\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 04, 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 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2005/04\hfil Positive solutions for elliptic equations]
{Positive solutions for elliptic equations with singular nonlinearity}

\author[J. Shi, M. Yao\hfil EJDE-2005/04\hfilneg]
{Junping Shi, Miaoxin Yao} % in alphabetical order

\address{Junping Shi \hfill\break
 Department of Mathematics, College of William and Mary \\
Williamsburg, VA 23187, USA \hfill\break
Department of Mathematics, Harbin Normal University,
 Harbin, Heilongjiang, China}
\email{shij@math.wm.edu}

\address{Miaoxin Yao\hfill\break
Department of Mathematics, Tianjin University\hfill\break
and Liu Hui Center for Applied Mathematics, Nankai University \&
Tianjin University\\
Tianjin, 300072, China}
\email{miaoxin@hotmail.com}

\date{}
\thanks{Submitted August 15, 2004. Published January 2, 2005.}
\thanks{J. Shi was supported by NSF grant DMS-0314736,
and by a grant from Science Council \hfill\break\indent
 of Heilongjiang Province, China }
\subjclass[2000]{35J25, 35J60}
\keywords{Singular nonlineararity; elliptic equation; positive solution;
\hfill\break\indent monotonic iteration}

\begin{abstract}
 We study an elliptic boundary-value problem with singular
 nonlinearity via the method of monotone iteration scheme:
 \begin{gather*}
 -\Delta u(x)=f(x,u(x)),\quad x \in \Omega,\\
 u(x)=\phi(x),\quad x \in \partial \Omega ,
 \end{gather*}
 where $\Delta$ is the Laplacian operator, $\Omega$ is a bounded domain
 in $\mathbb{R}^{N}$, $N \geq 2$, $\phi \geq 0$ may take the value $0$
 on $\partial\Omega$, and $f(x,s)$ is possibly singular near $s=0$.
 We prove the existence and the uniqueness of positive solutions
 under a set of hypotheses that do not make neither monotonicity nor
 strict positivity assumption on $f(x,s)$, which improvements of
 some previous results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^{N}$, $N \geq 2$. We assume that the boundary
$\partial \Omega$ of $\Omega$ is of $C^{2,\theta}$ for some $\theta \in (0,1)$.
Let $\phi (x)$ be a nonnegative function belonging to
$C^{2,\theta}(\partial \Omega)$
and $f(x,s)$ be a function defined on $\overline{\Omega} \times (0,+\infty )$
which is locally H\"older continuous with exponent $\theta$.
We consider the existence and the uniqueness of positive solutions for
the nonlinear boundary-value problem
\begin{gather}
-\Delta u(x)=f(x,u(x)),\quad x \in \Omega , \label{eq:pp1}\\
u(x)=\phi(x),\quad x \in \partial \Omega , \label{eq:pp2}
\end{gather}
where $\Delta$ is the Laplacian operator.

A positive solution of problem \eqref{eq:pp1}-\eqref{eq:pp2} is
a function $u(x) \in C^{0}(\overline{\Omega}) \cap C^{2}(\Omega)$ satisfying
\eqref{eq:pp1}-\eqref{eq:pp2} and $u(x) > 0$ for $x \in \Omega $.

Many articles treat the problem of
the existence and/or the uniqueness of positive solutions for
\eqref{eq:pp1}-\eqref{eq:pp2} under a variety of hypotheses on function $f(x,s)$.
When $f(x,s)$ is locally Lipschiz in $\Omega \times [0, +\infty )$,
the existence and uniqueness  of positive solutions (for some cases) are
well understood.
However, if there is a sequence $\{(x_{i},s_{i})\}$ in $\Omega \times
(0,+\infty)$, for which $x_{i}$ converges to some point in the set
$\{x\in\partial\Omega | \phi(x)=0\}$ and $s_{i}$ tends to $0$ as $i\to
+\infty$, such that $f(x_{i},s_{i})\to\infty$,
then problem \eqref{eq:pp1}-\eqref{eq:pp2}
is singular, it does not have a solution in $C^2(\overline{\Omega})$, and
the existence or uniqueness results
do not follow from the results obtained for nonsingular equations in the literature.

It is well-known that such singular elliptic problems arise in the
contexts of chemical heterogeneous catalysts, non-Newtonian fluids
and also the theory of heat conduction in electrically conducting
materials, see \cite{CK, CN, DMO, FM}  for a detailed discussion.

In \cite{FM}, the existence of a positive solution of such a
singular problem is established under a set of assumptions in
which $f(x,s)$ is assumed to be non-increasing in $s$. Thus if
$f(x,s)$ is defined, say, by
\begin{equation}
f(x,s)=g(x)\ln^{2}s,  \label{eq:ln}
\end{equation}
or by
\begin{equation}
f(x,s)=g(x)s^{-\alpha}+h(x)s^{\beta}-k(x)s^{\rho}, \label{eq:ap}
\end{equation}
where $\alpha>0, \beta\in (0,1), \rho\geq 1$ , $g$ and $k$ are
nonnegative H\"older continuous functions, then the existence of
positive solutions does not follow from the results in \cite{FM}.

the authors in \cite{St} and \cite{CRT} treat the singular problem with
no monotonicity assumption on $f(x,s)$, and
the results there may imply the existence of positive solutions even when $f(x,s)$
is given by (\ref{eq:ln}), (\ref{eq:ap}), in which $k(x)=0, g(x), h(x)>0$
for $x\in\overline{\Omega}$, or, by (see \cite{St})
\begin{equation} f(x,s)=1+\{1+\cos\frac{1}{s}\}s^{1/2}e^{1/s}. \label{eq:cos1}
\end{equation}
Some uniqueness results are also given in \cite{St} and
\cite{CRT}. However, the method of proof in \cite{St} and
\cite{CRT} requires that $f(x,s)$ be strictly positive near $s=0$,
i.e., $f(x,s)$ is bounded away from $0$ as $s\to0^{+}$,
for $x\in\overline{\Omega}$ (See $(H_{2}), (H_{2}')$ in
\cite{St} and $(g_{1})$ in \cite{CRT}). Therefore, if $f(x,s)$ is
given, say, by (\ref{eq:ln}), (\ref{eq:ap}), with $g(x)$ and
$h(x)$ vanishing on some non-empty subset of $\Omega$, or given by
\begin{equation}
f(x,s)=s^{1/2}e^{\frac{1}{s}(1+\cos\frac{1}{s})}, \label{eq:cos0}
\end{equation}
then no conclusion regarding the existence of  positive solutions
can be derived from the results in \cite{St} and \cite{CRT}.

For the special case where $f(x,s)=g(x)s^{-\alpha}$ in which $g$ is a
sufficiently regular function and is positive in $\Omega$, and $\alpha>0$,
\cite{LM} gives some results when $g(x)$ is vanishing or tending to $\infty$
near $\partial\Omega$ with a  suitable  rate, and the positivity of $f(x,s)$
for $x\in\Omega$ is still assumed.

Recently the case where $f(x,s)=g(x)s^{-\alpha}+h(x)s^p$ is
studied with $p\in(0,1)$ and the restriction that $\alpha\in
(0,\frac{1}{N})$, also assumed the positivity hypotheses on
functions $g(x)$ and $h(x)$ on whole $\Omega$.

In the present article, neither monotonicity nor positivity on
whole $\Omega$ is assumed for $f(x,s)$, and the results are more
general, implying the existence of positive solutions for
\eqref{eq:pp1}-\eqref{eq:pp2} even with $f(x,s)$ given by any of
(\ref{eq:ln})--(\ref{eq:cos0}), where $g(x)$ and $h(x)$ may be
$0$, and even $h(x)$ may  be negative, in some subset of $\Omega$
. Also a uniqueness result is obtained. If we assume that for each
$x\in \Omega$ either $s^{-1}f(x,s)$ is strictly decreasing in $s$
for $s>0$, or $f(x, s)$ and $s^{-1}f(x,s)$ are both nonincreasing
in $s$, and that function $f$ satisfies some certain conditions in
addition to the conditions for existence results, then we can
further prove that the solution is unique. When $f(x,s)$ is
locally Lipschiz in $ \Omega \times [0, +\infty )$ and hence not
singular, and  $s^{-1}f(x,s)$ is strictly decreasing in $s$ for
$s>0$ at every $x$ in $\Omega$, this kind of uniqueness result is
well-known (see for example, \cite{OS}), however, our result
extends it to include singular nonlinearity cases, which covers
the special case where $f$ is given by (\ref{eq:ap}), and also
applies to the case where $s^{-1}f(x,s)$ need'nt be strictly
decreasing in $s$ for all $x$ in $\Omega$.

The precise hypotheses and main results are stated in Section
\ref{sec2}, and the proof for the results is given in Section
\ref{sec3}. The proof for the existence results is based on a
monotone convergence argument with solutions of (\ref{eq:pp1})
corresponding to the boundary data $\phi(x)+\frac {1}{k}$, which
are obtained by using a monotone iteration scheme started with
certain supersolutions and subsolutions particularly chosen; the
proof for the uniqueness result makes use of a comparison lemma,
which stems from some idea of a lemma in \cite{ABC}.

\section{Hypotheses and Main Results}\label{sec2}

We assume that the function $f$ that defines the nonlinear term
in (\ref{eq:pp1}) satisfies the following conditions:
\begin{itemize}
\item [(F1)]
$f$: $\overline{\Omega}\times(0,+\infty)\to\mathbb{R}$ is H\"older continuous
with exponent $\theta \in (0,1)$
on each compact subset of $\overline{\Omega}\times(0,+\infty)$.

\item [(F2)]
$$ \limsup_{s\to+\infty}\Big(s^{-1} \max_{x\in\overline{\Omega}}f(x,s)\Big)
<\lambda_{1}, $$
where $\lambda_{1}$ is the first eigenvalue of $-\Delta$ on $\Omega$
with Dirichlet boundary value.

\item [(F3)]
For each $t>0$, there exists a constant $D(t)>0$ such
that
$$  f(x,r)-f(x,s)\geq -D(t)(r-s)
$$
for $x\in\overline{\Omega}$  and $r\geq s\geq t$.
(Without loss of generality we assume that $D(s)\leq D(t)$ for $s\geq t>0$.)
\end{itemize}

For the case in which $\phi(x)\not\equiv 0$ on $\partial\Omega$, we have
the following result.


\begin{theorem} \label{thm1}
Suppose that $f$ satisfies (F1)--(F3) and $\phi\in C^{2,\theta} (\partial\Omega)$. If $\phi(x)\geq 0$
and $\phi(x)\not\equiv 0$ on $\partial\Omega$, and if there exist $\gamma, \delta
 >0$ such that
\begin{equation} f(x,s)\geq -\gamma s, \quad\mbox{for}\quad
 x\in \overline{\Omega}  \; s\in (0,\delta) ,
\label{eq:t1} \end{equation}
then there exists at least one positive solution $u(x)$ of
problem~{\rm (\ref{eq:pp1})~(\ref{eq:pp2})} such that
for any
compact subset $G$ of
$\Omega\cup\{x\in\partial\Omega | \phi(x)>0\}$, $u(x)\in C^{2,\theta}(G)$.
\end{theorem}

For the general case where $\phi(x)$ may be $0$ for all $x\in\partial\Omega$,
we have the following theorems.

\begin{theorem}\label{thm2}
Suppose that $f$ satisfies (F1)--(F3) and $\phi\in C^{2,\theta}(\partial\Omega)$.
If $\phi(x)\geq 0$ on $\partial\Omega$ and if there exist positive numbers
$\delta, \gamma$ and a nonempty open subset $\Omega_{0}$ of $\Omega$ such that
\begin{gather}
f(x,s)\geq -\gamma s, \quad\mbox{for } x\in \overline{\Omega}  \; s\in (0,\delta),
\label{t21} \\
s^{-1}f(x,s)\to +\infty \quad\mbox{as }  s\to 0^{+} \mbox{ uniformly for }
x \in\Omega_{0},
\label{t22}
\end{gather}
then the conclusion of Theorem~{\rm \ref{thm1}} holds.
\end{theorem}

\begin{theorem}\label{thm3}
Suppose that $f$ satisfies (F1)--(F3) and $\phi\in C^{2,\theta}(\partial\Omega)$.
If $\phi(x)\geq 0$ on $\partial\Omega$ and if there exists $\delta >0$ such
that
\begin{equation} f(x,s)\geq \lambda_{1}s \quad\mbox{for } x\in\overline{\Omega} \;
 s\in(0,\delta) ,
\label{t3} \end{equation}
then the conclusion of Theorem~{\rm \ref{thm1}} holds.
\end{theorem}

The following theorem concerns to the uniqueness of positive solutions for
problem \eqref{eq:pp1}-\eqref{eq:pp2}. We use the hypotheses
\begin{itemize}
\item [(F4)]
Either $f(x,s)$ is nonincreasing in $s$ for each $x$ in $\Omega$,
or,
 $s^{-1}f(x, s)$ is strictly decreasing in $s$ for each $x$ in an open subset $\Omega_0$
 of $\Omega$
 and both $f(x, s)$ and $s^{-1}f(x, s)$ are nonincreasing in $s$ for all $x$
 in the remainder part $\Omega -\Omega_0$,

\item [(F5)] The function
$$
F(s, t)=\max_{d(x)=s}|f(x, t)|, \quad\mbox{with }
d(x)=\mathop{\rm dist}(x,\partial\Omega),
$$
either is bounded on $(0, \delta)\times(0, \delta)$, or is a sum of such
a bounded function and some function that
is decreasing in $t$ on $(0, \delta)$ for any $s\in (0, \delta)$, and
$$
\int_{0}^{\delta}F(s, c_0s)ds < +\infty, \quad \mbox{for  all } c_0 \in (0, 1).
$$
\end{itemize}

\begin{theorem}\label{thm4}
Under the assumption of any of Theorems \ref{thm1}--\ref{thm3}, if in addition
the function $f(x, s)$ satisfies (F4) and (F5),
then problem \eqref{eq:pp1}-\eqref{eq:pp2} has one and
only one positive solution in $C^{0}(\overline{\Omega})\cap C^{2,\theta}(\Omega)$.
\end{theorem}

\noindent{\bf Remarks.}
\begin{itemize}
\item[(1)]
Examples of $f(x, s)$ , at a point $x$, satisfying the condition
in (F4) that both $f(x, s)$ and $s^{-1}f(x, s)$ are non-increasing
in $s$, are $f(x, s)= f_1(x)s^{\rho_1}$ for $s>0$ with $\rho_1
\leq 0$ and $f_1(x)\geq 0$ , $f(x,s)=f_2(x)s^{\rho_2}$ for $s>0$
with $ \rho_2 \geq 1 $ and $f_2(x)\leq 0$, and so on.
\item[(2)]
If $f(x, s)$ is a sum of a function $f_1(x, s)$ that is bounded on $\Omega \times
(0, \delta)$ and some function $f_2(x, s)$ that is decreasing in
$s$ on $(0, \delta)$ for any $x\in\Omega$, and if for any $c_0 \in
(0, 1)$, there exists $\alpha_0 <1$ such that
$$|f_2\left(x, c_0d(x)\right)|=O\left((d(x))^{-\alpha_0}\right), \quad
\mathop{\rm as } d(x)\to 0,
$$
then (F5) is obviously satisfied.
\end{itemize}

By the above remarks, we can easily derive from Theorems
\ref{thm2} and \ref{thm4} the following corollary, in which $h^+$
and $h^-$ stand respectively for the positive part and the
negative part of $h$, i.e., $h^+(x)=\max\{h(x), 0\}$,
$h^-(x)=\max\{-h(x), 0\}$.


\begin{cor} \label{cor1}
The singular nonlinear elliptic problem
\begin{gather*}
\Delta u+g(x)u^{-\alpha}+h(x)u^{\beta}-k(x)u^{\rho}=0, \quad x\in\Omega,\\
u(x)=0, \quad x\in\partial\Omega,
\end{gather*}
with $\beta\in (0, 1)$, $\rho\geq 1$, and $\alpha >0$, possesses a
positive solution $u$ in $C^{0}(\overline{\Omega})\cap
C^{2,\theta}(\Omega)$, provided that functions $g, h$ and $k$ are
$\theta-$H\"older continuous on $\overline{\Omega}$ , $g, k$ are
nonnegative, $g+h^+$ is not identically zero, and
$h^-(x)\leq\sigma_0g(x),\forall x\in \Omega$, for some constant
$\sigma_0>0$.

If in addition the function $h$ is non-negative or non-positive on whole
$\Omega$, and for some $\alpha_0 <1$,
$$g(x)=O\left((d(x))^{\alpha-\alpha_0}\right), \quad \mbox{as }
d(x)\to 0, x\in \Omega,
$$
then the solution $u$ is unique.
\end{cor}

This is an example in which the behavior of a coefficient function
near the boundary affects the existence and uniqueness of
solutions. Moreover, the  result here makes improvement to some
results in the literature \cite{CRT} \cite{FM} \cite{St} and
\cite{SW}.

\section{Proof of Results}\label{sec3}

Let $(P_{k})$ denote the boundary-value problem:
\begin{equation} \label{eq:pk1-2}
\begin{gathered}
-\Delta u(x)=f(x,u(x)),\quad x\in\Omega , \\ % \label{eq:pk1}
u(x)=\phi(x)+\frac{1}{k},\quad  x\in\partial\Omega , %\label{eq:pk2}
\end{gathered}
\end{equation}
where $k$ is a positive integer. We say that a function $u$ is a supersolution,
or a subsolution, of \eqref{eq:pk1-2} if $u$ belongs to
$C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ and satisfies \eqref{eq:pk1-2} with sign
$=$ replaced by signs $\geq$, or $\leq$, respectively.

In this Section we first prove Theorem \ref{thm1} in detail, then we outline
the proofs for Theorems \ref{thm2}
and \ref{thm3}. After we state and prove a lemma we finally prove Theorem \ref{thm4}.


\begin{proof}[Proof of Theorem \ref{thm1}]
{\it Step 1.} Let $m,k$ be positive integers and denote by $\psi_{m,k}(x)$
(resp. $\psi_{m,\infty}(x)$) the unique solution in $C^{2}(\overline{\Omega})$
of problem
\begin{gather*}
-\Delta\psi(x)+\gamma\psi(x)=0,\quad x\in\Omega , \\
\psi(x)=\frac{1}{m}\phi(x)+\frac{1}{k},\quad x\in\partial\Omega , \\
(\mbox{\rm resp.}\quad  \psi(x)=\frac{1}{m}\phi(x),\quad x\in\partial\Omega .)
\end{gather*}
Then it follows from the estimates of Schauder type \cite{GT} and the maximum
principle for $-\Delta +\gamma$ that there exists a positive integer $m_{0}$
such that
\begin{gather*}
 0<\psi_{m_{0},\infty}(x)<\psi_{m_{0},k}(x),\quad x\in\Omega,\; k\geq m_{0}, \\
0<\psi_{m_{0},k+1}(x)<\psi_{m_{0},k}(x)<\delta,\quad x\in\overline{\Omega}, \;
k\geq m_{0}.
\end{gather*}
Hence, by (\ref{eq:t1}), $\psi_{m_{0},k}(x)$ is a subsolution of \eqref{eq:pk1-2} for
every $k\geq m_{0}$.
Let
$$
\delta_{k}=\min_{x\in\overline{\Omega}}\psi_{m_{0},k}(x),
$$
we have
$$
0<\delta_{k+1}<\delta_{k},\;\; k\geq m_{0}.
$$
By (F2) we may take $\lambda_{0}>0$ such that
$$
\limsup_{s\to+\infty}\big( s^{-1}\max_{x\in\overline{\Omega}}f(x,s)\big)
<\lambda_{0}<\lambda_{1},
$$
and then consider the problem
\begin{gather*}
-\Delta\xi(x)\geq \lambda_{0}\xi(x),\quad x\in\Omega , \\
 \xi(x)>0,\quad x\in\overline{\Omega}.
\end{gather*}
The existence of solutions to this problem is established in \cite{Se}.
Let $\xi(x)$ be such a function and $k_{0}$ be a positive integer sufficiently
large. Then it's easy to verify that $k_{0}\xi(x)$ is a supersolution of
\eqref{eq:pk1-2} for every $k\geq m_{0}$, and we may have
$$
k_{0}\xi(x)\geq \psi_{m_{0},k}(x)+\max_{x\in\overline{\Omega}}\phi(x),
\quad x\in\overline{\Omega},\;k\geq m_{0}.
$$

\noindent {\it Step 2.}
We define the iteration scheme below, as in the standard supersolution and
subsolution argument,
\begin{gather*}
-\Delta w_{n}(x)+D(\delta_{m_{0}})w_{n}(x)
=f(x,w_{n-1}(x))+ D(\delta_{m_{0}})w_{n-1}(x),\quad x\in\Omega , \\
w_{n}(x)=\phi(x)+\frac{1}{m_{0}},\quad x\in\partial \Omega ,
\end{gather*}
noting that (F3) implies that for each
$x\in\Omega, s\mapsto f(x,s)+D(\delta_{m_{0}})s$ is an increasing function
on $[\delta_{m_{0}},+\infty)$. Thus, as in the proof of Theorem 1 in \cite{Aman},
by setting $w_{0}(x)=\psi_{m_{0},m_{0}}(x)$ (or $k_{0}\xi(x)$) for $x\in\Omega $,
we obtain a monotonic sequence that converges to a solution
$u_{m_{0}}(x)\in C^{2}(\overline{\Omega})$ of $(P_{m_{0}})$ such that
$$
\psi_{m_{0},m_{0}}(x)\leq u_{m_{0}}(x)\leq k_{0}\xi(x),\quad x\in\overline{\Omega}.
$$
Using the same iteration scheme with $m_{0}$ replaced by $m_{0}+1$, and
setting $w_{0}(x)=\psi_{m_{0},m_{0}+1}(x)$ (or $u_{m_{0}}(x)$), we can obtain,
as in above, a positive solution $u_{m_{0}+1}(x)\in C^{2}(\overline{\Omega})$
of $(P_{m_{0}+1})$. Furthermore, by the maximum principle for
$-\Delta +D(\delta_{m_{0}+1})$, we have
$$
\psi_{m_{0},m_{0}+1}(x)\leq u_{m_{0}+1}(x)\leq u_{m_{0}}(x), \quad
x\in\overline{\Omega}.
$$
Hence, by repeating the above process, we obtain the sequence
$\{u_{k}(x)\}_{k\geq m_{0}}$ satisfying
\begin{equation}
\psi_{m_{0},\infty }(x)\leq u_{k+1}(x)\leq u_{k}(x)\leq k_{0}\xi(x),\quad
 x\in\overline{\Omega}, \;k\geq m_{0}. \label{uk}
 \end{equation}
and $u_{k}(x)$ solves \eqref{eq:pk1-2} for any $k\geq m_{0}$.

\noindent {\it Step 3.} We can define function $u$ by
$$
u(x)=\lim_{k\to +\infty}u_{k}(x),\quad x\in\overline{\Omega},
$$
because $\{u_{k}(x)\}_{k\geq m_{0}}$  is a decreasing sequence  uniformly
bounded from below by $\psi_{m_{0},\infty}(x)$  on $\overline{\Omega}$.
Now, we have from (\ref{uk}) that
$$
\psi_{m_{0},\infty}(x)\leq u(x)\leq k_{0}\xi(x),\quad x\in\overline\Omega.
$$
Thus, if $G$ is a compact subset of $\Omega\cup \{x\in\partial\Omega | \phi(x)>0\}$,
then there exist two positive constants $E_{1}(G)$ and $E_{2}(G)$ such that
$$
E_{1}(G)\leq u(x)\leq E_{2}(G),\quad x\in G,\;k\geq m_{1}.
$$
Therefore, using the same reasoning as that in \cite{St} and \cite{LM} and
the Schauder theory as stated in \cite{GT}, we conclude that $u(x)$
satisfies (\ref{eq:pp1}) and belong to $C^{2,\theta}(G)$.

On the other hand, by the hypotheses about function $f$, the number $$H:=\inf_{k\geq m_{0}}\{\min_{x\in\overline{\Omega}}f(x,u_{k}(x))\}$$
exists, hence by the maximum principle we have
$$
Q(x)\leq u_{k}(x), \quad x\in\overline{\Omega}, \;k\geq m_{0},
$$
and hence
$$ Q(x)\leq u(x),\quad  x\in\overline{\Omega},
$$
where $Q(x)$ is the solution of problem
\begin{gather*}
-\Delta Q(x)=H,\quad x\in\Omega , \\
Q(x)=\phi(x) , \quad x\in\partial\Omega .
\end{gather*}
Furthermore, it is easy to see that if $x_{0}\in\partial\Omega$,
then for any $\varepsilon>0$ there exist  $r_0>0$ and an integer
$m_{1}\geq m_{0}$ such that
$$
Q(x)\leq u_{k}(x)\leq \phi(x_{0})+\varepsilon,
$$
for  all $k\geq m_{1}$ and $x\in\Omega$ for which $|x-x_{0}| < r_0$.
Therefore, $u(x)$ is continuous on $\overline{\Omega}$ satisfying
(\ref{eq:pp2}). This completes the proof.
\end{proof}

Functions $\psi_{m_{0},\infty }(x)$ and $\psi_{m_{0},k}(x)$ play an
important role in the proof above.
For the proof of Theorem \ref{thm2} or \ref{thm3}, we only show the way for
obtaining these two functions, the remainder of the proof is almost the same as
that of Theorem \ref{thm1} and is omitted.


\begin{proof}[Proof of Theorem \ref{thm2}]
Choose an $\eta(x)\in C_{0}^{\infty}(\Omega)$ such that $0\leq \eta(x)\leq 1$
for $x\in\overline{\Omega}$, $\eta (x)\not\equiv 0$, and
$\mathop{\rm supp}\eta \subset\Omega_{0}$. By  (F1) and (\ref{t22}), there
exist $c_{1}, c_{2}>0$ such that $c_1<\delta$, hence
$f_{\gamma}(x, c_1)\geq 0, x \in \overline{\Omega}$, here
$f_{\gamma}(x, s)\equiv f(x, s)+\gamma s$, and
\begin{equation}
c_{1}\leq f_{\gamma}(x,c_{1})\leq c_{2},\;x\in\Omega_{0},  \label{eq:t17}
\end{equation}
then we denote by $\psi_{m,k}(x)$ ( resp. $\psi_{m,\infty}(x)$) the unique
solution in $C^{2}(\overline{\Omega})$ of the problem
\begin{gather*}
-\Delta\psi(x)+\gamma \psi(x)=\frac{1}{m}\eta(x)f_{\gamma}(x,c_{1}),\quad
x\in\Omega ,\\
\psi(x)=\frac{1}{k},\quad x\in\partial\Omega.\\
\mbox{(resp.}\quad   \psi(x)= 0 , \quad x\in\partial\Omega.)
\end{gather*}
We have for all $m,k\geq 1$ that
\begin{gather*}
\psi_{m,\infty}(x)=\frac{1}{m}\psi_{1,\infty}(x),\quad x\in\overline{\Omega} \\
\psi_{m,k}(x)\geq \psi_{m,\infty}(x) > 0,\quad x\in\Omega \\
\psi_{m,k}(x)\geq \psi_{m^{*},k^{*}}(x) > 0,\quad  x\in \overline{\Omega},
\mbox{if $m^{*}\geq m $ and $k^{*}\geq k$}.
\end{gather*}
Clearly there exist $d_{1}, d_{2}>0$ such that
\begin{equation}
d_{1}\leq \psi_{1,\infty}(x)\leq d_{2} \quad\mbox{for }
x\in \mathop{\rm supp}\eta. \label{eq:t18}
\end{equation}
By the Schauder estimates \cite{GT}, we can make $\psi_{m,k}(x)$,
uniformly for $x\in\overline{\Omega}$, as small as we want by taking $m$ and $k$
 both large enough. Hence there exists integer $m_{0}$ such that
 $$
\frac{f_{\gamma}( x, \psi_{m,k}(x))}{\psi_{m,k}(x)}\geq
\frac{c_{2}}{d_{1}}, \quad m, k\geq m_{0}, \; x\in \mathop{\rm supp}\eta ,
$$
by (\ref{t22}), now  by \eqref{t21},
$$
f_{\gamma}(x,\psi_{m,k}(x))\geq 0,\quad x\in\Omega .
$$
Therefore, if $x\in \mathop{\rm supp}\eta$ and $m,k\geq m_{0}$,
\begin{align*}
-\Delta\psi_{m,k}(x)-f( x,\psi_{m,k}(x))
&= \frac{1}{m}f_{\gamma}(x,c_{1})\big[\eta(x)-\frac{f_{\gamma}(x,\psi_{m,k}(x))}
{\psi_{m,k}(x)}\frac{\psi_{m,k}(x)}{\frac{1}{m}f_{\gamma}(x,c_{1})}\big]\\
&\leq \frac{1}{m}f_{\gamma}(x,c_{1})
\big[\eta(x)-\frac{c_{2}}{d_{1}} \frac{\psi_{1,\infty}(x)}{f_{\gamma}(x,c_{1})}\big]\\
&\leq 0;
\end{align*}
(by \eqref{eq:t17} and \eqref{eq:t18}).
If $x\in\Omega\backslash\mathop{\rm supp}\eta$,
$$
-\Delta\psi_{m,k}(x)-f\big((x,\psi_{m,k}(x)\big)
=-f_{\gamma}\big(x,\psi_{m,k}(x)\big) \leq 0.
$$
Thus, if $m,k\geq m_{0}$, then $\psi_{m,k}(x)$ is a subsolution of \eqref{eq:pk1-2}.
Therefore, the functions $\psi_{m_{0},k}(x)$ and $\psi_{m_{0},\infty}(x)$ meet
the needs.
\end{proof}

\begin{proof}{Proof of Theorem \ref{thm3}}
We point our that it suffices to let $\psi_{m,\infty}(x)$ be the
function $m^{-1}\Phi_{1}(x)$ and
$\psi_{m,k}(x)$ be the unique solution of the problem
\begin{gather*}
-\Delta\psi(x)=\frac{\lambda_{1}}{m}\Phi_{1}(x),\quad x\in\Omega , \\
 \psi(x)=\frac{1}{k},\quad x\in\partial\Omega\,
\end{gather*}
where $\Phi_1$ is the first eigenfunction of $-\Delta$ with zero
boundary value which satisfies  $\max_{x\in\Omega} \Phi_1(x)=1$.
\end{proof}

To prove Theorem \ref{thm4}, we need the following lemma,
which is an extension of a lemma in \cite{ABC}.


\begin{lemma}\label{lm1}
Let $\Omega$ be a domain  with a $C^2$ boundary $\partial \Omega$
or no boundary in $\mathbb{R}^{N}$, $N \geq 2$. Suppose that
$f:\Omega \times (0, +\infty )\to \mathbb{R}$ is a continuous
function such that the  assumption (F4) is
satisfied, and let $w,v \in C^2(\Omega)$ satisfy:
\begin{itemize}
\item[(a)] $\Delta w + f(x, w)\leq 0 \leq \Delta v + f(x, v)$ in
$\Omega$

\item[(b)] $w, v > 0$ in $\Omega$,
$\liminf_{|x|\to +\infty}\big(w(x)-v(x)\big)$, and
$\liminf_{x\to \partial\Omega}\big(w(x)-v(x)\big)\geq 0$

\item[(c)] $\Delta v \in L^1(\Omega)$.
\end{itemize}
Then $w(x)\geq v(x)$ for all $x \in \Omega$.
\end{lemma}

\begin{proof} The proof for the case where $f(x,s)$ is
non-increasing in $s$ at each $x$ in $\Omega$ is trivial, so we
only prove for the second case in assumption (F4).

Without loss of generality, we assume that
 $\Omega = \Omega_1 \cup \Omega_2$ in which
$$
\Omega_1 = \{x \in \Omega : f(x, s)\;{\rm and }\;s^{-1}f(x, s)
\mbox{ are nonincreasing in } s\},
$$
$\Omega_1\neq \Omega$ and
$$
\Omega_2 = \Omega_0-\Omega_1
$$
which is an anon-empty and open subset of $\Omega$, since $\Omega_1$ is a
relative closed subset of $\Omega$.

To prove the lemma by contradiction, we let $S_\delta$ be the set
$\{x \in \Omega \mid w(x)<v(x)-\delta\}$ for $\delta\geq 0$ and
suppose that $S_0\not=\emptyset$. Then by the condition (b), there
exists some $\sigma >0$ such that $S_{\sigma}\not=\emptyset$ and
$\overline{S_{\sigma}}\subset \Omega$ .

If $S_\sigma \cap\Omega_2=\emptyset$, then
$\overline{S_{\sigma}}\subset \Omega_1$. Noting that, at the
boundary of $S_{\sigma}, w(x)=v(x)-\sigma$, and that, for $x\in
s_\sigma$,
$$
\Delta ( w(x)-(v(x)-\sigma))\leq f(x, v(x))-f(x, w(x)\leq 0
$$
by the assumption on $f(x,s)$ for $x\in\Omega_1$ and the condition
$(a)$, one could have $ w(x)\geq v(x)-\sigma $ for all $x\in
s_\sigma$ by the aid of the maximum principle applied on
$S_{\sigma}$. But this is a contradiction to the definition of
$S_\sigma$.


If $S_{\sigma} \cap \Omega_2 \not=\emptyset$, then it is easily
seen from the assumption on $f(x,s)$ for $x\in \Omega_0$ that
there exist $\varepsilon_0 > 0 $ and a closed ball
$\overline{B} \subset (S_{\sigma} \cap \Omega_2)$ such that
\begin{equation}
v(x)-w(x)\geq \varepsilon_0,\; x \in {\rm B},\label{eqn1.1}
\end{equation}
and
\begin{equation}
\delta_0:=\int_{B}vw\big(\frac{f(x, w)}{w}-\frac{f(x,
v)}{v}\big)dx >0.\label{eqn1.2}
\end{equation}
Let
$$
M = \max \{1, \|\Delta v \|_{L^1(\Omega)}\}, \quad
\varepsilon = \min \{ 1, \varepsilon_0, \frac{\delta_0}{4M}\}.
$$
Let $\theta$ be a smooth function on $\mathbb{R}$ such that $\theta(t)=0$
if $t \leq 1/2$, $\theta (t)=0$ if $t \geq 1, \theta (t) \in (0, 1)$
if $t \in (\frac{1}{2}, 1)$, and $\theta'(t) \geq 0$ for
$t \in \mathbb{R}$. Then, for $\varepsilon > 0$, define the function
$\theta_{\varepsilon}(t)$ by
$$
\theta_{\varepsilon}(t)=\theta \big(\frac{t}{\varepsilon}\big),
\;t \in \mathbb{R}.
$$
It then follows from condition (a) and the fact
that $\theta_{\varepsilon}(t) \geq 0$ for $t \in \mathbb{R}$ that
$$
(w\Delta v - v\Delta w)\theta_{\varepsilon}(v-w)
\geq vw\big(\frac{f(x, w)}{w}-\frac{f(x, v)}{v}\big)\theta_{\varepsilon}(v-w),
\quad x \in \Omega.
$$
On the other hand, by the continuity of $w, v$ and
$\theta_{\varepsilon}$, and condition (b), we can take an open set
$D$ with a smooth boundary such that ${\overline{\rm B}}\subset D
\subset S_{\delta}$, here  $\delta = \min\{\sigma,
\frac{\varepsilon}{4}\}$, and $ v(x)-w(x) \leq
\frac{\varepsilon}{2}$, for all $x \in S_0-D$. Then we have
$$
\int_{D}(w\Delta v - v\Delta w)\theta_{\varepsilon}(v-w)dx
\geq \int_{D}vw\big(\frac{f(x, w)}{w}-\frac{f(x, v)}{v}\big)
\theta_{\varepsilon}(v-w)dx.
$$
Denote
$$
\Theta_{\varepsilon}(t)=\int_{0}^{t}s\theta'(s)ds,\;t \in \mathbb{R},
$$
then it is easy to verify that
\begin{equation}
0\leq \Theta_{\varepsilon}(t)\leq 2\varepsilon ,\quad
t \in \mathbb{R},\quad \mbox{and}\quad
\Theta_{\varepsilon}(t)=0, \;\mbox{if } t<\frac{\varepsilon}{2}.\label{eqn1.3}
\end{equation}
Therefore,
\begin{align*}
&\int_{D}(w\Delta v - v\Delta w)\theta_{\varepsilon}(v-w)dx\\
& = \int_{\partial D} w\theta_{\varepsilon}(v-w)
 \frac{\partial v}{\partial n}ds
 -\int_{D}(\nabla v \cdot \nabla w)\theta_{\varepsilon}(v-w)dx\\
&\quad -\int_{D}w\theta_{\varepsilon}'(v-w)\nabla v \cdot (\nabla v-\nabla w)dx
  -\int_{\partial D}v\theta_{\varepsilon}(v-w)\frac{\partial w}{\partial n}ds\\
&\quad+\int_{D}(\nabla w \cdot \nabla v)\theta_{\varepsilon}(v-w)dx
  +\int_{D}v\theta_{\varepsilon}'(v-w)\nabla w \cdot (\nabla v-\nabla w)dx\\
& = \int_{D}v\theta_{\varepsilon}'(v-w)(\nabla w - \nabla v)
\cdot (\nabla v-\nabla w)dx\\
&\quad  +\int_{D}(v-w)\theta_{\varepsilon}'(v-w)\nabla v \cdot (\nabla v-\nabla w)dx\\
& \leq \int_{D}\nabla v \cdot \nabla\left(\Theta_{\varepsilon}(v-w)\right)dx\\
& = \int_{\partial D}\Theta_{\varepsilon}(v-w)\frac{\partial v}{\partial n}ds
 -\int_{D}\Theta_{\varepsilon}(v-w)\Delta vdx\\
& \leq 2\varepsilon\int_{D}|\Delta v|dx\quad \mbox{( by \eqref{eqn1.3})} \\
& \leq 2\varepsilon M  < \frac{\delta_0}{2}\,.
\end{align*}
However,
\begin{align*}
 \int_{D}vw\big(\frac{f(x, w)}{w}-\frac{f(x, v)}{v}\big)\theta_{\varepsilon}(v-w)dx
& \geq \int_{{\rm B}}vw\big(\frac{f(x, w)}{w}-\frac{f(x, v)}{v}\big)
\theta_{\varepsilon}(v-w)dx\\
& = \int_{{\rm B}}vw\big(\frac{f(x, w)}{w}-\frac{f(x, v)}{v}\big)dx
\quad \mbox{(by \eqref{eqn1.1})}\\
& \geq  \delta_0 \quad \mbox{ by \eqref{eqn1.2})},
\end{align*}
which is a contradiction.
Thus $S_0$ must be empty, and the lemma is proved.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4}]
  Let $u_{1},u_{2}\in C^{0}(\overline{\Omega})\cap C^{2}(\Omega)$ be
two positive solutions of  problem~\eqref{eq:pp1}-\eqref{eq:pp2}.
We prove that $u_{1}(x)=u_{2}(x)$, $x\in\overline{\Omega}$.
 From the proofs of Theorems 1-3, we can easily see that if
$v=\psi_{m_{0},\infty}$ then
\begin{gather*}
\Delta v(x)+f(x,v(x))\geq 0,\quad x\in\Omega,\\
     v(x)> 0,\quad x\in\Omega, \\
 \phi(x) \geq v(x) \geq 0,\quad x\in\partial\Omega,
\end{gather*}
and
 $\Delta v \in L^{1}(\Omega)$.
Therefore it follows from Lemma \ref{lm1} that
$$
u_{i}(x)\geq v(x),\quad x\in\overline{\Omega},\; i=1,2.
$$
Moreover, by the Hopf's strong maximum principle,
we have $\frac{\partial v}{\partial n}<0$ on $\partial \Omega $,
hence there exists $c_{0}>0$ such that $u_{i}(x)\geq c_0d(x), \;x\in\overline{\Omega},\; i=1,2$,
where $d(x)=\mathop{\rm dist}(x, \partial\Omega)$.
Let $\Omega_{\varepsilon}=\{x\in\Omega : d(x) \leq \varepsilon \}$
for $\varepsilon >0 $ and $U_{i}(\delta )=\{x\in\Omega : u_{i}(x)\leq \delta \}$,
$i=1,2$.
Since $\partial\Omega\in C^{2,\theta }$, there exist $\varepsilon \in (0, \delta)$
such that if $x\in\Omega_{\varepsilon}$, then there is a unique
$y_{x}\in\partial\Omega $
such that dist$(x, y_{x})= d(x), c_0d(x)<\delta$.
Thus, for some $M>0$ only depending on $\partial\Omega $,
\begin{align*}
\int_{\Omega_{\varepsilon}}|f\left(x, c_0d(x)\right)|dx
 &\leq M\int_{\partial \Omega}\int_{0}^{\varepsilon}|f(y-sn_{y}, c_0s)|ds\,dy\\
&\leq M\int_{\partial \Omega } \int_{0}^{\varepsilon}F(s, c_0s)ds\,dy\\
&\leq M^{*} < +\infty ,
\end{align*}
where
$$
M^{*}=M\int_{\partial \Omega} \int_{0}^{\delta}F(s, c_0s)ds\,dy\,.
$$
By the hypothesis (F5), there exists $M_{0}>0$ such that
$$
0\leq F(r, s)\leq F(r, t)+M_{0}\quad \mbox{for }\delta\geq s\geq
t>0 \;  r\in(0, \delta).
$$
Therefore,
\begin{align*}
\int_{\Omega_{\varepsilon} \cap U_{i}(\delta )}|f(x, u_{i}(x))|dx
 &\leq  \int_{\Omega_{\varepsilon}}|f\big(x, c_0d(x)\big)|dx
 + M_{0}\mathop{\rm meas}(\Omega) \\
  &\leq M^{*}+M_{0}\mathop{\rm meas}(\Omega) < +\infty ,\quad i=1,2.
\end{align*}
Consequently,
\begin{align*}
\int_{\Omega}|f\big(x, u_{i}(x)\big)|dx
 &\leq \int_{\Omega_{\varepsilon}\cap U_{i}(\delta)}|f\big(x, u_{i}(x)\big)|dx
  + \int_{\Omega\backslash (\Omega_{\varepsilon }\cap U_{i}(\delta ))}|
  f\big(x, u_{i}(x)\big)|dx \\
 &\leq M^{*}+M_{0}\mathop{\rm meas}(\Omega) +M_{i}^{**}\mathop{\rm meas}(\Omega)
 <  +\infty ,
\end{align*}
where
$$
M_{i}^{**}=\max_{x\in\overline{\Omega},\,\delta\leq s\leq \delta_{i}^{*}}|f(x, s)|,
\quad \delta_{i}^{*}=\max_{x\in\overline{\Omega}}u_{i}(x),\quad i=1,2.
$$
Therefore,
$$
 \int_{\Omega}|\Delta u_{i}|dx =\int_{\Omega} |f(x,u_{i})|dx <  +\infty,
 \quad i=1,2.
$$
i.e., $\Delta u_{i}\in L^{1}(\Omega )$, $i=1,2$. Hence, it
follows from Lemma \ref{lm1} that
$$
u_{1}(x)=u_{2}(x),\quad x\in \overline{\Omega},
$$
and the theorem is proved.
\end{proof}

\begin{thebibliography}{00}

\bibitem{Aman} H. Amann;
\emph{Existence of multiple solutions of nonlinear elliptic boundary-value problems},
Indiana Univ. Math. J. {\bf 21} (1972), 925-935.

\bibitem{ABC} A. Ambrosetti, H. Br\'ezis and G. Cerami;
\emph{Combined effects of concave and convex nonlinearities in some
elliptic problems},
 J. Funct. Anal., {\bf 122}(1994), No.2, 519-543.

\bibitem{CK} D. S. Cohen and H. B. Keller;
\emph{Some positive problems suggested by nonlinear heat generators},
 J. Math. Mech., {\bf 16}(1967), 1361-76.

\bibitem{CN} A. Callegari and A. Nashman;
\emph{A nonlinear singular boundary-value problem in
the theory of psedoplastic fluids},  SIAM J. Appl. Math., {\bf 38}(1980), 275-281.

\bibitem{CRT} M. G. Crandall, P. H. Rabinowitz and L. Tartar;
\emph{On a Dirichlet problem with a singular nonlinearity},
{\sl Comm. Part. Diff. Eq.} {\bf 2}(2)(1977), 193-222.

\bibitem{DMO} Diaz, J. M. Morel and L. Oswald;
\emph{An elliptic equation with singular nonlinearaity},
Comm. Part. Diff. Eq., {\bf 12}(1987), 1333-44.

\bibitem{FM} W. Fulks and J. S. Maybee;
\emph{A singular nonlinear equation}, Osaka Math. J., {\bf 12}(1960), 1-19.

\bibitem{GT} D. Gilberg and N. S. Trudinger,
\emph{Elliptic Partial Differential Equations of Second Order}, 2nd ed.,
Springer-Verlag, Berlin(1983).

\bibitem{LM} A. C. Lazer and P. J. Mckenna;
\emph{On a singular nonlinear elliptic boundary value problem},
Proc. Amer. Math. Soc., {\bf 3}(1991), 720-730.

\bibitem{OS} Tiancheng Ouyang and Junping Shi;
\emph{Exact multiplicity of positive solutions for a class of semilinear
problem: II.},   J. Diff. Eqns. {\bf 158}, (1999), 94-151.

\bibitem{Se} J. Serrin:  \emph{A remark on the proceeding paper of Amann},
Arch. Rat. Mech. Analysis, {\bf 44}(1972), 182-186.

\bibitem{St} C. A. Staurt;
\emph{Existence and approximation of solutions
of nonlinear elliptic equations},  Math. Z. {\bf 147}(1976), 53-62.

\bibitem{SW} Sun Yijing and Wu Shaoping;
\emph{Iterative solution for a singular nonlinear elliptic problem},
Applied Mathematics and Computation, {\bf 118}(2001), 53-62.

\end{thebibliography}
\end{document}