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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 146, 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}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/146\hfil
Remarks on the Phragm\'en-Lindel\"of theorem]
{Remarks on the Phragm\'en-Lindel\"of theorem for $L^p$-viscosity
solutions of fully nonlinear PDEs with unbounded ingredients}

\author[S. Koike, K. Nakagawa\hfil EJDE-2009/146\hfilneg]
{Shigeaki Koike, Kazushige Nakagawa}  % in alphabetical order

\address{Shigeaki Koike \newline
Department of Mathematics, Saitama University,
255 Shimo-Okubo, Sakura, Saitama 338-8570, Japan}
\email{skoike@rimath.saitama-u.ac.jp}

\address{Kazushige Nakagawa \newline
Department of Mathematics, Saitama University,
255 Shimo-Okubo, Sakura, Saitama 338-8570, Japan}
\email{knakagaw@rimath.saitama-u.ac.jp}

\thanks{Submitted August 28, 2009. Published November 20, 2009.}
\subjclass[2000]{35B53, 35D40, 35B50}
\keywords{Phragm\'en-Lindel\"of theorem; $L^p$-viscosity
solution; \hfill\break\indent weak Harnack inequality}

\begin{abstract}
 The Phragm\'en-Lindel\"of theorem for $L^p$-viscosity solutions of
 fully nonlinear second order elliptic partial differential
 equations with unbounded coefficients and inhomogeneous terms is
 established.
\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}

The notion of $L^p$-viscosity solutions was introduced in  \cite{CCKS}
to study fully nonlinear second order elliptic partial differential
equations (PDEs for short) with unbounded  inhomogeneous terms.
We refer to \cite{C} (see also \cite{CC}) as a pioneering work
for the regularity theory of viscosity solutions of fully nonlinear
 PDEs.

It turned out that the Aleksandrov-Bakelman-Pucci (ABP for short)
maximum principle can be extended to $L^p$-viscosity solutions for
fully nonlinear second order elliptic PDEs  with unbounded coefficients
and inhomogeneous terms in \cite{KS2}.
See also \cite{N} for a generalization.

As an application of the ABP maximum principle in \cite{KS2}, it
is known that the (boundary) weak Harnack inequality for
$L^p$-viscosity solutions of the associated extremal PDEs in
\cite{KS3} (see also \cite{KS4}) holds, which implies H\"older
continuity for $L^p$-viscosity solutions of fully nonlinear
elliptic PDEs with unbounded ingredients. We also refer to
\cite{Si} for H\"older continuity estimates on $L^p$-viscosity
solutions by a different approach.

On the other hand, qualitative properties of viscosity solutions
of fully nonlinear elliptic PDEs have been investigated as
generalizations for classical elliptic PDE theory. For instance,
the ABP maximum principle in unbounded domains in \cite{CdLV} and
\cite{KS3}, the Liouville property in \cite{CuL,CdC},
the Hadamard principle in \cite{CdC}, and the
Phragm\'en-Lindel\"of theorem in \cite{CdV}. We refer to
references in \cite{CdV,CuL,CdC} for these
qualitative properties of strong/classical solutions.

Our aim here is to extend the Phragm\'en-Lindel\"of theorem in
\cite{CdV} when PDEs have unbounded coefficients (i.e. $\mu$ in
this paper). In view of the boundary weak Harnack inequality in
\cite{KS3}, it is natural to relax the hypotheses on coefficients
and inhomogeneous terms. However, for the weak Harnack inequality,
we need to suppose that the coefficient to the first derivatives
is small enough in $L^n$-norm. When we work in bounded domains,
this is not a restriction.
Since we are concerned with unbounded domains, we will need a bit
more delicate analysis than  those in \cite{CdV}.

Since our argument is essential to treat  domains of conical type
(i.e. the case for $\eta >0$ in our notation), we will mainly
discuss this case.
We will add corresponding results for domains of cylindrical
type (i.e. the case for $\eta=0$).

Our paper is organized as follows: section 2 is devoted to showing
the definitions and known results. In section 3, we present the
ABP type estimates on $L^p$-viscosity subsolutions of fully
nonlinear PDEs with unbounded ingredients under appropriate
geometric conditions. We  show  the Phragm\'en-Lindel\"of theorem
in our setting in section 4.
In section 5, we give a proof of an elementary geometric property, which
is needed in the proof of Lemma \ref{lem3-1}.


\section{Preliminaries}

We consider fully nonlinear second order PDEs
in unbounded domains $\Omega \subset \mathbb{R}^n$:
\begin{equation}\label{ell}
    G(x,u,Du,D^2u)=f(x)\quad\text{in }\Omega ,
\end{equation}
where $G : \Omega \times \mathbb{R} \times \mathbb{R}^n \times S^n\to \mathbb{R}$
and $f : \Omega \to \mathbb{R}$ are given measurable functions.
We also suppose that
$(r,p,M)\in \mathbb{R}\times \mathbb{R}^n\times S^n\to G(x,r,p,M)$ is
continuous for almost all $x\in\Omega$.
Here, $S^n$ denotes the set of symmetric matrices of order $n$
equipped with the standard order.

We will use the standard notation from \cite{GT}.
We denote by $L^p_{+}(\Omega)$ the set of all nonnegative
functions in $L^p(\Omega)$.

Throughout this paper, we assume that
$$
p>\frac{n}{2}.
$$
We recall two  facts: if $u\in W^{2,p}_{\rm loc}(\Omega)$ for
$p>\frac{n}{2}$,
then we may identify $u$ with a continuous function on $\Omega$,
and $u$ is twice differentiable for almost all $x\in\Omega$.

First of all, we recall the definition of $L^p$-viscosity
solutions of \eqref{ell}.

\begin{definition} \label{def2.1} \rm
We call $u\in C(\Omega)$ an $L^p$-viscosity subsolution
(resp., supersolution) of \eqref{ell} if
\begin{gather*}
 \mathop{\rm ess\,lim\, inf}_{x\to x_0}\{ G(x,u(x), D\phi (x),
 D^2\phi (x)) -f(x) \} \leq 0 \\
 \Big(\text{resp., } \mathop{\rm  ess\,lim\,sup}_{x\to x_0}
\{ G(x,u(x), D\phi(x), D^2\phi(x)) - f(x)\} \ge 0 \Big)
\end{gather*}
whenever $\phi \in W^{2,p}_{{\rm loc}}(\Omega)$ and
$x_0\in \Omega$ is a local maximum (resp., minimum) point of $u-\phi$.
A function $u\in C(\Omega)$ is called an $L^p$-viscosity solution of \eqref{ell} if it is
both an $L^p$-viscosity subsolution
and an $L^p$-viscosity supersolution of \eqref{ell}.
\end{definition}

To make easier recalling the right  inequality,
we will often say that $u$ is an $L^p$-viscosity solution of
\begin{gather}
    G(x,u,Du,D^2u)\leq f(x) \label{subsol}\\
\big(\text{resp.},\quad G(x,u,Du,D^2u)\geq f(x) \big)\label{supersol},
\end{gather}
if it is an $L^p$-viscosity subsolution (resp., supersolution)
of \eqref{ell}.

\begin{remark} \label{rmk2.2} \rm
If $u$ is an $L^p$-viscosity subsolution (resp., supersolution)
of \eqref{ell},
then it is also an $L^q$-viscosity subsolution (resp., supersolution) of
\eqref{ell} provided $q \geq p$.
\end{remark}

In what follows, instead of \eqref{ell},
we mainly consider PDEs which do not depend on $u$-variable:
\begin{equation}\label{PDE}
F(x,Du,D^2u)=f(x)\quad\text{in }\Omega.
\end{equation}

We will assume that $F$ is (degenerate) elliptic: 
\begin{equation}\label{C-ell}
\begin{gathered}
F(x,p,M) \leq F(x,p,N)\\
\text{for }(x,p,M,N)\in\Omega
\times\mathbb{R}^n\times S^n\times S^n\text{ provided }M\geq N.
\end{gathered}
\end{equation}
For fixed  ellipticity constants $0<\lambda \leq\Lambda$,
we assume that
\begin{equation}\label{C-SC}
\begin{gathered}
\text{there is }\mu\in L^q_+(\Omega)\text{ such that}\\
\mathcal{P}^-(M) -\mu (x) |p| \leq F(x,p,M)\quad
\text{for }(x,p,M)\in \Omega \times
\mathbb{R}^n\times S^n,
\end{gathered}
\end{equation}
where the Pucci operators $\mathcal{P}^\pm :S^n\to \mathbb{R}$ are defined by
$$
 \mathcal{P}^-(M)=\min\{ -\text{trace}(AM) :A\in S^n_{\lambda,\Lambda}\},\quad
 \mathcal{P}^+(M)=-\mathcal{P}^-(-M).
$$
Here,
$S_{\lambda,\Lambda}^n := \{ M\in S^n :\lambda I\leq M\leq \Lambda I\}$.
We refer the reader to \cite{CdV} for examples of PDEs which satisfy
(\ref{C-ell}) and (\ref{C-SC}).
We first recall a lemma concerning  change of unknown functions.

\begin{lemma}[{\cite[Lemma 1]{CdV}}] \label{lem2-1}
Assume  \eqref{C-ell} and \eqref{C-SC} with
$\mu\in L^q_{+}(\Omega)$ for $q>n$.
Then, there exist constants $h_j>0$ $(j=1,2)$ satisfying the
following property:
if $\xi \in C^2(\Omega)$ satisfies
$$
\xi (x)> 0,\quad \frac{|D\xi |}{\xi}(x) \leq k_1(x),
\quad \frac{|D^2\xi |}{\xi}(x)\leq k_2(x)
\quad\text{for }  x\in \Omega
$$
with some functions $k_j\in C(\Omega)$ $(j=1,2)$, then
for $L^p$-viscosity subsolution  $w\in C(\Omega)$ of \eqref{PDE}
with $f\in L^p_{+}(\Omega)$, $u:= \frac{w}{\xi}$
is an $L^p$-viscosity solution of
\begin{equation}
    \mathcal{P}^-(D^2u) - \gamma_1(x) |Du| - \gamma_2(x) u \leq \frac{f(x)}{\xi(x)}\quad
    \text{in }\Omega [u],
\end{equation}
where $\Omega [u] = \{x\in \Omega \ |\ u(x)> 0 \}$,
$\gamma_1(x) =  h_1 k_1(x) + \mu (x)$ and
$\gamma_2(x) =  h_2 k_2(x) + k_1(x) \mu (x)$.
\end{lemma}

We will use the constant $p_0=p_0(n,\lambda ,\Lambda)\in [\frac{n}{2},n)$,
for which we refer to \cite{E}.
It is known that for $p>p_0$, and $f\in L^p(B_r(z))$,
where $B_r(x)=\{ y\in\mathbb{R}^n :|x-y|<r\}$,
there exists a (unique) strong solution $u\in C(\overline{B}_r(z))
\cap W^{2,p}_{\rm loc}(B_r(z))$ of
$$
\mathcal{P}^-(D^2v(x))=f(x)\quad \text{a.e. in }B_r(z)$$
under $v(x)=0$ for $x\in \partial B_r(z)$ with estimates:
$$
-C\| f^-\|_{L^p(B_r(z))}\leq v(x)\leq C\| f^+\|_{L^p(B_r(z))}\quad
\text{in }B_r(z),
$$
where $C=C(n,\lambda,\Lambda,p)>0$ is a constant,
and for $0<s<r$,
$$
\| v\|_{W^{2,p}(B_s(z))}\leq C'\| f\|_{L^p(B_r(z))},
$$
where $C'=C'(n,\lambda,\Lambda ,p,r-s)>0$.

We remark that to prove the ABP maximum principle
\cite[Theorem 2.9]{KS2}, which implies the boundary weak Harnack
inequality \cite[Theorem 6.1]{KS3}, it suffices to obtain the
existence  of strong solutions of the above extremal equation only
in balls although  this fact is not clearly mentioned in
\cite{KS2,KS3}. In fact, this existence result holds with local
$W^{2,p}$-estimates for more general domains  satisfying the
uniform exterior cone property but the $p_0\in [\frac{n}{2},n)$
associated with general domains might be bigger than the above. We
also notice that we may replace cubes by balls in the (boundary)
weak Harnack inequality in \cite{KS3} by Cabr\'e's covering
argument, which we will see in the proof of Lemma \ref{lem3-1}
below.

Fix $R>0$ and $z\in\mathbb{R}^n$.
Let $T, \ T'\subset B_R(z)$ be domains  such that
$$
\overline{T}\subset T',\quad \text{and}\quad \theta_0
\leq \frac{|T|}{|T'|} \leq 1\quad\text{for some }\theta_0>0.
$$
When we apply our weak Harnack inequality below,
our choice of $T$ and $T'$ always satisfies the above condition.

For a given domain $A\subset \mathbb{R}^n$ and a function $v\in C(A)$,
we define $v^-_{T',A}$ on $T'\cup A$ by
\[
v^-_{T',A}(x) =  \begin{cases}
               \min\{v(x), m\} & \text{if }x\in A,\\
               m & \text{if }x\in T'\setminus A,
               \end{cases}
\]
where
\[
    m = \liminf _{x\to T'\cap \partial A} v(x).
\]
We note that if $T'\cap \partial A\neq \emptyset$, then
$v^-_{T',A}$ is a real-valued function and that if
$T'\cap\partial A\neq\emptyset$, $v$ is a nonnegative
$L^p$-viscosity supersolution of
(\ref{PDE}) and $f\leq 0$ in $T'\cap A$,
then $v^-_{T',A}$ is a nonnegative $L^p$-viscosity supersolution
of (\ref{PDE}) in $T'$.

Next, we recall the boundary weak Harnack inequality
when PDEs have unbounded coefficients and inhomogeneous terms.


\begin{lemma}[{\cite[Theorem 6.1]{KS3}}] \label{BwH}
Let $T$, $T'$, $A$ be as above.
Assume that $T\cap A\neq \emptyset$ and $T'\setminus A\neq \emptyset$
and that
\begin{equation}\label{C-pq}
q>n,\quad q\geq p>p_0.
\end{equation}
Then, there exist constants $\varepsilon_0=\varepsilon_0(n,\lambda,\Lambda)>0$,
$r=r(n,\lambda,\Lambda ,p,q)>0$ and
$C_0=C_0(n,\lambda,\Lambda,p,q)>0$
satisfying the following property: if $\mu\in L^q_+(T'\cap A)$,
$f\in L^p_+(T'\cap A)$, a nonnegative $L^p$-viscosity solution
$w\in C(T'\cap A)$  of
\[
    \mathcal{P}^+(D^2w) + \mu(x)|Dw| \geq -f(x) \quad \text{in }T'\cap A,
\]
and
\begin{equation}\label{e_0}
\| \mu\|_{L^n(T'\cap A)}\leq \varepsilon_0,
\end{equation}
then it follows that
\[
    \Big( \frac{1}{|T|}\int_{T} (w^-_{T',A})^r\,dx \Big)^{1/r} \leq
    C_0 \Big( \inf_{T}w^-_{T',A} +  R\| f\|_{L^n(T'\cap A)} \Big)
\]
provided that $q>n$ and $q\geq p\geq n$, and
\begin{align*}
&\Big(\frac{1}{|T|}\int_T (w^-_{T',A})^rdx \Big)^{1/r}\\
&\leq C_0\Big(\inf_Tw^-_{T',A}+R^{2-\frac{n}{p}}\| f\|_{L^p(T'\cap A)}
\sum_{k=0}^{M}
R^{(1-\frac{n}{q})k}\|\mu\|^k_{L^q(T'\cap A)} \Big)
\end{align*}
provided  that $q>n>p>p_0$,
where $M=M(n,p,q)\geq 1$ is an integer.
\end{lemma}

\begin{remark} \label{rmk2.5} \rm
We refer to \cite{KS4} for the (boundary) weak Harnack inequality
for $L^p$-viscosity supersolutions of fully nonlinear
PDEs with superlinear growth in the gradient and unbounded ingredients.
\end{remark}

In the next section, we will establish some local and global
ABP type estimates on  $L^p$-viscosity subsolutions for (\ref{PDE}).
To this end, we recall the notations concerning  the
shape of domains from \cite{CdV}.


\begin{definition}[Local geometric condition] \label{def1} \rm
Let $\sigma, \tau \in (0,1)$.
We call $y\in \Omega$ a $G_{\sigma, \tau}$ point in $\Omega$
if there exist  $R=R_y>0$ and $z=z_y\in \mathbb{R}^n$
such that
\begin{equation}\label{localG}
  y\in B_R(z),\quad\text{and}\quad |B_R(z)\backslash
 \Omega_{y, B_R(z), \tau}| \geq  \sigma |B_R(z)|,
\end{equation}
where $\Omega_{y, B_R(z), \tau}$ is the connected component of
$B_{\frac{R}{\tau}}(z)\cap \Omega$ containing $y$.
For $\sigma,\tau\in (0,1)$, and $R_0>0$, $\eta \geq 0$,
    we call $y\in \Omega$ a $G^{R_0, \eta}_{\sigma, \tau}$ point
    in $\Omega$ if  $y$ is a $G_{\sigma,\tau}$ point in $\Omega$ with
    $R=R_y>0$ and $z=z_y$ satisfying
\begin{equation}\label{rateR}
R \leq R_0 + \eta|y|.
\end{equation}
\end{definition}


\begin{remark} \label{rmk2.7} \rm
For the sake of simplicity of notations, for a $G_{\sigma,\tau}$ point
$y\in \Omega$,
we will write  $B_y$ for $B_{\frac{R_y}{\tau}}(z_y)$, where
$R_y>0$ and $z_y\in\mathbb{R}^n$ are from Definition \ref{def1}.
\end{remark}


\begin{definition}[Global geometric condition] \label{def2.8} \rm
We call $\Omega$ a $\hat G^{R_0,\eta}_{\sigma,\tau}$ domain
if any $y\in\Omega$ is a  $G^{R_0, \eta}_{\sigma, \tau}$ point
in $\Omega$.
\end{definition}

We refer the reader to \cite{Vit} and \cite{CdV}
for examples of domains $\Omega$ satisfying $G_{\sigma,\tau}^{R_0,\eta}$.
We also refer to \cite{ARV} for a generalization.


\section{ABP type estimates}


We present pointwise estimates on $L^p$-viscosity subsolutions of
(\ref{PDE}),
which is often referred as the  Krylov-Safonov growth lemma.

In what follows, we fix $\sigma,\tau \in (0,1)$ and $R_0>0$.
Let $y\in\Omega$ be a $G_{\sigma,\tau}^{R_0,\eta}$ point with $\eta \geq 0$.
It is possible to apply our weak Harnack inequality in
$B_y$, which is from Definition \ref{def1},
if $\|\mu\|_{L^n(B_y\cap\Omega)}\leq \varepsilon_0$.
Here and later, $\varepsilon_0>0$ is the constant from Lemma \ref{BwH}.


Even if $\| \mu\|_{L^n(B_y\cap \Omega)}>\varepsilon_0$, we may use
Cabr\'e's covering argument; we divide $B_y$ into small pieces so
that we may apply the weak Harnack inequality in each piece. We
then obtain the weak Harnack inequality in $B_y$ by combining all
the inequalities for small pieces.

However, since we need the estimates uniform in $y\in\Omega$,
this argument does not work immediately because of unboundedness
of $\{ R_y\}_{y\in\Omega}$ when $\eta >0$.

To avoid this difficulty, we will suppose a decay rate of $\mu$:
$\|\mu\|_{L^q(\Omega\setminus B_t(0))}=o(t^{-(1-\frac{n}{q})})$.
More precisely,  for fixed $q>n$, we suppose that
for all  $\delta>0$  there is $T_\delta>0$  such that
\begin{equation}\label{hypomu}
\|\mu\|_{L^q(\Omega\setminus B_t(0))}\leq \delta t^{-(1-\frac{n}{q})}
\quad \text{for }t\geq T_\delta.
\end{equation}

\begin{remark} \label{rmk3.1} \rm
It is assumed in \cite{CdV} that $\mu (x)=O (|x|^{-1})$ as $|x|\to\infty$,
which only implies
$\|\mu\|_{L^q(\Omega\setminus B_t(0))}=O(t^{-(1-\frac{n}{q})})$.
\end{remark}

Of course, if $\eta =0$ (hence  $R_y\leq R_0$), then we can
apply directly Cabr\'e's argument. 
%When we write $\eta$ in $G_{\sigma,\tau}^{R_0,\eta}$ etc., we always suppose $\eta>0$
%throughout this paper. When we discuss the case of $\eta=0$, we will write $G_{\sigma,\tau}^{R_0,0}$ etc.

\begin{lemma} \label{lem3-1}
Assume that \eqref{C-ell}, \eqref{C-pq} and \eqref{C-SC} hold
with $\mu\in L^q_{+}(\Omega)$.
Let $\eta>0$ and   $y\in \Omega$ be a $G^{R_0,\eta}_{\sigma,\tau}$
 point in $\Omega$  with $R=R_y>0$ and $z=z_y\in\mathbb{R}^n$.
Then, there exist
 $\kappa = \kappa
 (n, \lambda, \Lambda,  \sigma, \tau, R_0, \eta ) \in (0,1)$
and $\varepsilon=\varepsilon(n,\sigma, \eta)>0$
satisfying the following property:
if  $w\in C(\Omega)$ is an
$L^p$-viscosity subsolution of \eqref{PDE} with $f\in L^p_{+}(\Omega)$,
 then we have the following properties:
(i) Assume that $|y|\leq  R_0$. (a) If  $p\geq n$, then
$$
w(y) \leq \kappa \sup_{B_y\cap \Omega}w^+
+ (1-\kappa) \limsup_{x\to B_y\cap\partial\Omega} w^+
+R_0\|f\|_{L^n(B_y\cap \Omega)}.
$$
(b) If $p_0<p<n$, then
\begin{align*}
w(y) &\leq \kappa \sup_{B_y\cap \Omega}w^+
+ (1-\kappa) \limsup_{x\to B_y\cap\partial\Omega} w^+ \\
&\quad +R_0^{2-\frac{n}{p}}\|f\|_{L^p(B_y\cap \Omega)}\sum_{k=0}^M
R_0^{(1-\frac{n}{q})k}\|\mu\|^k_{L^q(B_y\cap \Omega)}.
\end{align*}
(ii) Assume that  \eqref{hypomu} is satisfied  and that $|y|>R_0$.
(a) If $p\geq n$, then
$$
w(y) \leq \kappa \sup_{B_y\cap \Omega}w^+
+ (1-\kappa) \limsup_{x\to B_y\cap\partial\Omega} w^+
+R\|f\|_{L^n(B_y\cap \Omega\setminus B_{\varepsilon R}(0))}.
$$
(b) If  $p_0<p<n$, then
\begin{align*}
w(y) &\leq \kappa \sup_{B_y\cap \Omega}w^+
+ (1-\kappa) \limsup_{x\to B_y\cap\partial\Omega} w^+\\
&\quad +R^{2-\frac{n}{p}}\|f\|_{L^p(B_y\cap
 \Omega\setminus B_{\varepsilon R}(0))}
\sum_{k=0}^MR^{(1-\frac{n}{q})k}\|\mu\|^k_{L^q(B_y\cap \Omega
\setminus B_{\varepsilon R}(0))}.
\end{align*}
Here $M=M(n,p,q)\geq 1$ is the integer in Lemma \ref{BwH}.
\end{lemma}

\begin{remark} \label{rmk3.3} \rm
To get the weak maximum principle (Lemma \ref{lem4-1} below),
it is important to have the term
$\| f\|_{L^p(B_y\cap \Omega\setminus B_{\varepsilon R}(0))}$
instead of $\| f\|_{L^p(B_y\cap \Omega)}$ in the estimates of
the assertion
(ii) above.
\end{remark}

\begin{proof}
First of all, we shall omit giving the proof in
the case of $\|\mu\|_{L^q(\Omega)}=0$ because
it is easy to do it, and we suppose that $\|\mu\|_{L^q(\Omega)}>0$.

It is enough to show the assertion when
$\hat C:=\limsup_{x\to B_y\cap\partial\Omega} w^+(x)= 0$.
In fact, after having established the assertion when $\hat C= 0$,
we may apply the result to $w-\hat C$ to prove the assertion in the
general case.


Due to (\ref{C-SC}), $w$ is an $L^p$-viscosity solution of
\[
    \mathcal{P}^-(D^2w) - \mu(x) |Dw| \leq f(x)\quad \text{in }\Omega.
\]
Setting $C_w=\sup_{B_y\cap \Omega}w^+$,
we immediately see that $v(x):= C_w -w(x)$ is
an $L^p$-viscosity solution of
\[
    \mathcal{P}^+(D^2v) +\mu(x)|Dv| \geq -f(x)\quad\text{in }\Omega.
\]

We shall first prove  (ii).

\textbf{Case (ii) $ |y|> R_0$:}
Fix $\varepsilon\in (0,\frac{1}{2}\min\{\frac{1}{1+\eta}, ( \frac{\sigma}{4} )^\frac{1}{n}\})$.
Note that $2\varepsilon <1/(1+\eta)$ and $(2\varepsilon)^n<\sigma/4$.
We set $T=B_R(z)\setminus \overline B_{2\varepsilon R}(0)$ and
$T'=B_y\setminus \overline B_{\varepsilon R}(0)$.
Observe that
$$
2\varepsilon R<\frac{R}{1+\eta}\leq \frac{R_0+\eta |y|}{1+\eta}<|y|
$$
and consequently
$y\in T=B_R(z)\setminus \overline{B}_{2\varepsilon R}(0)$.
Let  $A$ be the connected component of $T'\cap \Omega$ which contains $y$.
We have
\begin{align*}
|T\backslash A| &\geq |T \backslash \Omega_{y,B_R(z),\tau}|\\
 & \geq |B_R(z)\backslash \Omega_{y,B_R(z),\tau}|-
                    |B_{2\varepsilon R}(0)|\\
 & \geq \sigma |B_R(0)| - (2\varepsilon )^n|B_R(0)|\\
               & \geq \frac{\sigma}{2} |B_R(0)|\\
&\geq \frac{\sigma}{2} |T|.
\end{align*}
Since
\begin{equation}\label{setinclusion}
    T'\cap \partial A \subset T'\cap \partial (T'\cap \Omega)
                      \subset T' \cap (\partial T' \cup \partial \Omega)
                      = T'\cap \partial \Omega,
\end{equation}
in view of $\hat C\leq 0$, we have
\begin{equation}\label{4-3}
    \liminf_{x\to T'\cap \partial A} v(x) = C_w -
    \limsup_{x\to T'\cap \partial A} w(x)
 \geq C_w.
\end{equation}
Now, we verify (\ref{e_0}).
By (\ref{hypomu}), we can choose $T_\varepsilon>0$ such that
$$\|\mu\|_{L^q(\Omega\setminus B_t(0))}\leq
\frac{\varepsilon_0}{|B_1(0)|^{\frac{1}{n}(1-\frac{n}{q})}}
\left(\frac{\tau\varepsilon}{t}\right)^{1-\frac{n}{q}}
\quad\text{for }t\geq T_\varepsilon.
$$
Assume  $R\geq A_1:=T_\varepsilon \varepsilon^{-1}$.
Using the above, we see
$$
\|\mu\|_{L^n(T'\cap A)}\leq |B_1(0)|^{\frac{1}{n}(1-\frac{n}{q})}
\left(\frac{R}{\tau}\right)^{1-\frac{n}{q}}
\| \mu\|_{L^q(\Omega\setminus B_{\varepsilon R}(0))}\leq  \varepsilon_0.
$$
Setting $m=\liminf_{x\to T'\cap \partial A} v(x)$,
we use (\ref{4-3}) to show for any $r>0$,
$$
 \left( \frac{\sigma}{2} \right)^{1/r} C_w
\leq \Big( \frac{|T\backslash A|}{|T|} \Big)^{1/r} C_w
     \leq  \Big( \frac{1}{|T|}\int_{T\backslash A}m^r dx\Big)^{1/r}
     \leq  \Big( \frac{1}{|T|}\int_{T}(v^-_{T',A})^r dx\Big)^{1/r}.
$$
Since $y\in  A$,  we have
\begin{equation}\label{4-4}
    \inf_{T} v_{T',A}^- \leq v(y) = C_w -w(y).
\end{equation}
Thus, letting $r>0$ be the constant from Lemma \ref{BwH}, we have
$$
\left(\frac{\sigma}{2}\right)^{1/r}C_w
\leq C_0 \left( \inf_{T}v_{T',A}^- + R \|f\|_{L^n(T'\cap A)} \right)
\leq  C_0 \left( C_w -w(y) + R\|f\|_{L^n(T'\cap \Omega)}\right)%\nonumber
$$
if $p\geq n$, and
$$
\left( \frac{\sigma}{2} \right)^{1/r} C_w\leq
C_0\Big( C_w-w(y)+\| f\|_{L^p(T'\cap \Omega)}\sum_{k=0}^M
R^{(1-\frac{n}{q})k+2-\frac{n}{p}}
\|\mu\|^k_{L^q(T'\cap \Omega)}\Big)
$$
if $p\in (p_0,n)$.
Therefore, we conclude that the assertion (ii) holds with $\kappa
= 1-(\frac{\sigma}{2})^{1/r}\min\{C_0^{-1},1\}>0$ in the case
where $R\geq A_1$.

Next assume that $R<A_1$.
We can choose  constants
\[
\rho_0=\rho_0(n,q,\tau,\varepsilon_0,\varepsilon,A_1,
\|\mu\|_{L^q(\Omega)}),
\]
$\mu_0=\mu_0(n,q,\tau,\varepsilon_0,\varepsilon,A_1,\|\mu\|_{L^q(\Omega)})\in (0,1)$,
$N_0=N_0(n,q,\tau,\varepsilon_0,\varepsilon,A_1,\|\mu\|_{L^q(\Omega)})\in\mathbb{N}$ and
a finite sequence $\{ x_i\}_{i=1}^{N_0}\subset T'$
such that
\begin{gather}\label{cabre1}
\overline T\subset \cup_{i=1}^{N_0}B_{\rho_0 R}(x_i)\subset \cup_{i=1}^{N_0}
\overline B_{2\rho_0 R}(x_i)
\subset T', \\
\label{cabre2}
|B_{\rho_0 R}(x_i)\cap B_{\rho_0 R}(x_{i+1})|\geq \mu_0 |B_{\rho_0R}(0)|,
\end{gather}
where $B_{\rho_0R}(x_{N_0+1})=B_{\rho_0 R}(x_1)$,
and
\begin{equation}\label{cabre3}
\rho_0\leq \frac{1}{A_1|B_1(0)|^{1/n}}\Big(
\frac{\varepsilon_0}{\|\mu\|_{L^q(\Omega)}}\Big)^{\frac{q}{q-n}}.
\end{equation}
We see that
$$
\|\mu\|_{L^n(B_{\rho_0R}(x_i))}
\leq |B_{\rho_0R}(x_i)|^{\frac{1}{n}-\frac{1}{q}}
\|\mu\|_{L^q(B_y\cap \Omega)}\leq \varepsilon_0.
$$

For the reader's convenience, we recall Cabr\'e's covering
argument when $p\geq n$. Since $v^-_{T',A}$ is a nonnegative
$L^p$-viscosity supersolution of $\mathcal{P}^+(D^2u)+\mu (x)|Du|\geq
-f(x)$ in $T'$, in view of Lemma \ref{BwH}, we have
$$
\| v^-_{T',A}\|_{L^r(B_{\rho_0 R}(x_i))}\leq |B_{\rho_0R}(x_i)|^{1/r}
C_0\Big(\inf_{B_{\rho_0 R}(x_i)}v^-_{T',A}+\rho_0R\| f\|_{L^n(A)}\Big)
$$
for $i=1,2,\dots, N_0$, where $r,C_0>0$ are from Lemma \ref{BwH}.
Furthermore, for $i\in\{ 1,2,\dots ,N_0\}$, setting
$B_i=B_{\rho_0R}(x_i)$, we have
\begin{align*}
\inf_{B_i}v^-_{T',A}&\leq \inf_{B_i\cap B_{i+1}}v^-_{T',A}\\
&\leq \Big(\frac{1}{|B_i\cap B_{i+1}|}
\int_{B_i\cap B_{i+1}}(v^-_{T',A})^rdx\Big)^{1/r}\\
&\leq C_1\Big(\inf_{B_{i+1}}v^-_{T',A}+R\| f\|_{L^n(A)}\Big)
\end{align*}
for some $C_1\geq 1$. Thus, repeating this argument,
for $1\leq i<N_0$, we have
$$
\inf_{B_i}v^-_{T',A}\leq C_1^{N_0-1}
\Big(\inf_{B_{N_0}}v^-_{T',A}+N_0R\| f\|_{L^n(A)}\Big).
$$
Since we may assume  that $\inf_{T}v^-_{T',A}=\inf_{B_{N_0}}v^-_{T',A}$,
there is $C_2>0$ such that
$$
\| v^-_{T',A}\|_{L^r(T)}\leq \sum_{i=1}^{N_0}
\| v^-_{T',A}\|_{L^r(B_i)}\leq R^{\frac{n}{r}}
C_2\left(
\inf_{T}v^-_{T',A}+R\| f\|_{L^n(A)}\right) .
$$
When $p_0<p<n$, we can easily apply the above argument to show that
$$
\| v^-_{T',A}\|_{L^r(T)}
\leq  R^{\frac{n}{r}}
C_2\Big(\inf_{T}v^-_{T',A}+R^{2-\frac{n}{p}}\| f\|_{L^p(A)}
\sum_{k=0}^MR^{(1-\frac{n}{q})k}\|\mu\|^k_{L^q(A)}\Big) .
$$
What remains of the proof follows the same argument as in the case of
$R\geq A_1$.


\textbf{Case (i) $|y|\leq {R_0}$:}
Since  we have $R\leq (1+\eta)R_0$ in this case,
we may regard $\Omega$ as a bounded domain.
Thus, we can use the standard covering argument by Cabr\'e without
using (\ref{hypomu}).
Setting $T=B_R(z)$, $T'=B_{\frac{R}{\tau}}(z)$ and
$A=\Omega_{y, B_R(z), \tau}$,
 we have
$$
|T\backslash A|= |B_R (z)\setminus \Omega_{y, B_R(z), \tau}|
                   \geq \sigma |B_R(z)| \geq \frac{\sigma}{2}|T|.
$$
We shall only give a proof when
$\|\mu\|_{L^n(T'\cap A)}\leq\varepsilon_0$.

Following the same argument as in case (ii) with the above inequality,
and new $A,T,T'$, we have
$$
\left( \frac{\sigma}{2} \right)^{1/r} C_w
\leq C_0 \left( \inf_{T}v_{T',A}^- + R_0 \|f\|_{L^n(B_y\cap \Omega)} \right)
  \leq  C_0 \left( C_w -w(y) + R_0\|f\|_{L^n(B_y\cap\Omega)}\right)%\nonumber
$$
provided that $p\geq n$, and
$$
\left( \frac{\sigma}{2} \right)^{1/r} C_w
\leq
C_0\Big( C_w-w(y)+\| f\|_{L^p(B_y\cap \Omega)}
\sum_{k=0}^MR_0^{(1-\frac{n}{q})k+2-\frac{n}{p}}
\|\mu\|^k_{L^q(B_y\cap \Omega)}\Big)
$$
provided that $p\in (p_0,n)$.
Therefore, we conclude that the assertion holds with the same
$\kappa\in (0,1)$ as in case (ii).
\end{proof}

\begin{remark} \label{rmk} \rm
The above proof clearly shows that $\varepsilon$ can be 
any constant satisfying
$0<\varepsilon <\frac{1}{2}\min\{ \frac{1}{1+\eta},
(\frac{\sigma}{4})^{1/n}\}$.
In the above proof, we have stated  that $N_0$ can be chosen independently
of $z$ and $R$, which may not be trivial.
%It seems trivial once we write down all the cases of couples $(T,T')$.
We will give a proof of this fact in Appendix.
\end{remark}

The corresponding result for $\eta =0$ is as follows.


\begin{corollary}\label{lem3-12}
Assume that \eqref{C-ell}, \eqref{C-pq} and \eqref{C-SC}
with $\mu\in L^q_{+}(\Omega)$.
Let $y\in \Omega$ be a $G^{R_0,0}_{\sigma,\tau}$ point in
$\Omega$  with $R=R_y>0$ and $z=z_y\in\mathbb{R}^n$.
Then, there exist
 $\kappa = \kappa
 (n, \lambda, \Lambda,  \sigma, \tau, R_0 ) \in (0,1)$
and $\varepsilon=\varepsilon(n,\sigma)>0$
satisfying the following property:
if  $w\in C(\Omega)$ is an
$L^p$-viscosity subsolution of \eqref{PDE} with $f\in L^p_{+}(\Omega)$,
 then  the same estimates  as in Lemma \ref{lem3-1} (i) hold.
\end{corollary}

In the case of $\eta =0$, we always have $|y|\leq R_0$ unlike
 Lemma \ref{lem3-1}.
For the proof of the above corollary, we just follow the steps in
the proof of Lemma \ref{lem3-1} (i).

When $\Omega\subset\mathbb{R}^n$ is a $\hat G^{R_0,\eta}_{\sigma,\tau}
$ domain, we derive the ABP maximum principle for $L^p$-viscosity
subsolutions bounded from above of (\ref{PDE}).


\begin{theorem}[ABP maximum principle in unbounded domains] \label{ThABP}
Assume  \eqref{C-pq},\\ \eqref{C-ell} and \eqref{C-SC} with
$\mu\in L^q_{+}(\Omega)$ satisfying \eqref{hypomu}.
Let $\eta>0$ and $\Omega\subset\mathbb{R}^n$ be a
$\hat G^{R_0,\eta}_{\sigma,\tau}$ domain.
Assume also
\begin{equation}\label{f}
\begin{gathered}
\sup_{y\in\Omega,|y|>R_0}R_y\| f\|_{L^n(A_y\cap \Omega)}<\infty
\quad \text{if } p\geq n,\\
\sup_{y\in\Omega,|y|>R_0}R_y^{2-\frac{n}{p}}
\| f\|_{L^p(A_y\cap \Omega)}<\infty \quad\text{if } p_0<p<n.
\end{gathered}
\end{equation}
Let $0<\varepsilon<\min\{\frac{1}{1+\eta},(\frac{\sigma}{4})^{1/n}\}$.
Then, there exists
\[
C=C(n,\lambda,\Lambda,p,q,\varepsilon,\sigma,\tau,R_0,\eta)>0
\]
satisfying the following properties:
if  $w\in C(\Omega)$ is an $L^p$-viscosity subsolution bounded from above of
\eqref{PDE} with $f\in L^p_{+}(\Omega)$,
then it follows that
\begin{equation}\label{ABP1}
\begin{aligned}
\sup_\Omega w
&\leq \limsup_{x\to\partial \Omega}w^+(x)+ C
\sup_{y\in \Omega, |y| >R_0} R_y
\| f\|_{L^n(A_y\cap\Omega)}\\
&\quad + CR_0 \sup_{y\in \Omega, |y| \leq R_0}\| f\|_{L^n(B_y\cap\Omega)},
\end{aligned}
\end{equation}
provided that $p\geq n$, and
\begin{equation}\label{ABP2}
\begin{aligned}
\sup_\Omega w
&\leq \limsup_{x\to\partial \Omega}w^+(x)
  + C \sup_{y\in \Omega,|y|>R_0}
R_y^{2-\frac{n}{p}}\| f\|_{L^p(A_y\cap\Omega)}\sum_{k=0}^M
R_y^{(1-\frac{n}{q})k}\|\mu\|_{L^q(A_y\cap\Omega)}^k\\
&\quad +CR_0^{2-\frac{n}{p}}\sup_{y\in\Omega, |y|\leq R_0}
\| f\|_{L^p(B_y\cap\Omega)}
\sum_{k=0}^MR_0^{(1-\frac{n}{q})k}\|\mu\|_{L^q(B_y\cap\Omega)}^k
\end{aligned}
\end{equation}
provided that $p\in (p_0,n)$.
Here, $A_y=B_{\frac{R_y}{\tau}}(z_y)\setminus
B_{\varepsilon R_y}(0)$ and
$B_y=B_{\frac{R_y}{\tau}}(z_y)$.
\end{theorem}


\begin{proof}
We take the supremum over $y\in \Omega$ with the estimates in
Lemma \ref{lem3-1} to conclude
the inequalities (\ref{ABP1}) and (\ref{ABP2}).
\end{proof}

\begin{remark} \label{rmk3.7} \rm
By following  our proof of Lemma \ref{lem3-1} (ii),
it is easy to show that (\ref{hypomu}) implies
\begin{equation}\label{mu2}
\sup_{y\in\Omega,|y|>R_0}R_y^{1-\frac{n}{q}}
\|\mu\|_{L^q(A_y\cap\Omega)}<\infty .
\end{equation}
\end{remark}

To show the ABP maximum principle in unbounded domains corresponding to
the case $\eta =0$, we do not need to assume (\ref{f}) since $R_y\leq R_0$.


\begin{corollary}\label{ThABP2}
Assume  \eqref{C-pq}, \eqref{C-ell} and \eqref{C-SC} with
$\mu\in L^q_{+}(\Omega)$.
Let  $\Omega\subset\mathbb{R}^n$ be a $\hat G^{R_0,0}_{\sigma,\tau}$
domain.
Then, there exists $C=C(n,\lambda,\Lambda,p,q,\varepsilon,\sigma,\tau,R_0)>0$
satisfying the following properties:
if  $w\in C(\Omega)$ is an $L^p$-viscosity subsolution bounded from
above of \eqref{PDE} with $f\in L^p_{+}(\Omega)$,
then it follows that  \eqref{ABP1} holds provided $p\geq n$, and that
\eqref{ABP2} holds provided $p\in (p_0,n)$.
\end{corollary}


\section{Phragm\'en-Lindel\"of theorem}

In this section, we  show that the  weak maximum principle holds
for PDEs with  zero-order terms.
As before, assuming that $\Omega$ is a $\hat G_{\sigma,\tau}^{R_0,\eta}$ domain,
 for each $y\in\Omega$,
we use the notations  $R_y>0$ and $z_y\in\mathbb{R}^n$.
Also, $B_y$ and $A_y$, respectively,
denote $B_{\frac{R_y}{\tau}}(z_y)$ and $B_{\frac{R_y}{\tau}}(z_y)\setminus
B_{\varepsilon R_y}(0)$ for $\varepsilon\in (0,\frac{1}{2}
\min\{ \frac{1}{1+\eta},(\frac{\sigma}{4})^{1/n}\} )$.

\begin{lemma}\label{lem4-1}
Assume  \eqref{C-ell}, \eqref{C-pq}  and \eqref{C-SC} with
$\mu\in L^q_+(\Omega)$ satisfying \eqref{hypomu}.
Let $\eta>0$ and $\Omega$ be a $\hat G^{R_0,\eta}_{\sigma,\tau}$ domain.
Then, there exists $c_0=c_0(n,\lambda,\Lambda,p,q,\sigma,\tau,R_0,\eta)>0$
satisfying the following property:
if $c\in L^n_{+}(\Omega)$,
 $w\in C(\Omega)$ is an  $L^p$-viscosity solution bounded from above of 
\begin{equation}\label{F+c}
    F(x, Dw, D^2w) - c(x)w^+ \leq 0\quad \text{in }
    \Omega
\end{equation}
such that
\begin{equation}\label{Dirichlet}
   \limsup_{x\to \partial \Omega}w(x) \leq 0,
\end{equation}
and
\begin{equation}\label{c-c}
K_0:=\max\big\{\sup_{y\in\Omega,|y|>R_0}
\| \hat c\|_{L^n(A_y\cap \Omega)} ,\sup_{y\in\Omega,|y|\leq R_0}
\|c\|_{L^n(B_y\cap \Omega)}\big\} \leq c_0,
\end{equation}
where $\hat c(x)=(1+|x|^2)^{1/2}c(x)$,
then $w\leq 0$ in $\Omega$.
\end{lemma}

\begin{remark} \label{rmk4.2} \rm
Instead of \eqref{c-c}, it is assumed in  \cite{CdV} that
\begin{equation}\label{c-c2}
c(x)\leq \frac{c_0}{1+|x|^2}\quad\text{for }x\in\Omega .
\end{equation}
Set
$c(x)=\frac{1}{1+|x|^2}$.
We easily see by following an argument in the proof
 of Lemma \ref{BwH} (ii) that the
$K_0$ associated with this $c$ is finite.
\end{remark}

\begin{proof}
Note that by (\ref{C-SC}) together with Remark \ref{rmk2.2},
$w$ is an $L^n$-viscosity solution of
\[
    \mathcal{P}^-(D^2w) - \mu(x)|Dw| - c(x)w^+\leq 0.
\]
We apply Theorem \ref{ThABP} with $f=cw^+$
to obtain that when $|y| \leq R_0$,
\[
R_0 \|cw^+\|_{L^n(B_y\cap\Omega)}
      \leq R_0 \sup_{\Omega}w^+ \|c\|_{L^n(B_y\cap \Omega)}
      \leq R_0 K_0 \sup_{\Omega}w^+.
\]
On the other hand, when $|y|>R_0$,  we have
\begin{equation}\label{cw}
R_y \|c w^+\|_{L^n(A_y\cap \Omega)}
      \leq \frac{R_y}{\sqrt{1+(\varepsilon R_y)^2}} \sup_\Omega w^+
      \|\hat c\|_{L^n(A_y\cap\Omega)}
      \leq \frac{K_0}{\varepsilon} \sup_\Omega w^+.
\end{equation}
Choosing $\varepsilon_1=\frac{1}{4}\min\{ \frac{1}{1+\eta},
(\frac{\sigma}{4})^{1/n}\}$ for instance, we have
\[
    \sup_\Omega w \leq C_3\max\big\{ R_0,\frac{1}{\varepsilon_1}\big\} c_0 \sup_\Omega w^+
\]
for some constant $C_3>0$.
Taking $c_0<1/(C_3\max\{ R_0,\frac{1}{\varepsilon_1}\})$,
 we conclude the proof.
\end{proof}

The next Corollary can be proved exactly same as above by using
Corollary \ref{ThABP2} instead of Theorem \ref{ThABP}.


\begin{corollary}\label{lem4-12}
Assume  \eqref{C-ell}, \eqref{C-pq}  and \eqref{C-SC} with
$\mu\in L^q_+(\Omega)$.
Let $\Omega$ be a $\hat G^{R_0,0}_{\sigma,\tau}$ domain.
Then, there exists $c_0=c_0(n,\lambda,\Lambda,p,q,\sigma,\tau,R_0)>0$
satisfying the following property:
if $c\in L^n_{+}(\Omega)$ and
 $w\in C(\Omega)$ is an  $L^p$-viscosity solution bounded from above of
$(\ref{F+c})$
such that \eqref{Dirichlet}
and \eqref{c-c} hold, then $w\leq 0$ in $\Omega$.
\end{corollary}



\begin{theorem}[Phragm\'en-Lindel\"of theorem] \label{ThPL}
Assume  \eqref{C-ell}, \eqref{C-pq} and \eqref{C-SC} with
$\mu\in L_{+}^q(\Omega)$ satisfying \eqref{hypomu}.
Let $\eta>0$ and $\Omega$ be a $\hat{G}^{R_0,\eta}_{\sigma,\tau}$ domain.
If $w\in C(\Omega)$ is an $L^p$-viscosity solution of
\begin{equation}\label{F=0}
    F(x, Dw, D^2w) \leq 0\quad\text{in } \Omega
\end{equation}
such that \eqref{Dirichlet} holds and
\begin{equation}\label{decayw}
   w^+(x)=O ( \log  |x|)\quad\text{as }|x|\to\infty,
\end{equation}
then $w\leq 0$ in $\Omega$.
\end{theorem}

\begin{remark} \label{rmk4.5} \rm
In \cite{CdV},  it is assumed that
$w^+(x)=O(|x|^\alpha)$ with a constant $\alpha>0$ as $|x|\to\infty$,
which is weaker than (\ref{decayw}).
In fact, to deal with unbounded coefficients (i.e. $\mu$),
we will have to use a different function $\xi$ to apply Lemma \ref{lem2-1}.
This is the reason why we suppose a restrictive growth rate (\ref{decayw})
in comparison with that in \cite{CdV}.
\end{remark}

\begin{proof}[Proof of Theorem \ref{ThPL}]
Define a positive smooth function
\[
\xi(x) = \log (1+(1+|x|^2)^{\beta/2}),
\]
where $\beta>0$ will be fixed later, 
and  set $u=w / \xi$, which is bounded from above.
A straightforward calculation shows that
\begin{gather*}
\frac{|D\xi|}{\xi}(x)
      \leq \frac{\beta}{(1+|x|^2)^{1/2}\log(1+(1+|x|^2)
  ^{\beta/2})}=:k_1(x),\\
\frac{|D^2\xi|}{\xi}(x)
   \leq \frac{\beta C_4}{(1+|x|^2)\log (1+(1+|x|^2)^{\beta/2})}
      =:k_2(x)
\end{gather*}
for some $C_4>0$.
Thus, in view of  Lemma \ref{lem2-1}, we see that $u$ is an
$L^n$-viscosity solution of
\[
    \mathcal{P}^-(D^2u) - \gamma_1(x) |Du| - \gamma_2(x) u^+ \leq 0\quad\text{in }\Omega ,
\]
where
\begin{gather*}
\gamma_1(x)
      = \frac{h_1\beta}{(1+|x|^2)^{1/2}\log(1+(1+|x|^2)^{\beta/2})}
      + \mu (x)
      =:\gamma_{11}(x)+\gamma_{12}(x)\\
\begin{aligned}
\gamma_2(x)
      &= \frac{ h_2 \beta C_4}{(1+|x|^2)\log(1+(1+|x|^2)^{\beta/2})}
      + \frac{\beta \mu (x)}{(\log 2)(1+|x|^2)^{1/2}}\\
      &=: \gamma_{21}(x)+\gamma_{22}(x)
      \end{aligned}
\end{gather*}
We first show that $\gamma_1$ satisfies (\ref{hypomu}).
Note that we only need to show that $\gamma_{11}$
satisfies (\ref{hypomu}).
Setting $g(x)=(|x|\log |x|)^{-1}$ for $|x|>1$,
we easily show $\| g\|_{L^q(B_t^c(0))}=o(t^{-(1-\frac{n}{q})})$ as
$t\to\infty$, which implies that  $\gamma_{11}$ satisfies (\ref{hypomu}).

We next show  that \eqref{c-c} holds for $\gamma_2$.
We shall observe that
\begin{equation}\label{K0}
K_0':=\max\big\{\sup_{y\in\Omega,|y|>R_0}\| \hat\gamma_2\|_{L^n(A_y\cap\Omega)},
\sup_{y\in\Omega,|y|\leq R_0}\|\gamma_2\|_{L^n(B_y\cap\Omega)}\big\}
\end{equation}
is small when $\beta\to 0$, where
$\hat\gamma_2(x)=\sqrt{1+|x|^2}\gamma_2(x)$.

When $y\in\Omega$ satisfies $|y|\leq R_0$, we see that
$B_y\subset B_{R_0(2+\eta +\tau^{-1}(1+\eta))}(0)$.
Thus, the second term in (\ref{K0}) can be small when $\beta >0$
is small enough.

To estimate the first term of (\ref{K0}), we note
that $A_y=B_y\setminus B_{\varepsilon R_y}(0) \subset B_{\varepsilon R_y}(0)^c$
provided $\varepsilon <\frac{1}{2(1+\eta)}$.
Setting $\hat\gamma_{22}(x)=\sqrt{1+|x|^2}\gamma_{22}(x)$,
by (\ref{hypomu}), we can choose $T_0>1$ such that
$$
\|\hat\gamma_{22}\|_{L^q( \Omega\setminus B_t(0))}
\leq \beta t^{-(1-\frac{n}{q})}\quad\text{for }t\geq T_0.
$$
Hence,  for $R_y>A_2:=\frac{T_0}{\varepsilon}$, we have
$$
\|\hat\gamma_{22}\|_{L^n(A_y\cap\Omega)}
\leq C_5R_y^{1-\frac{n}{q}}\|\hat\gamma_{22}\|_{L^q(A_y\cap\Omega)}\leq C_5
\frac{\beta}{\varepsilon_1^{1-\frac{n}{q}}}
$$
for some $C_5>0$, where $\varepsilon_1=\frac{1}{4}\min\{\frac{1}{1+\eta},
(\frac{\sigma}{4})^{1/n}\}$.
If $R_y\leq A_2$, then we have
$$
\|\hat\gamma_{22}\|_{L^n(A_y\cap \Omega)}
\leq C_6\beta R_y^{1-\frac{n}{q}}\|\mu\|_{L^q(\Omega)}
\leq C_6\beta A_2^{1-\frac{n}{q}}\|\mu\|_{L^q(\Omega)}
$$
for some $C_6>0$. Thus, in this case, we may suppose that
$\|\hat\gamma_{22}\|_{L^n(A_y\cap\Omega)}$
is small by taking small $\beta>0$.

The remaining case is to prove that $\sup_{y\in\Omega,|y|>R_0}
\|\hat\gamma_{21}\|_{L^n(A_y\cap\Omega)}$ is small, where
$\hat\gamma_{21}(x)=
\sqrt{1+|x|^2}\gamma_{21}(x)$.
To this end, we shall show that for any $c_0>0$, there is
small $\beta>0$ such that
$\|\hat\gamma_{21}\|_{L^n(\mathbb{R}^n)}\leq c_0$.
Since
$$
\int_t^\infty \frac{1}{r (\log r)^n}dr
=\frac{1}{(n-1)(\log t)^{n-1}}\quad \text{for }t>1,
$$
we can choose $\hat T>1$ independent of $\beta >0$ such that
$\|\hat\gamma_{21}\|_{L^n(B_{\hat T}(0)^c)}\leq c_0/2$.
For this fixed $\hat T>0$,
we can find small $\beta>0$ such that
$\|\hat\gamma_{21}\|_{L^n(B_{\hat T}(0))} \leq  c_0/2$.
Therefore, using Lemma \ref{lem4-1} with $\mu=\gamma_1$ and
$c=\gamma_2$,
we get $u\leq 0$. This concludes the proof.
\end{proof}

Our Phragm\'en-Lindel\"of theorem for $\eta =0$ is as follows.


\begin{corollary}[Phragm\'en-Lindel\"of theorem] \label{ThPL2}
Assume  \eqref{C-ell}, \eqref{C-pq} and \eqref{C-SC} with
$\mu\in L_{+}^q(\Omega)$.
Let $\Omega$ be a $\hat{G}^{R_0,0}_{\sigma,\tau}$ domain.
If $w\in C(\Omega)$ is an $L^p$-viscosity solution of
\eqref{F=0} such that \eqref{Dirichlet} and \eqref{decayw} hold,
then $w\leq 0$ in $\Omega$.
\end{corollary}

\begin{proof}
The only difference from the proof of Theorem \ref{ThPL} is
how to estimate $\hat \gamma_{22}$.
However, since $R_y\leq R_0$, we can show it immediately.
\end{proof}


\section{Appendix: A proof of an elementary geometric property}

In the proof of Lemma \ref{lem3-1}, the integer $N_0$ might depend on
$y\in\Omega$ such that $|y|>R_0$ and $R:=R_y<A_1$.
We shall show that the integer $N_0$ has an upper bound independent of
such $y\in\Omega$.
To this end, we recall our domains $T$ and $T'$ in this case:
$T=B_R(z)\setminus \overline B_{2\varepsilon R}(0)$ and
$T'=B_{\frac{R}{\tau}}(z)\setminus \overline B_{\varepsilon R}(0)$.

We note that  the position of $(T,T')$ varies depending on the distance of
two centers; $|z|$.


For $t\in [0,1]$, we denote by $(T_t,T_t')$ the couple
$(T,T')$ when $|z|=(1-t)(\frac{1}{\tau}+2\varepsilon )$.
For instance,  $T_1$ and $T'_1$ are annuli with the common center at $z=0$
while $T_0=B_R(z)$ and  $T_0'=B_{\frac{R}{\tau}}(z)$.
All the possible positions of $(T,T')$ can be found in $\{ (T_t,T_t') :
t\in [0,1]\}$.
For each $(T_t,T_t')$,
it is easy to find an integer $N_{0,t}\in\mathbb{N}$ satisfying
(\ref{cabre1}), (\ref{cabre2}),
(\ref{cabre3}) with $N_0=N_{0t}$.

For any fixed $t\in [0,1]$, we can choose
$\{ x_{i,t}\}_{i=1}^{N_{0,t}}\subset
T_t'$
such that (\ref{cabre1}), (\ref{cabre2}), (\ref{cabre3}) with
$N_0=N_{0,t}$, $x_i=x_{i,t}$, $T=T_t$ and $T'=T_t'$.
We can find $\delta_t>0$ such that
(\ref{cabre1}) holds for $T=T_s$ and $T'=T_s'$ for
$s \in I_t:=(t-\delta_t,t+\delta_t)\cap [0,1]$ because $(T_t,T_t')$
changes continuously in $t$.
Since $[0,1]\subset \cup_{t\in [0,1]} I_t$,
we can choose a finite set $\{ t_k\in [0,1]\}_{k=1}^L$ such that
$[0,1]\subset \cup_{k=1}^L I_{t_k}$.
Therefore, we can take $\hat N:=\max\{ N_{0,t_k} :k=1,2,\dots, L\}$ as
an upper bound for $N_0$.

\subsection*{Acknowledgements}
The authors  want to thank the anonymous referee for several
 suggestions and comments on the first draft of this article.

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