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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 148, pp. 1--7.\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-2006/148\hfil Dirichlet problem]
{Dirichlet problem for a second order singular
differential equation}

\author[W. Zhou\hfil EJDE-2006/148\hfilneg]
{Wenshu Zhou}

\address{Wenshu Zhou \newline
 Department of Mathematics, Jilin University, Changchun
130012, China} 
\email{wolfzws@163.com}

\thanks{Submitted July 31, 2006. Published December 5, 2006.}
\thanks{Supported by grants 10626056 from Tianyuan Youth Foundation
 and 420010302318 \hfill\break\indent
 from  Young Teachers Foundation of Jilin University}
\subjclass[2000]{34B15}
\keywords{Singular differential equation; positive solution; existence}

\begin{abstract}
 This article concerns the existence of positive solutions to
 the Dirichlet problem  for a  second order singular
 differential equation.  To prove existence, we use the classical
 method of elliptic regularization.
\end{abstract}

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

\section{Introduction}

 We study the existence of positive solutions for the second order singular differential equation
\begin{equation}
  u''+\lambda\frac{u'}{t-1}-\gamma \frac{|u'|^2}{u}+f(t)=0,
 \quad 0<t<1,
\label{e1}
\end{equation}
with the Dirichlet boundary conditions
\begin{equation}
u(1)=u(0)=0, \label{e2}
\end{equation}
where $\lambda \in\mathbb{R}$, $\gamma>0$, $f(t) \in C^1[0,1]$ and
$f(t)>0$ on $[0,1]$.

It is well known that boundary value problems for singular
 ordinary differential equations arise in the field of
gas dynamics, flow mechanics, theory of boundary layer, and so on.
In recent years, singular second order ordinary differential
equations with dependence on the first order derivative have been
studied extensively,  see for example \cite{a1, b4, j1, o1, o2,
s1, t1, w1}  and references therein where some general existence
results were obtained.  We point out that the case considered here
is not in their considerations since it does not satisfy some
sufficient conditions of those papers. Our considerations were
motivated by the model, which arises in the studies of a
degenerate parabolic equation (see for instance \cite{b1, b2,b3}),
considered by Bertsch and Ughi \cite{b3} in which they
studied \eqref{e1} with $\lambda=0$ and $f\equiv1$ and the
boundary boundary conditions: $ u(1)=u'(0)=0$. By ordinary
differential equation theories, they obtained a decreasing
positive solution. However, it is easy to see from the boundary
conditions \eqref{e2} that any positive solution to the Dirichlet
problem for \eqref{e1} must not be decreasing. Recently, in
\cite{z1} the authors studied the Dirichlet problem for \eqref{e1}
with $\lambda=0$, and proved that if $\gamma>0$, then the problem
has a positive solution $u$; moreover, if $\gamma>\frac{1}{2}$,
then $u$ satisfies also $u'(1)=u'(0)=0$. Note that the equation
considered here is more general since it is also singular at $t=1$
for $\lambda\neq0$. Thus the existence result obtained here is not
a simple extension of \cite{b3, z1}.

We say $u \in C^2(0,1) \cap  C[0,1]$ is  a solution to the
Dirichlet problem \eqref{e1}, \eqref{e2} if it is positive in
$(0,1)$ and satisfies \eqref{e1} and \eqref{e2}.

 The main purpose of this paper is to prove the following theorem.

\begin{theorem}\label{thm1}
Let $\lambda>-1, \gamma>\frac{1}{2}(1+\lambda)$,
$f(t) \in C^1[0,1]$ and $f>0$ on $[0,1]$. Then the Dirichlet problem
\eqref{e1}, \eqref{e2} has a solution $u$. Moreover, $u$ satisfies
$u'(1)=0$. If in addition we assume that $\lambda$ is
non-negative, then $u$ satisfies also $u'(0)=0$.
\end{theorem}

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

We will use  the classical method of elliptic regularization to
prove Theorem \ref{thm1}. For this, we consider the following
regularized problem:
\begin{gather*}
u''+\lambda\frac{u'}{t-1-\varepsilon^{1/2}}-\gamma
\frac{|u'|^2\mathop{\rm sgn}{}_\varepsilon(u)}{I_\varepsilon(u)}+f(t)=0,
 \quad 0<t<1, \\
 u(1)=u(0)=\varepsilon,
\end{gather*}
  where $\varepsilon \in (0,1)$,
$I_\varepsilon(s)$ and $\mathop{\rm sgn}{}_\varepsilon(s)$ are defined as
follows:
\[
I_\varepsilon(s)=\begin{cases}
   s, &  s \geqslant \varepsilon,\\
\frac{s^2+\varepsilon^2}{2\varepsilon}, &|s| <\varepsilon,\\
-s, & s\leqslant -\varepsilon,
\end{cases}
\quad  \mathop{\rm sgn}{}_\varepsilon(s)
=\begin{cases}
  1,& s \geqslant \varepsilon,\\
  \frac{2s}{\varepsilon}-\frac{s^2}{\varepsilon^2},&
 0\leqslant s < \varepsilon,\\
 \frac{2s}{\varepsilon}+\frac{s^2}{\varepsilon^2},&
  -\varepsilon \leqslant s <0,\\
 -1,& s <-\varepsilon.
\end{cases}
\]
Clearly, $I_\varepsilon(s), \mathop{\rm sgn}{}_\varepsilon(s) \in
C^1(\mathbb{R}),$ and $ I_\varepsilon(s) \geqslant \varepsilon/2, 1
\geqslant |\mathop{\rm sgn}_\varepsilon(s)|,
 \mathop{\rm sgn}_\varepsilon(s)\mathop{\rm sgn}(s) \geqslant0$
in $\mathbb{R}$.

It follows from  \cite[Theorem 4.1, Chapter 7]{c1} that for any
fixed $\varepsilon \in (0,1)$, the above regularized problem
admits a classical solution
 $u_\varepsilon \in C^2(0,1) \cap C^1[0,1] $.  By
the maximal principle, it is easy to see that
  $u_\varepsilon(t) \geqslant \varepsilon$ on $[0,1]$.
  Thus $u_\varepsilon$ satisfies
  \begin{equation}
\begin{gathered}
 u_\varepsilon''+\lambda\frac{u_\varepsilon'}{t-1-\varepsilon^{1/2}} -\gamma
\frac{|u_\varepsilon'|^2}{u_\varepsilon
}+f(t)=0,  \quad 0<t<1, \\
u_\varepsilon(0)=u_\varepsilon(1)=\varepsilon.
\end{gathered} \label{e3}
\end{equation}
Note that this system is equivalent to
\begin{equation}
\begin{gathered}
\Big[(1+\varepsilon^{1/2}-t)^{\lambda}u_\varepsilon'\Big]'-\gamma
\frac{(1+\varepsilon^{1/2}-t)^{\lambda}|u_\varepsilon'|^2}{u_\varepsilon }\\
+(1+\varepsilon^{1/2}-t)^{\lambda}f(t) = 0,\quad \quad 0<t<1.
\end{gathered}\label{e4}
\end{equation}

\begin{lemma}\label{lem2.1}
Under the assumptions of Theorem \ref{thm1}, we  have
$$
\big|(1+\varepsilon^{1/2}-t)^\lambda u_\varepsilon'(t)\big|
\leqslant
\frac{(1+\varepsilon^{1/2})^{1+\lambda}\max_{[0,1]}f}{1+\lambda}.
$$
In particular, for any $\delta \in (0,1)$ there
 exists  a positive constant $C_\delta$ independent of $\varepsilon$ such
 that
$|u_\varepsilon'(t)| \leqslant C_\delta$, for $0 \leqslant t \leqslant
\delta$.
\end{lemma}

\begin{proof}
By $u_\varepsilon(1)=u_\varepsilon(0)=\varepsilon $ and
$u_\varepsilon(t) \geqslant\varepsilon$ for all $t \in [0, 1],$ we
have
\begin{equation}
\begin{gathered}
u_\varepsilon'(0)=\lim\limits_{t \to
0^+}{\frac{u_\varepsilon(t)-\varepsilon}{t}}\geqslant0,\\
u_\varepsilon'(1)=\lim\limits_{t \to
1^-}{\frac{u_\varepsilon(t)-\varepsilon}{t-1}}\leqslant0.
\end{gathered}\label{e5}
\end{equation}
 From \eqref{e4} we obtain
 \[
\big[(1+\varepsilon^{1/2}-t)^{\lambda}u_\varepsilon'\big]'
+ A (1+\varepsilon^{1/2}-t)^{\lambda}
\geqslant 0,\quad  0 <t<1,
\]
where $A=\max_{[0,1]}f$, i.e.
\[
\Big[(1+\varepsilon^{1/2}-t)^{\lambda}u_\varepsilon'
-\frac{A(1+\varepsilon^{1/2}-t)^{1+\lambda}}{1+\lambda} \Big]'
\geqslant 0,\quad  0 <t<1.
\]
  Therefore, the function
$(1+\varepsilon^{1/2}-t)^{\lambda}u_\varepsilon'
-\frac{A(1+\varepsilon^{1/2}-t)^{1+\lambda}}{1+\lambda} $ is
non-decreasing on $[0,1]$, and then, noticing $\lambda>-1$ and
using \eqref{e5}, we have
\begin{align*}
0 \geqslant& \varepsilon^{\lambda/2}u_\varepsilon'(1)
-\frac{A\varepsilon^{(1+\lambda)/2}}{1+\lambda}  \\
\geqslant& (1+\varepsilon^{1/2}-t)^{\lambda}u_\varepsilon'(t)
 -\frac{ A(1+\varepsilon^{1/2}-t)^{1+\lambda}}{1+\lambda}\\
\geqslant&(1+\varepsilon^{1/2})^{\lambda}u_\varepsilon'(0)
- \frac{A(1+\varepsilon^{1/2})^{1+\lambda}}{1+\lambda}\\
\geqslant & - \frac{A(1+\varepsilon^{1/2})^{1+\lambda}}{1+\lambda},\quad
t \in [0,1],
\end{align*}
and hence
\[
\frac{A(1+\varepsilon^{1/2}-t)^{1+\lambda}}{1+\lambda} \geqslant
 (1+\varepsilon^{1/2}-t)^{\lambda}u_\varepsilon'(t)
 \geqslant -\frac{A(1+\varepsilon^{1/2})^{1+\lambda}}{1+\lambda},
\quad t \in [0,1].
\]
 This completes the proof of Lemma \ref{lem2.1}.
\end{proof}

 Obviously,  we have
\begin{gather}
-u_\varepsilon''-\lambda\frac{u_\varepsilon'}{t-1-\varepsilon^{1/2}}+
\gamma \frac{|u_\varepsilon'|^2}{u_\varepsilon}-\min_{[0,1]}f
\geqslant 0,\quad  t \in (0,1),\label{e6}\\
-u_\varepsilon''-\lambda\frac{u_\varepsilon'}{t-1-\varepsilon^{1/2}}+
\gamma \frac{|u_\varepsilon'|^2}{u_\varepsilon} -
\max_{[0,1]}f\leqslant 0,\quad  t \in (0,1).\label{e7}
\end{gather}

   To obtain  uniform  bounds of $u_\varepsilon$,
the following comparison theorem will be proved to be very useful.

 \begin{proposition}\label{prop1}
 Let $u_i \in C^2(0,1)\cap C[0,1]$ and
$u_i>0$ on $[0,1] (i=1,2)$. If $u_2 \geqslant u_1$ for $t=0,1$, and
\begin{gather}
-u_2''-\eta\frac{u_2'}{t-1-\rho}+\varrho\frac{|u_2'|^2}{u_2}-\theta
\geqslant 0,\quad  t \in (0,1),\label{e8}\\
-u_1''-\eta\frac{u_1'}{t-1-\rho}+\varrho\frac{|u_1'|^2}{u_1}-\theta
\leqslant 0,\quad  t \in (0,1),\label{e9}
\end{gather}
where $\rho, \varrho, \theta>0, \eta \in \mathbb{R}$, then
$ u_2(t)\geqslant u_1(t)$, $t \in [0,1]$.
\end{proposition}

\begin{proof} From \eqref{e8} and \eqref{e9}, we have
\begin{gather*}
 \Big(\frac{u_2^{1-\varrho}}{1-\varrho}\Big)''
 +\frac{\eta}{t-1-\rho}\Big(\frac{u_2^{1-\varrho}}{1-\varrho}\Big)'
 \leqslant-\frac{\theta}{u_2^{\varrho}},
 \quad (\varrho \neq 1)\\
 \Big({\rm ln} (u_2)\Big)''
 +\frac{\eta}{t-1-\rho}\Big({\rm ln} (u_2)\Big)'
  \leqslant-\frac{\theta}{u_2},
 \quad (\varrho =1)
\end{gather*}
and
\begin{gather*}
 \Big(\frac{u_1^{1-\varrho}}{1-\varrho}\Big)''
 +\frac{\eta}{t-1-\rho}\Big(\frac{u_1^{1-\varrho}}{1-\varrho}\Big)'
 \geqslant-\frac{\theta}{u_1^{\varrho}},
 \quad (\varrho \neq 1)
\\
 \Big({\rm ln} (u_1)\Big)''
 +\frac{\eta}{t-1-\rho}\Big({\rm ln} (u_1)\Big)'
  \geqslant-\frac{\theta}{u_1}.
 \quad (\varrho =1)
\end{gather*}
 Combining the above inequalities, we obtain
\begin{equation}
w''+\frac{\eta}{t-1-\rho} w'
 \leqslant \theta
\Big(\frac{1}{u_1^{\varrho}}-\frac{1}{u_2^{\varrho}}\Big),\quad 0<t<1,
\label{e10}
\end{equation}
where $w: [0,1]\to \mathbb{R}$ is defined by
\[
w=\begin{cases}
\frac{u_2^{1-\varrho}}{1-\varrho}-\frac{u_1^{1-\varrho}}{1-\varrho},&
(\varrho\neq 1)\\
 \ln (u_2)-\ln (u_1) &  (\varrho=1)
\end{cases}
\]
Clearly,  $w \in C^2(0,1) \cap C[0,1]$.
To prove the proposition, we argue by contradiction and assume that
there exists a point $t_0$ of $(0,1)$ such that
$u_2(t_0)-u_1(t_0)<0$.  From the assumption, it is easy
to see that $w$ reaches a minimum at some point $t_* $ of $(0,1)$
such that
\begin{gather}
w(t_*)=\min\limits_{t \in [0, 1]}w(t)<0, \label{e11}\\
w'(t_*)=0,\quad   w''(t_*) \geqslant0.\label{e12}
\end{gather}
Combining \eqref{e12} with  \eqref{e10} we have
\begin{gather*}
\theta\Big(\frac{1}{u_1^{\varrho}(t_*)}-\frac{1}{u_2^{\varrho}(t_*)}\Big)
\geqslant 0.
\end{gather*}
 This implies $u_2(t_*)\geqslant u_1(t_*)$. However,
from \eqref{e11} we find that  $u_2(t_*)<u_1(t_*)$, a
contradiction. Thus the proof of Proposition \ref{prop1} is
completed.
\end{proof}

\begin{lemma}\label{lem2.3}
Under the assumptions of Theorem \ref{thm1}, for all
$\varepsilon \in (0,1)$ there exist positive constants $C_i$ ($i=1,2$)
independent of $\varepsilon$ such that
\[
C_2(1+\varepsilon^{1/2}-t)^2\geqslant u_\varepsilon(t)
\geqslant C_1 [t(1-t)+\varepsilon^{1/2}]^2,\quad  t \in [0,1].
\]
\end{lemma}

 \begin{proof}
 We shall first show the right-hand side of the above inequalities.
Let $w_\varepsilon=C[t(1-t)+\varepsilon^{1/2}]^2$, where $C \in
(0,1]$ will be determined later.  By Proposition \ref{prop1} and
noticing \eqref{e6}, it suffices to show that
\begin{equation}
-w_\varepsilon''-\lambda\frac{w_\varepsilon'}{t-1-\varepsilon^{1/2}}+
\gamma \frac{|w_\varepsilon'|^2}{w_\varepsilon}-\min_{[0,1]}f
\leqslant 0,\quad  t \in (0,1), \label{e13}
\end{equation}
 for some sufficiently small positive constant $C$ independent of
$\varepsilon$.
Simple calculations show  that
\begin{gather*}
w_\varepsilon'=2C[t(1-t)+\varepsilon^{1/2}](1-2t),\\
 w_\varepsilon''=2C(1-2t)^2-4C[t(1-t)+\varepsilon^{1/2}],
 \end{gather*}
and
\begin{align*}
-&w_\varepsilon''-\lambda\frac{w_\varepsilon'}{t-1-\varepsilon^{1/2}}+
\gamma
\frac{|w_\varepsilon'|^2}{w_\varepsilon}-\min_{[0,1]}f\\
&=-2C(1-2t)^2+4C[t(1-t)+\varepsilon^{1/2}]\\
&\quad -2C\lambda\frac{[t(1-t)+\varepsilon^{1/2}](1-2t)}{t-1-\varepsilon^{1/2}}
+ 4C\gamma (1-2t)^2- \min_{[0,1]}f \\
&\leqslant 4C( 2+|\lambda|+ \gamma)- \min_{[0,1]}f ,\quad 0<t<1.
\end{align*}
Choosing a positive constant $C$ such that
\[
C \leqslant \min\Big\{1,\frac{\min_{[0,1]}f}{4(2
+|\lambda|+\gamma)}\Big\},
\]
 we find that \eqref{e13} holds.

Next we show the left-hand side of the inequalities. Let
$v_{\varepsilon}=C(1+\varepsilon^{1/2}-t)^2$, where $C \geqslant
1$ will be determined later. A calculation shows that
\[
-v_{\varepsilon}''-\lambda\frac{v_{\varepsilon}'}{t-1-\varepsilon^{1/2}}+ \gamma
\frac{|v_{\varepsilon}'|^2}{v_{\varepsilon}}-\max_{[0,1]}f
=2C(2\gamma -1-\lambda)-\max_{[0,1]}f,\quad 0<t<1.
\]
Choosing a positive constant $C$ such that
\[
C \geqslant \max\Big\{1, \frac{\max_{[0,1]}f
}{2(2\gamma -1-\lambda)}\Big\}
\]
and noticing $\gamma>\frac{1}{2}(1+\lambda)$, we find that
\[
-v_{\varepsilon}''-\lambda\frac{v_{\varepsilon}'}{t-1-\varepsilon^{1/2}}+ \gamma
\frac{|v_{\varepsilon}'|^2}{v_{\varepsilon}}-\max_{[0,1]}f
\geqslant 0,\quad 0<t<1,
\]
and then, by Proposition \ref{prop1} and noticing \eqref{e7}, we
obtain $u_{\varepsilon} \leqslant v_{\varepsilon}$
 on $[0,1]$.  This completes the proof of Lemma \ref{lem2.3}.
\end{proof}

 From \eqref{e6}, \eqref{e7}, Lemma \ref{lem2.1} and Lemma
\ref{lem2.3}, we derive that for any $ \delta\in (0,\frac12)$
there exists a positive constant $C_\delta$ independent of
$\varepsilon$ such that
\begin{equation}
|u_\varepsilon''(t)| \leqslant C_\delta,\quad  \delta \leqslant t
\leqslant 1- \delta. \label{e14}
\end{equation}
Differentiating \eqref{e3} with respect to $t$ we get
\[
u_\varepsilon'''
=\lambda\frac{u_\varepsilon'-(t-1-\varepsilon^{1/2})
u_\varepsilon''}{(t-1-\varepsilon^{1/2})^2}+
\gamma\frac{2u_\varepsilon
u_\varepsilon'u_\varepsilon''-(u_\varepsilon')^3}{u_\varepsilon^2}
-f'(t),\quad 0<t<1.
\]
By \eqref{e14}, Lemma \ref{lem2.1} and Lemma \ref{lem2.3}, we
derive that for any $\delta \in (0, \frac12)$, there exists a
positive constant $C_\delta$ independent of $\varepsilon$ such
that
\[
|u_{\varepsilon}'''(t)| \leqslant C_\delta,\quad \delta\leqslant t
\leqslant 1- \delta.
\]
 From this and Lemma \ref{lem2.1} and using Arzel\'a-Ascoli
theorem and diagonal sequential process, we see that there exist a
subsequence $\{u_{\varepsilon_j}\}$ of $\{u_\varepsilon \}$ and a
function $u \in C^2(0,1)\cap C[0,1)$ such that,  as
$\varepsilon_j \to 0$,
\begin{gather*}
u_{\varepsilon_j} \to u,\quad \text{uniformly in } C[0,1-\delta],\\
 u_{\varepsilon_j} \to u,\quad \text{uniformly in } C^2[\delta, 1-\delta].
\end{gather*}
Combining this with \eqref{e3}  and
$u_{\varepsilon_j}(0)=\varepsilon_j$,
 we find that $u$ satisfies (1) and $u(0)=0$. By Lemma \ref{lem2.3}, we have
\begin{equation}
C_2(1-t)^2\geqslant u(t) \geqslant C_1[t(1-t)]^2,\quad t\in[0,1),
\label{e15}
\end{equation}
 therefore $u>0$ in $(0, 1)$, and
 $\lim_{t\to 1^-}u(t)=0$.
 Define $u(1)=0$. Thus $u$ is a solution to the Dirichlet
problem \eqref{e1}, \eqref{e2},
  and it follows from \eqref{e15} that $u'(1)=0$.

It remains to show that for  $\lambda \geqslant 0$,   $u $
satisfies $u'(0)=0$. Let
$h_{\varepsilon_j}=C(t+\varepsilon_j^{1/2})^2$, where $C \geqslant
\max\Big\{1, \frac{\max_{[0,1]}f }{2(2\gamma -1)}\Big\}$. Noticing
$\lambda \geqslant 0$ and $\gamma>\frac{1}{2}(1+\lambda)$, we have
\begin{align*}
&-h_{\varepsilon_j}''-\lambda\frac{h_{\varepsilon_j}'}{t-1-\varepsilon_j^{1/2}}+
\gamma
\frac{|h_{\varepsilon_j}'|^2}{h_{\varepsilon_j}}-\max_{[0,1]}f \\
&=2C(2\gamma -1)-2C\lambda\frac{t+\varepsilon_j^{1/2}}{t-1-\varepsilon_j^{1/2}}-\max_{[0,1]}f\\
&\geqslant 2C(2\gamma -1)-\max_{[0,1]}f
\geqslant 0,\quad 0<t<1.
\end{align*}
By Proposition \ref{prop1} and noticing \eqref{e7}, we obtain
$u_{\varepsilon_j} \leqslant h_{\varepsilon_j}$ on $[0,1]$.
Letting $\varepsilon_j \to 0$, we have
$$
u(t) \leqslant Ct^2, \quad t \in [0,1].
$$
Combining this with the right-hand side of \eqref{e15}, we obtain
$u'(0)=0$. This completes the proof of Theorem \ref{thm1}.

 \subsection*{Acknowledgement} The author would like to
thank the anonymous referee for his/her important comments.

\begin{thebibliography}{00}

\bibitem {a1} R. P. Agarwal, D. O'Regan; \emph{Singular boundary value
problems for superlinear second order ordinary and delay
differential euqations}, J. Differential Equations, 130(1996),
333-355.

\bibitem {b1} G. I. Barenblatt,  M. Bertsch,  A. E. Chertock,
V. M. Prostokishin;
\emph{Self-similar intermediate asymptotic for a degenerate parabolic
filtration-absorption equation}. Proc. Nat. Acad. Sci. (USA),
18(97)(2000), 9844-9848.

\bibitem {b2} M. Bertsch, R. D. Passo and M. Ughi;
\emph{Discontinuous viscosity solutions of a degenerate parabolic equation},
 Trans. Amer. Math. Soc., 320(2)(1990), 779-798.

\bibitem {b3}  M. Bertsch,  M. Ughi; \emph{Positivity properties of viscosity
 solutions of a degenerate parabolic equation}, Nonlinear Anal., 14(1990)
 571-592.

\bibitem {b4} D. Bonheure,  J. M. Gomes,  L. Sanchez;
\emph{Positive solutions of a second order singular
ordinary differential euqation}, Nonlinear Anal. TMA. 61(2005),
1383-1399.

\bibitem {c1} Y. Chen,  L. Wu; \emph{Second Order Elliptic Equations and
Elliptic Systems} (B. Hu. Trans.), Science Press, Beijing, 1997.
(Original work  published in 1991, in Chinese)  Translations of
Mathematical Monographs, 174. American Mathematical Society,
Providence, RI, 1998.

\bibitem {j1} D. Jiang; \emph{Upper and lower solutions method and
a singular boundary value problem},
 Z. Angew. Math. Mech. 82(7) (2002), 481-490.

 \bibitem {o1} D. O'Regan; \emph{Theory of Singular Boundary Value Problems},
 World Scientific, Singapore, 1994.

\bibitem {o2} D. O'Regan; \emph{Existence Theory for Nonlinear Differential
Equations}, Kluwer Acad., Dordrecht/Boston/London 1997.

\bibitem {s1} S. Stan\v ek;
\emph{Positive solutions for  singular semipositone boundary value
porblems}, Math. Comput. Model, 33(2001), 341-351.

\bibitem {t1} A. Tineo; \emph{Existence theorems for a singular
two point Drichlet porblem}, Nonlinear Anal. 19(1992), 323-333.

\bibitem {w1} J. Wang, J. Jiang; \emph{The existence of positive solutions
to a singular nonlinear boundary value porblem},
J. Math. Anal. Appl. 176(1993), 322-329.

 \bibitem {z1}  W. Zhou,  X. Wei;
\emph{Positive solutions to BVPs for a singular differential equation},
Nonlinear  Anal. TMA. in press.

\end{thebibliography}

\end{document}