\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 122, pp. 1--14.\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 2003 Texas State University-San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/122\hfil Rate of convergence for solutions]
{Rate of convergence for solutions to Dirichlet problems of quasilinear equations}

\author[Zhiren Jin\hfil EJDE--2003/122\hfilneg]
{Zhiren Jin}

\address{Zhiren Jin \hfill\break
Department of Mathematics and Statistics \\
Wichita State University \\
Wichita, Kansas, 67260-0033, USA}
\email{zhiren@math.wichita.edu}

\date{}
\thanks{Submitted November 2, 2002. Published December 9, 2003.}
\subjclass[2000]{35J25, 35J60, 35J70}
\keywords{Elliptic boundary value problems, asymptotic
 behavior of solutions, \hfill\break\indent
 unbounded domains, barriers }

\begin{abstract}
  We obtain rates of convergence for solutions to Dirichlet problems of
  quasilinear elliptic (possibly degenerate) equations in slab-like
  domains.  The rates found depend on the convergence of the boundary
  data and of the coefficients of the operator. These results are obtained
  by constructing appropriate barrier functions based on the structure
  of the operator and on the convergence of the boundary data.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{example}[theorem]{Example}


\section{Introduction and statement of Main Results}

Let $\Omega$ be a slab-like domain in $\mathbb{R}^n$ ($n\geq 2$)
defined by
$$
\Omega = \{ ({\bf{x}} , y )\in \mathbb{R}^n : |y|<M , \; |{\bf{x}}|> N_{1}\}
$$
where ${\bf{x}}=(x_1,\dots,x_{n-1})$, $N_{1}$ and $M$ are fixed positive constants.
For a continuous function $\phi$ on $\partial \Omega$,
we consider a Dirichlet problem
\begin{equation}
\begin{gathered}
Qf=0 \quad\mbox{in }\Omega , \\
f=\phi \quad\mbox{on }\partial\Omega\,,
\end{gathered}\label{eq:problem1}
\end{equation}
where $Q$ is a second-order quasilinear operator of the form
\begin{equation}
Qf =
\sum_{i,j=1}^{n}\ a_{ij}({\bf{x}}, y,f, Df)D_{ij} f +B({\bf{x}} ,y,f,Df)\,.
\label{eq:operator}
\end{equation}
Here  $(a_{ij}({\bf{x}},y,t,P))$ is a positive
semi-definite matrix in which each entry (and $B$) is a $C^{1}$
function on $ {\bf{R}}^{n}\times \mathbb{R} \times
\mathbb{R}^{n}$.

We shall investigate the asymptotic behavior of bounded solutions of
(\ref{eq:problem1}). That is, if there is a function
$\Phi \in C(S^{n-1}\times [-M,M])$ and
a decreasing function $g_{1}(t)$, such that $g_{1}(t)\to 0$ as $t\to \infty $,
and that
\begin{equation}
|\phi ({\bf{x}},\pm M)-\Phi({\bf x}/|{\bf x}|,\pm M)|\leq g_{1}( |{\bf{x}}| )
 \quad\mbox{for } |{\bf x}|> N_{1} \,.
\label{eq:rateofdata111}
\end{equation}
We want to see how fast $f({\bf{x}},y)$ approaches  a limiting function.
Specifically, we want to find a function $k({\bf{x}}/|{\bf{x}}|, y)$ and a
decreasing function $d(t)$ such that
\begin{equation}
|f({\bf{x}},y)-k({\bf x}/|{\bf x}|,y)|\leq d(|{\bf{x}}|)
\quad \mbox{for } ({\bf{x}},y)\in \Omega .
\label{eq:target1}
\end{equation}
Apparently the function $d(t)$ can not approach zero faster than $g_{1}(t)$.
In general, $d(t)$ will approach zero slower than $g_{1}(t)$ as
illustrated in the following example.


\begin{example}[{\cite[example 3]{JL2}}]  \label{ex1} \rm
Let
\begin{gather*}
\Omega=\{(x_{1},x_{2},x_{3})\in \mathbb{R}^{3}:
x_{1}^2+x_{2}^{2}> 1,\ |x_{3}|<1\}, \\
Qu=(1/3)\Delta u, \\
\phi(x_{1},x_{2},\pm 1)
=\frac{2x_{2}(x_{1}^{2}+x_{1}^{2}+1)^{1/2}}{x_{1}^2+x_{2}^{2}}
 - \frac{x_{2}}{(x_1^2+x_2^2+1)^{3/2}}.
\end{gather*}
Then
$$
f(x_{1},x_{2},x_{3})=\frac{2x_{2}\sqrt{x_1^2+x_2^2+x_3^2}}{x_{1}^2+x_{2}^{2}}
-\frac{x_{2}}{(x_1^2+x_2^2+x_3^2)^{3/2}}.
$$
is a bounded solution to $Qf=0$ in $\Omega $, $f=\phi $ on $\partial \Omega $
(see  \cite[pp. 165-1666]{Hobson}).
When
$$
\Phi(\omega)=2\omega_{2}\quad
\mbox{for} \quad \omega=(\omega_{1},\omega_{2})\in S^{1},
$$
a short calculation shows that
$|\phi({\bf x},\pm 1)-\Phi({\bf x}/|{\bf x}|)|=O(|{\bf x}|^{-4})$
as $|{\bf x}|\to\infty$. From the results in \cite{JL1} or \cite{JL3},
we see that $k({\bf x}/|{\bf x}|,y)$
in (\ref{eq:target1}) must be $\Phi ({\bf x}/|{\bf x}|)$.
However we can calculate that
$$
|f({\bf x},y)-\Phi({\bf x}/|{\bf x}|)|=O(|{\bf x}|^{-2})\quad
\mbox{as } |{\bf x}|\to\infty .
$$
Thus in this case $g_{1}(t)$ behaves like $t^{-4}$ and $d(t)$ behaves like
$t^{-2}$. That is, $d(t)$ approaches zero much slower than $g_{1}(t)$.
\end{example}

Although $d(t)$ can not go to zero faster than $g_{1}(t)$ in general, there are
a lot of cases that $d(t)$ will go to zero at the same rate as $g_{1}(t)$
(the best case we can expect).
When $g_{1}(t)$ has one of the special forms like $t^{-\alpha }$,
$e^{-t^{\alpha }}$, and when the lower order term $B$ is zero,
in \cite{JL2}, it is proved that $d(t)$ can be chosen as a function of the same
form as $g_{1}(t)$. Thus in this case
$d(t)$ and $g_{1}(t)$ go to zero in the same rate.
In \cite{JL4}, when the lower order term $B$ and boundary limit $\Phi $ are smooth
enough, $d(t)$ also approaches zero in the same rate as $g_{1}(t)$ if $g_{1}(t)$ approaches zero
slower than $t^{-1/2}$, or $t^{-1}$, or $t^{-2}$ (depending on the structure of the operator and smoothness of
the data).

In this paper, we want to investigate when $d(t)$ will go to zero in the same rate as $g(t)$ even when
$g(t)$ approaches zero faster than $t^{-2}$ and the lower order term $B$ is not zero. From above example, we see that
it is clear some condition on $\Phi $ is necessary even for the Laplace operator $Q$.
Comparing to the assumptions used in \cite{JL4}, we mainly add a new assumption that
$\Phi (\omega, y)$ and $k(\omega ,y)$ are
independent of $\omega$. We will obtain fast rate of convergence for bounded
solutions of (\ref{eq:problem1}) that improves the results in \cite{JL4}.

The spatial decay estimates for solutions of partial differential equations
have applications in fluid mechanics, extensible films and Saint-Venant's principle
of elasticity theory. For extensive reviews of the research in this area, we refer
the readers to \cite{Horgan1,Horgan2,HK}.  Here we just mention some of the closely
related results.
In \cite{BR}, an exponential decay estimate was obtained when $\Omega $ is a
cylinder, $B$ is a quadratic function of $Df$ and $\phi =0$;  In \cite{HO},
an exponential decay estimate for energy function was considered when
$n=2$, $\phi =0$. In \cite{HP}, an exponential decay estimate for energy function
was obtained for equations modelling the constant mean
curvature equation on a strip (n=2) with $\phi =0$.
In \cite{KP}, Phragm{\'e}n-Lindel{\"o}f type results were obtained
for equation modelling constant mean curvature equation on a semi-infinite strip
with $\phi =0$; and finally in \cite{JL4}, for general boundary data $\phi $,
the rates of convergence for solutions of (\ref{eq:problem1}) were obtained in
terms of the structure of $Q$ and the rate of convergence (\ref{eq:rateofdata111}).
The result in this paper will do better in either
dealing with general boundary data, or general equation,  or obtaining better
 estimates on the rate of convergence.

Now we state the assumptions to be used in this paper.
We assume that the coefficients of $Q$  are normalized so
that
\begin{equation}
\mathop{\rm Trace}(a_{ij})= \sum_{i=1}^{n}a_{ii} =1
\label{eq:traceis1}
\end{equation}
We assume $\phi ({\bf{x}},y)$ has a limit in the following sense.
\begin{itemize}

\item[(C1)] There exists a function $\Phi (y)$ defined on $[-M,M]$ and
a decreasing function $g_{1}(t)$, $g_{1}(t)\to 0$ as $t \to +\infty $, such that
\begin{equation}
|\phi ({\bf{x}},\pm M)-\Phi(\pm M)|\leq g_{1}( |{\bf{x}}| )
 \quad \mbox{for }   |{\bf x}|\geq N_{1} .
\label{eq:conditionofboundary1}
\end{equation}
\end{itemize}
We assume the term $a_{nn}$ satisfies the assumption.
\begin{itemize}
\item[(C2)]
For any fixed positive numbers $a$, $b$, there is a positive number
$\mu (a,b)$  such that
\begin{equation}
a_{nn}({\bf{x}},y, z,{\bf{v}})\geq \mu (a,b)
\label{eq:mu(a,b)}
\end{equation}
for all $({\bf{x}},y)\in \Omega,$  $z\in R$ , ${\bf{v}}\in
\mathbb{R}^{n}$ with $|z|\leq a$, $|{\bf{v}}|\leq b$.
\end{itemize}
We assume that the term $B({\bf{x}},y,z,{\bf{p}},q)$ satisfies:
\begin{itemize}
\item[(C3)]
There is a $C^{1}$ function $E(y, z, q)$
on $[-M,M]\times R^{2}$ and for each fixed bounded set
$D$ in $R^{2}$, there are positive constants $C$, $\alpha_{0}\geq 1$ and a
decreasing function $g_{2}(t)$, $g_{2}(t)\to 0$ as $t\to +\infty$,
satisfying
$$
\big|\frac{B({\bf{x}},y,z,{\bf{p}},q)}{a_{nn}({\bf{x}},y,z,{\bf{p}},q)}
- E(y, z, q)\big|
\leq g_{2}(|{\bf{x}}|) + C |{\bf{p}}|^{\alpha_{0}}
$$
for $({\bf x},y)\in\overline{\Omega}$, $(z,q)\in D$ and
$|{\bf{p}}|\leq 1$.
\end{itemize}
We assume, as in \cite{JL4}, that an ODE involving $E$ is solvable.
\begin{itemize}
\item[(C4)]
There is a function
$k(y)\in C^{1}([-M,M])\cap C^{2}((-M,M))$, such that
\begin{equation}
k''(y)+E(y,k,k')=0 \quad\mbox{on } |y|\le M, \quad
k(\pm M)=\Phi (\pm M).
\label{eq:equationsoforiginalk}
\end{equation}
\item[(C5)] $E(y,z,q)$ is non-increasing on $z$.
\end{itemize}
Then we have the following theorem on the rate of convergence.

\begin{theorem} \label{thm1}
Assume (C1)--(C5) and that
$f\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$
is a bounded solution of (\ref{eq:problem1}).
Then for each integer $J$, there is a number $C_{J}$, such that
\begin{equation}
\big|f({\bf{x}},y)-k(y)\big|\leq
C_{J}g_{1}(\frac{1}{2^{J}}|{\bf{x}}| ) +
C_{J} g_{2}(\frac{1}{2^{J}}|{\bf{x}}|) +
\frac{C_{J}}{|{\bf{x}}|^{J\beta }} \quad\mbox{on } \Omega .
\label{eq:conclusion}
\end{equation}
where $\beta =\min \{ \alpha_{0} , 2\}$.
\end{theorem}

As an application of this result, we give the following example.

\begin{example} \label{ex2} \rm
 Consider the Dirichlet problem for
the prescribed mean curvature equation
\begin{gather*}
\sum_{i,j=1}^{n} \frac{(1+|Df|^{2})\delta_{ij} -D_{i}fD_{j}f }{n+(n-1)|Df|^{2}}
D_{ij}f = n \Lambda \frac{({\bf{x}},y) (1+|Df|^{2})^{3/2}}{n+(n-1)|Df|^{2}}
\quad \mbox{in } \Omega \\
f({\bf{x}}, \pm M) =\phi ({\bf{x}}, \pm M) \quad for \quad |{\bf{x}}|>N_{1}.
\end{gather*}
If there are functions
$\Lambda_{0}(y)$, $\Phi (y )$ and  $k(y)$ satisfying
that for $|{\bf{x}}|>N_{1}$, $|y|\leq M$,
\begin{gather*}
|\phi ({\bf{x}}, \pm M) - \Phi (\pm M )| \leq g_{1}(|{\bf{x}}|),
\quad
|\Lambda ({\bf{x}},y)- \Lambda_{0} (y)| \leq g^{*}_{2}(|{\bf{x}}|),\\
k''-n \Lambda_{0}(y)(1+(k')^{2})^{3/2} =0 \quad\mbox{on } |y|<M,
\; k( \pm M) = \Phi (\pm M ),
\end{gather*}
then in $(C2)$, we can choose $\mu (a,b)=\frac{1}{n+(n-1)b^{2}}$.
In $(C3)$ we can choose $E(y,z,q)=n\Lambda_{0}(1+q^{2})^{3/2}$,
$g_{2}(t)=c^{*}g^{*}_{2}(t)$ and $\alpha_{0}=2$.
Then from (\ref{eq:conclusion}), for a bounded solution $f({\bf{x}},y)$,
we have that for any integer $J$, there is a constant $C_{J}$ such
that
\begin{equation}
|f({\bf{x}},y)-k(y)|\leq
C_{J}g_{1}(\frac{1}{2^{J}}|{\bf{x}}| ) +
C_{J} g_{2}(\frac{1}{2^{J}}|{\bf{x}}|) +
\frac{C_{J}}{|{\bf{x}}|^{2J}} \quad \mbox{on } \Omega .
\end{equation}
\end{example}
The main idea in the proof of the theorem is to use the barriers constructed
in \cite{JL4} repeatedly. The construction in \cite{JL4} was adapted from
\cite{JL1} which in turn was inspired on \cite{Finn} and \cite{Serrin}.

\section{The barrier functions}

 From (C2), for fixed positive numbers $K_{0}$  and $K_{1}$,
there is a constant $c_{1}$, $0<c_{1}<1$,  such that
\begin{equation}
a_{nn}({\bf x},y,t, {\bf{v}}) \ge c_{1}
\label{eq:coeffbound}
\end{equation}
for $({\bf x},y)\in \Omega$, $t\in R$ with $|t|\leq 40K_{0}+20$,
${\bf v}\in \mathbb{R}^{n}$ with $|{\bf{v}}|\leq K_{1}+2$. We
define a new operator on functions $u({\bf{x}},y)\in
C^{2}(\Omega)$ with parameters $t\in R$ and ${\bf{v}}\in
\mathbb{R}^{n}$  by
$$
Q_{1} u =\sum_{i,j=1}^{n} a_{ij}({\bf{x}}, y, t, Du +{\bf{v}})D_{ij}u .
$$
Then we can prove that
there are positive decreasing functions  $\chi (t)$ (depending on $c_{1}$ only),
$h_{a}(t)$  and a positive increasing function $A(t)$ (depending on $c_{1}$
and $M$ only)
such that for any number $K$, $0<K\leq 3K_{0}+1$,
there is a number $H_{0}$ (depending only on $K_{0}$, $M$ and $c_{1}$),
such that when $H>H_{0}$
\begin{equation}
0< \chi (H)<1; \quad
\frac{22MH}{c_{1}}\leq A(H)e^{\chi (H)} \leq \frac{66MH}{c_{1}};
\label{eq:boundofae}
\end{equation}
and the function
\begin{equation}
z=\gamma +A(H)e^{\chi (H)} -\{ (h_{a}^{-1}(y+M))^2-|{\bf x}-{\bf x}_{0}|^2 \}
^{1/2} \label{eq:barrier111}
\end{equation}
satisfies the following conditions for $|t|\leq 40K_{0}+20$,
$|{\bf v}|\leq K_{1}+1$, $0\leq \gamma <1$:
\begin{gather}
\label{eq:upperbarrier1}
Q_{1}z  \le  \frac{-3c_{1}}{22eMH}  \quad\mbox{in }
\Omega_{{\bf x}_{0},H,K} \cap \Omega    \\
\label{eq:upperbarrier2}
\gamma \le z  \le  \gamma +\frac{4M}{H}+4K \quad \mbox{on  }
\overline{\Omega}_{{\bf x}_{0},H,K}  \\
\label{eq:upperbarrier4}
z \ge   \gamma + K \quad \mbox{on }
\partial\Omega_{{\bf x}_{0},H,K}\cap \{ |y|<M \} \\
 \label{eq:upperbarrier5}
z({\bf x}_{0},y) \le \gamma +\frac{2M}{H}  \quad\mbox{for }  |y|\le M\\
 \label{eq:upperbarrier6}
|D_{x}z({\bf x},y)| \le 2\big(\frac{c_{1}K}{M}\big)^{1/2}\frac{1}{\sqrt{H}},
\quad |D_{y}z({\bf x},y)| \le \frac{1}{H}
 \quad \mbox{on }  \Omega_{{\bf x}_{0},H,K},
\end{gather}
where
\begin{equation}
\Omega_{{\bf x}_{0},H,K}
=\big\{({\bf x},y):|y|< M,|{\bf x}-{\bf x}_{0}|<
\sqrt{\frac{2K}{A(H)e^{\chi (H)}}}h_{a}^{-1}(y+M) \big\} .
\label{eq:domain1}
\end{equation}

To make this paper self-contained, we include the following section.

\section{Construction of barrier functions \cite{JL4}}

Set $\Phi_{1}(\rho)=\rho^{-2}$  if $0<\rho<1$  and
$\Phi_{1}(\rho)=11/c_{1}$ if $\rho \geq 1$, and define a function
$$
\chi(\alpha)=\int_{\alpha}^{\infty}\ \frac{d\rho}{\rho^{3}\Phi_{1}(\rho)}
\quad\mbox{for }\alpha>0.
$$
It is clear that $\chi (\alpha )$ is a decreasing function with
range $(0,\infty)$.  Let $\eta$  be the inverse of $\chi.$
Then $\eta$  is a positive, decreasing function with range $(0,\infty)$.

Let $c_{2}=11/c_{1}$.  For $\alpha >1$,  we have
\begin{equation}
\chi(\alpha)=\int_{\alpha}^{\infty}\ \frac{d\rho}{\rho^{3}\Phi_{1}(\rho)}
=\int_{\alpha}^{\infty}\ \frac{d\rho}{c_{2}\rho^{3}}
= \frac{1}{2c_{2}} \alpha^{-2} .
\label{eq:chi2}
\end{equation}
Thus
\begin{equation}
\eta (\beta ) = (2c_{2}\beta)^{-1/2}\quad\mbox{for } 0<\beta <(2c_{2})^{-1}.
\label{eq:beta1}
\end{equation}
Let $H\ge 2$. Since $\eta(\chi(H))=H$ and $\eta $ is decreasing, we have
$\eta(\beta)> H$ for $0<\beta< \chi(H)$. We define a function
\begin{equation}
A(H) = 2M (\int_{1}^{e^{\chi(H)}}\ \eta (\ln t)  dt)^{-1} .
\label{eq:ah}
\end{equation}
 For the rest of this article, we set $a=A(H)$ and define
\begin{equation}
h_{a}(r)=\int_{r}^{ae^{\chi(H)}}
\eta (\ln \frac{t}{a} )\ dt \hspace{5 mm}\mbox{for}\ \
a\le r\le ae^{\chi(H)}.
\end{equation}
Then
\begin{equation}
h_{a}(ae^{\chi (H)})=0, \quad h_{a}(a) =h_{A(H)}(A(H))= 2M.
\label{eq:chi3}
\end{equation}
For $a<r\le ae^{\chi(H)}$,
\begin{equation}
\begin{gathered}
h_{a}'(r)=-\eta(\ln \frac{r}{a} )<0,\quad  |h_{a}'(r)|>H, \\
h_{a}''(r)=\frac{1}{r}(\eta(\ln \frac{r}{a} ))^{3}\Phi_{1}
(\eta(\ln \frac{r}{a} )).
\end{gathered}
\label{eq:derivativelarge}
\end{equation}
Thus for $a<r\le ae^{\chi(H)},$
\begin{equation}
\frac{h_{a}''(r)}{(h_{a}'(r))^2}=-\frac{h_{a}'(r)}{r}\Phi_{1}(-h_{a}'(r)) .
\label{eq:hequ}
\end{equation}
Let $h_{a}^{-1}$ be the inverse of $h_{a}.$  Then $h_{a}^{-1}$ is decreasing and
\begin{equation}
h_{a}^{-1}(0)=A(H)e^{\chi (H)}, \quad
h_{a}^{-1}(2M) = A(H) .
\label{eq:valueofinverseh}
\end{equation}
Further for $-M\leq y\leq M$,
$$
(h_{a}^{-1})'(y+M) =\frac{1}{h'_{a}(h_{a}^{-1}(y+M))}
$$
\begin{align*}
(h_{a}^{-1})''(y+M) &=(\frac{1}{h'_{a}(h_{a}^{-1}(y+M))})' \\
&=- \frac{h''_{a}(h_{a}^{-1}(y+M))(h_{a}^{-1})'
(y+M)}{ (h'_{a}(h_{a}^{-1}(y+M)))^{2}} \\
&= - \frac{h''_{a}(h_{a}^{-1}(y+M))}{ (h'_{a}(h_{a}^{-1}(y+M)))^{3}}\\
&= \frac{1}{h_{a}^{-1}(y+M)} \Phi_{1} (-h'_{a}(h_{a}^{-1}(y+M))) .
\end{align*}
Thus
\begin{equation}
(h_{a}^{-1})''(y+M)h_{a}^{-1}(y+M) =\Phi_{1} (-h'_{a}(h_{a}^{-1}(y+M))) .
\label{eq:equationofinverse}
\end{equation}
Now we choose an $H_{0}>2$ such that for $H\geq H_{0}$,
\begin{equation}
H_{0}>\frac{1}{\sqrt{2c_{2}}} +3M+4+\frac{24nc_{1}K_{0}}{M},
\quad
\sqrt{\frac{4K_{0}}{A(H)e^{\chi (H)}}}\leq \frac{1}{\sqrt{2}}.
\label{eq:Hlarge}
\end{equation}
For $H>H_{0}$,
by (\ref{eq:chi2}), (\ref{eq:beta1}), we have
\begin{align*}
A(H)^{-1} &= (2M)^{-1} \int_{1}^{e^{\chi(H)}}\ \eta (\ln t)\, dt\\
&=(2M)^{-1} \int_{0}^{\chi(H)}\ \eta (m)e^{m} \, dm\\
&=(2M)^{-1} \int_{0}^{\chi(H)}\ \frac{e^{m}}{\sqrt{2c_{2}m}} \, dm\,.
\end{align*}
From
$$
\frac{1}{\sqrt{2c_{2}}}\int_{0}^{\chi(H)}\ \frac{1}{\sqrt{m}}   dm
\leq
\int_{0}^{\chi(H)}\ \frac{e^{m}}{\sqrt{2c_{2}m}}   dm
\leq
\frac{e^{\chi (H)}}{\sqrt{2c_{2}}}
\int_{0}^{\chi(H)}\ \frac{1}{\sqrt{m}}  dm,
$$
we have
$$
\frac{1}{c_{2}H} =
\frac{2\sqrt{\chi (H)}}{\sqrt{2c_{2}}} \leq
\int_{0}^{\chi(H)}\ \frac{e^{m}}{\sqrt{2c_{2}m}}   dm
\leq \frac{2e^{\chi (H)}\sqrt{\chi (H)}}{\sqrt{2c_{2}}}
=\frac{e^{\frac{1}{2c_{2}H^{2}}}}{c_{2} H} .
$$
Thus
\begin{equation}
2Mc_{2}H \geq
A(H)\geq 2Mc_{2}He^{-\chi (H)} = 2Mc_{2}H e^{-\frac{1}{2c_{2}H^{2}}} .
\label{eq:ah2}
\end{equation}

For ${\bf x}_{0}\in \mathbb{R}^{n-1}$, a constant $\gamma$ with
$0\leq \gamma <1$ and a fixed constant $K$ with $0<K\leq
3K_{0}+1$, we define a domain $\Omega_{{\bf x}_{0},H,K}$ in
$({\bf{x}},y)$ space by (\ref{eq:domain1}) and define a function
$z=w_{{\bf x}_{0},\gamma,H}({\bf{x}}, y)$ by
(\ref{eq:barrier111}). Since $h_{a}^{-1}(y+M)\geq 0$ for $|y|\leq
M$, $({\bf{x}}_{0}, y)\in \Omega_{{\bf x}_{0},H,K}$ for $|y|< M$.
Further it is clear that the function
$z=w({\bf{x}},y)=w_{{\bf x}_{0},\gamma,H}({\bf{x}}, y)$ is well
defined on $\Omega_{{\bf x}_{0},H,K}$.


Now we verify (\ref{eq:upperbarrier2}). Since
$h_{a}^{-1}$ is a decreasing function,
$h_{a}^{-1}(y+M)\leq h_{a}^{-1}(0)=A(H)e^{\chi (H)}$
for
$y\ge -M$. Thus
$$
z \ge \gamma +A(H)e^{\chi (H)} - h_{a}^{-1}(y+M) \geq  \gamma .
$$
 From (\ref{eq:domain1}) and
(\ref{eq:valueofinverseh}), we have
that on  ${\Omega}_{{\bf x}_{0},H,K}$,
\begin{align*}
z&=\gamma +A(H)e^{\chi (H)} -\{ (h_{a}^{-1}(y+M) )^2-|{\bf x}
  -{\bf x}_{0}|^2\}^{1/2}\\
&\leq \gamma +A(H)e^{\chi (H)} -\{ (h_{a}^{-1}(y+M))^2-
  \frac{2K}{A(H)e^{\chi (H)}}(h_{a}^{-1}(y+M))^{2} \}^{1/2}\\
&= \gamma + A(H)e^{\chi (H)} -
  h_{a}^{-1}(y+M) (1- \frac{2K}{A(H)e^{\chi (H)} } )^{1/2}\\
&\leq \gamma + A(H)e^{\chi (H)} -
  h_{a}^{-1}(2M) (1- \frac{2K}{A(H)e^{\chi (H)} } )^{1/2}\\
&= \gamma + A(H)e^{\chi (H)} - A(H) (1- \frac{2K}{A(H)e^{\chi (H)} } )^{1/2}\\
&\leq \gamma +A(H)(1+e \chi (H))-A(H) (1-2\frac{2K}{A(H)e^{\chi (H)} }).
\end{align*}
Since $e^{t}\leq 1+et$ for $0< t <1$, and $\sqrt{1-t} \geq 1-2t$ for
$0\leq t \leq \frac{1}{2}$, the above expression is equal to
$$
\gamma +e A(H)\chi (H) +\frac{4K}{e^{\chi (H)} }
\leq \gamma +e\frac{2Mc_{2}H}{2c_{2}H^{2}})+4K
\leq \gamma + \frac{4M}{H}+4K\,.
$$
This because of (\ref{eq:chi2}),
(\ref{eq:Hlarge}) and (\ref{eq:ah2}). This is (\ref{eq:upperbarrier2}).

For (\ref{eq:upperbarrier4}),
on $\partial \Omega_{{\bf x}_{0},H,K} \cap \{ ({\bf{x}},y): |y|<M \}$,
$$
|{\bf x}-{\bf x}_{0}|=
\sqrt{\frac{2K}{A(H)e^{\chi (H)}}}h^{-1}(y+M) .
$$
Then from (\ref{eq:valueofinverseh}), we have
\begin{align*}
z&=\gamma +A(H)e^{\chi (H)} -\{ (h_{a}^{-1}(y+M))^2-|{\bf x}
-{\bf x}_{0}|^2 \}^{1/2}\\
&= \gamma + A(H)e^{\chi (H)} - h_{a}^{-1}(y+M)
(1- \frac{2K}{A(H)e^{\chi (H)} } )^{1/2} \\
&\geq \gamma + A(H)e^{\chi (H)} -
A(H)e^{\chi (H)} (1- \frac{2K}{A(H)e^{\chi (H)} } )^{1/2} \\
&\geq \gamma + A(H)e^{\chi (H)}(1 - (1- \frac{2K}{2A(H)e^{\chi (H)} } ))\\
&=\gamma + K.
\end{align*}
Here we have used (\ref{eq:Hlarge}) and
the fact that $\sqrt{1-t}\leq 1-\frac{1}{2}t $ for $0<t<1$.

For (\ref{eq:upperbarrier5}),
since
$h_{a}^{-1}(r)$ and $\eta $ are decreasing functions, we have
\begin{equation}
\begin{aligned}
\frac{-1}{h_{a}'(h_{a}^{-1}(y+M))}
&=\frac{1}{ \eta(\ln(\frac{1}{a}h_{a}^{-1}(y+M))) }\\
&\leq \frac{1}{\eta (\ln e^{\chi (H)})}\\
&=\frac{1}{\eta(\chi(H))}= \frac{1}{H},
\quad \mbox{for } |y|\le -M.
\end{aligned}\label{eq:neededagain}
\end{equation}
Then by (\ref{eq:barrier111}), we have
$$
\frac{\partial z}{\partial y}({\bf x}_{0},y)
=\frac{-1}{h_{a}'(h_{a}^{-1}(y+M))}= \frac{1}{H},
\quad \mbox{for }  |y|\le -M.
$$
Then (\ref{eq:upperbarrier5}) follows from this and
$$
z({\bf x}_{0},-M)=\gamma +A(H)e^{\chi (H)}  -h_{a}^{-1}(0) =\gamma +A(H)e^{\chi (H)}
 -A(H)e^{\chi (H)} =\gamma .
$$


For (\ref{eq:upperbarrier1}) and (\ref{eq:upperbarrier6}),
set $S= \{(h_{a}^{-1}(y+M))^2-|{\bf x}-{\bf x}_{0}|^2 \}^{1/2}$,
we have that for $1\le i\le n-1 $,
$$
\frac{\partial z}{\partial x_i}=
\frac{1}{S}(x_i-x_{0i}),\quad
\frac{\partial z}{\partial y}=
- \frac{1}{S}h_{a}^{-1} (h_{a}^{-1})'.
$$
By (\ref{eq:Hlarge}) and (\ref{eq:ah2}), on $\Omega_{{\bf{x}}_{0}, H, K}$, we have
\begin{gather*}
\frac{1}{2}
h_{a}^{-1}(y+M)
\leq S \leq h_{a}^{-1}(y+M),\\
\frac{|{\bf{x}}-{\bf{x}}_{0}|}{S}
\leq 2(\frac{2K}{A(H)e^{\chi (H)}})^{1/2}
\leq 2(\frac{2K}{2Mc_{2}H} )^{1/2} .
\end{gather*}
Thus, by (\ref{eq:neededagain}), we have
\begin{equation}
\big|\frac{\partial z}{\partial x_{i}}\big| \leq 2\big(\frac{c_{1}K}{MH}\big)^{1/2},
\quad
\big|\frac{\partial z}{\partial y}\big|
\leq \frac{h_{a}^{-1}(y+M)}{S |h'_{a}(h_{a}^{-1}(y+M)|}
\leq \frac{2}{H} .
\label{eq:gradient2}
\end{equation}
This gives (\ref{eq:upperbarrier6}).
Hence from (\ref{eq:Hlarge}),
for any positive semi-definite matrix $(d_{ij})$  with $trace(d_{ij})=1$
(hence all eigenvalues of
$(d_{ij})$ are less than or equal to 1), we have
\begin{equation}
\big|\sum_{i,j=1}^{n} d_{ij} \frac{\partial z}{\partial x_{i}}
\frac{\partial z}{\partial x_{j}}\big| \leq |Dz|^2 \leq 1
\label{eq:normofgradient}
\end{equation}

For $t$ with $|t|\leq 40K_{0}+20$, $|{\bf{v}}| \leq K_{1}$,
from (\ref{eq:normofgradient}), we have $ |Dz|+|{\bf{v}}|\leq K_{1}+1$. Then
from (\ref{eq:traceis1}) and (\ref{eq:coeffbound}),
for $S= \{(h_{a}^{-1}(y+M))^2-|{\bf x}-{\bf x}_{0}|^2\}^{1/2}$, we have
\begin{align*}
Q_{1}z&=\sum_{i,j=1}^{n}\ a_{ij}({\bf x},y,t,Dz+{\bf{v}})D_{ij}z\\
&= \frac{1}{S}\sum_{i=1}^{n-1} a_{ii}
+\frac{1}{S^{3}}\sum_{i,j=1}^{n-1} a_{ij}(x_{i}-x_{i}^{0})(x_{j}-x_{j}^{0})
-\frac{1}{S^{3}}\sum_{i=1}^{n-1} a_{in}(x_{i}-x_{i}^{0})h_{a}^{-1} (h_{a}^{-1})'\\
&\quad -\frac{1}{S}a_{nn}((h_{a}^{-1})^{2} + h_{a}^{-1} (h_{a}^{-1})'')
 +\frac{1}{S^{3}}a_{nn}(h_{a}^{-1})^{2} ((h_{a}^{-1})')^{2}\\
&=\frac{1}{S}\{1-a_{nn}+\sum_{i,j=1}^{n} a_{ij} \frac{\partial z}{\partial x_{i}}
\frac{\partial z}{\partial x_{j}}-a_{nn} ((h_{a}^{-1})^{2}
+ h_{a}^{-1} (h_{a}^{-1})'') \}\\
&\leq \frac{1}{S}\{1+\sum_{i,j=1}^{n} a_{ij} \frac{\partial z}{\partial x_{i}}
\frac{\partial z}{\partial x_{j}}-a_{nn} h_{a}^{-1} (h_{a}^{-1})'' \}\,.
\end{align*}
(By (\ref{eq:coeffbound}), (\ref{eq:equationofinverse})), (\ref{eq:ah2}) and
(\ref{eq:normofgradient}) this expression is less than or equal to
$$
\frac{-9}{S} \leq \frac{-9} {h_{a}^{-1}(y+M)}
\leq  \frac{-9}{A(H)e^{\chi (H)}} \leq \frac{-9}{2Mc_{2}He^{\frac{1}{2c_{2}H^{2}}}}
\leq \frac{-3c_{1}}{22eMH}
$$
which implies (\ref{eq:upperbarrier1}).


\section{Proof of Main Theorem }


For the proof of Theorem \ref{thm1}, we need  the following result on $E$ and $k$.

\begin{proposition}[{\cite[Lemma 4.1]{JL3}}] \label{prop1}
Under assumptions (C4) and C5), there exist numbers $\gamma_{1}>0$ and $c_{3}$
(depending only on $k$, $E$), such that for
any constant $\delta_{1}$ with $|\delta_{1}|<\min \{ \gamma_{1}, 1\}$,
there is a (unique) function $k_{\lambda }(y)=k_{\lambda }(y)$ in
$C^{1}([-M,M])\cap C^{2}((-M,M))$ satisfying
$$
k''_{\lambda }(y) +E(y, k_{\lambda }(y), k'_{\lambda }(y))
=-\frac{3}{4c_{3}}\delta_{1}, \quad
k_{\lambda }(\pm M)=k(\pm M),
$$
and on $|y|\leq M$,
$$
|k(y)-k_{\lambda }(y)|\leq |\delta_{1}|, \quad
|k'(y)-k'_{\lambda }(y)|\leq |\delta_{1}|,\quad
|k''(y)-k''_{\lambda }(y)|\leq |\delta_{1}|.
$$
\end{proposition}

\begin{proof}[Proof of Theorem \ref{thm1}]
We assume that there exists a non-increasing function $g(t)$ such that
\begin{equation}
|f({\bf{x}},y)-k(y)|\leq g(|{\bf{x}}|)\quad \mbox{for } ({\bf{x}},y)\in \Omega .
\label{eq:assumedecay}
\end{equation}
(since $f({\bf{x}},y)$ and $k(y)$ are bounded, (\ref{eq:assumedecay}) holds
for $g(t)$ to be some appropriate constant. $g(t)$ will also take other
forms as we shall explain later).

For a small positive number $\delta_{1}$ (to be chosen later),
let $k_{\lambda }(y)$ be the function defined in the Proposition.
We will use the barrier function $u({\bf{x}},y) +  k_{\lambda } (y)$.
Let
\begin{gather*}
K_{0}=\sup \{ |f({\bf{x}},y)|: ({\bf{x}},y)\in \Omega \}
+\sup \{ |k(y)| : |y|\leq M \}+g(0),\\
K_{1} = 2 \sup \{ |k'(y)| :  |y|\leq M \} +10\,.
\end{gather*}
and $c_{1}$ be a number such that
\begin{equation}
c_{1} \leq a_{nn}({\bf{x}},y,t,{\bf{v}})
\label{eq:coeffboundthm2}
\end{equation}
for $({\bf{x}},y)\in \Omega$, $|t|\leq 40K_{0}+20$,
${\bf{v}} \in R^{n}$ with $|{\bf{v}}|\leq K_{1} +2$.


Since $g_{1}(t)\to 0$ as $t\to \infty$, there is a number
$H_{1}$ such that $g_{1}(\frac{1}{2}|{\bf{x}}|)\leq \frac{1}{2} $
for $|{\bf{x}}|\geq H_{1}$ and $H_{1}>800MK_{0}/c_{1}$.
We fixed an ${\bf{x}}_{0}$ with $|{\bf{x}}_{0}|\geq H_{0}+H_{1}$
($H_{0}$ is given in (\ref{eq:barrier111})).
Let $u({\bf{x}},y)=z({\bf{x}},y)$ defined on $\Omega_{{\bf{x}}_{0}, H,K}$
be given by (\ref{eq:barrier111})
with the choice of parameters:
$$
\gamma = g_{1}(\frac{1}{2}|{\bf{x}}_{0}|)+\delta_{1},
\quad H=\frac{c_{1}|{\bf{x}}_{0}|^{2}}{800MK },
\quad K=2g(\frac{1}{2}|{\bf{x}}_{0}|) .
$$

>From (\ref{eq:boundofae}),  (\ref{eq:upperbarrier6}),
$h_{a}^{-1}(y+M)\leq A(H)e^{\chi (H)}$ and the choice of $H$, there is a number
$c_{6}$ independent of $\delta_{1}$, such that on $\Omega_{{\bf{x}}_{0}, H,K}$,
$$
|D_{x}u| \leq  \frac{c_{6}}{|{\bf{x}}_{0}|} g(\frac{1}{2}|{\bf{x}}_{0}|),
\quad
|D_{y}u| \leq  \frac{c_{6}}{|{\bf{x}}_{0}|^{2}} g(\frac{1}{2}|{\bf{x}}_{0}|),
$$
and
\begin{equation}
|{\bf{x}} - {\bf{x}}_{0}|
\leq \sqrt{\frac{2K}{A(H)e^{\chi (H)}}} h^{-1}_{a}(y+M)
\leq \sqrt{2KA(H)e^{\chi (H)}}
\leq \frac{1}{2}|{\bf{x}}_{0} | .
\end{equation}
Then for $|{\bf{x}}_{0}|$ large, on $\Omega_{{\bf{x}}_{0}, H,K} $, (where $c_{9}$
is independent of $\delta_{1}$)
\begin{equation}
\frac{1}{2}|{\bf{x}}_{0}| \leq |{\bf{x}}|
\leq \frac{3}{2}|{\bf{x}}_{0}|,  \quad
|D_{x}u|^{\alpha _{0}} \leq c_{9}
\frac{(g(\frac{1}{2}|{\bf{x}}_{0}|))^{\alpha_{0}}}{|{\bf{x}}|^{\alpha_{0}}},
\quad
|D_{y}u| \leq c_{9} \frac{g(\frac{1}{2}|{\bf{x}}_{0}|)}{|{\bf{x}}|^{2}} .
\label{eq:estimateofux3}
\end{equation}
Then from (\ref{eq:upperbarrier2}),
for $|{\bf{x}}_{0}|\geq H_{0}+H_{1}$ and
any positive constant $b$, $b<10K_{0}+1$,
on $\Omega_{{\bf{x}}_{0}, H,K}$, we have
\begin{gather*}
u({\bf{x}},y) + k_{\lambda } (y) + b \leq 40K_{0}+20,
|Du({\bf{x}},y)| + |k'_{\lambda } (y)|  \leq K_{1} +1.
\end{gather*}
Set
$$
M_{3}=\sup \big\{ \frac{\partial E}{\partial q}(y,z,q):
 |y|\leq M, \ |z|\leq 40K_{0}+20 , \ |q|\leq K_{1}+1 \big\}
$$
>From (\ref{eq:upperbarrier1}), for $0<b<10K_{0}+1$, we have
(note that $k_{\lambda }$ depends on $y$ only)
\begin{align*}
&\sum_{i,j=1}^{n}\ a_{ij}({\bf{x}},y,u+k_{\lambda } +b,D(u + k_{\lambda } ) )
 D_{ij} (u + k_{\lambda } )\\
&+B({\bf{x}},y,u+k_{\lambda }  +b,D(u + Dk_{\lambda } )) \\
&< a_{nn}({\bf{x}},y, u+k_{\lambda }  +b  ,D(u + k_{\lambda } ) )k_{\lambda }''(y)
+ B({\bf{x}},y,u+k_{\lambda }+b ,D(u + k_{\lambda } ) ) .
\end{align*}
By (C3) and(C5), Proposition \ref{prop1},
(\ref{eq:coeffboundthm2}), (\ref{eq:estimateofux3}) and $b>0$, $u>0$, we have
\begin{align*}
&a_{nn}({\bf{x}},y,u+k_{\lambda }  +b,D(u + k_{\lambda } ))k_{\lambda }''(y)
+B({\bf{x}},y,u+k_{\lambda }  +b,D(u + k_{\lambda }) )\\
&=a_{nn}({\bf{x}},y,u+k_{\lambda }  +b,D(u + k_{\lambda } ))\big\{ k_{\lambda }''(y)
+\frac{B({\bf{x}},y,u+k_{\lambda }
+b,D(u + k_{\lambda } ))}{a_{nn}({\bf{x}},y,u+k_{\lambda }
+b,D(u+k_{\lambda }))} \big\}\\
&=a_{nn}({\bf{x}},y,u+k_{\lambda }  +b,D(u +k_{\lambda }) )\Big\{ k_{\lambda }''(y)
 + E(y, k_{\lambda },k'_{\lambda })
 + E(y, k_{\lambda },D_{y}u+k'_{\lambda })\\
&\quad - E(y, k_{\lambda },k'_{\lambda })
 +E(y, k_{\lambda }+u+b,D_{y}u+k'_{\lambda })
 -E(y, k_{\lambda },D_{y}u+k'_{\lambda })\\
&\quad +\frac{B({\bf{x}},y,u+k_{\lambda }
 +b,D(u+k_{\lambda } ))}{a_{nn}({\bf{x}},y,u+ k_{\lambda }
 +b,D(u+Dk_{\lambda } ))}-E(y, k_{\lambda }+u+b,D_{y}u+k'_{\lambda } ) \Big\}\\
&\leq a_{nn}({\bf{x}},y,u+k_{\lambda }  +b,D(u+k_{\lambda }))
 \big\{ -\frac{3}{4c_{3}}\delta_{1}+ M_{3} |D_{y}u|
+g_{2}(|{\bf{x}}|)   +C |D_{x}u|^{\alpha _{0}} \big\} \\
&\leq -\frac{3c_{1}}{4c_{3}}\delta_{1} + M_{3} |D_{y}u|+
C |D_{x}u|^{\alpha _{0}} + g_{2}(|{\bf{x}}|) \\
& \leq \frac{Cc_{9}(g(\frac{1}{2}|{\bf{x}}_{0}|))^{\alpha_{0}}}{|{\bf{x}}|^{\alpha_{0}}}
+ \frac{c_{9}M_{3}g(\frac{1}{2}|{\bf{x}}_{0}|)}{|{\bf{x}}|^{2}} +g_{2}(|{\bf{x}}|)
-\frac{3c_{1}}{4c_{3}} \delta_{1}
\end{align*}
Set $d=\frac{1}{2}|{\bf{x}}_{0}|$,
we have that on $\Omega_{{\bf{x}}_{0}, H, K} $,
$|{\bf{x}}|\geq d$ (by (\ref{eq:estimateofux3})).
Now we fixed a $d$ such that $d > H_{2}\geq H_{0}+H_{1}$ and
choose $\delta_{1}$ by
\begin{equation}
\frac{3c_{1}}{4c_{3}} \delta_{1}
=
\frac{Cc_{9}(g(\frac{1}{2}|{\bf{x}}_{0}|))^{\alpha_{0}} }{d^{\alpha_{0}}}
+ \frac{c_{9}M_{3}g(\frac{1}{2}|{\bf{x}}_{0}|)}{d^{2}} +g_{2}(d) .
\label{eq:definitionofdelta12}
\end{equation}
Then on $\Omega_{{\bf{x}}_{0}, H, K}$, we have
(since $|{\bf{x}}|\geq d$ and $g_{2}$ is non-increasing)
\begin{equation}
\begin{aligned}
&\sum_{i,j=1}^{n}\ a_{ij}({\bf{x}},y,u+k_{\lambda }
+b,D(u+k_{\lambda } ) )D_{ij}(u+ k_{\lambda } )\\
&+B({\bf{x}},y,,u+k_{\lambda }  +b, D(u+k_{\lambda } ) )\\
&<  \frac{Cc_{9}(g(\frac{1}{2}|{\bf{x}}_{0}|))^{\alpha_{0}} }{|{\bf{x}}|^{\alpha_{0}}}
+ \frac{c_{9}M_{3}g(\frac{1}{2}|{\bf{x}}_{0}|)}{|{\bf{x}}|^{2}}  +g_{2}(|{\bf{x}}|)
- \frac{3c_{1}}{4c_{3}} \delta_{1} \leq 0.
\end{aligned}\label{eq:somethingtouse3}
\end{equation}
For such an ${\bf{x}}_{0}$, on $\Omega_{{\bf{x}}_{0}, H, K}\cap \Omega $,
we will compare the function $f({\bf{x}},y)$ with the function
$u({\bf{x}},y) + k_{\lambda }(y)$.
On $\partial \Omega_{{\bf{x}}_{0}, H, K}\cap \Omega $, (\ref{eq:upperbarrier4})
and Proposition \ref{prop1} imply
\begin{align*}
u({\bf{x}},y) + k_{\lambda }(y) &\geq \gamma + K + k_{\lambda }(y)\\
&\geq g_{1}(\frac{1}{2}|{\bf{x}}_{0}|)+ \delta_{1}
+ 2g(\frac{1}{2}|{\bf{x}}_{0}|)+ k_{\lambda }(y)\\
&\geq 2g(\frac{1}{2}|{\bf{x}}_{0}|)+ k(y)
\geq f({\bf{x}},y)
\end{align*}
(by (\ref{eq:assumedecay}) and $|{\bf{x}}|\geq \frac{1}{2}|{\bf{x}}_{0}|$
on $\Omega_{{\bf{x}}_{0}, H,K}$).
On $\Omega_{{\bf{x}}_{0}, H, K}\cap \partial \Omega$,
$y=\pm M$ and
$\Phi (\pm M)=k (\pm M) = k_{\lambda }(\pm M)$. Then
from ({\bf{C1}}) and (\ref{eq:upperbarrier2}),
we have
\begin{align*}
\phi ({\bf{x}},\pm M) &= \phi ({\bf{x}},\pm M) - \Phi (\pm M) + \Phi (\pm M)
\leq g_{1}(|{\bf{x}}|) +k(\pm M)\\
&\leq g_{1}(\frac{1}{2}|{\bf{x}}_{0}|)+\delta_{1}+ k_{\lambda } (\pm M)
=\gamma +k_{\lambda }(\pm M)\\
&\leq u({\bf{x}},\pm M) +k_{\lambda } (\pm M).
\end{align*}
Let
$$
\Omega_{1} = \{ ({\bf{x}},y)\in \Omega_{{\bf{x}}_{0}, H,K}\cap \Omega :
f({\bf{x}},y)> u ({\bf{x}},y) + k_{\lambda }(y)\} .
$$
Since
$f({\bf{x}},y)\leq  u ({\bf{x}},y) + k_{\lambda }$
on $\partial (\Omega_{{\bf{x}}_{0}, H,K}\cap \Omega )$,
$\Omega_{1}$ is in the interior of $\Omega_{{\bf{x}}_{0}, H,K}$.
If $({\bf{x}}_{2},y_{2})\in \Omega_{1}$, we let
$b= f({\bf{x}}_{2},y_{2})-( u ({\bf{x}}_{2},y_{2}) + k_{\lambda }  (y_{2}))$, then
$0<b<f({\bf{x}}_{2},y_{2})\leq K_{0} <10K_{0}+1$. Using this $b$ in
(\ref{eq:somethingtouse3}) and evaluating the formula at $({\bf{x}}_{2},y_{2})$,
we get
$$
\sum_{i,j=1}^{n}\ a_{ij}({\bf{x}}_{2},y_{2},f({\bf{x}}_{2},y_{2}), D(u + k_{\lambda }) )
D_{ij}(u+ k_{\lambda } )
+B({\bf{x}}_{2},y_{2},f({\bf{x}}_{2},y_{2}), D(u + k_{\lambda } ))
<0 .
$$
Since $({\bf{x}}_{2},y_{2})\in \Omega_{1}$ is arbitrary, on
$\Omega_{1}$, we have
\begin{equation}
\sum_{i,j=1}^{n}\ a_{ij}({\bf{x}},y,f({\bf{x}},y), D(u + k_{\lambda }) )
D_{ij}(u+ k_{\lambda } )
+B({\bf{x}},y,f({\bf{x}},y), D(u + k_{\lambda } ))<0 .
\label{eq:degenerate3}
\end{equation}
Now we can apply a comparison principle  \cite[Theorem10.1]{GT}
to conclude that $\Omega_{1}$ is empty. Thus
$$
f({\bf{x}},y)\leq u ({\bf{x}},y) + k_{\lambda }  (y)
\quad \mbox{on } \Omega_{{\bf{x}}_{0}, H,K}\cap \Omega  .
$$
In particular, from (\ref{eq:upperbarrier5})
$$
f({\bf{x}}_{0},y)\leq u ({\bf{x}}_{0},y)+ k_{\lambda } (y)
\leq \gamma + \frac{2M}{H} + k_{\lambda } (y).
$$
Then from Proposition 2 and the choices of $\gamma $, $H$ and $\delta_{1}$, we have
\begin{align*}
f({\bf{x}}_{0},y)
&\leq g_{1}(\frac{1}{2}|{\bf{x}}_{0}| ) + \delta_{1}
+ \frac{c_{11}g(\frac{1}{2} |{\bf{x}}_{0}|)}{ |{\bf{x}}_{0}|^{2} }+k_{\lambda } (y)\\
&\leq g_{1}(\frac{1}{2}|{\bf{x}}_{0}| ) + \frac{c_{11}g(\frac{1}{2}
|{\bf{x}}_{0}|)}{ |{\bf{x}}_{0}|^{2} }+k (y) +2\delta_{1}
\end{align*}
\begin{align*}
f({\bf{x}}_{0},y)-k(y)&\leq g_{1}(\frac{1}{2}|{\bf{x}}_{0}| )
+ \frac{c_{11}g(\frac{1}{2} |{\bf{x}}_{0}|)}{ |{\bf{x}}_{0}|^{2} }+2\delta_{1}\\
&=g_{1}(\frac{1}{2}|{\bf{x}}_{0}| ) + \frac{c_{11}g(\frac{1}{2}
|{\bf{x}}_{0}|)}{ |{\bf{x}}_{0}|^{2} }
+\frac{c_{12}(g(\frac{1}{2}|{\bf{x}}_{0}|))^{\alpha_{0}}}{d^{\alpha_{0}}}
+ \frac{c_{12}g(\frac{1}{2}|{\bf{x}}_{0}|)}{d^{2}}+g_{2}(d)
\end{align*}
However $d=\frac{1}{2}|{\bf{x}}_{0}|$, thus
$$
f({\bf{x}}_{0},y)-k(y)\leq
g_{1}(\frac{1}{2}|{\bf{x}}_{0}| ) +
\frac{c_{13}(g(\frac{1}{2}|{\bf{x}}_{0}|))^{\alpha_{0}} }{|{\bf{x}}_{0}|^{\alpha_{0}}}
+ \frac{c_{13}g(\frac{1}{2}|{\bf{x}}_{0}|)}{|{\bf{x}}_{0}|^{2}} +g_{2}(\frac{1}{2}|{\bf{x}}_{0}|).
$$
Since $d$ is arbitrary as long as it is greater than
$H_{2}$, ${\bf{x}}_{0}$ is arbitrary as long as $|{\bf{x}}_{0}|$ is greater than
$2H_{2}$. Then we have that when $|{\bf{x}}|$ is large,
$$
f({\bf{x}},y)-k(y)\leq
g_{1}(\frac{1}{2}|{\bf{x}}| ) +
\frac{c_{13}(g(\frac{1}{2}|{\bf{x}}|))^{\alpha_{0}} }{|{\bf{x}}|^{\alpha_{0}}}
+ \frac{c_{13}g(\frac{1}{2}|{\bf{x}}|)}{|{\bf{x}}|^{2}} + g_{2}(\frac{1}{2}|{\bf{x}}|).
$$
Similarly using $-u({\bf{x}},y)+k_{\lambda }(y)$ (now with $\delta_{1}<0$ and choosen appropriately),
we can prove that
$$
f({\bf{x}},y)-k(y)\geq -\big\{
g_{1}(\frac{1}{2}|{\bf{x}}| ) +
\frac{c_{13}(g(\frac{1}{2}|{\bf{x}}|))^{\alpha_{0}} }{|{\bf{x}}|^{\alpha_{0}}}
+ \frac{c_{13}g(\frac{1}{2}|{\bf{x}}|)}{|{\bf{x}}|^{2}}+g_{2}(\frac{1}{2}|{\bf{x}}|)
\big\} .
$$
Thus on $\Omega $, we have
$$
|f({\bf{x}},y)-k(y)|\leq
g_{1}(\frac{1}{2}|{\bf{x}}| ) +
\frac{c_{13}(g(\frac{1}{2}|{\bf{x}}|))^{\alpha_{0}} }{|{\bf{x}}|^{\alpha_{0}}}
+ \frac{c_{13}g(\frac{1}{2}|{\bf{x}}|)}{|{\bf{x}}|^{2}}+g_{2}(\frac{1}{2}|{\bf{x}}|).
$$
Since $\alpha_{0}\geq 1$ and $g(t)$ is non-increasing, we have
$g(t)^{\alpha_{0}} \leq g(1)^{\alpha_{0} -1} g(t)$, thus for some constant $c_{14}$,
we have ($\beta=\min \{ \alpha_{0}, 2\}$)
\begin{equation}
|f({\bf{x}},y)-k(y)|\leq
g_{1}(\frac{1}{2}|{\bf{x}}| ) + g_{2}(\frac{1}{2}|{\bf{x}}|)+
\frac{c_{14}}{|{\bf{x}}|^{\beta }}  g(\frac{1}{2}|{\bf{x}}|) \quad \mbox{on }\Omega .
\label{eq:endofestimate}
\end{equation}
Now we choose $g(t)=K_{0}$ in (\ref{eq:assumedecay}), then (\ref{eq:endofestimate}) becomes
\begin{equation}
|f({\bf{x}},y)-k(y)|\leq
g_{1}(\frac{1}{2}|{\bf{x}}| ) + g_{2}(\frac{1}{2}|{\bf{x}}|)+
\frac{c_{14}K_{0}}{|{\bf{x}}|^{\beta }}   \quad \mbox{on } \Omega .
\label{eq:1stresult}
\end{equation}

shen we choose
$$
g(t) =
g_{1}(\frac{1}{2}t) + g_{2}(\frac{1}{2}t)+
\frac{c_{14}K_{0}}{t^{\beta }},
$$
that (\ref{eq:1stresult}) implies (\ref{eq:assumedecay}) is still true.
Then for this new $g(t)$,
we can apply (\ref{eq:endofestimate}) to obtain
\begin{equation}
|f({\bf{x}},y)-k(y)|\leq
c_{15}g_{1}(\frac{1}{4}|{\bf{x}}| ) + c_{15}g_{2}(\frac{1}{4}|{\bf{x}}| )+
\frac{c_{15}}{|{\bf{x}}|^{2\beta } }\quad\mbox{on } \Omega .
\label{eq:2ndround}
\end{equation}
for some constant $c_{15}$ (since $g_{1}$, $g_{2}$ are non-increasing).
Then once again, in (\ref{eq:assumedecay}), we can reset the function $g(t)$ as
\begin{equation}
g(t)= c_{15}g_{1}(\frac{1}{4}t) + c_{15}g_{2}(\frac{1}{4}t ) +
\frac{c_{15}}{t^{2\beta }} .
\end{equation}
and apply (\ref{eq:endofestimate}) to conclude that
there is a constant $c_{16}$ such that
\begin{equation}
|f({\bf{x}},y)-k(y)| \leq
c_{16}g_{1}(\frac{1}{8}|{\bf{x}}| ) + c_{16}g_{2}(\frac{1}{8}|{\bf{x}}| )
+\frac{c_{16}}{|{\bf{x}}|^{3\beta }}
\quad\mbox{on } \Omega .
\end{equation}
We can repeat this procedure to conclude that for any integer $J$, there is a
number $C_{J}$, such that
\begin{equation}
|f({\bf{x}},y)-k(y)|\leq
C_{J}g_{1}(\frac{1}{2^{J}}|{\bf{x}}| ) + C_{J}g_{2}(\frac{1}{2^{J}}|{\bf{x}}|)  +
\frac{C_{J}}{|{\bf{x}}|^{J\beta }} \quad\mbox{on }\Omega .
\end{equation}
\end{proof}

\begin{thebibliography}{00}

\bibitem{BR} S. Breuer \& J. Roseman,
{\em Decay theorems for nonlinear Dirichlet problems
in semi-infinite cylinders},
Arch. Rational Mech. Anal., {\bf{94}} (1986), 363-371.


\bibitem{Finn} R. Finn,
{\em New Estimates for equations of minimum surface type},
Arch. Rational Mech. Anal., {\bf{14}} (1963), 337-375.


\bibitem{GT} D. Gilbarg \& N. Trudinger,
{\em Second order elliptic partial differential equations},
second edition, Springer 1983.

\bibitem{Hobson} E.W. Hobson,
{\em The theory of spherical and ellipsoidal harmonics}, Cambridge University
Press (1931)


\bibitem{Horgan1} C. Horgan,
{\em Recent developments concerning
 Saint-Venant's principle: an update}, Appl. Mech. Rev. {\bf 42} (1989), 295-303.

\bibitem{Horgan2} C. Horgan,
{\em Recent developments concerning
 Saint-Venant's principle: a second update}, Appl. Mech. Rev.
 {\bf 49} (1996), S101--S111.

\bibitem{HK} C. Horgan \& J. K. Knowles,
{\em Recent developments concerning  Saint-Venant's principle"}
in advances in Applied Mechanics,
TY Wu and JW Hutchinson (eds),
Vol 23 (1983), Academic Press, New York, 179-269.

\bibitem{HO} C. Horgan \& W. E. Olmstead,
{\em Exponential Decay Estimates for a class of Nonlinear Dirichlet Problems},
Arch. Rational Mech. Anal., {\bf{71}} (1979), 221-235.

\bibitem{HP} C. Horgan \& L. E. Payne,
{\em On the Asymptotic behavior of solutions of inhomogeneous
second order quasilinear partial differential equations},
Quart. Appl. Math, {\bf{47}} (1989), 753-771.

\bibitem{KP} R. J. Knops \& L. E. Payne,
{\em Phragm{\'e}n-Lindel{\"o}f principle for the equation
of a surface of constant mean curvature},
Proc. Royal Soc. Edinburgh, {\bf{124A}} (1994), 105-119.

\bibitem{JL1} Z. Jin \& K. Lancaster,
{\em Phragm{\'e}n-Lindel{\"o}f Theorems and the
 Asymptotic Behavior of solutions of quasilinear elliptic equations in slabs},
 Proc. Roy. Soc. Edinburgh {\bf{130A}} (2000), 335-373.

\bibitem{JL2} Z. Jin \& K. Lancaster,
{\em The convergence rate of solutions of quasilinear elliptic equations in slabs},
Comm. PDE, {\bf{25}} (2000), 957--985.

\bibitem{JL3} Z. Jin \& K. Lancaster,
{\em Phragm{\'e}n-Lindel{\"o}f theorems  and the behavior at infinity
of solutions of degenerate elliptic equations}, Pacific J. of Math. {\bf{211}} (2003), 101-121. 

\bibitem{JL4} Z. Jin \& K. Lancaster,
{\em The convergent rate of solutions of Dirichlet problems for quasilinear
equations} (2002), submitted for publication.

\bibitem{Serrin} J. Serrin,
{\em The problem of Dirichlet for quasilinear equations with many independent
variables}, Phil. Trans. Royal Soc. Lond. A {\bf264} (1969), 413-496.

\end{thebibliography}

\end{document}