\documentclass[english,11pt]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath,amsthm,amssymb}
%\usepackage{graphicx}
\usepackage{graphics,epsfig,graphicx}
\renewcommand{\arraystretch}{1.7}
\textwidth13cm \textheight22cm \hoffset-16mm \voffset-20mm
\font\sans=cmss10
\def\proof{{\it Proof.}\ }
\def\endproof{\nolinebreak\hfill $\square$ \par\vskip3mm}
\def\pt#1{{\it Proof of Theorem \ref{#1}.}}
\def\eq#1{(\ref{#1})}
\def\th#1{Theorem \ref{#1}}
\def\neweq#1{\begin{equation}\label{#1}}
\def\endeq{\end{equation}}
\def\weak{\rightharpoonup}
\def\ep{\varepsilon}
\def\la{\lambda}
\def\half{{1\over2}}
\def\RR{{\mathbb R} }
\def\NN{{\mathbb N} }
\def\supp{{\rm supp}}
\def\di{\displaystyle}
\def\ri{\rightarrow}
\def\intom{\int_\Omega}
\def\ii{\^\i }
\newtheorem{theorem}{THEOREM}[section]
\newtheorem{lem}{LEMMA}[section]
\newtheorem{corollary}{COROLLARY}[section]
\newtheorem{rem}{REMARK}[section]
\title{\bf COMBINED EFFECTS IN NONLINEAR SINGULAR ELLIPTIC PROBLEMS WITH CONVECTION\thanks{Abbreviated title: ``NONLINEAR SINGULAR ELLIPTIC PROBLEMS"}}
\author{
Vicen\c tiu R\u adulescu$\,^{a,b}$\\ \small $^a\,$Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,\\
\small P.O. Box 1-764, 014700 Bucharest, Romania\\
\small $^b\,$Department of Mathematics, University of Craiova, 200585
Craiova, Romania\\
\small E-mail:
{\tt vicentiu.radulescu@imar.ro}}
\date{}

\begin{document}
\baselineskip16pt \maketitle
%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}

\begin{center}{Dedicated with esteem to Professor Philippe G. Ciarlet on his 70th birthday}\end{center}

\begin{abstract}
We establish some bifurcation results for the boundary value
problem $-\Delta u=\la u^{-a}|\nabla u|^2+ \mu$ in $\Omega,$
$u>0$ in $\Omega,$ $u=0$ on $\partial\Omega,$ where $\Omega$ is a
smooth bounded domain in $\RR^N,$ while $a,\,\la\,\mu> 0$.  Our approach relies on finding explicit sub-
and super-solutions combined with various techniques related to
the maximum principle for elliptic equations with singular nonlinearities.

\noindent{\bf Key words}: singular elliptic equation,
nonlinear perturbation,
 bifurcation problem,
maximum principle.

\noindent{\bf 2000 Mathematics Subject Classification}: 35A20,
35B25, 35B50, 35J60, 58J55, 58K55.
\end{abstract}

\section{Introduction}
This paper is motivated by recent advances in the study of nonlinear elliptic problems involving
both singular nonlinearities and convection (gradient) terms. We start with the following elementary example. Consider the nonlinear boundary value problem
$$\left\{\begin{tabular}{ll}
$\Delta u=u^p$ \quad & $\mbox{\rm in}\ \Omega,\ $\\
$u>0$ \quad & $\mbox{\rm in}\ \Omega,$\\
$u=+\infty$ \quad & $\mbox{\rm on}\ \partial\Omega\,,$
\end{tabular} \right.$$
where $\Omega\subset\RR^N$ is a smooth bounded domain and $p>1$.
Then the Gelfand transform $v=u^{-1}$ yields
\begin{equation}\label{eq1}\left\{\begin{tabular}{ll}
$-\Delta v=v^{2-p}-\displaystyle  \frac 2v\,|\nabla v|^2$ \quad & $\mbox{\rm in}\ \Omega,$\\
$v>0$ \quad & $\mbox{\rm in}\ \Omega,$\\
$v=0$ \quad & $\mbox{\rm on}\ \partial\Omega.$\\
\end{tabular} \right.\end{equation}
The above equation contains both singular nonlinearities (like
$v^{-1}$ or $v^{2-p}$, if $p>2$) and a convection term (denoted by
$|\nabla v|^2$). These nonlinearities make more difficult to
handle problems like \eq{eq1}. We
recall the pioneering paper \cite{crt} that contains one of the
first existence results for singular elliptic problems. In fact,
it is proved in \cite{crt} that the boundary value problem
$$
 \left\{\begin{tabular}{ll}
$-\Delta u-u^{-\alpha}= -u$ \quad & ${\rm in}\
\Omega,$\\
$u>0$ \quad & ${\rm in}\ \Omega,$\\
$u=0$ \quad & ${\rm on}\ \partial\Omega$
\end{tabular} \right.
$$
has a solution, for any $\alpha>0$. Let us now consider the
problem
\begin{equation} \label{unuunu}
 \left\{\begin{tabular}{ll}
$-\Delta u-u^{-\alpha}=\lambda u^p$ \quad & ${\rm in}\
\Omega,$\\
$u>0$ \quad & ${\rm in}\ \Omega,$\\
$u=0$ \quad & ${\rm on}\ \partial\Omega,$\\
\end{tabular} \right.
\end{equation}
where $\lambda\geq 0$ and $\alpha,p\in(0,1).$ In \cite{cp} it is
proved that problem \eq{unuunu} has at least one solution for
all $\lambda\geq 0$ and $0<p<1$. Moreover, if $p\geq 1,$ then
there exists $\lambda^*$ such that problem \eq{unuunu} has a
solution for $\lambda\in[0,\lambda^*)$ and no solution for
$\lambda>\lambda^*$. In \cite{cp} it is also proved a related
non--existence result. More exactly, the  problem
$$
 \left\{\begin{tabular}{ll}
$-\Delta u+u^{-\alpha}= u$ \quad & ${\rm in}\
\Omega,$\\
$u>0$ \quad & ${\rm in}\ \Omega,$\\
$u=0$ \quad & ${\rm on}\ \partial\Omega$\\
\end{tabular} \right.
$$
has no solution, provided that $0<\alpha <1$ and $\lambda_1\geq 1$
(that is, if $\Omega$ is ``small''), where $\lambda_1$ denotes the
first eigenvalue of $(-\Delta)$ in $H^1_0(\Omega)$.

Problems related to multiplicity and uniqueness become difficult
even in simple cases. In \cite{sy1} it is studied the existence of
radial symmetric solutions to the problem
$$
 \left\{\begin{tabular}{ll}
$\displaystyle \Delta u+\lambda(u^p-u^{-\alpha})=0$ \quad & ${\rm
in}\ B_1,$\\
$u>0$ \quad & ${\rm in}\ B_1,$\\
$u=0$ \quad & ${\rm on}\ \partial B_1,$\\
\end{tabular} \right.
$$
where $\alpha>0$, $0<p<1,$ $\lambda>0$, and $B_1$ is the unit ball
in $\RR^N$. Using a bifurcation theorem of Crandall and
Rabinowitz, it has been shown in \cite{sy1}  that there exists
$\lambda_1>\lambda_0>0$ such that the above problem has no
solutions for $\lambda<\lambda_0,$ exactly one solution for
$\lambda=\lambda_0$ or $\lambda>\lambda_1,$  and two solutions for
$\lambda_0 < \lambda\leq\lambda_1$.

Singular elliptic problems have been intensively studied in the last decades.
Stationary problems involving singular nonlinearities,
 as well as the associated evolution equations,
describe naturally several physical phenomena. At our best
knowledge, the first study in this direction is due to Fulks and
Maybee \cite{ful}, who proved existence and uniqueness results by
using a fixed point argument; moreover, they showed that solutions
of the associated parabolic problem tend to the unique solution of
the corresponding elliptic equation. A different approach (see
Coclite and Palmieri \cite{cp}, Crandall, Rabinowitz, and Tartar
\cite{crt}, Stuart \cite{stu}) consists in approximating the
singular equation with a regular problem, where the standard
techniques (e.g., monotonicity methods) can be applied and then
passing to the limit to obtain the solution of the original
equation. Nonlinear singular boundary value problems arise in the
context of chemical heterogeneous catalysts and chemical catalyst
kinetics, in the theory of heat conduction in electrically
conducting materials, singular minimal surfaces, as well as in the
study of non-Newtonian fluids, boundary layer phenomena for
viscous fluids (we refer for more details to Caffarelli, Hardt,
and L. Simon \cite{caf}, Callegari and Nachman \cite{cn},
D\'iaz \cite{diaz}, D\'iaz,  Morel, and Oswald \cite{dmo} and the
more recent papers by Ghergu and R\u adulescu \cite{gr,grroyal,grjmaa}, Haitao \cite{hai}, Hern\'andez, Mancebo, and
 Vega \cite{her}, Meadows \cite{mea}, Shi and Yao \cite{sy1}).
We also point out that, due to the meaning of the unknowns
(concentrations, populations, etc.), only the positive solutions
are  relevant in most cases. For instance, problems of this type
characterize some reaction-diffusion processes where $u\geq 0$ is
viewed as the density of a reactant and the region where $u=0$ is
called the {\it dead core}, where no reaction takes place (see
Aris \cite{a} for the study of a single, irreversible steady-state
reaction). Nonlinear singular elliptic equations are also
encountered in glacial advance, in transport of coal slurries down
conveyor belts and in several other geophysical and industrial
contents (see Callegari and Nachman \cite{cn}  for the case of the
incompressible flow of a uniform stream past a semi-infinite flat
plate at zero incidence).

\section{The main result}
In the present paper we continue the bifurcation analysis
developed in our previous works \cite{grroyal,grbook,radbirkh,hbook} for a large class of semilinear elliptic equations
with singular nonlinearity and Dirichlet boundary condition.
In these works we have been concerned with various boundary value problems involving singular nonlinearities and we have been concerned with the existence or nonexistence of solutions under
various assumptions.

Let $\Omega\subset\RR^N$ ($N\geq 3$) be a bounded domain with smooth boundary.
Consider the nonlinear singular problem
\neweq{P}
\left\{\begin{tabular}{ll}
$-\Delta u=u^{-a}+\la|\nabla u|^2+\mu $ \quad & $\mbox{\rm in}\ \Omega,$\\
$u>0$ \quad & $\mbox{\rm in}\ \Omega,$\\
$u=0$ \quad & $\mbox{\rm on}\ \partial\Omega,$\\
\end{tabular} \right.
\endeq
where $a>0$ and $\la,\,\mu\geq 0.$ If $\lambda_1$ denotes the first eigenvalue of the Laplace operator in $H^1_0(\Omega)$, we have established in \cite{grroyal,radbirkh,hbook} the following results:

(i) Problem \eq{P} has a solution if and only if $\la\mu<\la_1;$

(ii) Assume $\mu>0$ is fixed and let
$\la^*=\la_1/\mu.$ Then problem \eq{P} has a unique solution
$u_\la$ for every $\la<\la^*$ and the sequence
$(u_\la)_{\la<\la^*}$
is increasing with respect to $\la.$
Moreover, if $a<1$, then the sequence of solutions
$(u_\la)_{0<\la<\la^*}$ has the following properties

(ii1) For all $0<\la<\la^*$ there exist two positive
constants $c_1,c_2$ depending on $\la$ such that $c_1\,{\rm
dist}(x,\partial \Omega)\leq u_\la\leq c_2\,{\rm dist}(x,\partial
\Omega)$ in $\Omega;$

(ii2) $u_\la\in C^{1,1-\alpha}(\overline\Omega)\cap
C^2(\Omega);$

(ii3) $u_{\la}\longrightarrow +\infty$ as
$\la\nearrow \la^*$, uniformly on compact subsets of $\Omega.$

\smallskip
Our purpose in the present paper is to study a related nonlinear elliptic equation
involving both a singular nonlinearity and a quadratic convection (gradient) term. 

We are concerned with the boundary value problem
\neweq{pnew}
\left\{\begin{tabular}{ll}
$-\Delta u=\la u^{-a}\,|\nabla u|^2+\mu $ \quad & $\mbox{\rm in}\ \Omega,$\\
$u>0$ \quad & $\mbox{\rm in}\ \Omega,$\\
$u=0$ \quad & $\mbox{\rm on}\ \partial\Omega\,,$
\end{tabular} \right.
\endeq
where $a$, $\la$, and $\mu$ are positive real numbers.

Our main result is the following.

\begin{theorem}\label{t1} There exists
$\lambda_0>0$ such that problem \eq{pnew} has a solution $u\in
C^2(\Omega)\cap C(\partial\Omega)$ for any
$0<\lambda\leq\lambda_0$.
\end{theorem}

As remarked in \cite{choquet,kazdan}, the requirement that the nonlinearity grows at most quadratically in $|\nabla u|$ is natural in order to apply the maximum principle.

The proof of Theorem \ref{t1} relies on comparison arguments which are based on the
method of lower and upper solutions for elliptic equations
involving both singular nonlinearities and gradient terms. The roots of the method of
lower and upper solutions go back to Picard, who applied, in the early 1880s, the method of successive approximations to argue the existence of solutions for nonlinear elliptic equations that are suitable perturbations of uniquely solvable linear problems. Picard's techniques
were applied later by Poincar\'e \cite{poincare} in connection with problems arising in astrophysics. In particular, Poincar\'e established that the nonlinear equation $\Delta u=e^{u}$ has at least one solution. Because of the major contributions of Lyapunov (1906),
Schmidt (1908), and Leray and Schauder (1934), the method of lower and upper solutions became a powerful tool in the modern nonlinear analysis. Due to the singular character of problem \eq{pnew}, we apply in this paper a recent version of the lower and upper solutions method which is due to Cui \cite{cui} and reads as follows: suppose that $\underline u,\, \overline u\in C^2(\Omega )\cap C(\partial\Omega)$ satisfy $0<\underline u\leq\overline u$ in $\Omega$,
$\underline u=\overline u=0$ on $\partial\Omega$ and
$$-\Delta \underline u\leq\la \underline u^{-a}\,|\nabla \underline u|^2+\mu  \qquad  \mbox{\rm in}\ \Omega$$
$$-\Delta \overline u\geq\la \overline u^{-a}\,|\nabla \overline u|^2+\mu  \qquad  \mbox{\rm in}\ \Omega\,.$$
Then problem \eq{pnew} has at least one solution $u\in C^2(\Omega )\cap C(\partial\Omega)$ and, moreover, $\underline u\leq u\leq\overline u$ in $\Omega$.

\section {A related singular differential inequality}
As in \cite{dgr},
we are first concerned with the inequality problem
\neweq{p1}
\left\{\begin{tabular}{ll}
$-\Delta u\geq\la u^{-a}\,|\nabla u|^2+\mu $ \quad & $\mbox{\rm in}\ B_R,$\\
$u>0$ \quad & $\mbox{\rm in}\ B_R,$\\
$u=0$ \quad & $\mbox{\rm on}\ \partial B_R\,,$
\end{tabular} \right.
\endeq
where $R>0$ is fixed.

The main result of this section is the following.

\begin{theorem}\label{t2} Fix $\mu > 0$ and $R>0$. Then there
exists $\lambda^{*}>0$ depending on $\mu$ and $R$ such that
problem \eqref{p1} has at least one radially symmetric solution in
$C^2(B_R)\cap C(\partial B_R)$, for any $0<\lambda\leq\lambda^*$.
\end{theorem}

We start with the study of the related nonlinear differential
equation with initial boundary conditions
\neweq{p2}
\left\{\begin{tabular}{llll}
$-(r^{N-1}z'(r))'=r^{N-1}[z^{-a}(r)z'(r)^2+\alpha]$ \quad & $\mbox{\rm if}\ 0<r<T,$\\
$z(0)=\beta$ \quad & \\
$z'(0)=0$ \quad &\\
$z>0$ \quad & $\mbox{\rm in}\ (0,T)\,,$
\end{tabular} \right.
\endeq
where $T,\alpha$ and $\beta$ are positive numbers.

\begin{lem}\label{l1} Fix the positive numbers $\alpha$ and
$\beta$. Then there exists $T>0$ such that problem \eqref{p2} has
a solution $z\in C^2[0,T)\cap C[0,T]$. Moreover, $z$ is decreasing
on $[0,T]$ and $z(T)=0$.
\end{lem}

\proof
Problem \eqref{p2} may be written, equivalently, in terms of the
following nonlinear integral equation
\neweq{p3}
z(r)=\beta-\frac{\alpha}{2N}r^2-\int_{0}^{r}t^{1-N}\int_{0}^{t}s^{N-1}z^{-a}(s)z'(s)^2dsdt,
\endeq
for $r\in[0,T)$.

Fix $\varepsilon>0$. Consider the function space
$C^1[0,\varepsilon]$ endowed with the standard norm
$$\|z\|=\max_{t\in[0,\varepsilon]}|z(t)|+\max_{t\in[0,\varepsilon]}|z'(t)|.$$
Let $F_{\ep}\subset C^1[0,\varepsilon]$ be the closed ball defined
by
$$F_{\ep}=\{z\in C^1[0,\varepsilon];\quad\|z-\beta\|\leq\beta/2\}$$
and consider the nonlinear integral operator $T$ defined by
$$Tz(r)=\beta-\frac{\alpha}{2N}r^2-\int_{0}^{r}t^{1-N}\int_{0}^{t}s^{N-1}z^{-a}(s)z'(s)^2dsdt,$$
for any $r\in[0,\ep]$.

A standard straightforward computation shows that $T$ maps
$F_{\ep}$ into $F_{\ep}$ and $T$ is a contraction, that is,
$$\|Tz_1-Tz_2\|\leq\frac{1}{2}\|z_1-z_2\|\quad\mbox{for all }z_1,z_2\in F_{\ep},$$
provided $\ep>0$ is small enough.

Thus, by the Banach contraction principle, $T$ has a fixed point
$z\in F_{\ep}$, which is a solution of problem \eqref{p3}, hence
of \eqref{p2}.

Set
$$T=\sup\{\ep>0;\;\; \mbox{problem \eqref{p2} has a solution in }(0,\ep)\}.$$
Using now the definition of $T$ we deduce that $z'(r)<0$ for any
$r\in(0,T)$.

We argue in what follows that $z(T)=0$. For this purpose we first
observe that since $z'<0$ on $(0,T)$, then there exists
$\ell=\lim_{r\rightarrow T^-}z(r)\geq 0$. It suffices to show that
$\ell=0$. We argue by contradiction and assume that $\ell>0$.

Fix $\ep>0$ and consider the closed set
$$F_{\ep}(\ell)=\{z\in C^1[T,T+\ep];\quad\|z-z_T\|\leq l/2\},$$
where $z_T$ is defined by
$$z_T(r)=\ell+\frac{\alpha}{2N}(T^2-r^2)\quad \mbox{for all }r\in [T,T+\ep]$$
and $\|\,.\,\|$ denotes the norm on the space $C^1[T,T+\ep]$.

For any $z\in F_{\ep}(\ell)$, define the nonlinear operator
$$S_z(r)=z_T(r)-\int_{T}^{r}t^{1-N}\int_{T}^{t}s^{N-1}z^{-a}(s)z'(s)^2dsdt,$$
for all $r\in [T,T+\ep]$. Then $S$ maps $F_{\ep}(\ell)$ into itself
and
$$\|Sz_1-Sz_2\|\leq\frac{1}{2}\|z_1-z_2\|\quad\mbox{for any }z_1,z_2\in F_{\ep}(\ell).$$
Again, by Banach's contraction principle, $S$ has a fixed point
$z\in F_{\ep}(\ell)$, that is,
$$z(r)=z_T(r)-\int_{T}^{r}t^{1-N}\int_{T}^{t}s^{N-1}z^{-a}(s)z'(s)^2dsdt,$$
for $r\in [T,T+\ep]$. This shows that problem \eq{p2} has a
solution on $[0,T+\ep]$, which contradicts the definition of $T$.
Thus, $\ell=0$, hence $z(T)=0$.

It remains to argue the regularity of $z$. Integrating \eq{p2} on
$[0,r]$ we find
$$-z'(r)=\frac{1}{r^{N-1}}\int_{0}^{r}t^{N-1}z(t)^{-a}z'(t)^2 dt +\frac{\alpha}{N}r.$$

Therefore
$$z''(r)=-\frac{\alpha}{N}+(N-1)r^{-N}\int_{0}^{r}t^{N-1}z(t)^{-a}z'(t)^2 dt -z(r)^{-a}z'(r)^2,$$
hence $z\in C^2(0,T)\cap C[0,T]$. We also deduce that
$$z''(0)=\lim_{r\rightarrow 0}z''(r)=-\frac{\alpha}{N},$$
which concludes that $z\in C^2[0,T)\cap C[0,T]$.
\endproof
$$$$

\textbf{\emph{Proof of Theorem \ref{t2} concluded.}} Fix $0<\beta<\alpha
R^2/(2N)$ and consider the corresponding solution $z$ of problem
\eq{p2}.

The local existence theorem applied to \eq{p2} implies $\ep\leq
T\leq (2N\beta/\alpha)^{1/2}$. The above choice of $\alpha$ and
$\beta$ implies that $T<R$. For any $r\in [0,R)$ define the
mapping
$$w(r)=\frac{R^2}{T^2}\;z\left(\frac{rT}{R}\right).$$

Then $w(0)=R^2\beta/T^2$, $w(R)=0$, and $w>0$ on $[0,R)$. A
straightforward computation shows that $w$ satisfies the nonlinear
differential equation
$$-(r^{N-1}w'(r))'=r^{N-1}\left[z\left(\frac{rT}{R}\right)^{-a}z'\left(\frac{rT}{R}\right)^{2}+\alpha\right].$$
Since $z$ decreases, we deduce that
$$-(r^{N-1}w'(r))'\geq r^{N-1}\left[\left(\frac{T}{R}\right)^2w(r)^{-a}w'(r)^2+\alpha\right].$$

Set $\la^*=T^2/R^2$. Thus, for any $\la\in[0,\la^*]$,
$$-(r^{N-1}w'(r))'\geq r^{N-1}\left[\la w(r)^{-a}w'(r)^2+\alpha\right],$$
for all $r\in(0,r)$. Equivalently, $w$ is a solution of the
inequality problem \eq{p1}, provided $\alpha$ is chosen so that
$\alpha\geq \mu$.
\endproof

\section{An auxiliary approximate problem}
Inspired by the techniques developed by Crandall, Rabinowitz and
Tartar, we consider the nonlinear problem for any $\ep>0$
\neweq{p4}
\left\{\begin{tabular}{ll}
$-\Delta u=\la u^{-a}|\nabla u|^2+\mu+\ep $ \quad & $\mbox{\rm in}\ \Omega,$\\
$u>\ep$ \quad & $\mbox{\rm in}\ \Omega$\\
$u\geq\ep$ \quad & $\mbox{\rm on}\ \partial\Omega$\,.\\
\end{tabular} \right.
\endeq

The main result of this section establishes that problem \eq{p4}
has at least one solution, provided $\ep$ and $\la$ are small
enough. More precisely, the following property holds true.

\begin{lem}\label{l2} There exists $\ep_0>0$ and $\Lambda>0$ such
that for any $\ep\in(0,\ep_0)$ and $\la\in(0,\Lambda)$, problem
\eq{p4} has a solution in $C^2(\overline{\Omega})$.
\end{lem}


\proof Our arguments are based on the method of lower and upper
solutions. So, we construct functions $\overline{u}$ and
$\underline{u}$ such that $\overline{u}\geq\underline{u}>\ep$ in
$\Omega$ and
$$-\Delta \underline{u}-\la \underline{u}^{-a}|\nabla \underline{u}|^2-\mu-\ep\leq0
\leq -\Delta \overline{u}-\la \overline{u}^{-a}|\nabla
\overline{u}|^2-\mu-\ep\quad\mbox{in }\Omega.$$

Set $d=\sup_{x\in\Omega}|x|$. For $R=3d/2$, denote
$\overline{u}=z+\ep$, where $z$ is the solution of problem \eq{p1}
given by Theorem \ref{t2}. So, $\overline{u}\in
C^2(\overline{\Omega})$ is an upper solution of problem \eq{p4}.

In order to find a lower solution, fix $p>N$ and consider the
unique solution $u_0\in C^{\infty}(\overline{\Omega})$ of the
linear problem
$$\left\{\begin{tabular}{ll}
$-\Delta u_0=\mu$ \quad & $\mbox{\rm in}\ \Omega,\ $\\
$u_0>0$ \quad & $\mbox{\rm in}\ \Omega,$\\
$u_0=0$ \quad & $\mbox{\rm on}\ \partial\Omega\,.$
\end{tabular} \right.$$
Then the function $\underline{u}=u_0+\ep$ is a linear solution of
\eq{p4}.

It remains to argue that $\underline{u}\leq\overline{u}$ in
$\Omega$. For this purpose we observe that
$\underline{u}\leq\overline{u}$ on $\partial \Omega$ and
$-\Delta(\overline{u}-\underline{u})\geq 0$ in $\Omega$. Thus, by
the maximum principle, $\underline{u}\leq\overline{u}$ in
$\Omega$. It follows that problem \eq{p4} has a smooth solution
$u_{\ep}$ such that $\underline{u}\leq u_{\ep}\leq\overline{u}$ in
$\Omega$.

\section{Proof of Theorem \ref{t1}}

We first prove that if $0<\ep_1<\ep_2$ then $u_{\ep_1}\geq
u_{\ep_2} $, where $u_\ep$ denotes the solution of problem \eq{p4}. Arguing by contradiction, let $x_0\in\Omega$ be such
that  $u_{\ep_2}(x_0)< u_{\ep_1} (x_0)$. Moreover, since
$$u_{\ep_1}- u_{\ep_2}=\ep_1-\ep_2<0\qquad \mbox{on $\partial \Omega$,}$$ we
deduce that we can assume, without loss of generality, that
$x_0\in\Omega$ is a global maximum point of the function
$u_{\ep_1}- u_{\ep_2}$ in $\overline{\Omega}$. Hence
$$\nabla u_{\ep_1}(x_0)=\nabla u_{\ep_2}(x_0)\quad\mbox{ and }\quad \Delta(u_{\ep_1}-u_{\ep_2})(x_0)\leq 0.$$
Therefore
$$\Delta u_{\ep_1}(x_0)-\Delta u_{\ep_2}(x_0)=\la \left(u_{\ep_2}(x_0)^{-a}-u_{\ep_1}(x_0)^{-a}\right)|\nabla u_{\ep_1}(x_0)|^2+\ep_2-\ep_1>0.$$
This contradiction shows that $u_{\ep_1}\geq u_{\ep_2}$ in
$\Omega$, provided $\ep_1<\ep_2$.

For any $x\in\Omega$, denote
$$u(x)=\lim_{\ep\rightarrow 0} u_{\ep}(x).$$

To conclude the proof, we show that $u$ is a solution of problem
\eq{pnew}, that is, $u\in C^2(\Omega)\cap C(\partial\Omega)$ and
$$\begin{tabular}{ll}
$-\Delta u=\la u^{-a}\,|\nabla u|^2+\mu $ \quad & $\mbox{\rm in}\ \overline{\Omega},$\\
$u=0$ \quad & $\mbox{\rm on}\ \partial\Omega\,.$
\end{tabular}
$$

To obtain the regularity of $u$ we apply bootstrap arguments. We
first observe that the definition of $u$ implies $u\in
L^{\infty}(\Omega)$. Let $u_n$ be a solution of the problem
$$-\Delta u_n=\la u_n^{-a}\,|\nabla u_n|^2+\mu+\frac{1}{n}\quad\mbox{in }\Omega. $$
Then $\underline{u}\leq u_n\leq C$ in $\overline{\Omega}$ for all
$n\geq 1$, where $C>0$ is a constant. Moreover, $u_n\in
W^{2,p}(\Omega)$ for any $p>N$ and for all $\varphi\in
C_{0}^{\infty}(\Omega)$,
$$-\int_{\Omega}u_n\Delta\varphi dx=\int_{\Omega}\la u_{n}^{-a}\,|\nabla u_n|^2\varphi dx+\left(\mu+\frac{1}{n}\right)\int_{\Omega} \varphi dx.$$

Consider a sequence $(\Omega_m)_{m\geq1}$ of open sets with smooth
boundary such that $\Omega=\bigcup_{m=1}^{\infty}\Omega_m$. Fix a
positive integer $k\geq 1$ and a function $\varphi\in
C_{0}^{\infty}(\Omega)$ satisfying $0\leq\varphi\leq 1$ and
$\varphi \equiv 1$ on $\overline{\Omega}_k$. We deduce that
$$\la\int_{\Omega_k} u_{n}^{-a}\,|\nabla u_n|^2\varphi dx\leq -\int_{\Omega}u_n\Delta\varphi dx-\left(\mu+\frac{1}{n}\right)\int_{\Omega} \varphi dx,$$
hence
$$\la C^{-a}\int_{\Omega_k} |\nabla u_n|^2dx\leq C\int_{\Omega}|\Delta\varphi| dx+(\mu+1)|\Omega|.$$
It follows that the sequence $(u_n)_{n\geq 1}$ is bounded in
$H^1(\Omega_k)$. Thus, up to a subsequence,

\begin{tabular}{ll}
$u_n\rightharpoonup v$ \quad & $\mbox{\rm in}\ H^1(\Omega_k),$\\
$u_n\rightarrow v$ \quad & $\mbox{\rm in}\ L^q(\Omega_k),\;\quad\mbox{for all } 1\leq q<\frac{2N}{N-2},$\\
$u_n\rightarrow v$ \quad & $\mbox{\rm a.e. in}\ \Omega_k.$
\end{tabular}

Hence, by the definition of $u$, we have $u=v$ a.e. in $\Omega_k$,
for any $k\geq 1$. This shows that, in fact,

\begin{tabular}{ll}
$u_n\rightharpoonup u$ \quad & $\mbox{\rm in}\ H_{loc}^1(\Omega),$\\
$u_n\rightarrow u$ \quad & $\mbox{\rm in}\
L_{loc}^q(\Omega),\;\quad\mbox{for any } 1\leq q<\frac{2N}{N-2}.$
\end{tabular}

By Schauder estimates we deduce that $(u_n)$ is bounded in
$W_{loc}^{2,p}(\Omega)$, so in $C_{loc}^{1,\alpha}(\Omega)$ for
any $\alpha\in (0,1)$. It follows that for any compact set
$K\subset\Omega$ we have
$$u_n\rightarrow u\quad\mbox{in } C^2(K)$$
and
$$-\Delta u=\la u^{-a}\,|\nabla u|^2+\mu  \quad \mbox{in } \Omega.$$

To show that $u\in C(\partial\Omega)$, let $x_0\in\partial\Omega$
and consider a sequence of points $(x_j)_{j\geq 1}\subset\Omega$
such that $x_j\rightarrow x_0$. Then
$$0\leq u(x_j)\leq u_n (x_j),$$
hence
$$0\leq\overline{\lim}_{j\rightarrow\infty} u(x_j)\leq\lim_{j\rightarrow\infty}  u_n(x_j)=u_n(x_0)=\frac{1}{n}\,.$$
We obtain that $u$ is continuous at $x_0\in\partial\Omega$ and
$u(x_0)=0$.

The proof of theorem is now complete.
\endproof


\medskip
{\bf Acknowledgments.} The author has been supported by Grants GAR 315/2008 and CNCSIS PNII-79/2007.


\begin{thebibliography}{99}  {\footnotesize
\bibitem{a} R. Aris, {\it The Mathematical Theory of
Diffusion and Reaction in Permeable Catalysts}. Clarendon Press,
Oxford, 1975.

\bibitem{caf} L. Caffarelli, R. Hardt and L. Simon, {\it  Minimal surfaces with isolated singularities}. Manuscripta Math. {\bf 48} (1984), 1-18.

\bibitem{cn} A. Callegari and A. Nachman, {\it A nonlinear
singular boundary value problem in the theory of pseudoplastic
fluids}. SIAM J. Appl. Math. {\bf 38} (1980), 275-281.

\bibitem{choquet} Y. Choquet-Bruhat and J. Leray, {\it Sur le probl\`eme de Dirichlet quasilin\'eaire d'ordre $2$}.  C. R. Acad. Sci. Paris S\'er. A-B {\bf 274}  (1972), A81-A85.

\bibitem{cp} M. M. Coclite and G. Palmieri, {\it On a
singular nonlinear Dirichlet problem}. Comm. Partial
Differential Equations {\bf 14} (1989), 1315-1327.

\bibitem{crt} M. G. Crandall, P. H. Rabinowitz and L. Tartar, {\it On a Dirichlet problem
with a singular nonlinearity}. Comm. Partial Differential
Equations {\bf 2} (1977), 193-222.

\bibitem{cui} S. Cui, {\it Existence and nonexistence of positive solutions for singular semilinear elliptic boundary value problems}. Nonlinear Anal. {\bf 41}  (2000),  149-176.

\bibitem{diaz} J. I. D\'iaz,  {\it Nonlinear Partial Differential Equations and Free
Boundaries. Vol. I. Elliptic Equations}. Research Notes in
Mathematics, vol.~106, Pitman (Advanced Publishing Program),
Boston, MA, 1985.

\bibitem{dmo} J. I. D\'iaz, J. M. Morel and L. Oswald,
{\it An elliptic equation with singular nonlinearity}. Comm. Partial Differential
Equations {\bf 12} (1987), 1333-1344.

\bibitem{dgr} L. Dupaigne, M. Ghergu and V. R\u adulescu, {\it Lane-Emden-Fowler equations with convection and singular potential}. J.~Math. Pures Appl. {\bf 87} (2007), 563-581.

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

\bibitem{gr} M. Ghergu and V. R\u adulescu, {\it Sublinear
singular elliptic problems with two parameters}. J.~Differential Equations {\bf 195} (2003), 520-536.

\bibitem{grroyal} M. Ghergu and V. R\u adulescu, {\it Multiparameter bifurcation and asymptotics for the
singular Lane-Emden-Fowler equation with a convection term}. Proc. Royal Soc. Edinburgh
Sect. A {\bf 135} (2005), 61-84.

\bibitem{grjmaa} M. Ghergu and V. R\u adulescu, {\it Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term}.  J. Math. Anal. Appl. {\bf 333} (2007), 265-273.

\bibitem{grbook} M. Ghergu and V. R\u adulescu, {\it  Singular Elliptic Problems. Bifurcation and Asymptotic Analysis}. Oxford Lecture Series in Mathematics and Its Applications, vol. 37, Oxford University Press, 2008.

\bibitem{gt} D. Gilbarg and N. S. Trudinger, {\it Elliptic Partial
Differential Equations of Second Order}. 2nd ed., Springer-Verlag,
Berlin Heidelberg New York, 1983.

\bibitem{hai} Y. Haitao, {\it Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic
problem}. J.~Differential Equations {\bf 189} (2003), 487-512.

\bibitem{her} J. Hern\'andez, F. J. Mancebo and J. M. Vega, {\it On the linearization
of some singular nonlinear elliptic problems and applications}.
Ann. Inst. H.~Poincar\'e, Anal. Non Lin\'eaire {\bf 19}
(2002), 777-813.

\bibitem{kazdan} J. Kazdan and F. W. Warner, {\it Remarks on some
quasilinear elliptic equations}. Comm. Pure Appl. Math. {\bf
28} (1975), 567-597.

\bibitem{mea} A. Meadows, {\it Stable and singular solutions of the equation $\Delta u=1/u$}.  Indiana Univ. Math.~J.
{\bf 53} (2004),  1681-1703.

\bibitem{poincare} H. Poincar\'e, (1898). {\it Les fonctions fuchsiennes et l'\'equation
$\Delta u=e^u$}.
J.~Math. Pures Appl. {\bf 4} (1898), 137-230.

\bibitem{radbirkh} V. R\u adulescu, {\it Bifurcation and asymptotics for elliptic problems
with singular nonlinearity}. In: C.~Bandle, H.~Berestycki, B.~Brighi, A.~Brillard, M.~Chipot,
J.-M.~Coron, C.~Sbordone, I.~Shafrir, V.~Valente, G.~Vergara
Caffarelli (Eds.), {\it Studies in Nonlinear Partial
Differential Equations: In Honor of Haim Brezis, Fifth European
Conference on Elliptic and Parabolic Problems: A special tribute
to the work of Haim Brezis, Gaeta, Italy, May 30--June 3, 2004}, pp. 349-362. Birkh\"auser Verlag, Basel, 2005.

\bibitem{hbook} V. R\u adulescu, {\it Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities}. In: M.~Chipot (Ed.),   {\it Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 4}, pp. 483-591, Elsevier, North-Holland, Amsterdam, 2007.

\bibitem{sy1} J. Shi and M. Yao, {\it On a singular
nonlinear semilinear elliptic problem}. Proc. Royal Soc.
Edinburgh Sect. A {\bf 128} (1998), 1389-1401.

\bibitem{stu} C. A. Stuart, {\it Existence and approximation of solutions of nonlinear elliptic
equations}. Math.~Z. {\bf 147} (1976), 53-63.

 }


\end{thebibliography}












\end{document}
