(x)\leq Hg(x)$ for each $x\in \Omega $.
\end{theorem}

Note that if $q\in K(\Omega )$ and $\varphi (x,t) \leq q(x)t$ locally on 
$t$, then the solution $u$ satisfies $cHg(x)\leq u(x)\leq Hg(x)$, for 
$c\in (0,1)$.

This result follows up the one of Lair and Wood in \cite{l2}, who have
considered the equation 
\begin{equation*}
\Delta u=q(x)f(u),
\end{equation*}
in both bounded and unbounded domains of $\mathbb{R}^n$ $( n\geq 2)$ in the
case $f(u)=u^{\gamma }$, $0<\gamma \leq 1$. They studied the existence and
nonexistence of positive large solutions and positive bounded ones under
adequate hypothesis on $q$. The result of Lair and Wood have been
generalized later by Bachar and Zeddini \cite{b1} to more general functions 
$f$ and $q$ satisfying some restrictive conditions.

To simplify our statements, we define some convenient notation:

\noindent (i) $B(\Omega )$ denotes the set of Borel measurable functions in 
$\Omega $ and $\mathcal{B}^{+}(\Omega )$ the set of nonnegative functions.

\noindent (ii) $C_{0}(\Omega ):=\{ w\in C(\Omega ) :\lim_{x\to \partial
\Omega }w(x)=0\} $. We recall that this space is Banach with the uniform
norm 
\begin{equation*}
\| w\| _{\infty}=\sup_{x\in \Omega }| w(x)|. 
\end{equation*}

\noindent (iii) For $q\in \mathcal{B}(\Omega )$, we put 
\begin{equation*}
\| q\| :=\sup_{x\in \Omega }\int_{\Omega }\frac{ \delta (y)}{\delta (x)}
G(x,y)| q(y)| dy.
\end{equation*}

\noindent (iv) Let $f$ and $g$ be two nonnegative functions on a set $S$. We
call $f\preceq g$, if there is $c>0$ such that 
\begin{equation*}
f(x)\leq cg(x)\quad \text{for all }x\in S.
\end{equation*}
We call $f\sim g$, if there is $c>0$ such that 
\begin{equation*}
\frac{1}{c}g(x)\leq f(x)\leq cg(x)\quad \text{for all }x\in S.
\end{equation*}

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

The existence results to prove, suggest collecting some estimates on the
Green function $G$ and some properties of functions belonging to the Kato
class $K(\Omega )$. The proofs of the following estimates and inequalities
of $G$ can be found in \cite{m5} for $n\geq 3$ and \cite{z1} for $n=2$.

\begin{proposition} \label{prop1}
For each $x,y\in \Omega $, we have
\begin{equation} \label{e2.1}
G(x,y)\sim \begin{cases}
\frac{\delta (x)\delta (y)}{| x-y| ^{n-2}\big(
| x-y| ^{2}+\delta (x)\delta (y)\big)} &\text{if }n\geq 3, \\
\log\big(1+\frac{\delta (x)\delta (y)}{| x-y| ^{2}}\big)
&\text{if }n=2.
\end{cases}
\end{equation}
\end{proposition}

\begin{corollary} \label{coro1}
For $x,y\in \Omega $,
\begin{equation} \label{e2.2}
\delta (x)\delta (y)\preceq G(x,y).
\end{equation}
\end{corollary}

\begin{theorem}[3G-Theorem] \label{thm3}
There exists $C_{0}>0$ depending only on $\Omega$, such that
for $x,y,z\in \Omega $, we have
\begin{equation} \label{e2.3}
\frac{G(x,z)G(z,y)}{G(x,y)}\leq C_{0}\big[ \frac{\delta
(z)}{\delta (x)} G(x,z)+\frac{\delta (z)}{\delta
(y)}G(y,z)\big] .
\end{equation}
\end{theorem}

To recall some properties of the class $K(\Omega )$, we first give the
following examples:

\begin{enumerate}
\item By \cite[Proposition 4]{m5}, the function $q(x)=1/ (\delta
(x))^{\lambda }$ is in $K(\Omega )$ if and only if $\lambda <2$.

\item By \cite[Proposition 3]{t1}, if $p>n/2$ and $\lambda <2-\frac{n}{p}$,
then $L^{p}(\Omega )/(\delta (.))^{\lambda } \subset K(\Omega)$.
\end{enumerate}

The proof of the following Proposition can be found in \cite{m5,z1}.

\begin{proposition} \label{prop2}
Let $q$ be a nonnegative function in $K(\Omega )$. Then
\begin{itemize}
\item[(i)] $\| q\| <\infty $.

\item[(ii)] The function $x\mapsto \delta (x) q(x)$ is in
$L^{1}(\Omega )$.

\item[(iii)] We have
\begin{equation} \label{e2.4}
\delta (x)\preceq Vq(x).
\end{equation}
\end{itemize}
\end{proposition}

For a fixed nonnegative function $q$ in $K(\Omega )$, we put 
\begin{equation*}
\mathcal{M}_{q}:=\{ \varphi \in B(\Omega ),\text{ \ }| \varphi | \preceq q\}
.
\end{equation*}

\begin{proposition} \label{prop3}
Let $q$ be a nonnegative function in $K(\Omega )$, then the family
of functions
\begin{equation*}
V(\mathcal{M}_{q})=\{ V\varphi :\varphi \in \mathcal{M}_{q}\}
\end{equation*}
is uniformly bounded and equicontinuous in $C_{0}(\Omega )$, and
consequently it is relatively compact in $C_{0}(\Omega )$.
\end{proposition}

\begin{proof}
The result holds by similar arguments as in \cite[proposition 3]{m5} and 
\cite[Proposition 8]{z1}.
\end{proof}

In the sequel, we use the notation 
\begin{equation*}
\alpha _{q}:=\sup_{x,y\in \Omega }\int_{\Omega} \frac{ G(x,z)G(z,y)}{G(x,y)}%
| q(z)| dz.
\end{equation*}

\begin{proposition}\label{prop4} %2.7
Let $q$ be a function in $K(\Omega )$ and $v$ be a
nonnegative superharmonic function in $\Omega $. Then for each
$x\in \Omega$,
\begin{equation} \label{e2.5}
\int_{\Omega }G(x,y)v(y)| q(y)| dy\leq \alpha
_{q}v(x)
\end{equation}
and consequently, $\| q\| \leq \alpha _{q}\leq 2C_{0}\| q\|$,
where $C_{0}$ is the constant given in \eqref{e2.3}.
\end{proposition}

For the proof of the above proposition, we refer the reader to 
\cite[Proposition 2]{t1}.

\begin{corollary} \label{coro2} %\label{coro2.8}
Let $q$ be a nonnegative function in $K(\Omega )$
and $v$ be a nonnegative superharmonic function in $\Omega $, then
for each $x\in \Omega $ such that $v(x)<\infty $, we have
\begin{equation*}
\exp (-\alpha _{q})v(x)\leq (v-V_{q}(qv))(x)\leq v(x).
\end{equation*}
\end{corollary}

\begin{proof}
The upper inequality is trivial. For the lower one, we consider the function 
$\gamma (\lambda )=v(x)-\lambda V_{\lambda q}( qv)(x)$ for $\lambda \geq 0$.
The function $\gamma $ is completely monotone on $[0,\infty )$ and so 
$\log\gamma $ is convex in $[0,\infty )$. This implies 
\begin{equation*}
\gamma (0)\leq \gamma (1)\exp (-\frac{\gamma ^{\prime }(0)}{\gamma (0)}).
\end{equation*}
That is, 
\begin{equation*}
v(x)\leq (v-V_{q}(qv))(x)\exp (\frac{V(qv)(x)}{v(x) }).
\end{equation*}
So, the result holds by \eqref{e2.5}.
\end{proof}

\section{First existence result}

\begin{proof}[Proof of Theorem \protect\ref{thm1}]
Assume (H1)-(H3). Using the Schauder fixed point theorem, we are going to
construct a solution to problem \eqref{e1.4}. We note that by \eqref{e2.2}
there exists a constant $\alpha _{1}>0$ such that for each $x,y\in \Omega $ 
\begin{equation}  \label{e3.1}
\alpha _{1}\delta (x)\delta (y)\leq G(x,y).
\end{equation}
Now, using (H3), there exists a compact $K$ of $\Omega $ such that 
\begin{equation*}
0<\alpha :=\int_{K}\delta (y)p(y)dy<\infty .
\end{equation*}
We put $\beta :=\min \{ \delta (x):x\in K\} $. Then from \eqref{e1.6}, we
conclude that there exists $a>0$ such that 
\begin{equation}  \label{e3.2}
\alpha _{1}\alpha h(a\beta )\geq a.
\end{equation}
Furthermore, since $q\in K(\Omega )$, then by Proposition \ref{prop3} we
have obviously that $\| Vq\| _{\infty }<\infty $. So taking $0<\eta
<1/\|Vq\| _{\infty }$, we deduce by \eqref{e1.7} that there exists $\rho >0$
such that for $t\geq \rho $ we have $f(t)\leq \eta t$. Put $\gamma
=\sup_{0\leq t\leq \rho }f(t)$. So we have that 
\begin{equation}  \label{e3.3}
0\leq f(t)\leq \eta t+\gamma ,\quad t\geq 0.
\end{equation}
Next by (2.4), we note that there exists a constant $\alpha _{ \text{2}}>0$
such that 
\begin{equation}  \label{e3.4}
\alpha _{2}\delta (x)\leq Vq(x),\quad \forall x\in \Omega .
\end{equation}
>From (H2) and Proposition \ref{prop3}, we have that $\| V\varphi (.,a\delta
)\| _{\infty }<\infty $. Hence, put 
\begin{equation*}
b=\max \{ \frac{a}{\alpha_{2}}, \frac{\eta \| V\varphi (.,a\delta ) \|
_{\infty }+\gamma }{1-\eta \| Vq\| _{\infty}}\} 
\end{equation*}
and consider the closed convex set 
\begin{equation*}
\Lambda =\{u\in C_{0}(\Omega ):a\delta (x)\leq u(x)\leq V\varphi (.,a\delta
)(x)+bVq(x),\forall x\in \Omega \}.
\end{equation*}
Obviously, by \eqref{e3.4} we have that the set $\Lambda $ is nonempty.
Define the integral operator $T$ on $\Lambda $ by 
\begin{equation*}
Tu(x)=\int_{\Omega }G(x,y)[ \varphi (y,u(y))+\psi (y,u(y))] dy,\quad \forall
x\in \Omega .
\end{equation*}
Let us prove that $T\Lambda \subset \Lambda $. Let $u\in \Lambda $ and 
$x\in \Omega $, then by (H1), (H3) and \eqref{e3.3} we have 
\begin{align*}
Tu(x) & \leq V\varphi (.,a\delta )(x) +\int_{\Omega }G(x,y)q(y)f(u(y))dy \\
& \leq V\varphi (.,a\delta )(x)+\int_{\Omega }G(x,y)q(y)[ \eta u(y)+\gamma ]
dy \\
&\leq V\varphi (.,a\delta )(x) +\int_{\Omega }G(x,y)q(y)[ \eta (\| V\varphi
(.,a\delta )\| _{\infty }+b\| Vq\| _{\infty })+\gamma ] dy \\
& \leq V\varphi (.,a\delta )(x)+bVq(x).
\end{align*}
Moreover from the monotonicity of $h$, \eqref{e3.1} and \eqref{e3.2}, we
have 
\begin{align*}
Tu(x) & \geq \int_{\Omega }G(x,y)\psi (y,u(y))dy \\
& \geq \alpha _{1}\delta (x)\int_{\Omega }\delta (y)p(y)h(a\delta (y))dy \\
& \geq \alpha _{1}\delta (x)h(a\beta )\int_{K}\delta (y)p(y)dy \\
& \geq \alpha _{1}\alpha h(a\beta )\delta (x) \\
& \geq a\delta (x).
\end{align*}
On the other hand, we have that for each $u\in \Lambda $, 
\begin{equation}  \label{e3.5}
\varphi (.,u)\leq \varphi (.,a\delta )\text{ and } \psi (.,u)\leq [ \eta (\|
V\varphi ( .,a\delta )\| +b\| Vq\| _{\infty })+\gamma ] q.
\end{equation}
This implies by Proposition \ref{prop3} that $T\Lambda $ is relatively
compact in $C_{0}(\Omega )$. In particular, we deduce that $T\Lambda \subset
\Lambda $.

Next, we prove the continuity of $T$ in $\Lambda $. Let $(u_{k})_{k}$ be a
sequence in $\Lambda $ which converges uniformly to a function $u$ in $%
\Lambda $. Then since $\varphi $ and $\psi $ are continuous with respect to
the second variable, we deduce by the dominated convergence theorem that 
\begin{equation*}
\forall x\in \Omega , \quad Tu_{k}(x)\to Tu(x)\quad \text{as } k\to \infty .
\end{equation*}
Now, since $T\Lambda $ is relatively compact in $C_{0}(\Omega )$, then we
have the uniform convergence. Hence $T$ is a compact operator mapping from 
$\Lambda $ to itself. So the Schauder fixed point theorem leads to the
existence of a function $u\in \Lambda $ such that 
\begin{equation}  \label{e3.6}
u(x)=\int_{\Omega }G(x,y)[ \varphi (y,u(y))+\psi (y,u(y))] dy,\quad \forall
x\in \Omega .
\end{equation}
Finally, we need to prove that $u$ is solution of the problem \eqref{e1.4}.
Since $q$ and $\varphi (.,a\delta )$ are in $K(\Omega )$, we deduce by 
\eqref{e3.5} and Proposition \ref{prop2}, that $y\mapsto \varphi (y,u(y))+\psi
(y,u(y))\in L^{1}( \Omega ) $. Moreover, since $u\in C_{0}(\Omega )$, we
deduce from \eqref{e3.6}, that $V(\varphi (.,u)+\psi (.,u))\in L^{1}(\Omega )$.
Hence $u$ satisfies in the sense of distributions the elliptic equation 
\begin{equation*}
\Delta u+\varphi (.,u)+\psi (.,u)=0,\quad \text{in }\Omega .
\end{equation*}
This completes the proof.
\end{proof}

\begin{example} \label{exa3} \rm
Let $\alpha ,\beta \geq 0$ such that $0\leq \alpha
+\beta <1$, $\gamma >0$ and $p,q\in K^{+}(\Omega )$. Then the
problem
\begin{equation} \label{e3.7}
\begin{gathered}
\Delta u+p(x)(u(x))^{-\gamma }(\delta (x))^{\gamma
}+q(x)(u(x))^{\alpha }\log(1+(u(x))
^{\beta })=0,\quad \text{in }\Omega  \\
u>0,\quad \text{in }\Omega
\end{gathered}
\end{equation}
has a solution $u\in C_{0}(\Omega )$ satisfying
$a\delta (x)\leq u(x)\leq Vp(x)+bVq(x)$, where $a,b>0$.
\end{example}

\begin{remark} \label{rmk1} \rm
Taking in Example \ref{exa3} $\lambda <2$,
$$
p(x)=q(x)=\frac{1}{(\delta (x) ) ^{\lambda}},
$$
we deduce from \cite{m5} that the solution of \eqref{e3.7}
satisfies  the following:
\begin{itemize}
\item[(i)] $u(x)\preceq (\delta (x))^{2-\lambda }$, if $1<\lambda <2$.

\item[(ii)] $u(x)\preceq \delta (x)\log\frac{ (\sqrt{5}+1)
d}{2\delta (x)}$,  if $\lambda =1$,

\item[(iii)] $u(x)\preceq \delta (x)$,  if $\lambda<1$,
where $d=\mathop{\rm diam}(\Omega )$.
\end{itemize}

Note that in Example \ref{exa3}, we have the result obtained by
 Shi and Yao \cite{s1}.
\end{remark}

\section{Second existence result}

In this section, we shall prove Theorem \ref{thm2}. The proof is based on a
comparison principle given by the following Lemma. For $u\in B(\Omega )$,
put $u^{+}=\max (u,0)$.

\begin{lemma} \label{lem1}
Let $\varphi $ and $\psi $ satisfying (H4)-(H6). Assume
that $\varphi \leq \psi $ on $\Omega \times \mathbb{R} _{+}$ and
there exist continuous functions $u,v$ on $\Omega $
satisfying
\begin{itemize}
\item[(a)] $\Delta u-\varphi (.,u^{+})\leq \Delta v-\psi (.,v^{+})$ in
$\Omega $ (in the distributional sense)

\item[(b)] $u,v\in C_{b}(\Omega )$

\item[(c)] $u\geq v$ on $\partial \Omega $.
\end{itemize}
Then $u\geq v$ in $\Omega $.
\end{lemma}

\begin{proof}
Suppose that the open set $D=\{ x\in \Omega :u(x)<v(x)\}$ is nonempty. 
Put $z=u-v$. Then $z$ satisfies 
\begin{gather*}
\begin{aligned} \Delta z&=\varphi (.,u^{+})-\psi (.,v^{+})\\ &=(\varphi
(.,u^{+}) -\psi(.,u^{+}))+(\psi ( .,u^{+})-\psi (.,v^{+}))\leq 0\quad \text{
in }D \end{aligned} \\
z\geq 0 \quad\text{on }\partial D \\
z\in C_{b}(D).
\end{gather*}
Hence from the maximum principle, we conclude that $z\geq 0$ in $D$.
Therefore, we get a contradiction with the definition of $D$. This completes
the proof.
\end{proof}

In the sequel, we recall that for each function $q\in \mathcal{B}^{+}(\Omega
)$ such that $Vq<\infty $, we denote by $V_{q}$ the unique kernel which
satisfies the following resolvent equation (see \cite{m1,n1}): 
\begin{equation}  \label{e4.1}
V=V_{q}+V_{q}(qV)=V_{q}+V(qV_{q}).
\end{equation}
So for each $u\in \mathcal{B}(\Omega )$ such that $V(q| u| )<\infty $, we
have 
\begin{equation}  \label{e4.2}
(I-V_{q}(q.))(I+V(q.))u=(I+V(q.))( I-V_{q}(q.))u=u.
\end{equation}

\begin{proof}[Proof of Theorem \protect\ref{thm2}]
As consequence of the comparison principle in Lemma \ref{lem1}, we deduce
that problem \eqref{e1.8} has at most one solution. The existence of a such
solution is assured by the Schauder fixed point Theorem. Indeed, we consider
the convex set 
\begin{equation*}
\Lambda =\{ u\in C_{b}(\Omega ):u\leq \| g\| _{\infty }\} .
\end{equation*}
We define the integral operator $T$ on $\Lambda $ by 
\begin{equation*}
Tu(x)=Hg(x)-V(\varphi (.,u^{+}))(x).
\end{equation*}
Since $Hg(x)\leq \| g\| _{\infty }$, for $x\in \Omega $, we deduce that for
each $u\in \Lambda $, 
\begin{equation*}
Tu\leq \| g\| _{\infty } \quad\text{in }\Omega .
\end{equation*}
Furthermore, putting $q=\varphi (.,\|g\| _{\infty })$, we have by (H4)  and
(H6) that $q$ is in $K(\Omega )$ and $V(\varphi (.,u^{+}))$ is in 
$V(\mathcal{M}_{q})$. This together with the fact that $Hg$ is in $C_{b}(\Omega
)$ imply by Proposition \ref{prop3} that $T\Lambda $ is relatively compact
in $C_{b}(\Omega )$ and in particular $T\Lambda \subset \Lambda $.

>From the continuity of $\varphi $ with respect to the second variable, we
deduce that $T$ is continuous in $\Lambda $ and so it is a compact operator
from $\Lambda $ to itself. Then by the Schauder fixed point Theorem, we
deduce that there exists a function $u\in \Lambda $ satisfying 
\begin{equation*}
u(x)=Hg(x)-V(\varphi (.,u^{+}))(x).
\end{equation*}
Finally, since $\varphi (.,u^{+})\in \mathcal{M}_{q}$, we conclude by
Proposition \ref{prop2} that $u$ satisfies in the sense of distributions the
following 
\begin{gather*}
\Delta u-\varphi (.,u^{+})=0 \\
\lim_{x\to \partial \Omega }u(x)=g.
\end{gather*}
Hence by (H5) and Lemma \ref{lem1}, we conclude that $u\geq 0$ in $\Omega $
and so it is a solution of \eqref{e1.8}.
\end{proof}

\begin{corollary} \label{coro3}
Suppose that $\varphi $ satisfies (H4)-(H6) and $g$ is a
nontrivial nonnegative continuous function in $\partial \Omega$.
Suppose that there exists a function $q\in K(\Omega )$ such that
\begin{equation} \label{e4.3}
0\leq \varphi (x,t)\leq q(x)t\quad \text{on }\Omega \times
[0,\| g\| _{\infty }] .
\end{equation}
Then the solution $u$ of \eqref{e1.8} given by Theorem \ref{thm2} satisfies
\begin{equation*}
e^{-\alpha _{q}}Hg(x)\leq u(x)\leq Hg(x).
\end{equation*}
\end{corollary}

\begin{proof}
Since $u$ satisfies the integral equation 
\begin{equation*}
u(x)=Hg(x)-V(\varphi (.,u))(x),
\end{equation*}
using \eqref{e4.1}, we obtain 
\begin{align*}
u-V_{q}(qu)&=(Hg-V_{q}(qHg))-( V(\varphi (.,u))-V_{q}(qV(\varphi (.,u))) \\
&=(Hg-V_{q}(qHg))-V_{q}(\varphi (.,u)).
\end{align*}
That is, 
\begin{equation*}
u=(Hg-V_{q}(qHg))+V_{q}(qu-\varphi (.,u)).
\end{equation*}
Now since $0<u\leq \| g\| _{\infty }$ then by \eqref{e4.3}, we have that $%
u\geq Hg-V_{q}(qHg)$. Consequently, the result holds from Corollary \ref%
{coro2}.
\end{proof}

\begin{example} \label{exa4} \rm
Let $g$ be a nontrivial nonnegative continuous function in
$\partial \Omega$. Let $\sigma >0$ and $q\in K^{+}(\Omega )$.
Then the  problem (in the sense of distributions)
\begin{gather*}
\Delta u-q(x)u^{\sigma }=0,\quad \text{in }\Omega \\
u=g\quad \text{on } \partial \Omega
\end{gather*}
has a positive bounded continuous solution $u$ satisfying, in
$\Omega $,
\begin{equation*}
0\leq Hg(x)-u(x)\leq \| g\| _{\infty }^{\sigma }Vq(x).
\end{equation*}
Furthermore, if $\sigma \geq 1$,  by Corollary \ref{coro3}, for
each $x\in \Omega $,
\begin{equation*}
e^{-\alpha _{q}}Hg(x)\leq u(x)\leq Hg(x).
\end{equation*}
\end{example}

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\end{document}