\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 $$\Omega = \{ ({\bf{x}} , y )\in \mathbb{R}^n : |y| N_{1}\}$$ 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 $$\begin{gathered} Qf=0 \quad\mbox{in }\Omega , \\ f=\phi \quad\mbox{on }\partial\Omega\,, \end{gathered}\label{eq:problem1}$$ where $Q$ is a second-order quasilinear operator of the form $$Qf = \sum_{i,j=1}^{n}\ a_{ij}({\bf{x}}, y,f, Df)D_{ij} f +B({\bf{x}} ,y,f,Df)\,. \label{eq:operator}$$ 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 $$|\phi ({\bf{x}},\pm M)-\Phi({\bf x}/|{\bf x}|,\pm M)|\leq g_{1}( |{\bf{x}}| ) \quad\mbox{for } |{\bf x}|> N_{1} \,. \label{eq:rateofdata111}$$ 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 $$|f({\bf{x}},y)-k({\bf x}/|{\bf x}|,y)|\leq d(|{\bf{x}}|) \quad \mbox{for } ({\bf{x}},y)\in \Omega . \label{eq:target1}$$ 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 $$\mathop{\rm Trace}(a_{ij})= \sum_{i=1}^{n}a_{ii} =1 \label{eq:traceis1}$$ 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 $$|\phi ({\bf{x}},\pm M)-\Phi(\pm M)|\leq g_{1}( |{\bf{x}}| ) \quad \mbox{for } |{\bf x}|\geq N_{1} . \label{eq:conditionofboundary1}$$ \end{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 $$a_{nn}({\bf{x}},y, z,{\bf{v}})\geq \mu (a,b) \label{eq:mu(a,b)}$$ 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 $$k''(y)+E(y,k,k')=0 \quad\mbox{on } |y|\le M, \quad k(\pm M)=\Phi (\pm M). \label{eq:equationsoforiginalk}$$ \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 $$\big|f({\bf{x}},y)-k(y)\big|\leq C_{J}g_{1}(\frac{1}{2^{J}}|{\bf{x}}| ) + C_{J} g_{2}(\frac{1}{2^{J}}|{\bf{x}}|) + \frac{C_{J}}{|{\bf{x}}|^{J\beta }} \quad\mbox{on } \Omega . \label{eq:conclusion}$$ 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|H_{0}$$$0< \chi (H)<1; \quad \frac{22MH}{c_{1}}\leq A(H)e^{\chi (H)} \leq \frac{66MH}{c_{1}}; \label{eq:boundofae}$$ and the function $$z=\gamma +A(H)e^{\chi (H)} -\{ (h_{a}^{-1}(y+M))^2-|{\bf x}-{\bf x}_{0}|^2 \} ^{1/2} \label{eq:barrier111}$$ 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|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 $$\chi(\alpha)=\int_{\alpha}^{\infty}\ \frac{d\rho}{\rho^{3}\Phi_{1}(\rho)} =\int_{\alpha}^{\infty}\ \frac{d\rho}{c_{2}\rho^{3}} = \frac{1}{2c_{2}} \alpha^{-2} . \label{eq:chi2}$$ Thus $$\eta (\beta ) = (2c_{2}\beta)^{-1/2}\quad\mbox{for } 0<\beta <(2c_{2})^{-1}. \label{eq:beta1}$$ 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 $$A(H) = 2M (\int_{1}^{e^{\chi(H)}}\ \eta (\ln t) dt)^{-1} . \label{eq:ah}$$ For the rest of this article, we set a=A(H) and define $$h_{a}(r)=\int_{r}^{ae^{\chi(H)}} \eta (\ln \frac{t}{a} )\ dt \hspace{5 mm}\mbox{for}\ \ a\le r\le ae^{\chi(H)}.$$ Then $$h_{a}(ae^{\chi (H)})=0, \quad h_{a}(a) =h_{A(H)}(A(H))= 2M. \label{eq:chi3}$$ For aH, \\ h_{a}''(r)=\frac{1}{r}(\eta(\ln \frac{r}{a} ))^{3}\Phi_{1} (\eta(\ln \frac{r}{a} )). \end{gathered} \label{eq:derivativelarge} Thus for a2 such that for H\geq H_{0}, $$H_{0}>\frac{1}{\sqrt{2c_{2}}} +3M+4+\frac{24nc_{1}K_{0}}{M}, \quad \sqrt{\frac{4K_{0}}{A(H)e^{\chi (H)}}}\leq \frac{1}{\sqrt{2}}. \label{eq:Hlarge}$$ 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} . $$Thus $$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}$$ For {\bf x}_{0}\in \mathbb{R}^{n-1}, a constant \gamma with 0\leq \gamma <1 and a fixed constant K with 00 and c_{3} (depending only on k, E), such that for any constant \delta_{1} with |\delta_{1}|<\min \{ \gamma_{1}, 1\}, there is a (unique) function k_{\lambda }(y)=k_{\lambda }(y) in C^{1}([-M,M])\cap C^{2}((-M,M)) satisfying$$ 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}|. $$\end{proposition} \begin{proof}[Proof of Theorem \ref{thm1}] We assume that there exists a non-increasing function g(t) such that $$|f({\bf{x}},y)-k(y)|\leq g(|{\bf{x}}|)\quad \mbox{for } ({\bf{x}},y)\in \Omega . \label{eq:assumedecay}$$ (since f({\bf{x}},y) and k(y) are bounded, (\ref{eq:assumedecay}) holds for g(t) to be some appropriate constant. g(t) will also take other forms as we shall explain later). For a small positive number \delta_{1} (to be chosen later), let k_{\lambda }(y) be the function defined in the Proposition. We will use the barrier function u({\bf{x}},y) + k_{\lambda } (y). Let \begin{gather*} K_{0}=\sup \{ |f({\bf{x}},y)|: ({\bf{x}},y)\in \Omega \} +\sup \{ |k(y)| : |y|\leq M \}+g(0),\\ K_{1} = 2 \sup \{ |k'(y)| : |y|\leq M \} +10\,. \end{gather*} and c_{1} be a number such that $$c_{1} \leq a_{nn}({\bf{x}},y,t,{\bf{v}}) \label{eq:coeffboundthm2}$$ for ({\bf{x}},y)\in \Omega, |t|\leq 40K_{0}+20, {\bf{v}} \in R^{n} with |{\bf{v}}|\leq K_{1} +2. Since g_{1}(t)\to 0 as t\to \infty, there is a number H_{1} such that g_{1}(\frac{1}{2}|{\bf{x}}|)\leq \frac{1}{2} for |{\bf{x}}|\geq H_{1} and H_{1}>800MK_{0}/c_{1}. We fixed an {\bf{x}}_{0} with |{\bf{x}}_{0}|\geq H_{0}+H_{1} (H_{0} is given in (\ref{eq:barrier111})). Let u({\bf{x}},y)=z({\bf{x}},y) defined on \Omega_{{\bf{x}}_{0}, H,K} be given by (\ref{eq:barrier111}) with the choice of parameters:$$ \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}|), $$and $$|{\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} | .$$ Then for |{\bf{x}}_{0}| large, on \Omega_{{\bf{x}}_{0}, H,K} , (where c_{9} is independent of \delta_{1}) $$\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}$$ Then from (\ref{eq:upperbarrier2}), for |{\bf{x}}_{0}|\geq H_{0}+H_{1} and any positive constant b, b<10K_{0}+1, on \Omega_{{\bf{x}}_{0}, H,K}, we have \begin{gather*} u({\bf{x}},y) + k_{\lambda } (y) + b \leq 40K_{0}+20, |Du({\bf{x}},y)| + |k'_{\lambda } (y)| \leq K_{1} +1. \end{gather*} Set$$ 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 00, 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 $$\frac{3c_{1}}{4c_{3}} \delta_{1} = \frac{Cc_{9}(g(\frac{1}{2}|{\bf{x}}_{0}|))^{\alpha_{0}} }{d^{\alpha_{0}}} + \frac{c_{9}M_{3}g(\frac{1}{2}|{\bf{x}}_{0}|)}{d^{2}} +g_{2}(d) . \label{eq:definitionofdelta12}$$ Then on \Omega_{{\bf{x}}_{0}, H, K}, we have (since |{\bf{x}}|\geq d and g_{2} is non-increasing) \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} 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