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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 93, pp. 1--8.\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-2006/93\hfil Asymptotic behaviour of the solution]
{Asymptotic behaviour of the solution for the singular
Lane-Emden-Fowler equation with nonlinear convection terms}

\author[Z. Zhang\hfil EJDE-2006/93\hfilneg]
{Zhijun Zhang}  

\address{Zhijun Zhang \newline
Department of mathematics and informational science,
Yantai university, Yantai 264005,  China}
\email{zhangzj@ytu.edu.cn}

\date{}
\thanks{Submitted December 23, 2005. Published August 18, 2006.}
\thanks{Supported by Grant 10071066 from the National Natural Science Foundation of China}
\subjclass[2000]{35J65, 35B05, 35O75, 35R05}
\keywords{ Semilinear elliptic equations;  Dirichlet problem;  singularity;
\hfill\break\indent  nonlinear convection terms;
 Karamata regular variation theory; unique solution; \hfill\break\indent
 exact asymptotic behaviour}

\begin{abstract}
 We show the exact asymptotic behaviour near the
 boundary for the classical solution to the Dirichler
 problem
 $$
 -\Delta u =k(x)g(u)+\lambda |\nabla u|^q,  \quad u>0,\;
 x\in \Omega,\quad u\big|_{\partial{\Omega}}=0,
 $$
 where $\Omega$ is a bounded domain with smooth boundary in $\mathbb R^N$.
 We use the Karamata regular varying theory, a perturbed argument,
 and constructing comparison functions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction and statement of the main results}

 Let $\Omega$ be a bounded domain with smooth boundary in
$\mathbb R^N$ ($N\geq 1$). Consider the singular Dirichlet
problem for the Lane-Emden-Fowler equation
\begin{equation}
-\Delta u=k(x)g(u)+\lambda |\nabla u|^q, \quad  u>0, \;  x \in \Omega,
\quad  u|_{\partial \Omega}=0,
\label{e1.1}
\end{equation}
where  $\lambda\in\mathbb R$, $q\in [0,2]$,  and the functions
$g$, $k$ satisfy the hypotheses:
\begin{itemize}
\item[(H1)]   $g\in C^1((0,\infty),(0,\infty))$, $g'(s)\leq 0$ for all
$s>0$, $\lim_{s \to 0^+}g(s)=\infty$

\item[(H2)]  $k \in C^{\alpha}(\bar{\Omega})$  for  some $\alpha \in (0,1)$, is
non-negative and non-trivial on $\Omega$.
\end{itemize}

The problem above arises in the study of non-Newtonian fluids,
 boundary layer phenomena for viscous fluids,  chemical heterogeneous catalysts,
 as well as in the theory of heat conduction in electrical materials
 \cite{c3,f1,g2,n1,s2}.

  The main feature of this paper is the presence of the three
  terms: the singularity term $g(u)$  which  is regular varying at zero of index
  $-\gamma$  with $\gamma \in (0, 1)$, the weight   $k(x)$
  which may be vanishing at
the boundary, the both of them include a large class of
 functions, and the nonlinear convection terms $\lambda |\nabla u|^q$.

This problem was discussed in a number of works; see, for instance,
\cite{b1,c1,c2,c3,c4,c5,f1,g1,g2,g3,g5,l1,l2,m1,s2,s3,u1,y1,z1,z2,z3,z4,z5}.
When  $\lambda =0$, i.e., problem \eqref{e1.1} becomes
\begin{equation}
-\Delta u=k(x)g(u),\quad  u>0, \;  x\in\Omega,\quad
 u|_{\partial\Omega}=0.
 \label{e1.2}
\end{equation}
  For $k\equiv 1$ on $\Omega$. Fulks and Maybee \cite{f1}, Stuart \cite{s2},
Crandall, Rabinowitz and Tartar \cite{c3} showed that \eqref{e1.2}
  has a unique solution $u\in C^{2+\alpha}(\Omega) \cap C(\bar{\Omega})$.
   Moreover, Crandall, Rabinowitz and   Tartar \cite[Theorems 2.2 and 2.5]{c3}
showed that if $p\in C[0,a]\cap C^2(0,a]$ is the local solution
  to the problem
$$
  -p''(s)=g(p(s)),\quad  p(s)>0,\;  0<s<a,\quad  p(0)=0,%\label{e1.3}
$$
then there exist positive constants  $C_1$ and $C_2$  such that
\begin{itemize}
\item[(i)]
 $C_1 p(d(x))\leq u(x)\leq C_2p(d(x))$  near $\partial\Omega$, where
    $d(x)= \emph{\emph{dist}}(x,\partial\Omega)$

\item[(ii)]
  $|\nabla u(x)|\leq C_2[d(x)g(C_1p(d(x)))+p(d(x))/{d(x)}]$ near
  $\partial\Omega$.
\end{itemize}
In particular,  $u$ is Lipschitz continuous on $\bar{\Omega}$ if
and only if $\int _0^1 g(s)ds<\infty$. Recently,   Ghergu and
R\v{a}dulescu \cite{g1} showed that if $g$ satisfies (H1) and
\begin{itemize}
\item[(H3)] There exist positive constants $C_0$, $\eta_0$ and
$\gamma\in(0,1)$ such that $g(s)\leq C_0s^{-\gamma}$, for all
$s\in (0,\eta_0)$

\item[(H4)] There exist $\theta>0$ and $t_0\geq 1$ such that
$g(\xi t)\geq\xi^{-\theta}g(t)$ for all $\xi \in (0,1)$ and $0<t\leq
t_0\xi$

\item[(H5)] The mapping $\xi \in (0,\infty)\to T(\xi) = \lim_{t\to
0^+}\frac{g(\xi t)}{\xi g(t)}$ is a continuous function; and $k$
satisfies (H2) and the following assumptions: there exist
$\delta_0>0$ and a positive non-decreasing function $h\in
C(0,\delta_0)$ such that

\item[(H6)]  $\lim_{d(x)\to 0}\frac{k(x)}{h(d(x))} =c_0$

\item[(H7)]  $\lim_{t\to 0^+}h(t)g(t)=+\infty$.
\end{itemize}
Then  \eqref{e1.2} has a unique solution $u\in
C^{1,1-\alpha}(\bar{\Omega})\cap C^2(\Omega)$ satisfying
\begin{equation}
\lim_{d(x)\to 0}\frac{u(x)}{p(d(x))}=\xi_0, \label{e1.4}
\end{equation}
where $T(\xi_0)=c_0^{-1}$, and $p\in C^1 [0,a]\cap C^2(0,a](a\in
(0,\delta_0))$ is the local solution to the problem
\begin{equation}
-p''(s)=h(s)g(p(s)), \quad  p(s)>0, \;  0<s<a, \quad   p(0)=0. \label{e1.5}
\end{equation}

The exact asymptotic behaviour of the unique solution  to
 \eqref{e1.2} with  $\int_0^1 g(s)ds=\infty$ has been studied in
\cite{z3}.

For $\lambda \neq 0$,  existence and  uniqueness of solutions to
problem \eqref{e1.1}, see \cite{g2,g3,z2,z4},  and the exact asymptotic
behaviour of the unique solution to  \eqref{e1.1} with
$\int_0^1 g(s)ds=\infty$, see \cite{g3,z4,z5}.

In this paper,  we generalize the Ghergu and R\v{a}dulescu's results
\cite{g1}   to  problem \eqref{e1.1}, and  we showed that the
asymptotic behaviour \eqref{e1.4} of the unique solution $u_\lambda$ to
problem \eqref{e1.1} is independent on $\lambda|\nabla u_\lambda|^q$.

First we recall a basic definition (see \cite{m2,r1,s1}).

\begin{definition} \label{def1.1} \rm
 A positive measurable function $g$ defined on
some neighborhood $(0,a)$  for some $a>0$, is called regular
varying at zero with index $\beta$, written $g \in RVZ_\beta$ if
for each $\xi>0$ and some $\beta \in \mathbb{R}$,
$$
\lim_{t \to 0^+} \frac{g(\xi t)}{g(t)}= \xi^{\beta}. %\label{e1.6}
$$
\end{definition}

Our main result is summarized in the following theorem.


\begin{theorem} \label{thm1.1}
 Let  $g$  satisfy (H1)
and $g \in RVZ_{-\gamma}$ with $\gamma \in (0,1)$ and  $k$ satisfy
(H2), (H6) and $h \in RVZ_\beta$ with $\beta \in [0,1)$. If
$\beta<\gamma$, then  the unique solution $u_\lambda \in
C^{1,1-\alpha}(\bar{\Omega})\cap C^2(\Omega)$ to problem \eqref{e1.1}
satisfies
$$
\lim_{d(x)\to 0}\frac{u_\lambda (x)}{p(d(x))}=\xi_0, %\label{e1.7}
$$
where $\xi_0=c_0 ^{1/(1+\gamma)}$, and $p\in C^1[0,a]\cap
C^2(0,a]$ is the  local solution to problem \eqref{e1.5}.
\end{theorem}

\begin{remark} \label{rmk1.1}  \rm
 By (H1) and the proof of
the maximum principle \cite[Theorems 10.1 and 10.2]{g4} we
see that \eqref{e1.1} has at most one solution in $ C^{2}(\Omega)
\cap C(\bar{\Omega})$ for each fixed $\lambda$.
\end{remark}

\begin{remark} \label{rmk1.2}  \rm
In section 2, we will see that
$g \in RVZ_{-\gamma}$ with $\gamma >0 $ implies
$\lim_{s \to 0^+}g(s)=\infty$ and $h \in RVZ_\beta$ with
$\beta >0$ implies $\lim_{t\to 0^+}h(t)=0$.
\end{remark}

\begin{remark} \label{rmk1.3} \rm
 For the existence of solutions
to  \eqref{e1.5} with $a\in (0, 1)$, see  \cite[Corollary 2.1]{a1}.
\end{remark}

The outline of this article is as follows. In section 2, we
 recall some basic definitions and the properties to
Karamata regular varying theory. In section 3, we prove the
asymptotic behaviour of the unique solution   in
Theorem \ref{thm1.1}.


\section{Karamata regular varying theory}

Let us recall some basic definitions and the properties
to Karamata regular varying theory, which is a basic tool in
probability theory (see \cite{m2,r1,s1}).

\begin{definition} \label{def2.1} \rm
A positive measurable function $f$ defined on $[a,\infty)$, for
some $a>0$, is called regular varying at infinity with index $\rho$,
written $f \in RV_\rho$, if for each $\xi>0$ and some $\rho \in \mathbb{R}$,
\begin{equation}
\lim_{t \to \infty} \frac{f(\xi t)}{f(t)}= \xi^\rho. \label{e2.1}
\end{equation}
The real number $\rho$ is called the index of regular variation.
\end{definition}

\begin{definition} \label{def2.2} \rm
When $\rho=0$, a positive measurable function $L$
defined on $[a,\infty)$, for some $a>0$, is called slowly varying
at infinity, if for each $\xi>0$
\begin{equation}
\lim_{t \to \infty }\frac{L(\xi t)}{L(t)}=1.  \label{e2.2}
\end{equation}
\end{definition}
It follows by the definition that if  $f \in RV_\rho$ it can be
represented in the form
$$
f(t)=t^\rho L(t).        % \label{e2.3}
$$

Some basic  examples of slowly varying functions are:
\begin{itemize}
\item[(i)] $\lim_{t\to \infty}L(t)=c\in (0, \infty);$

\item[(ii)] $ L(t)=\prod _{m=1}^{m=n}(\log_{m}(t))^{\alpha_m}$,
$\alpha_m\in \mathbb R;$

\item[(iii)] $ L(t)=e^{(\prod _{m=1}^{m=n}(\log_{m}(t))^{\alpha_m})}$,
$0<\alpha_m<1;$

\item[(iv)] $\displaystyle L(t)=\frac {1}{t}\int _a ^{t}\frac {ds}{\ln
s};$

\item[(v)] $L(t)=\displaystyle e^{((\ln t)^{1/3}\cos((\ln t)^{1/3} ))}$,
where $\lim_{t\to\infty}\inf L(t)=0,$
$\lim_{t\to\infty}\sup L(t)=+ \infty$.
\end{itemize}

\begin{lemma}[Uniform convergence theorem] \label{lem2.1}
If $f \in RV_\rho$, then  \eqref{e2.1} (and so \eqref{e2.2}) holds
uniformly for $\xi \in [a,b]$ with $0<a<b$.
\end{lemma}


\begin{lemma}[Representation theorem] \label{lem2.2}
  A function $L$ is slowly varying at infinity if and only if it may
be written in the form
$$
L(t)=c(t) \exp  \big( \int_a^t \frac {y(s)}{s} ds \big), \quad
 t \geq a, %\label{e2.4}
$$
for some $a>0$, where $c(t)$ and $y(t)$ are measurable and for
$t \to \infty$, $y(t) \to 0$ and $c(t) \to c$, with $c>0$.
\end{lemma}

\begin{lemma} \label{lem2.3}
If  functions $L,  L_1$ are slowly varying at infinity, then
\begin{itemize}
\item[(i)] $L^\alpha$ for every $\alpha\in \mathbb R$, $L(t)+L_1(t)$,
$L(L_1(t))$ (if $L_1(t)\to \infty$ as
$t\to\infty$), are also slowly varying at infinity;

\item[(ii)] for every $\theta >0$ and $t\to \infty$,
$$
t^{\theta} L(t)\to \infty, \quad t^{-\theta} L(t)\to 0; %\label{e2.5}
$$

\item[(iii)] for $t\to \infty$, $\ln (L(t))/{\ln t}\to 0$.
\end{itemize}
\end{lemma}

\begin{definition} \label{def2.3} \rm
A positive measurable function $H$ defined on some neighborhood $(0,a)$
for some $a>0$, is called slowly varying at zero, if for each $\xi>0$
$$
\lim_{t \to 0^+}\frac{H(\xi t)}{H(t)}=1. %\label{e2.6}
$$
\end{definition}
It follows by  Definitions \ref{def1.1} and \ref{def2.3} that if
$g \in RVZ_{-\gamma}$  it can be represented in the form
$g(t)=t^{-\gamma} H(t)$.        % \label{e2.7}


\begin{lemma} \label{lem2.4}
Definition \ref{def1.1} is equivalent to saying
that $f^*(t)=g(1/t)$  is regular varying at infinity of index
$-\beta$.
\end{lemma}

 Thus we transfer  our attention from infinity to the  origin.

\begin{corollary}[Representation theorem] \label{coro2.1}
 A function $H$ is slowly varying at zero  if and only if it may be
written in the form
$$
H(t)=c(t)\exp  \big( \int_t^a \frac {y(s)}{s} ds \big), \quad
0<t<a, %\label{e2.8}
$$
for some $a>0$, where $c(t)$ and $y(t)$ are measurable and for
$t \to 0^+$, $y(t) \to 0$ and $c(t) \to c$,
with $c>0$.
\end{corollary}

\begin{corollary} \label{coro2.2}
If a function $H$ is slowly varying at zero, then for every $\theta >0$
and $t\to 0^+$,
$t^{-\theta}H(t)\to \infty$,  $t^{\theta} H(t)\to 0$. %\label{e2.9}
\end{corollary}


\begin{corollary} \label{coro2.3} If $g$ satisfies (H1),
$g \in RVZ_{-\gamma}$ with $\gamma \in (0,1)$, and
 $k$ satisfies (H2), (H6),  $h \in RVZ_{\beta}$ with $\beta\in (0, 1)$,
 then
$$
 g(t)=t^{-\gamma} c_1(t)\exp \big( \int_t^a
\frac{y_1(s)}{s}ds\big), \quad
h(t)=t^{\beta} c_2(t)\exp \big(\int_t^a \frac{y_2(s)}{s}ds\big),
% \label{e2.10}
$$
where $y_1,y_2, c_1, c_2 \in C[0,a]$, $y_1(0)=y_2(0)=0$,
$ c_1(0)>0$, $c_2(0)>0 $.
\end{corollary}

\section{Asymptotic behaviour}

First we give some preliminary considerations.

\begin{lemma} \label{lem3.1}
If  $g$  satisfies (H1) and $g \in RVZ_{-\gamma}$ with $\gamma \in (0,1)$,
then
 $$
\int_0^1 g(t)dt<\infty. % \label{e3.1}
$$
\end{lemma}

\begin{proof}
We see by corollaries \ref{coro2.2} and \ref{coro2.3} that there exists
$\gamma_1\in (\gamma,1)$ such that
$$
\lim_{t\to 0^+}t^{\gamma_1}g(t)=\lim_{t\to
0^+}t^{\gamma_1-\gamma} c_1(t)\exp \big( \int_t^a
\frac{y_1(s)}{s}ds\big)=0.
$$
It follow that there exists $\delta\in (0,1)$ such that
$g(t)<t^{-\gamma_1}$, for all $t \in (0,\delta) $ and so  $g\in L^1(0,1)$.
\end{proof}

\begin{lemma} \label{lem3.2}
Under the assumptions in Theorem \ref{thm1.1},
 the local solution $p$ to problem \eqref{e1.5} has the following properties
\begin{itemize}
\item[(i)]  $p\in C^1[0,a]$;

\item[(ii)] $ \lim_{s\to0^+}p''(s)=-\infty;$

\item[(iii)] $\lim_{s\to 0^+}\frac {(p'(s))^q}{p''(s)}=0$ for $q\in [0, 2]$.
\end{itemize}
\end{lemma}

\begin{proof}  (i) Since
 $-((p'(s))^2)'=2h(s)g(p(s))p'(s)$ for $s\in (0,a]$,
 and  $p(s)$   is  a positive concave
 on $(0,a]$, $p(0)=0 $,    $p''(s)<0$,
 we see that $p'(s)$ is decreasing
 and $p'(s)>0$ on $(0,a]$, so $p(s)$ is increasing.
Since $h$ is non-decreasing, multiplying  \eqref{e1.5} by  $p'(s)$ and
integrating on $[t,a]$, $0<t<a$,
 we get by Lemma \ref{lem3.1} that
\begin{align*}
(p'(a))^2+2h(t)\int_t^a  g(p(s))p'(s)ds
&=(p'(a))^2+2h(t)\int_{p(t)}^{p(a)} g(y)dy\\
&\leq (p'(a))^2+2\int_t^a  h(s)g(p(s))p'(s)ds\\
&=(p'(t))^2\\
&\leq (p'(a))^2+2h(a)\int_{p(t)}^{p(a)}g(y)dy,\\
&\leq (p'(a))^2+2h(a)\int_0^{p(a)}g(y)dy
<\infty.
\end{align*}
Thus $p'(0)\in (0,\infty)$, i.e., $p\in C^1[0,a]$.

(ii)  Let $b=p'(0)$. Since $p'(s)$ is decreasing on $[0,a]$, it
follows by the Lagrange mean value theorem that there exists
$\tau_s \in(0,s)$ such that
$$
p(s)/s=(p(s)-p(0))/s=p'(\tau_s)<b, \quad \forall s\in (0,a].
$$
Thus $p(s)<bs$ for all $s\in (0,a]$ and so $g(p(s))\geq g(bs)$,
 for all $s\in (0,a]$. Since $\gamma>\beta$, we see by corollaries
\ref{coro2.2}  and \ref{coro2.3} that
$$
\lim_{t\to 0^+} h(t)g(t)=\lim_{t\to 0^+}
t^{-(\gamma-\beta)} c_1(t) c_2(t)\exp \big( \int_t^a
\frac{y_1(s)}{s}ds\big) \exp \big( \int_t^a
\frac{y_2(s)}{s}ds\big)=\infty,
$$
and
$$
\lim_{t\to 0^+}\frac{g(bt)}{g(t)}=b^{-\gamma}.
$$
Thus
\begin{gather*}
-p''(t)=h(t)g(p(t))\geq h(t)g(t)\frac {g(bt)}{g(t)}, \quad  \forall t\in (0,a],
\\
\lim_{s\to0^+}p''(s)=-\infty.
\end{gather*}
(iii) is follows by (i) and (ii). The proof  is complete.
\end{proof}



\begin{proof}[Proof of the asymptotic behaviour in Theorem \ref{thm1.1}]
Set $\xi_0=c_0^{1/(1+\gamma)}$ and for a fix $\varepsilon\in(0,1/4)$ let
$$
\xi_{1\varepsilon}=\big(\frac {c_0}{1-2\varepsilon}\big)^{1/(1+\gamma)},\quad
\xi_{2\varepsilon} =\big(\frac {c_0}{1+2\varepsilon}\big)^{1/(1+\gamma)},
$$
we see that
 $$
\big(\frac {c_0}{2}\big)^{1/(1+\gamma)}
<\xi_{2\varepsilon}<\xi_{1\varepsilon}< (2c_0)^{1/(1+\gamma)}.
$$
For any $\delta > 0$,
we define $\Omega_{\delta} = \{x\in\Omega: d(x)\leq \delta\}$. By
the regularity of $\partial\Omega$ and lemma \ref{lem3.2}, we can choose
$\delta$ sufficiently small such that
\begin{itemize}
\item[(i)]  $d(x)\in C^2(\Omega_\delta)$;

\item[(ii)] $|\frac{p'(s)}{p''(s)}\Delta d(x)+\lambda
\xi_{i\varepsilon}^{q-1}\frac{(p'(s))^q}{p''(s)}|< \varepsilon$,
 for all $  (x,s) \in \Omega_\delta \times (0,\delta)$,   $i=1, 2$ and fixed
  $\lambda;$

\item[(iii)]
$\frac{\xi_{2\varepsilon}h(d(x))g(p(d(x)))}{g(p(d(x))\xi_{2\varepsilon})}
(1+\varepsilon)<k(x)<\frac{\xi_{1\varepsilon}h(d(x))
g(p(d(x)))}{g(p(d(x))\xi_{1\varepsilon})}(1-\varepsilon)$   in
$\Omega_{\delta}$.
\end{itemize}
For any $x\in \Omega_{\delta}$, define
$\bar{u}=\xi_{1\varepsilon}p(d(x))$, and
$\underline{u}=\xi_{2\varepsilon} p(d(x))$. It follows from
$|\nabla d(x)|=1$ that
\begin{align*}
 &\Delta\overline u(x)+k(x)g(\overline u(x))+\lambda |\nabla \overline u(x)|^q\\
 &= k(x)g(\xi_{1\varepsilon}p(d(x)))+\xi_{1\varepsilon}p'(d(x))\Delta d(x)
    +\xi_{1\varepsilon}p''(d(x))+\lambda \xi_{1\varepsilon}^q (p'(d(x)))^q\\
 &=\xi_{1\varepsilon}h(d(x))g(p(d(x)))\Big[\frac{k(x)g(\xi_{1\varepsilon}p(d(x)))}
 {\xi_{1\varepsilon}h(d(x))g(p(d(x)))}
   -1-\frac{p'(d(x))}{p''(d(x))}\Delta d(x) \\
 &\quad  -\lambda \xi_{1\varepsilon}^{q-1}\frac{(p'(d(x))^q}{p''(d(x))}\Big]\\
 &\leq \xi_{1\varepsilon}h(d(x))g(p(d(x)))\Big[(1-2\varepsilon)-1
   -\frac{p'(d(x))}{p''(d(x))}\Delta d(x)-
  \lambda \xi_{1\varepsilon}^{q-1}\frac{(p'(d(x))^q}{p''(d(x))}\Big]
 \leq 0;
\end {align*}
and
\begin{align*}
 &\Delta\underline{u}(x)+k(x)g(\underline {u}(x))
 +\lambda |\nabla \underline {u}(x)|^q\\
 &=k(x)g(\xi_{2\varepsilon}p(d(x)))+\xi_{2\varepsilon}p'(d(x))\Delta d(x)
    + \xi_{2\varepsilon}p''(d(x))+\lambda \xi_{2\varepsilon}^q (p'(d(x)))^q\\
&= \xi_{2\varepsilon}h(d(x))g(p(d(x)))\Big[\frac{k(x)g(\xi_{2\varepsilon}
 p(d(x)))}{\xi_{2\varepsilon}h(d(x))g(p(d(x)))}
   -1-\frac{p'(d(x))}{p''(d(x))}\Delta d(x)\\
&\quad -   \lambda \xi_{2\varepsilon}^{q-1}\frac{(p'(d(x))^q}{p''(d(x))}\Big]\\
 &\geq \xi_{2\varepsilon}h(d(x))g(p(d(x)))\Big[(1+2\varepsilon)-1
   -\frac{p'(d(x))}{p''(d(x))}\Delta d(x)-
   \lambda \xi_{2\varepsilon}^{q-1}\frac{(p'(d(x))^q}{p''(d(x))}\Big]
 \geq 0.
\end {align*}
Let $u_\lambda \in C(\bar{\Omega})\cap C^{2+\alpha}(\Omega)$ be
the unique solution to problem \eqref{e1.1}.  We assert
$$
\xi_{2\varepsilon} p(d(x))=\underline{u}(x)\leq u_\lambda(x)\leq
\bar{u}(x)=\xi_{1\varepsilon} p(d(x)), \quad  \forall x\in
\Omega_\delta .
$$
In fact, denote $\Omega_\delta=\Omega_{\delta
+}\cup\Omega_{\delta -}$, where $\Omega_{ \delta +}=\{x\in
\Omega_\delta: u_\lambda(x)\geq \underline{u}(x)\}$ and
$\Omega_{\delta -}=\{x\in \Omega_\delta: u_\lambda(x)< \underline{u}(x)\}$.
We need to show $\Omega_{\delta -}=\emptyset$. Assume the contrary, we
see that there exists $x_0\in \Omega_{\delta -}$ such that
$$
0<\underline{u}(x_0)-u_\lambda(x_0)=\max_{x\in\bar{\Omega}_{\delta -} }
(\underline{u}(x)-u_\lambda(x)),
$$
and
$$
\nabla \underline{u}(x_0)=\nabla u_\lambda(x_0),\quad \Delta
(\underline{u}(x_0)-u_\lambda(x_0))\leq 0.
$$
On the other hand, we see by (H1) that
$$
-\Delta (u_\lambda-\underline{u})(x_0)= k(x_0)(g(\underline{u}(x_0))
-g(u_\lambda(x_0)))<0,
$$
which is a  contradiction. Hence $\Omega_{\delta -}=\emptyset$, i.e.,
$u_\lambda(x)\geq \underline{u}(x)$ in $\Omega_\delta.$  As the
same way, we can see that $ u_\lambda(x)\leq \bar{u}(x)$,
for all $x\in \Omega_\delta $.
  Let $\varepsilon\to 0$, we see that
$\lim_{d(x)\to0}\frac{u_\lambda(x)}{p(d(x))}=\xi_0$.  As the same
proof as in \cite{g1,g2}, we see that $u_\lambda\in
C^{1,1-\alpha}(\bar{\Omega})\cap C^2(\Omega)$.  The proof is
complete.
\end{proof}

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\end{document}