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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 144, pp. 1--29.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/144\hfil Global subsonic flow]
{Global subsonic flow in a 3-D infinitely long curved nozzle}

\author[W. Chen, G. Xu, Q. Xu \hfil EJDE-2017/144\hfilneg]
{Wenxia Chen, Gang Xu, Qin Xu}

\address{Wenxia Chen \newline
Faculty of Science, 
Jiangsu University, 
Zhenjiang, Jiangsu 212013, China}
\email{chenwx@ujs.edu.cn}

\address{Gang Xu (corresponding author)\newline
Faculty of Science,
Jiangsu University,
Zhenjiang, Jiangsu 212013, China}
\email{gxu@ujs.edu.cn}

\address{Qin Xu \newline
Faculty of Science,
Jiangsu University,
Zhenjiang, Jiangsu 212013, China}
\email{qinxu\_math@163.com}  

\dedicatory{Communicated by Goong Chen}

\thanks{Submitted April 27, 2017. Published June 17, 2017.}
\subjclass[2010]{35L70, 35L65, 35L67, 76N15}
\keywords{Subsonic flow; potential flow equation; 
weighted H\"older space; 
\hfill\break\indent global solution}

\begin{abstract}
 In this article, we focus on the  existence and stability of a
 subsonic global solution in an infinitely long curved nozzle for the
 three-dimensional steady potential flow equation.
 By introducing some suitably weighted H\"older spaces and establishing
 a series of a priori estimates on the solution to second order
 linear elliptic equation in an unbounded strip domain with two
 Neumann boundary conditions and one periodic boundary condition
 with respect to some variable, we show that the global subsonic
 solution of potential flow equation in a 3-D nozzle exists
 uniquely when the state of subsonic flow at negative infinity is
 given. Meanwhile, the asymptotic state of the subsonic solution at
 positive infinity as well as the asymptotic behavior at minus
 infinity are also studied.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction and statement of main results}

The  existence of global subsonic flows in infinitely long nozzles or
past obstacles is a fundamental problem in fluid dynamics. Such a
problem has been extensively studied by many authors
(see \cite{1,2,4,5, 7,8,9,10,14,15,16} and the references therein).
 For examples,
for 2-D or 3-D potential flow equations, if the speed of the gas is
assumed to be suitably low and the gas passes an obstacle, then it
is shown in \cite{4} and \cite{9,10} respectively that the whole subsonic
flows outside 2-D or 3-D obstacles exist uniquely. Very recently,
such a 2-D result on potential flow equation has been extended into
the 2-D full Euler system case in \cite{7} when it is supposed that the
low velocity gas hits a symmetric obstacle. With respect to the 2-D
subsonic potential flows in the infinitely long nozzles, those authors in
\cite{14,15,16} have shown the global existence and stability, in
particular, those authors in \cite{16} established the monotonicity of the
maximum of the flow speed with respect to the incoming mass flux. In
this paper, our focus is on the 3-D subsonic potential flow equation
in a 3-D infinitely long nozzle.

Now we use the potential flow equation to describe the motion of the
subsonic gas in a 3-D nozzle. Let $\varphi(x)$ be the potential of
velocity $u=(u_1,u_2,u_3)$, i.e., $u_i=\partial_i\varphi$, then it follows
from the Bernoulli's law that
\begin{equation} \label{e1.1}
\frac{1}{2}|\nabla\varphi|^2+h(\rho)=C_0,
\end{equation}
here $\nabla=(\partial_1,\partial_2,\partial_3)$, $h(\rho)=c^2(\rho)/(\gamma-1)$ is the
specific enthalpy for the polytropic gas with the state equation
$P=A\rho^{\gamma}$ $(1<\gamma<3)$ and the sonic speed
$c(\rho)=\sqrt{P'(\rho)}$, $C_0=\frac{1}{2}q_0^2+h(\rho_0)$ stands
for the Bernoulli's constant, where the far velocity field $(q_0,
0, 0)$ at minus infinity of the nozzle is subsonic, i.e.,
$q_0<c(\rho_0)$ holds true.

By use of \eqref{e1.1} and the implicit function theorem, the density
function $\rho(x)$ of gas can be expressed as
\begin{equation} \label{e1.2}
\rho=h^{-1}(C_0-\frac{1}{2}|\nabla\varphi|^2) \equiv H(\nabla\varphi).
\end{equation}

Substituting \eqref{e1.2} into the mass conservation equation
$\sum_{j=1}^3\partial_j(\rho u_j)=0$ of gas yields
\begin{equation} \label{e1.3}
\begin{aligned}
&\big((\partial_1\varphi)^2-c^2\big)\partial_1^2\varphi
+\big((\partial_2\varphi)^2-c^2\big)\partial_2^2\varphi
+\big((\partial_3\varphi)^2-c^2\big)\partial_3^2\varphi \\
&+2\partial_1\varphi\partial_2\varphi\partial_{12}^2\varphi
+2\partial_1\varphi\partial_3\varphi\partial_{13}^2\varphi+2\partial_2\varphi\partial_3\varphi\partial_{23}^2\varphi =0,
\end{aligned}
\end{equation}
 here $c=c(H(\nabla\varphi))$.

We assume that the 3-D infinitely nozzle $\Omega_0$(Figure \ref{fig1})
is bounded by the walls:
$\gamma_1=\{x: x_2=\varepsilon \tilde f_1(x_1,x_3), x_1\in \mathbb{R}, x_3\in(0,\frac 12)\}$,
$\gamma_2=\{x: x_2=1+\varepsilon \tilde f_2(x_1,x_3),  x_1\in \mathbb{R}, x_3\in(0,\frac 12)\}$,
$\gamma_3=\{x: x_3=0\}$ and $\gamma_4=\{x: x_3=\frac 12\}$, here
$\tilde f_i(x_1,x_3)\in C_0^{\infty}((-X_0,X_0)\times(0,\frac 12))$ for some fixed positive
constant $X_0$, and $\varepsilon>0$ is a suitably small constant, for example,
we can choose
\[
\tilde f_1(x_1,x_3)=\begin{cases}
-\exp(\frac{1}{x_1^2+(x_3-\frac14)^2-\frac{1}{25}})
 &\text{for }  \sqrt{x_1^2+(x_3-\frac14)^2}\le 1/5\\
0 &\text{for }\sqrt{x_1^2+(x_3-\frac14)^2}> 1/5
\end{cases}
\]
and $\tilde f_2(x_1,x_3)=-\tilde f_1(x_1,x_3)$. As
illustrated in \cite[(3.8), (3.9)]{6}, by the anti-symmetric extension
in the $x_3$-direction with
respect to the function $\tilde f_i(x_1,x_3) (i=1,2)$ and the
potential function $\varphi(x)$, then the domain $\Omega_0$ can be changed
 to
\[
\Omega_1=\{x: \varepsilon f_1^0(x_1,x_3)\le x_2\le 1+\varepsilon f_2^0(x_1,x_3),
x_1\in \mathbb{R}, x_3\in [0,1]\}
\]
(Figure \ref{fig2}),  where
\[
f_i^0(x_1,x_3)=\begin{cases}
\tilde f_i(x_1,x_3),& 0\le x_3\le 1/2 \\
\tilde f_i(x_1,1-x_3),&1/2 \le x_3\le 1.
\end{cases}
\]
 Then by using periodic extension in the $x_3-$direction, we can use the following
unbounded strip domain $\Omega$ instead of $\Omega_1$ to consider our
problem \eqref{e1.1} with $\varphi(x_1,x_2,x_3)=\varphi(x_1,x_2,x_3+1)$, where
$\Omega$ is bounded by
$\Gamma_1=\{x: x_2=\varepsilon f_1(x_1,x_3),-\infty<x_1,x_3<+\infty\}$ and
$\Gamma_2=\{x: x_2=1+\varepsilon f_2(x_1,x_3),-\infty<x_1,x_3<+\infty\}$, here
$f_i(x_1,x_3+1)=f_i(x_1,x_3)$ and
$f_i(x_1,x_3)\in C_0^{\infty}((-X_0,X_0)\times(-\infty,+\infty))$.
 More concretely,
$f_i(x_1,x_3)=f_i^0(x_1,x_3-l)$ for $l< x_3\le l+1$ with
$l\in\mathbb{Z}$.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\end{center}
\caption{Domain $\Omega_0$} \label{fig1}
\end{figure}


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2}
\end{center}
\caption{Anti-symmetric extension $\Omega_0$ to $\Omega_1$ with respect to $x_3=\frac12$}
\label{fig2}
\end{figure}


Since the flow is tangent to the nozzle walls, then one has
\begin{equation} \label{e1.4}
(\partial_1\varphi,\partial_2\varphi,\partial_3\varphi)\cdot (\varepsilon\partial_1 f_i, -1, \varepsilon\partial_3 f_i)=0\quad\text{on }
\Gamma_i,\quad i=1,2.
 \end{equation}
In addition, we suppose that the the state of subsonic flow at minus
infinity satisfies
\begin{equation} \label{e1.5}
\lim_{x_1\to-\infty}(\varphi(x)-q_0x_1)=0.
\end{equation}
On the other hand, from the physical point of view (see \cite{4,5,7,8,9,10}
 and the references therein), when a subsonic flow in an
unbounded domain is called to be stable, it should admit a
determined state at infinity. Namely,
\begin{equation} \label{e1.6}
\lim_{x_1\to+\infty}\nabla\varphi(x)\quad \text{ exists for }x\in\Omega.
\end{equation}

Our result read as follows.

\begin{theorem} \label{thm1.1}
If the 3-D unbounded strip domain $\Omega$ is defined
by $\Gamma_1=\{x: x_2=\varepsilon f_1(x_1,x_3),-\infty<x_1,x_3<+\infty\}$ and
$\Gamma_2=\{x: x_2=1+\varepsilon f_2(x_1,x_3),-\infty<x_1,x_3<+\infty\}$, here
$f_i(x_1,x_3+1)=f_i(x_1,x_3)$ and
$f_i(x_1,x_3)\in C_0^{\infty}((-X_0,X_0)\times(-\infty,+\infty))$ for some fixed
constant $X_0>0$, then there exists a small constant $\varepsilon_0>0$
such that the problem \eqref{e1.3}-\eqref{e1.6} has a global smooth solution
$\varphi(x)$ as $\varepsilon<\varepsilon_0$, which admits
\begin{itemize}
\item[(i)] $\varphi(x_1,x_2,x_3)=\varphi(x_1,x_2,x_3+1)$.

\item[(ii)] $|\nabla\varphi|<c(H(\nabla\varphi))$. Namely, the flow is globally subsonic
in the whole domain $\Omega$.

\item[(iii)] For $x_1<0$ and $x\in\Omega$, there exist a suitable constant
$\delta_0>0$ and a constant $C_0>0$ such that
$$
|\varphi(x)-q_0x_1|+|\nabla(\varphi(x)-q_0x_1)|\le C_0\varepsilon e^{-\delta_0|x_1|}.
$$

\item[(iv)] For $x_1>0$ and $x\in\Omega$, there exists a constant $C_0>0$
such that
$$
|\varphi(x)-q_0x_1|\le C_0\varepsilon (1+x_1).
$$

\item[(v)] $\lim_{x_1\to+\infty, x\in\Omega}\nabla\varphi(x)=(q_0,0,0)$
holds. Moreover, for $x_1>0$ and $x\in\Omega$, there exists a
constant $C_0>0$ such that
$$|\nabla_{x_2,x_3}\varphi(x)|\le C_0\varepsilon e^{-\delta_0x_1},$$
here $\delta_0>0$ is given in (iii).
\end{itemize}
\end{theorem}

Besides the estimates described by Theorem \ref{thm1.1}, we can give more
detailed asymptotic properties on the subsonic solution $\varphi(x)$
and its derivatives in $\Omega$ when $x_1\to\pm\infty$. This will be stated
more precisely in Theorem \ref{thm2.3}.

 Although there have been many results on the
weighted $W^{2,p}(\Omega)$ ($1<p<\infty$) estimates of solution to the
second order linear elliptic equation in an unbounded strip domain
$\Omega$ or half-space $\Omega$ (see \cite{3,12,13} and the references
therein), it is difficult for us to use these results to treat the
existence of solution to the quasilinear elliptic equation \eqref{e1.3} as
well as the asymptotic state and asymptotic behavior at minus or
positive infinity of solution since the related weighted Sobolev
spaces in \cite{3,12,13} can not be imbedded into the suitable
H\"older space $C^{\delta}(\bar\Omega)$ with some positive constant
$\delta>0$.

Now we mention some works which are related to this paper.
In \cite{4, 9, 10}, for the case of  the gas past an obstacle, by using
the Kelvin transformation, those authors have reduced the exterior
domain problem on the 2-D or 3-D potential flow equation into a
boundary value problem in a bounded domain. From this, together
with the maximum principle, some a priori estimates on the
solutions to second order linear elliptic equations and Schauder
fixed point theorem, those authors have shown that the global subsonic
flow field exists uniquely outside the obstacle. Although it seems
that the subsonic nozzle flow problem is perhaps similar to or
even simpler than the one for the subsonic flow past a profile,
which is also roughly described in \cite[page 75]{5} as ``the
problem of finding a subsonic flow in a given channel is
mathematically simpler than that of finding the flow past an
airfoil'', we find that these two problems have actually some
differences:

(i) One is that the Kelvin transformation used in \cite{4} and
\cite{9,10} can not be applied directly to our nozzle problem due to
the different geometric properties between the exterior domain and
the infinitely long nozzle.

(ii) Another one is that the asymptotic
properties of subsonic flow at minus infinity and positive
infinity are very different for the nozzle problem (however, the
far fields of subsonic flow at infinity are uniform for the
subsonic flow past an obstacle, one can see \cite{4,9,10}).

There also have some essential differences between 2-D and 3-D
subsonic nozzle flow. With
respect to the 2-D subsonic nozzle flow, those authors in \cite{16} use
the stream function $\psi$ to reduce the 2-D potential flow
equation into a second order quasilinear elliptic equation on
$\psi$, meanwhile, the fixed nozzle wall conditions are
correspondingly changed into the Dirichlet boundary value
conditions on $\psi$. By use of this kind of crucial reduction in
2-D case (at this time, the maximum principle and comparison
principle can be directly applied due to the appearance of
Dirichlet boundary value), together with some a priori uniform
estimates on the solutions to the suitably modified nonlinear
equations in some well-chosen bounded domains with the suitable
Dirichlet boundary values, those authors established the global
existence, stability and the monotonicity of the maximum of the
flow speed with respect to the incoming mass flux. However, for
the 3-D subsonic nozzle flows, the streamline function method does
not work (this is also illustrated in Chapter VI of \cite{8}), we have
to directly treat the 3-D potential flow equation with the fixed
nozzle wall condition, which is described by the Neumann boundary
value condition. In this case, the crucial comparison principle on
second order elliptic equations can not be used and further the
$L^{\infty}$ norm estimate of $\varphi -q_0x_1$ can not be obtained
directly. Therefore, we have to use some new ingredients to
overcome this essential difficulty.

Next we comment on the proof of the main result in this paper. By
introducing some suitable coordinate transformation and
linearizing the nonlinear equation \eqref{e1.3}, we can actually get a
Laplacian equation $\Delta u=f$ in an unbounded strip domain
$\tilde\Omega=\{(z_1,z_2,z_3): -\infty<z_1<\infty, 0<z_2<1,
-\infty<z_3<\infty\}$ with two Neumann boundary conditions on
$z_2=0$ and $z_2=1$, one periodic boundary condition on the
variable $z_3$, one Dirichlet boundary value condition at minus
infinity (i.e., $z_1\to -\infty$) and one restriction condition on
the existence of $\lim_{z_1\to +\infty}\nabla_zu(z)$. In order
to solve such a Laplacian equation in $\tilde\Omega$, our ingredient is to
use the separation variable method to write out the formal
expression of $u(z)$. From this, together with some delicate
analysis, we can show that this formal expression is actually a
solution of $\Delta u=f$ and its derivatives will decay at the
rate of $e^{-\delta_0|z_1|}$ ($\delta_0>0$ is a suitable constant) for
$z_1<0$; on the other hand, for $z_1>0$, the solution $u(z)$
increases at the rate of $(1+z_1)$ meanwhile its partial
derivative $\partial_{z_1}u$ is bounded and the partial derivatives
$(\partial_{z_2}u, \partial_{z_3}u)$ decay at the rate of $e^{-\delta_0z_1}$. In
terms of these properties, some inhomogeneous weighted H\"older spaces will be introduced by us and further be used to
treat the regularity and existence of solution to the second order
nonlinear elliptic problem in an unbounded strip domain. In this
procedure, some detailed analysis on the expression of solution
will be required, moreover, a priori estimates with different
weighted norms are required to be established. Subsequently, by using the continuity method, we can complete the proof of
Theorem \ref{thm1.1}.

This article is organized as follows. In $\S 2$,  we reformulate
the problem \eqref{e1.3} with \eqref{e1.4}-\eqref{e1.6}, and then
give a more precise descriptions on Theorem \ref{thm1.1} in some suitably
weighted H\"order spaces. In $\S 3$, we will linearize the nonlinear problem
\eqref{e1.3} with \eqref{e1.4}-\eqref{e1.6}.
By such a linearization, we essentially
obtain the Laplacian equation $\Delta u=\tilde f(z)$ in the strip
domain $\tilde\Omega=\{z=(z_1, z_2, z_3): -\infty<z_1<\infty, 0<z_2<1,
-\infty<z_3<\infty\}$ with two Neumann boundary conditions on
$z_2=0$ and $z_2=1$, one periodic condition on $z_3$ together with
$\lim_{z_1\to -\infty}u(z)=0$ and the requirement on the
existence of $\lim_{z_1\to\infty}\nabla_z u(z)$. By use of
Sturm-Liouville theorem and the separation variable method, we can
derive the formal expression of $u(z)$ in $\tilde\Omega$. Subsequently, it
follows from some detailed estimates that we can obtain the
existence and regularity of $u(z)$ in $\tilde\Omega$. In $\S 4$, based on
the crucial estimates and properties given in $\S 3$, by using
the suitable iteration scheme, we can complete the proof of
Theorem \ref{thm1.1} and further obtain the asymptotic behavior of
$\nabla_x\varphi$ at negative and positive infinity in the strip domain
$\Omega$ respectively.

\section{Reformulation on
\eqref{e1.3}-\eqref{e1.6} and more precise descriptions on Theorem \ref{thm1.1}}

In this section, we first introduce some notation and weighted
H\"older norms so that Theorem \ref{thm1.1} can be given a more
precise description.

Let $\Omega\subset\mathbb{R}^3$ be an open set including the origin
$O=(0,0,0)$, if $u\in C^{m,\alpha}(\Omega)$ with $0\le\alpha<1$, then we
define the following weighted H\"older norms for $x,
y\in\Omega$, some positive constant $\delta>0$ and $m\in\mathbb{N}\cup\{0\}$:
\begin{gather*}
[u]_{m,0;\Omega}^{(\delta)}\equiv \sum_{|\beta|=m}
\sup_{x\in\Omega} e^{\delta |x_1|}|D^{\beta}u(x)|; \\
[u]_{m,\alpha;\Omega}^{(\delta)}\equiv\sum_{|\beta|=m}\sup_{x,y\in\Omega}
e^{\delta
d_{x,y}}\frac{|D^{\beta}u(x)-D^{\beta}u(y)|}{|x-y|^{\alpha}},\quad
\text{here }d_{x,y}=\min(|x_1|,|y_1|); \\
|u|_{m,\alpha;\Omega}^{(\delta)}\equiv\sum_{0\le k\le
m}[u]_{k,0;\Omega}^{(\delta)}+[u]_{m,\alpha;\Omega}^{(\delta)}; \\
\begin{aligned}
\|u\|_{m,\alpha;\Omega}^{(\delta)}
&\equiv\sup_{x\in\Omega ; x_1<0}e^{\delta |x_1|}|u(x)|
+\sup_{x\in\Omega ; x_1>0}(1+x_1)^{-1}|u(x)|\\
&\quad +\sup_{x\in\Omega ; x_1<0}e^{\delta |x_1|}|\partial_{x_1}u(x)|
 +\sup_{x\in\Omega ; x_1>0}|\partial_{x_1}u(x)| \\
&\quad  +\sup_{x\in\Omega}e^{\delta |x_1|}(|\partial_{x_2}u(x)|+|\partial_{x_3}u(x)|)
 +\sum_{2\le k\le m}[u]_{k,0;\Omega}^{(\delta)}+[u]_{m,\alpha;\Omega}^{(\delta)},
\end{aligned}
\end{gather*}
and the corresponding function spaces are defined as
\begin{gather*}
H_{m,\alpha}^{(\delta)}(\Omega)=\{u(x)\in C^{m,\alpha}(\Omega):
|u|_{m,\alpha}^{(\delta)}<+\infty\}, \\
\mathbb{H}_{m,\alpha}^{(\delta)}(\Omega)=\{u(x)\in C^{m,\alpha}(\Omega):
\|u\|_{m,\alpha}^{(\delta)}<+\infty\}.\nonumber
\end{gather*}


\begin{lemma} \label{lem2.1}
 For $u(x)\in C^{m,\alpha}(\bar\Omega)$, one has
\begin{itemize}
\item[(i)] $H_{m,\alpha}^{(\delta)}(\Omega)\subset \mathbb{H}_{m,\alpha}^{(\delta)}(\Omega)$.

\item[(ii)] $|\partial_{x_i}u|_{m-1,\alpha;\Omega}^{(\delta)}\le\|u\|_{m,\alpha;\Omega}^{(\delta)}$ for
$i=2,3$ and $m\ge 1$.

\item[(iii)] $|D^2u|_{m-2,\alpha;\Omega}^{(\delta)}\le\|u\|_{m,\alpha;\Omega}^{(\delta)}$
for $m\ge 2$.
\end{itemize}
\end{lemma}

Since these properties can be directly verified  by using the definitions of the norms $|\cdot|_{m,\alpha}^{(\delta)}$ and
$\|\cdot\|_{m,\alpha}^{(\delta)}$, then we omit their proof.


By using of the weighted H\"older norms introduced
above, Theorem \ref{thm1.1} can be stated more precisely as follows.

\begin{theorem} \label{thm2.2}
 Under the assumptions of Theorem \ref{thm1.1}, in
the domain $\Omega=\{x: -\infty<x_1<+\infty, \varepsilon
f_1(x_1,x_3)<x_2<1+\varepsilon f_2(x_1,x_3), -\infty<x_3<+\infty\}$,
problem \eqref{e1.3}-\eqref{e1.6} has a unique solution $\varphi(x)\in
C^{6,\alpha}(\Omega)$ (any fixed constant $0<\alpha<1$), which satisfies
\begin{itemize}
\item[(i)] $\|\varphi(x)-q_0x_1\|_{6,\alpha;\Omega}^{(\delta_0)}\leq \tilde{C}\varepsilon$,
here $\delta_0>0$ is some suitable constant.

\item[(ii)] $\lim_{x\in\Omega; x_1\to+\infty}\nabla\varphi(x)=(q_0,0,0)$.
\end{itemize}
\end{theorem}

\begin{remark} \rm
From the results on the interior regularities
and boundary regularities of solutions to second order elliptic
equations (see \cite[Chapter 6]{11}), we know that $\varphi(x)\in
C^{\infty}(\bar\Omega)$ holds in Theorem \ref{thm2.2}.
\end{remark}

For the requirements to show Theorem \ref{thm2.2}, we intend to introduce
the following transformation so that the domain $\Omega$ can be
changed into a standard strip domain $\tilde\Omega\equiv\{z=(z_1,z_2,z_3):
-\infty<z_1<\infty, 0<z_2<1, -\infty<z_3<\infty\}$:
\begin{equation} \label{e2.1}
\begin{gathered}
z_1=x_1,\\
z_2=\frac{x_2-\varepsilon f_1(x_1,x_3)}{1+\varepsilon f_2(x_1,x_3)-\varepsilon f_1(x_1,x_3)},\\
z_3=x_3.
\end{gathered}
\end{equation}

In this case, for  notational convenience, we still denote by
$\varphi(z)$ as the solution instead of $\varphi(x)$ under the
transformation \eqref{e2.1}. It follows from a direct computation that
the problem \eqref{e1.3}-\eqref{e1.6} can be changed into
\begin{equation} \label{e2.2}
\begin{gathered}
\sum_{i,j=1}^3A_{ij}(z,\nabla_z\varphi)\partial_{z_iz_j}^2\varphi
+B(z,\nabla_z\varphi)\partial_{z_2}\varphi=0\quad \text{in }\tilde{\Omega}\\
b_{11}(z)\partial_{z_1}\varphi+\partial_{z_2}\varphi+b_{13}(z)\partial_{z_3}\varphi=0
\quad\text{on }z_2=0,\\
b_{21}(z)\partial_{z_1}\varphi+\partial_{z_2}\varphi+b_{23}(z)\partial_{z_3}\varphi=0
\quad\text{on }z_2=1,\\
\varphi(z_1,z_2,z_3+1)=\varphi(z_1,z_2,z_3),\\
\lim_{z_1\to-\infty}(\varphi(z)-q_0z_1)=0,\\
\lim_{z\in\tilde\Omega;
z_1\to+\infty}\nabla_z\varphi(z)\text{ exists},
\end{gathered}
\end{equation}
where
\begin{gather*}
A_{11}(z,\nabla_z\varphi)=c^2(H(\nabla_x\varphi))-(\partial_{x_1}\varphi)^2, \\
A_{22}(z,\nabla_z\varphi)
=\sum_{i=1}^3\big(c^2(H(\nabla_x\varphi))-(\partial_{x_i}\varphi)^2\big)(\frac{\partial z_2}{\partial x_i})^2
-2\sum_{1\le i<j\le3}\partial_{x_i}\varphi\partial_{x_j}\varphi\frac{\partial z_2}{\partial
x_i}\frac{\partial z_2}{\partial x_j}, \\
A_{33}(z,\nabla_z\varphi)=c^2(H(\nabla_x\varphi))-(\partial_{x_3}\varphi)^2, \\
\begin{aligned}
A_{12}(z,\nabla_z\varphi)
&=A_{21}(z,\nabla_z\varphi) \\
&=(c^2(H(\nabla_x\varphi))-(\partial_{x_1}\varphi)^2)\frac{\partial z_2}{\partial x_1}
 -\partial_{x_1}\varphi\partial_{x_2}\varphi\frac{\partial z_2}{\partial x_2}
 -\partial_{x_1}\varphi\partial_{x_3}\varphi\frac{\partial z_2}{\partial x_3},
\end{aligned} \\
A_{13}(z,\nabla_z\varphi)=A_{31}(z,\nabla_z\varphi)=-\partial_{x_1}\varphi\partial_{x_3}\varphi, \\
\begin{aligned}
A_{23}(z,\nabla_z\varphi)
&=A_{32}(z,\nabla_z\varphi) \\
&=(c^2(H(\nabla_x\varphi))-(\partial_{x_3}\varphi)^2)\frac{\partial z_2}{\partial x_3}
 -\partial_{x_1}\varphi\partial_{x_3}\varphi\frac{\partial z_2}{\partial x_1}
 -\partial_{x_2}\varphi\partial_{x_3}\varphi\frac{\partial z_2}{\partial x_2}, 
\end{aligned}\\
 B(z,\nabla_z\varphi)=\sum_{i=1}^3\big(c^2(H(\nabla_x\varphi))-(\partial_{x_i}\varphi)^2\big)
\frac{\partial^2z_2}{\partial x_i^2}-2\sum_{1\le i<j\le3}\partial_{x_i}\varphi\partial_{x_j}
 \varphi\frac{\partial^2z_2}{\partial x_i\partial x_j}, \\
b_{ij}(z)=\frac{\varepsilon\partial_{x_j}f_i}{\varepsilon\partial_{x_1}f_i\frac{\partial z_2}{\partial
x_1}-\frac{\partial z_2}{\partial x_2}+\varepsilon\partial_{x_3}f_i\frac{\partial z_2}{\partial x_3}},\quad
i=1,2;\; j=1,3\
\end{gather*}
with
$$
\partial_{x_1}\varphi=\partial_{z_1}\varphi+\partial_{z_2}\varphi\frac{\partial z_2}{\partial x_1},\quad
\partial_{x_2}\varphi=\partial_{z_2}\varphi\frac{\partial z_2}{\partial x_2},\quad
\partial_{x_3}\varphi=\partial_{z_3}\varphi +\partial_{z_2}\varphi\frac{\partial z_2}{\partial x_3}.
$$

By the transformation \eqref{e2.1}, together with the properties of
$f_i(x_1,x_3)$ ($i=1,2$) and the definition of the norm
$\|\cdot\|_{m,\alpha}^{(\delta)}$,  to show Theorem \ref{thm2.2}, we only
need to establish the following theorem.

\begin{theorem}  \label{thm2.3}
Under the assumptions in Theorem \ref{thm1.1},
problem \eqref{e2.2} has a unique solution $\varphi(z)\in C^{6,\alpha}(\tilde\Omega)$
which satisfies
\begin{itemize}
\item[(i)] $\|\varphi(z)-q_0z_1\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}\leq \tilde{C}\varepsilon$.

\item[(ii)] $\lim_{z\in\tilde\Omega; z_1\to+\infty}\nabla_z\varphi(z)=(q_0,0,0)$.
\end{itemize}
\end{theorem}

In next sections, we will focus on the proof of Theorem \ref{thm2.3}.

\section{Solvability and a
priori estimates for the linearized problem of \eqref{e2.2}}

To solve the  nonlinear problem \eqref{e2.2}, we first consider
its linearized case, which corresponds to a mixed boundary value
problem of a second order linear elliptic equation in an infinitely long
strip domain $\tilde\Omega$. In terms of the smallness of perturbed nozzle
walls and by using direct computations, the linearized problem of
\eqref{e2.2} can be essentially expressed as:
% \label{e3.1}
\begin{gather}
\begin{aligned}
\bar{L}(v)\dot{u}&\equiv\sum_{i,j=1}^3a_{ij}(z,\nabla_zv)\partial_{z_iz_j}^2\dot{u}\\
&\equiv\sum_{i=1}^3\big(c^2(H(\nabla_z
v))-(\partial_{z_i}v)^2\big)\partial_{z_i}^2\dot{u}-2\sum_{1\le
i<j\le3}\partial_{z_i}v\partial_{z_j}v\partial_{z_iz_j}^2\dot{u} \\
&=\dot{f}\quad  \text{in } \tilde\Omega,
\end{aligned} \nonumber \\
\partial_{z_2}\dot{u}=\dot{g}_1\quad\text{on } z_2=0, \nonumber \\
\partial_{z_2}\dot{u}=\dot{g}_2\quad\text{on } z_2=1, \nonumber \\
\dot{u}(z_1,z_2,z_3+1)=\dot{u}(z_1,z_2,z_3),\nonumber \\
\lim_{z_1\to-\infty,\, z\in\tilde\Omega}\dot{u}(z)=0,\nonumber \\
\lim_{z_1\to+\infty,\, z\in\tilde\Omega}\nabla_z\dot{u}(z) \text{ exists}, \label{e3.1}
\end{gather}
where $v(z_1,z_2,z_3), \dot f(z_1,z_2,z_3)$ and
$\dot{g}_i(z_1,z_3)$ are all $1$-periodic functions with respect
to the variable $z_3$, and $v\in \mathbb{H}_{6,\alpha}^{(\delta_0)}(\tilde\Omega)$
with $\|v-q_0z_1\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}<\varepsilon$ and $\delta_0>0$ a
suitably fixed constant.

It is easy to verify that the coefficients of  problem \eqref{e3.1}
satisfy the following uniformly elliptic condition in
$\tilde{\Omega}$:
\begin{equation} \label{e3.2}
\lambda|\xi|^2\le\sum_{i,j=1}^3a_{ij}(z,\nabla_zv)\xi_i\xi_j\le\Lambda|\xi|^2,
\end{equation}
for all $\xi=(\xi_1,\xi_2,\xi_3)\in \mathbb{R}^3$ and $z\in \tilde{\Omega}$,
here $\lambda$ and $\Lambda$ are two appropriate constants.

Next, we study the solvability of problem \eqref{e3.1} as well as the
regularity and a priori estimates of solution $\dot u(z)$ to
\eqref{e3.1}. To this end, we first study the Laplacian equation in
$\mathbb{R}^3$ with the following boundary conditions:
\begin{equation} \label{e3.3}
\begin{gathered}
L_0u\equiv\Delta u=\tilde f \quad   \tilde\Omega,\\
\partial_{z_2}u=\tilde{g}_1\quad \text{on } z_2=0,\\
\partial_{z_2}u=\tilde{g}_2 \quad \text{on } z_2=1,\\
u(z_1,z_2,z_3+1)=u(z_1,z_2,z_3),\\
\lim_{z_1\to-\infty}u(z)=0,\\
\lim_{z_1\to+\infty}\nabla_z u(z)\text{ exists}.
\end{gathered}
\end{equation}
where $\tilde f(z)\in H_{4,\alpha}^{(\delta_0)}(\tilde\Omega)$ and
$\tilde g_i(z_1,z_3)\in H_{5,\alpha}^{(\delta_0)}(\tilde\Omega)$ ($i=0,1$) are all
$1$-periodic functions with respect to the variable $z_3$.

For the later uses, we now give a lemma on the function $\tilde f(z)$.

\begin{lemma} \label{lem3.1}
 For $\tilde{f}\in H_{4,\alpha}^{(\delta_0)}(\tilde\Omega)$, if we set
\begin{gather*}
f_{m0}(z_1)=2\int_0^1\int_0^1\tilde{f}(z)\cos (m\pi z_2)dz_2dz_3, \\
f_{0n}^1(z_1)=2\int_0^1\int_0^1\tilde{f}(z)\cos (2n\pi z_3)dz_2dz_3, \\
f_{0n}^2(z_1)=2\int_0^1\int_0^1\tilde{f}(z)\sin (2n\pi z_3)dz_2dz_3, \\
f_{mn}^1(z_1)=4\int_0^1\int_0^1\tilde{f}(z)\cos(m\pi z_2)\cos(2n\pi z_3)dz_2dz_3, \\
f_{mn}^2(z_1)=4\int_0^1\int_0^1\tilde{f}(z)\cos(m\pi
z_2)\sin(2n\pi z_3)dz_2dz_3\nonumber
\end{gather*}
for $m,n\in \mathbb{N}$, then
\begin{gather*}
|f_{m0}(z_1)|\le\frac{C}{m^2}|\tilde{f}|_{4,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}, \\
|f_{0n}^i(z_1)|\le\frac{C}{n^2}|\tilde{f}|_{4,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|},\quad
i=1,2, \\
|f_{mn}^i(z_1)|\le\frac{C}{m^2n^2}|\tilde{f}|_{4,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|},\quad
i=1,2.
\end{gather*}
\end{lemma}

\begin{proof}
 Integrating by parts, we arrived at
\begin{align*}
 f_{m0}(z_1)
&= 2\int_0^1\Big(\frac{1}{m\pi}\tilde{f}\sin(m\pi z_2)|_{z_2=0}^{z_2=1}
-\frac{1}{m\pi}\int_0^1\partial_{z_2}\tilde{f}\sin (m\pi z_2)dz_2\Big)dz_3 \\
&= 2\int_0^1\Big(\frac{1}{m^2\pi^2}\partial_{z_2}\tilde{f}\cos(m\pi
z_2)|_{z_2=0}^{z_2=1}
-\frac{1}{m^2\pi^2}\int_0^1\partial_{z_2}^2\tilde{f}\cos(m\pi
z_2)dz_2\Big)dz_3.\nonumber
\end{align*}

Because $\tilde{f}\in H_{4,\alpha}^{(\delta_0)}(\tilde\Omega)$, we have
$|D^{\beta}\tilde{f}(z)|\le
|\tilde{f}|_{4,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}$ for $|\beta|\le
4$. From this, we derive that
$$
|f_{m0}(z_1)|\le\frac{C}{m^2}|\tilde{f}|_{4,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}.
$$
Analogously, by using the periodic property of $\tilde{f}$ with
respect to $z_3$ and integration by parts, we have
\begin{equation} \label{e3.4}
\int_0^1\tilde{f}\cos (2n\pi z_3)dz_3=-\frac{1}{4n^2\pi^2}\int_0^1\partial_{z_3}^2
\tilde{f}\cos(2n\pi z_3)dz_3.
\end{equation}
This yields
$$
|f_{0n}^1(z_1)|\le\frac{C}{n^2}|f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}.
$$
Analogously, $|f_{0n}^2(z_1)|\le\frac{C}{
n^2}|f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}$ also holds.

Next, we estimate $f_{mn}^i(z_1)$ for $i=1,2$.
It follows from \eqref{e3.4} and integration by parts with respect to
$z_2$ that
\begin{align*}
f_{mn}^1(z_1)
&=-\frac{1}{n^2\pi^2}\int_0^1\int_0^1 \partial_{z_3}^2\tilde{f}\cos(m\pi z_2)
   \cos (2n\pi z_3)dz_2dz_3 \\
&=\frac{1}{m^2n^2\pi^4}\Big(\int_0^1\big(\cos(m\pi
z_2)\partial_{z_2}\partial_{z_3}^2\tilde{f}\big)|_{z_2=0}^{z_2=1} \cos(2n\pi
z_3)dz_3 \\
&\quad -\int_0^1\int_0^1\partial_{z_2}^2\partial_{z_3}^2\tilde{f}\cos(m\pi
z_2)\cos(2n\pi z_3)dz_2dz_3\Big).\nonumber
\end{align*}
This yields
$$
|f_{mn}^1(z_1)|\le\frac{C}{m^2 n^2}|f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}.
$$
Analogously, $|f_{mn}^2(z_1)|\le\frac{C}{
m^2n^2}|f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}$ holds.
\end{proof}

\begin{lemma} \label{lem3.2}
If $\tilde f\in H_{4,\alpha}^{(\delta_0)}(\tilde\Omega)$
and $ \tilde{g}_i\in H_{5,\alpha}^{(\delta_0)}(\tilde\Omega)$ with
$0<\delta_0<\pi$, then the equation \eqref{e3.3} has a solution
$u\in C^2(\overline{\tilde{\Omega}})$, which satisfies the
estimate
\begin{equation} \label{e3.5}
\|u\|_{2,0;\tilde\Omega}^{(\delta_0)}\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}).
\end{equation}
\end{lemma}

\begin{proof}
 We intend to use the method of separation variables
to study the solvability and regularities of solution $u$ to
\eqref{e3.3}. To this end, we firstly focus on its corresponding
homogeneous problem.

Let us consider the nontrivial solutions of the  problem
\begin{equation} \label{e3.6}
\begin{gathered}
\Delta u=0 \quad\text{in }  \tilde\Omega,\\
\partial_{z_2}u=0\quad\text{on } z_2=0,\\
\partial_{z_2}u=0 \quad\text{on } z_2=1,\\
u(z_1,z_2,z_3+1)=u(z_1,z_2,z_3).
\end{gathered}
\end{equation}
Set $u(z)=X(z_1)Y(z_2)Z(z_3)$, then from \eqref{e3.6}  it follows that
\begin{equation} \label{e3.7}
\begin{gathered}
Y''(z_2)+\lambda Y(z_2)=0,\\
Y'(0)=0,\\
Y'(1)=0,
\end{gathered}
\end{equation}
and
\begin{equation} \label{e3.8}
\begin{gathered}
Z''(z_3)+\mu Z(z_3)=0,\\
Z(z_3+1)=Z(z_3),
\end{gathered}
\end{equation}
 and
\begin{equation} \label{e3.9}
X''(z_1)-(\lambda+\mu)X(z_1)=0,
\end{equation}
here $\lambda, \mu\in\mathbb{R}$.

By a simple computation, we can show that the eigenvalues of \eqref{e3.7} are
$\lambda_m=(m\pi)^2$ ($m=0,1,2,\ldots$), and the corresponding
eigenfunctions are $\cos (m\pi z_2)$. In addition, we can compute
that the eigenvalues of \eqref{e3.8} are $\mu_n=(2n\pi)^2$
($n=0,1,2,\ldots$), and the corresponding eigenfunctions are $\cos
(2n\pi z_3)$ and $\sin (2n\pi z_3)$ respectively.

We now solve equation \eqref{e3.6} by using the eigenfunction expansion
method in terms of the complete orthogonal basis
$\{\cos m\pi z_2\cos 2n\pi z_3, \cos m\pi z_2\sin 2n\pi z_3\}_{m,n=0}^{+\infty}$.

Set $h(z)=\frac{1}{2}(\tilde g_2(z_1,z_3)-\tilde g_1(z_1,z_3))z_2^2+\tilde
g_1(z_1,z_3)z_2$ and $v(z)=u(z)-h(z)$, then it follows from \eqref{e3.3}
that $v(z)$ satisfies
\begin{equation} \label{e3.10}
\begin{gathered}
\Delta v=\tilde f-\Delta h\equiv f \quad \quad in\quad \tilde\Omega,\\
\partial_{z_2}v=0\quad  \text{on } z_2=0,\\
\partial_{z_2}v=0 \quad \text{on }  z_2=1,\\
v(z_1,z_2,z_3+1)=v(z_1,z_2,z_3),\\
\lim_{z_1\to-\infty}v(z)=0,\\
\lim_{z_1\to+\infty}\nabla_z v(z)\text{ exists}.
\end{gathered}
\end{equation}
Let
\begin{equation} \label{e3.11}
\begin{aligned}
v(z)
&=X_{00}(z_1)+\sum_{m=1}^{\infty}X_{m0}(z_1)\cos(m\pi z_2) \\
&\quad +\sum_{n=1}^{\infty}\bigl(X_{0n}^1(z_1)\cos(2n\pi z_3)
+X_{0n}^2(z_1)\sin(2n\pi z_3)\bigr) \\
&\quad +\sum_{m,n=1}^{\infty}\bigl(X_{mn}^1(z_1)\cos(m\pi z_2)\cos(2n\pi z_3) \\
&\quad +X_{mn}^2(z_1)\cos(m\pi z_2)\sin(2n\pi z_3)\bigr)
\end{aligned}
\end{equation}
 and
\begin{align*}
 f(z)&=f_{00}(z_1)+\sum_{m=1}^{\infty}f_{m0}(z_1)\cos(m\pi z_2) \\
&\quad +\sum_{n=1}^{\infty}\bigl(f_{0n}^1(z_1)\cos(2n\pi z_3)
+f_{0n}^2(z_1)\sin(2n\pi z_3)\bigr) \\
&\quad +\sum_{m,n=1}^{\infty}\bigl(f_{mn}^1(z_1)\cos(m\pi
z_2)\cos(2n\pi z_3) +f_{mn}^2(z_1)\cos(m\pi z_2)\sin(2n\pi
z_3)\bigr),
\end{align*}
 where
\begin{gather*}
f_{00}(z_1)=\int_0^1\int_0^1 f(z)dz_2dz_3\\
f_{m0}(z_1)=2\int_0^1\int_0^1 f(z)\cos(m\pi z_2)dz_2dz_3,\\
f_{0n}^1(z_1)=2\int_0^1\int_0^1 f(z)\cos(2n\pi z_3)dz_2dz_3,\\
f_{0n}^2(z_1)=2\int_0^1\int_0^1 f(z)\sin(2n\pi z_3)dz_2dz_3,\\
f_{mn}^1(z_1)=4\int_0^1\int_0^1 f(z)\cos(m\pi z_2)\cos(2n\pi z_3)dz_2dz_3,\\
f_{mn}^1(z_1)=4\int_0^1\int_0^1 f(z)\cos(m\pi z_2)\sin(2n\pi
z_3)dz_2dz_3.
\end{gather*}

Next, we  determine the terms $X_{00}(z_1)$, $X_{m0}(z_1)$,
$X_{0n}^i(z_1)$ and $X_{mn}^i(z_1)$ $(i=1,2)$ in \eqref{e3.11}.
It follows from \eqref{e3.10} and \eqref{e3.11} that we can formally obtain
\begin{equation} \label{e3.12}
\begin{gathered}
X''_{00}(z_1)=f_{00}(z_1),\\
\lim_{z_1\to-\infty}X_{00}(z_1)=0,\quad \lim_{z_1\to+\infty}X'_{00}(z_1)
\text{ exists},
\end{gathered}
\end{equation}
\begin{equation} \label{e3.13}
\begin{gathered}
X''_{m0}(z_1)-m^2\pi^2X_{m0}(z_1)=f_{m0}(z_1),\\
\lim_{z_1\to-\infty}X_{m0}(z_1)=0, \quad \lim_{z_1\to+\infty}X'_{m0}(z_1)
\text{ exists},
\end{gathered}
\end{equation}
\begin{equation} \label{e3.14}
\begin{gathered}
(X_{0n}^i)''(z_1)-4n^2\pi^2X_{m0}^i(z_1)=f_{0n}^i(z_1),\\
\lim_{z_1\to-\infty}X_{0n}^i(z_1)=0,\quad
\lim_{z_1\to+\infty}(X_{0n}^i)'(z_1) \text{ exists},
\end{gathered}
\end{equation}
\begin{equation} \label{e3.15}
\begin{gathered}
(X_{mn}^i)''(z_1)-(m^2+4n^2)\pi^2X_{mn}^i(z_1)=f_{mn}^i(z_1),\\
\lim_{z_1\to-\infty}X_{mn}^i(z_1)=0,\quad
\lim_{z_1\to+\infty}(X_{mn}^i)'(z_1) \text{ exists}.
\end{gathered}
\end{equation}
Solving these ordinary differential equations directly yield
\begin{gather} \label{e3.16}
X_{00}(z_1)=\int_{-\infty}^{z_1}\int_{-\infty}^tf_{00}(\xi)d\xi dt,\\
\label{e3.17}
X_{m0}(z_1)=e^{m\pi z_1}\int_{+\infty}^{z_1}e^{-2m\pi t}
\int_{-\infty}^te^{m\pi\xi}f_{m0}(\xi)d\xi dt,\quad m\ge 1,\\
\label{e3.18}
X_{0n}^i(z_1)=e^{2n\pi z_1}\int_{+\infty}^{z_1}e^{-2n\pi t}
\int_{-\infty}^te^{2n\pi\xi}f_{0n}^i(\xi)d\xi dt,\quad n\ge 1,\\
\label{e3.19} 
\begin{aligned}
X_{mn}^i(z_1)&=e^{\sqrt{m^2+4n^2}\pi
z_1}\int_{+\infty}^{z_1}e^{-2\sqrt{m^2+4n^2}\pi t}
\int_{-\infty}^te^{\sqrt{m^2+4n^2}\pi\xi}f_{mn}^i(\xi)d\xi dt,\\
&m,n\ge 1.
\end{aligned}
\end{gather}
We now analyze the expressions in \eqref{e3.16}-\eqref{e3.19}. This will be
divided into three parts.
\smallskip

\noindent{\bf Part 1. Estimate of $X_{00}(z_1)$.}
By using the expression of $X_{00}(z_1)$ in \eqref{e3.16} and
integrating by parts, one has
\begin{equation} \label{e3.20}
X_{00}(z_1)=t\int_{-\infty}^tf_{00}(\xi)d\xi\big|_{-\infty}^{z_1}
-\int_{-\infty}^{z_1}tf_{00}(t)dt.
\end{equation}
By $f(z)\in H_{3,\alpha}^{(\delta_0)}(\tilde\Omega)$, we have
\begin{equation} \label{e3.21}
|f_{00}(z_1)|\le |f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}.
\end{equation}
Thus
$$
\lim_{t\to-\infty}t\int_{-\infty}^{t}f_{00}(\xi)dt=0.
$$
This and \eqref{e3.20}, yield
\begin{equation} \label{e3.22}
X_{00}(z_1)=z_1\int_{-\infty}^{z_1}f_{00}(t)dt
-\int_{-\infty}^{z_1}tf_{00}(t)dt.
\end{equation}
For $z_1<0$, it follows from \eqref{e3.16} and \eqref{e3.21} that
$$
|X_{00}(z_1)|\le\int_{-\infty}^{z_1}\int_{-\infty}^t|f_{00}(\xi)|d\xi
dt\le|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}\int_{-\infty}^{z_1}\int_{-\infty}^te^{\delta_0\xi}d\xi
dt\le\frac{1}{\delta_0^2}|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{\delta_0z_1}.
$$
For $z_1>0$, by using \eqref{e3.21}-\eqref{e3.22}, we have
\begin{align*}
 |X_{00}(z_1)|
&\le z_1\int_{-\infty}^{z_1}|f_{00}(t)|dt+\int_{-\infty}^{z_1}|tf_{00}(t)|dt \\
&\le |f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}\Big(z_1\int_{-\infty}^0e^{\delta_0t}dt
 +z_1\int_0^{z_1}e^{-\delta_0t}dt -\int_{-\infty}^{0}te^{\delta_0t}dt
 +\int_{0}^{z_1}te^{-\delta_0t}dt\Big) \\
&\le C(1+z_1)|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.\nonumber
\end{align*}
This means
\begin{equation} \label{e3.23}
\|X_{00}(z_1)\|_{0,0}^{(\delta_0)}\le C|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{equation}
Next we estimate $X'_{00}(z_1)$. Note that
$$
X'_{00}(z_1)=\int_{-\infty}^{z_1}f_{00}(t)dt.
$$
If $z_1<0$, then one has
$$
|X'_{00}(z_1)|\le |f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}\int_{-\infty}^{z_1}e^{\delta_0t}dt
\le\frac{1}{\delta_0}|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{\delta_0z_1}.
$$
If $z_1>0$, then
\begin{equation} \label{e3.24}
|X'_{00}(z_1)|\le |f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}\Big(\int_{-\infty}^0e^{\delta_0t}dt
+\int_{0}^{z_1}e^{-\delta_0t}dt\Big)
\le\frac{2}{\delta_0}|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{equation}
Thus, we arrive at
$$
\|X'_{00}(z_1)\|_{0,0}^{(\delta_0)}\le
C|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}
$$
and
\begin{equation} \label{e3.25}
\lim_{z_1\to+\infty}X'_{00}(z_1)
=\int_{-\infty}^{+\infty}\int_0^1\int_0^1f(t,z_2,z_3)dz_2dz_3dt.
\end{equation}
\smallskip

\noindent\textbf{Part 2. Estimate of $X_{m0}(z_1)$ with $m\ge 1$.}
By \eqref{e3.17} and integration by parts, we have
\begin{equation} \label{e3.26}
\begin{aligned}
X_{m0}(z_1)
&=-\frac{1}{2m\pi}e^{m\pi z_1}\Big(e^{-2m\pi t}
\int_{-\infty}^te^{m\pi\xi}f_{m0}(\xi)d\xi\Big|_{+\infty}^{z_1}
-\int_{+\infty}^{z_1}e^{-m\pi t}f_{m0}(t)dt\Big) \\
&=-\frac{1}{2m\pi}\Big(e^{-m\pi z_1}\int_{-\infty}^{z_1}e^{m\pi t}
f_{m0}(t)dt+e^{m\pi z_1}\int_{z_1}^{+\infty}e^{-m\pi
t}f_{m0}(t)dt\Big),
\end{aligned}
\end{equation}
 where we have used that
\[
\lim_{t\to+\infty}\frac{\int_{-\infty}^te^{m\pi\xi}f_{m0}(\xi)d\xi}
{e^{2m\pi t}} =\lim_{t\to+\infty}\frac{f_{m0}(t)}{
2m\pi e^{m\pi t}}=0.
\]
It is noted that by Lemma \ref{lem3.1} and the proof  of  Lemma \ref{lem3.1}, we have

(i) If $z_1<0$, then
\begin{equation} \label{e3.27}
\begin{aligned}
\big|e^{-m\pi z_1}\int_{-\infty}^{z_1}e^{m\pi t}f_{m0}(t)dt\big|
&\leq \frac{C}{m^2}|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)} e^{-m\pi
z_1}\int_{-\infty}^{z_1}e^{(m\pi+\delta_0)t}dt \\
&=\frac{C}{m^2(m\pi+\delta_0)}|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{\delta_0z_1}
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.28}
\begin{aligned}
&\big|e^{m\pi z_1}\int_{z_1}^{+\infty}e^{-m\pi t}f_{m0}(t)dt\big| \\
&\le \frac{C}{m^2}|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{m\pi
z_1}\Big(\int_{z_1}^0e^{(\delta_0-m\pi)t}dt
+\int_0^{+\infty}e^{-(m\pi+\delta_0)t}dt\Big) \\
&\le \frac{C}{m^2(m\pi-\delta_0)} |f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{\delta_0z_1}.
\end{aligned}
\end{equation}

(ii) If $z_1>0$, then
\begin{equation} \label{e3.29}
\begin{aligned}
&\big|e^{-m\pi z_1}\int_{-\infty}^{z_1}e^{m\pi t}f_{m0}(t)dt\big| \\
&\le \frac{C}{m^2}|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-m\pi z_1}
\Big(\int_{-\infty}^0e^{(m\pi+\delta_0)t}dt+\int_0^{z_1}e^{(m\pi-\delta_0)t}dt\Big) \\
&\le \frac{C}{m^2(m\pi-\delta_0)}
|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0z_1}
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.30}
\begin{aligned}
\big|e^{m\pi z_1}\int_{z_1}^{+\infty}e^{-m\pi t}f_{m0}(t)dt\big|
&\le\frac{C}{m^2}|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{m\pi z_1}\int_{z_1}^{+\infty}
e^{-(m\pi+\delta_0)t}dt \\
&\le\frac{C}{m^2(m\pi+\delta_0)}|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0z_1}.
\end{aligned}
\end{equation}
Substituting \eqref{e3.27}-\eqref{e3.28} and \eqref{e3.29}-\eqref{e3.30}
in \eqref{e3.26} yields
$$
|X_{m0}(z_1)|\le\frac{C}{m^3(m\pi-\delta_0)}
|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}.
$$
Namely,
\begin{equation} \label{e3.31}
 |X_{m0}(z_1)|_{0,0}^{(\delta_0)}\le\frac{C}{m^3(m\pi-\delta_0)}
|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{equation}
Next, we estimate $X'_{m0}(z_1)$.
Since
$$
X'_{m0}(z_1)=\frac{1}{2}\Big(e^{-m\pi z_1}\int_{-\infty}^{z_1}e^{m\pi t}f_{m0}(t)dt
-e^{m\pi z_1}\int_{z_1}^{+\infty}e^{-m\pi t}f_{m0}(t)dt\Big),
$$
by using \eqref{e3.27}-\eqref{e3.28} and \eqref{e3.29}-\eqref{e3.30}, we have
$$
|X'_{m0}(z_1)|
\le \frac{C}{m^2(m\pi-\delta_0)}|f|_{3,\alpha;\tilde\Omega}^{\delta_0}e^{-\delta_0|z_1|}.
$$
This means
\begin{gather} \label{e3.32}
|X'_{m0}(z_1)|_{0,0}^{(\delta_0)}
\le \frac{C}{m^2(m\pi-\delta_0)}|f|_{3,\alpha;\tilde\Omega}^{\delta_0}, \\
 \label{e3.33} \lim_{z_1\to\infty}X'_{m0}(z_1)=0.
\end{gather}
\smallskip

\noindent\textbf{Part 3. Estimates of $X_{0n}^i(z_1)$ and $X_{mn}^i(z_1)$
with $i=1,2$ and $m, n\ge 1$.}
As in Step 2, it follows from a direct computation that
\begin{gather*}
X_{0n}^i(z_1)=-\frac{1}{2n\pi}\Big(\int_{-\infty}^{z_1}
e^{2n\pi(t-z_1)}f_{0n}^i(t)dt+\int_{z_1}^{+\infty}e^{2n\pi(z_1-t)}
f_{0n}^i(t)dt\Big), \\
\begin{aligned}
X_{mn}^i(z_1)&=-\frac{1}{2\pi\sqrt{m^2+4n^2}}\Big(\int_{-\infty}^{z_1}
e^{\pi\sqrt{m^2+4n^2}(t-z_1)}f_{mn}^i(t)dt \\
&\quad + \int_{z_1}^{+\infty}e^{\pi\sqrt{m^2+4n^2}(z_1-t)}f_{mn}^i(t)dt\Big).
\end{aligned}
\end{gather*}

Similar to the estimates on $X_{m0}^i(z_1)$, we can arrive at
\begin{equation} \label{e3.34}
\begin{gathered}
|X_{0n}^i(z_1)|\le\frac{C}{n^3(2n\pi-\delta_0)}
|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|},\\
\begin{aligned}
|X_{mn}^i(z_1)|
&\le\frac{C}{m^2n^2\sqrt{m^2+4n^2}(\pi\sqrt{m^2+4n^2}-\delta_0)} \\
&\quad\times \Big(|\tilde
f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}+\sum_{i=1}^2|\tilde
g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}\Big)e^{-\delta_0|z_1|}.
\end{aligned}
\end{gathered}
\end{equation}
Namely,
\begin{gather} \label{e3.35}
|X_{0n}^i(z_1)|_{0,0}^{(\delta_0)}\le\frac{C}{n^3(2n\pi-\delta_0)}
|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)},\\
\begin{aligned}
|X_{mn}^i(z_1)|_{0,0}^{(\delta_0)}
&\le\frac{C}{m^2n^2\sqrt{m^2+4n^2}(\pi\sqrt{m^2+4n^2}-\delta_0)}\\
&\quad\times \Big(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}+\sum_{i=1}^2|\tilde
g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}\Big). 
\end{aligned}\label{e3.36}
\end{gather}
Analogously,
\begin{equation} \label{e3.37}
\begin{gathered}
|(X_{0n}^i)'(z_1)|_{0,0}^{(\delta_0)}\le\frac{C}{
n^2(2n\pi-\delta_0)}
|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)},\\
|(X_{mn}^i)'(z_1)|_{0,0}^{(\delta_0)}\le\frac{C}{
m^2n^2(\pi\sqrt{m^2+4n^2}-\delta_0)} \Big(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}\Big)
\end{gathered}
\end{equation}
and
\begin{equation} \label{e3.38}
\lim_{z_1\to\infty}(X_{0n}^i)'(z_1)=0,\quad
\lim_{z_1\to\infty}(X_{mn}^i)'(z_1)=0.
\end{equation}

Based on Parts 1--3, we now show that the formal solution
\eqref{e3.11} is actually a classical solution of \eqref{e3.10}.
 For convenience, we set
\begin{equation} \label{e3.39}
v(z)=X_{00}(z_1)+I(z),
\end{equation}
where $I(z)=\sum_{k=1}^5I_k(z)$ with
\begin{gather*}
I_1(z)\equiv I_1(z_1,z_2)=\sum_{m=1}^{\infty}X_{m0}(z_1)\cos(m\pi z_2), \\
I_2(z)\equiv I_2(z_1,z_3)=\sum_{m=1}^{\infty}X_{0n}^1(z_1)\cos(2n\pi z_3), \\
I_3(z)\equiv I_3(z_1,z_3)=\sum_{m=1}^{\infty}X_{0n}^2(z_1)\sin(2n\pi z_3), \\
I_4(z)=\sum_{m,n=1}^{\infty}X_{mn}^1(z_1)\cos(m\pi z_2)\cos(2n\pi z_3), \\
I_5(z)=\sum_{m,n=1}^{\infty}X_{mn}^2(z_1) \cos(m\pi z_2)\sin(2n\pi z_3).
\end{gather*}

Next, we show that $I_k(z)$ $(1\le k\le 5)$ is convergent for
$(z_1,z_2,z_3)\in(-\infty,+\infty)\times[0,1]\times(-\infty,+\infty)$.
Indeed, by using  \eqref{e3.31}, we have
\begin{gather} \label{e3.40}
|I_1(z)|\le\sum_{m=1}^{+\infty}\frac{C}{m^3(m\pi-\delta_0)}
|f|_{3,\alpha}^{(\delta_0)}e^{-\delta_0z_1} \le
C|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0z_1}, \\
\label{e3.41}
|I_1(z)|_{0,0}^{(\delta_0)}\le C|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{gather}

In  terms of \eqref{e3.35}-\eqref{e3.36}, we have
\begin{gather} \label{e3.42}
|I_2(z)|+|I_3(z)|
\le |\le\sum_{n=1}^{+\infty}\frac{C}{n^3(2n\pi-\delta_0)}
|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}
\le C|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|},\\
\begin{aligned}
|I_4(z)|+|I_5(z)|
&\le\sum_{m,n=1}^{+\infty}
\frac{C}{m^2n^2\sqrt{m^2+4n^2}(\pi\sqrt{m^2+4n^2}-\delta_0)} \\
&\quad\times \Big(|\tilde
f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}+\sum_{i=1}^2|\tilde
g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}\Big)e^{-\delta_0|z_1|} \\
&\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}+\sum_{i=1}^2
 |\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})e^{-\delta_0|z_1|}.
\end{aligned}
\end{gather}
This means
\begin{equation} \label{e3.44}
|I_k(z)|_{0,0}^{(\delta_0)}
\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}+\sum_{i=1}^2
|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})\quad for\quad k=2,3,4,5.
\end{equation}
Thus, the series $I(z)$ and further $v(z)$ are continuous because of
the uniform convergence of $I_k(z)$ in any compact subset of
$\tilde\Omega=(-\infty,+\infty)\times[0,1]\times(-\infty,+\infty)$.

Next, we show $I(z)\in C^1(\tilde\Omega)$ and further $v(z)\in C^1(\tilde\Omega)$.
It is noted that
\begin{align*}
|\partial_{z_1}I_1(z)|
&\le\sum_{m=1}^{+\infty}|X'_{m0}(z_1)|\\
&\le \sum_{m=1}^{+\infty}\frac{C}{m^2(m\pi-\delta_0)}
|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|} \\
\le C|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}
\end{align*}
and
\begin{align*}
|\partial_{z_2}I_1(z)|
&\le\sum_{m=1}^{+\infty}m\pi|X_{m0}(z_1)| \\
&\le \sum_{m=1}^{+\infty}\frac{C}{m^2(m\pi-\delta_0)}
|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}
\le C|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}e^{-\delta_0|z_1|}.
\end{align*}
Therefore, $I_1(z)\in C^1(\tilde\Omega)$ holds, and satisfies the estimate
$$
|\nabla_z I_1(z)|_{0,0}^{(\delta_0)}
\le C|f|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
$$
Analogously, $I_k(z)\in C^1(\tilde\Omega)$ $(k=2,3,4)$ holds.
Moreover, we have
\[
|\nabla_zI(z)|\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}+\sum_{i=1}^2|\tilde
g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})e^{-\delta_0|z_1|}
\]
and
\begin{equation} \label{e3.45}
|\nabla_zI(z)|_{0,0}^{(\delta_0)}\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}).
\end{equation}

Finally, we show $I(z)\in C^2(\tilde\Omega)$ and further $v(z)\in C^2(\tilde\Omega)$.
By using the expression of $I_1(z)$ and \eqref{e3.13}, we have
$$
\partial_{z_1}^2I_1(z)=\sum_{m=1}^{+\infty}X''_{m0}(z_1)\cos(m\pi z_2)
=\sum_{m=1}^{+\infty}(m^2\pi^2X_{m0}(z_1)+f_{m0}(z_1))\cos(m\pi z_2).
$$
It follows from Lemma \ref{lem3.1} and \eqref{e3.31} that
\begin{align*}
|\partial_{z_1}^2I_1(z)|
&\le\sum_{m=1}^{+\infty}\big(\frac{C}{m(m\pi-\delta_0)}
+\frac{C}{m^2}\big)(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})e^{-\delta_0|z_1|} \\
&\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}+\sum_{i=1}^2|\tilde
g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})e^{-\delta_0|z_1|}.
\end{align*}
Analogously, one has
\begin{align*}
|\partial_{z_1}^2I_2(z)|+|\partial_{z_1}^2I_3(z)|
&\le\sum_{n=1}^{+\infty}\big(\frac{C}{n(2n\pi-\delta_0)}
+\frac{C}{n^2}\big)(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
 +\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})e^{-\delta_0|z_1|} \\
&\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}+\sum_{i=1}^2|\tilde
g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})e^{-\delta_0|z_1|}
\end{align*}
and
\begin{align*}
&|\partial_{z_1}^2I_4(z)|+|\partial_{z_1}^2I_5(z)| \\
&\le\sum_{m,n=1}^{+\infty}\big(\frac{C\sqrt{m^2+4n^2}}{m^2n^2(\pi\sqrt{m^2+4n^2}-\delta_0)}
+\frac{C}{m^2n^2}\big)(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})e^{-\delta_0|z_1|} \\
&\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})
 e^{-\delta_0|z_1|}.
\end{align*}
This implies
\begin{equation} \label{e3.46}
|\partial_{z_1}^2I(z)|_{0,0}^{(\delta_0)}
\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}).
\end{equation}
Similarly, we can arrive at
\begin{equation} \label{e3.47}
|\partial_{z_iz_j}^2I(z)|_{0,0}^{(\delta_0)}
\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
 +\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})\quad\text{for }
1\le i\le 3,\;  1<j\le 3.
\end{equation}
Combining \eqref{e3.23}-\eqref{e3.25}, \eqref{e3.41} and
\eqref{e3.44}-\eqref{e3.47} yield \eqref{e3.5}.

On the other hand, it follows from \eqref{e3.25}, \eqref{e3.33}, \eqref{e3.38}
and the uniform convergence of $\partial_{z_1}I(z)$ with respect to $z_1$ that
\begin{gather} \label{e3.48}
\begin{aligned}
\lim_{z_1\to+\infty}\partial_{z_1}u(z)
&= \int_{-\infty}^{+\infty}\int_0^1\int_0^1\tilde f(t,z_2,z_3)dz_2dz_3dt \\
&\quad -\int_{-\infty}^{+\infty}\int_0^1 (\tilde g_2(z_1,z_3)-\tilde g_1(z_1,z_3))dz_1dz_3,
\end{aligned} \\
\label{e3.49}
\lim_{z_1\to+\infty}\partial_{z_k}u(z)=0\quad \text{for } k=2,3.
\end{gather}
The proof is complete.
\end{proof}

To obtain the higher regularities  and higher order norm
(i.e., $\|u\|_{6,\alpha}^{(\delta_0)}$) estimates  of $u(z)$ to \eqref{e3.3}
and further treat the nonlinear problem \eqref{e3.1} in the unbounded
strip domain $\tilde\Omega$, we have to overcome the difficulty induced by
the exponent weight $e^{\delta_0|z_1|}$ in the spaces
$H_{6,\alpha}^{(\delta_0)}$ or $\mathbb{H}_{6,\alpha}^{(\delta_0)}$ (it is noted
that in the general case, the weighted H\"older space with
the weight $|d_x|^{\nu} (\nu\in\mathbb{R})$ is only used to obtain a
priori estimates of solutions to second order elliptic equations,
here $d_x$ stands for the distance of the point $x$ to the
boundary or some parts of boundary. One can be referred to
\cite[Chapter 6]{11}). For this end, first we take a suitable transformation
(see \eqref{e3.53} below) to change the unbounded domain $\tilde\Omega$ into an
unbounded domain $Q$ which is bounded by two cones
$\{y:y_3=\mu_1\sqrt{y_1^2+y_2^2}\}$ and
$\{y: y_3=\mu_2\sqrt{y_1^2+y_2^2}\}$ with two suitable fixed constants
$\mu_1>\mu_2>0$. In this case, the exponent weight
$e^{\delta_0|z_1|}$ in the $z$-coordinates is equivalent to the
weight $|y|^{\delta_0}$ in the $y$-coordinates. From this, as in
\cite{17,18}, the estimate of solution in the weighted H\"older
space with the weight $|y|^{\delta_0}$ can be obtained. On the other
hand, due to the different properties of $u(z)$ as
$z_1\to -\infty$ or $z_1\to \infty$, we have to introduce another
transformation (see \eqref{e3.72} below) such that the estimate of
solution in the weighted H\"older space with the weight $|\tilde
y|^{-\delta_0}$ can be also obtained. Combining these two cases,
together with some delicate analysis, we can finally obtain the
estimates of $\|u\|_{6,\alpha}^{(\delta_0)}$. One can see the details
below.

For notational convenience, we use a weighted H\"older norm
which is introduced in \cite[Chapter 6]{11} and the references therein
as follows:

Let $D\subset \mathbb{R}^3$ be an open set, for $x, y\in D$, we define
$r_{x,y}=\min(|x|,|y|)$. For $m\in\mathbb{N}\cup\{0\}$,
$\alpha\in \Bbb{R^+}$, $\mu\in \Bbb{R^+}$,  $\mu_1, \mu_2\in \mathbb{R}$
and $v\in C^{m,\alpha}(D)$, we define
\begin{gather*}
[v]]_{m,0;D}^{(\mu)}\equiv\sum_{|\beta|=m}\sup_{x\in
D}|x|^{m+\mu}|D^{\beta}v(x)|, \\
[v]]_{m,\alpha;D}^{(\mu)}\equiv\sum_{|\beta|=m}
 \sup_{x, y\in D; x\neq y}r_{x,y}^{m+\alpha+\mu}\frac{|D^{\beta}v(x)
 -D^{\beta}v(y)|}{|x-y|^{\alpha}}, \\
|v\|_{m,\alpha;D}^{(\mu)}\equiv\sum_{0\le i\le m}[v]]_{i,0;D}^{(\mu)}
 +[v]]_{m,\alpha;D}^{(\mu)}, \\
[[v]]_{m,0;D}^{(\mu_1,\mu_2)}\equiv\max\Big\{\sup_{|x|<1}\sum_{|\beta|=m}
 |x|^{m+\mu_1}|D^{\beta}v(x)|,
\sup_{|x|>1}\sum_{|\beta|=m}|x|^{m+\mu_2}|D^{\beta}v(x)|\Big\}, 
\\
\begin{aligned}
[[v]]_{m,\alpha;D}^{(\mu_1,\mu_2)}
\equiv\max\Big\{&\sup_{0<r_{x,y}<1}\sum_{|\beta|=m}
r_{x,y}^{\mu_1+m+\alpha}\frac{|D^{\beta}v(x)-D^{\beta}v(y)|}{|x-y|^{\alpha}}, \\
&\sup_{r_{x,y}>1}\sum_{|\beta|=m}
r_{x,y}^{\mu_2+m+\alpha}\frac{|D^{\beta}v(x)-D^{\beta}v(y)|}{|x-y|^{\alpha}}\Big\},
\end{aligned} \\
\||v|\|_{m,\alpha;D}^{(\mu_1,\mu_2)}\equiv\sum_{0\le i\le m}
 [[v]]_{i,0;D}^{(\mu_1,\mu_2)}+[[v]]_{m,\alpha;D}^{(\mu_1,\mu_2)}.
\end{gather*}
Now let's consider the equation
\begin{equation} \label{e3.50}
\begin{gathered}
\Delta w=\hat{f} \quad \text{in }\tilde\Omega\\
w(z_1,0,z_3)=w(z_1,1,z_3)=0,\\
w(z_1,z_2,z_3+1)=w(z_1,z_2,z_3),\\
\lim_{z_1\to-\infty}w(z)=\lim_{z_1\to+\infty}w(z)=0,
\end{gathered}
\end{equation}
where $\hat{f}\in H_{3,\alpha}^{(\delta_0)}(\tilde\Omega)$ with
$\hat{f}(z_1,z_2,z_3+1)=\hat{f}(z_1,z_2,z_3)$.

\begin{lemma} \label{lem3.3}
If $w\in H_{5,\alpha}^{(\delta_0)}(\tilde\Omega)$ is a
solution of \eqref{e3.50}, which satisfies
\begin{equation} \label{e3.51}
\sup_{z\in\tilde\Omega} (e^{\delta_0|z_1|}|w(z)|)
\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)},
\end{equation}
then we have
$$
|w|_{5,\alpha;\tilde\Omega}^{(\delta_0)}\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
$$
\end{lemma}

\begin{proof}
First, we introduce  a coordinate transformation:
\begin{equation} \label{e3.52}
\begin{gathered}
y_1=e^{z_1}\cos(2\pi z_3)\sin(\frac{\pi z_2}{4}+\frac{\pi}{8}),\quad
y_2=e^{z_1}\sin(2\pi z_3)\sin(\frac{\pi z_2}{4}+\frac{\pi}{8}),\\
 y_3=e^{z_1}\cos(\frac{\pi z_2}{4}+\frac{\pi}{8}).
\end{gathered}
\end{equation}
In this case, the strip domain $\tilde\Omega$ is changed into an unbounded
domain $D$ which is bounded by two infinitely long cones
$\{y: y_3=\cot\frac{\pi}{8}\sqrt{y_1^2+y_2^2}\}$ and
$\{y: y_3=\cot\frac{3\pi}{8}\sqrt{y_1^2+y_2^2}\}$.

It follows from the transformation \eqref{e3.52} that \eqref{e3.50} can be changed
into the problem
\begin{equation} \label{e3.53}
\begin{gathered}
\sum_{i,j=1}^3\tilde{a}_{ij}(y)\partial_{ij}w
+\sum_{i=1}^3\tilde b_i(y)\partial_iw=F(y)\\
 \text{in }D\equiv\{y:\cot\frac{3\pi}{8}\sqrt{y_1^2+y_2^2}<y_3
<\cot\frac{\pi}{8}\sqrt{y_1^2+y_2^2}\},\\
w(y_1,y_2,y_3)=0\quad \text{on } cot\frac{3\pi}{8}\sqrt{y_1^2+y_2^2}=
y_3,\\
w(y_1,y_2,y_3)=0\quad \text{on } cot\frac{\pi}{8}\sqrt{y_1^2+y_2^2}=
y_3,\\
w(0,0,0)=\lim_{r\to+\infty}w(z_1,z_2,z_3)=0,
\end{gathered}
\end{equation}
where
\begin{gather*}
\tilde{a}_{11}=\frac{1}{|y|^2}\Bigl(y_1^2+(2\pi)^2y_2^2
+(\frac{\pi}{4})^2\frac{y_1^2y_3^2}{y_1^2+y_2^2}\Bigr),\\
\tilde{a}_{22}=\frac{1}{|y|^2}\Bigl(y_2^2+(2\pi)^2y_1^2
+(\frac{\pi}{4})^2\frac{y_2^2y_3^2}{y_1^2+y_2^2}\Bigr), \\
\tilde{a}_{33}=\frac{1}{|y|^2}\Bigl(y_3^2+(\frac{\pi}{4})^2(y_1^2+y_2^2)\Bigr),\quad
\tilde{a}_{12}=\frac{y_1y_2}{|y|^2}\Bigl(1+(\frac{\pi}{4})^2\frac{y_3^2}{y_1^2+y_2^2}
-(2\pi)^2\Bigr), \\
\tilde{a}_{13}=\frac{y_1y_3}{|y|^2}\big(1-(\frac{\pi}{4})^2\big),\quad
\tilde{a}_{23}=\frac{y_2y_3}{|y|^2}\big(1-(\frac{\pi}{4})^2\big),\\
\tilde{b}_1=\frac{y_1}{|y|^2}\big(1-(\frac{\pi}{4})^2-(2\pi)^2\big), \quad
\tilde{b}_2=\frac{y_2}{|y|^2}\big(1-(\frac{\pi}{4})^2-(2\pi)^2\big),\\
\tilde{b}_3=\frac{y_3}{|y|^2}\big(1-(\frac{\pi}{4})^2\big),\quad
F=\frac{1}{|y|^2}\hat{f}.
\end{gather*}
In addition,  from \eqref{e3.51} and the transformation \eqref{e3.52}
it follows that
\begin{equation} \label{e3.54}
\sup |y|^{\delta_0}|w|\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{equation}

Next, we show the  estimate
\begin{equation} \label{e3.55}
|w\|_{5,\alpha;D}^{(\delta_0)}
\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{equation}
To this end, for any fixed point $y_0=(y_1^0,y_2^0,y_3^0)\in D$,
we set $d_0=\mu|y_0|$ with  $0<\mu<<1$ and
$B_{d_0}(y_0)\equiv B(y_0,d_0)$, and define the map
$T: B_{d_0}(y_0)\to B_1(0)$ by
$T(y)=\frac{y-y_0}{d_0}$ for $y\in B_{d_0}(y_0)$.
In order to estimate $w(y)$ in $D$, we distinguish two cases:
\begin{itemize}
\item[(i)] $B_{d_0}(y_0)\subset\subset D$, and

\item[(ii)] $B_{d_0}(y_0)\bigcap\partial D\neq\emptyset$.
\end{itemize}
In case (i), set $\tilde{w}(x)=\frac{1}{d_0}w(y_0+d_0x)$ for $x\in
B_1(0)$, then it follows from a direct computation that $\tilde{w}(x)$
satisfies
\begin{equation} \label{e3.56}
\begin{aligned}
&\sum_{i,j=1}^2\tilde{a}_{ij}(y_0+d_0x)\partial_{ij}\tilde{w}(x)
+\sum_{i=1}^2d_0\tilde{b}_i(y_0+d_0x)\partial_{i}\tilde{w}(x) \\
&=d_0F(y_0+d_0x).
\end{aligned} \end{equation}
By the Schauder interior estimate (for example, see \cite[Chapter 6]{11}), one has
\begin{equation} \label{e3.57}
\|\tilde{w}\|_{5,\alpha;B_{\frac{1}{2}}(0)} \le C(\|\tilde{w}\|_{0;B_1(0)}
+\|d_0F\|_{3,\alpha;B_1(0)}).
\end{equation}
where the positive constant
$C$ depends only on $\alpha$.

For $y\in B_{\frac{d_0}{2}}(y_0)$, then $(\frac{1}{\mu}-\frac{1}{2})
d_0\le|y| \le(\frac{1}{2}+\frac{1}{\mu}) d_0$, and \eqref{e3.57} implies
\begin{equation} \label{e3.58}
\begin{aligned}
&\sum_{m=1}^5|y|^{m+\delta_0}|D_y^mw| \\
&\le C(\frac{1}{2}+\frac{1}{\mu})^{1+\delta_0}\bigl(d_0^{\delta_0}|w|_{0;B_{d_0}(y_0)}
+\sum_{m=1}^3d_0^{2+m+\delta_0}|D_y^mF(y)|_{0;B_{d_0}(y_0)} \\
&\quad +d_0^{5+\delta_0}[D_y^3F(y)]_{0,\alpha;B_{d_0}(y_0)}\bigr).
\end{aligned}
\end{equation}
Combining \eqref{e3.54}  with \eqref{e3.58} and noting
\[
\||F(y)|\|_{3,\alpha;D}^{(2-\delta_0,2+\delta_0)}
\le\||\hat{f}(y)|\|_{3,\alpha;D}^{(-\delta_0,\delta_0)}
\]
yield
\begin{equation} \label{e3.59}
\begin{aligned}
|w\|_{5,0;B_{\frac{d_0}{2}}(y_0)}^{(\delta_0)}
&\le C(|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}+\||F(y)|\|_{3,\alpha;D}^{(2-\delta_0,2+\delta_0)})\\
&\le C(|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}
 +\||\hat f(y)|\|_{3,\alpha;D}^{(-\delta_0,\delta_0)}).
\end{aligned}
\end{equation}

In case (ii), set $\tilde{w}(x)=\frac{1}{d_0}w(y_0+d_0x)$ for $x\in
M\equiv T(B_{d_0}(y_0)\bigcap D)$. As in Case (i), but it follows
from the Schauder boundary estimate that
\begin{equation} \label{e3.60}
\|\tilde{w}\|_{5,\alpha;B_{\frac{1}{2}(0)}\bigcap M}
\le C(\|\tilde{w}\|_{0;M}+\|d_0F\|_{3,\alpha;M}).
\end{equation}
Similar to \eqref{e3.57} and \eqref{e3.58}, we can arrive at
\begin{equation} \label{e3.61}
|w\|_{5,0;B_{\frac{d_0}{2}}(y_0)\bigcap D}^{(\delta_0)}
\le C(|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}+\||\hat
f(y)|\|_{3,\alpha;D}^{(-\delta_0,\delta_0)}).
\end{equation}
Therefore, by \eqref{e3.59} and \eqref{e3.61}, we have
\begin{equation} \label{e3.62}
|w\|_{5,0;D}\le  C(|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}
+\||\hat f(y)|\|_{3,\alpha;D}^{(-\delta_0,\delta_0)}).
\end{equation}

Next, we estimate $[D^5w]]_{0,\alpha;D}^{(\delta_0)}$.
Let $y,y'$ be distinct points in $D$ with $|y|\le|y'|$. We now
consider the following two cases:
\begin{itemize}
\item[(a)] $\operatorname{dist}(y,y')\le\frac{d}{2}$;

\item[(b)] $\operatorname{dist}(y,y')>\frac{d}{2}$, here $d=\mu|y|$.
\end{itemize}
In case (a), \eqref{e3.54}, \eqref{e3.57} and \eqref{e3.60} imply 
\begin{equation} \label{e3.63}
\begin{aligned}
|y|^{5+\delta_0+\alpha}\frac{|D^5w(y)-D^5w(y')|}{|y-y'|^{\alpha}}
&\le C(d^{\delta_0}|w|_{0;B_d(y)}+d^{2+\delta_0}\|F\|_{3,\alpha;B_d(y)}) \\
&\le C(|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}
 +\||\hat f(y)|\|_{3,\alpha;D}^{(-\delta_0,\delta_0)}).
\end{aligned}
\end{equation}
In case (b), \eqref{e3.62} implies
\begin{equation} \label{e3.64}
\begin{aligned}
&|y|^{5+\delta_0+\alpha}\frac{|D^5w(y)-D^5w(y')|}{|y-y'|^{\alpha}} \\
&\le Cd^{5+\delta_0}(|D^5w(y)|+|D^5w(y')|) \\
&\le C(\frac{1}{2}+\frac{1}{\mu})^{-1}
(|y|^{5+\delta_0}|D^5w(y)|+|y'|^{5+\delta_0}|D^5w(y')|)\\
&\le C(|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}
 +\||\hat f(y)|\|_{3,\alpha;D}^{(-\delta_0,\delta_0)}).
\end{aligned}
\end{equation}
Taking the supremum with respect to $y$ and $y'$ in \eqref{e3.63} and
\eqref{e3.64} respectively, we obtain
\begin{equation} \label{e3.65}
[D^5w]]_{0,\alpha;D}^{(\delta_0)}\le C(|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}+\||\hat
f(y)|\|_{3,\alpha;D}^{(-\delta_0,\delta_0)}).
\end{equation}

Next we show that
\begin{equation} \label{e3.66} \||\hat
f(y)|\|_{3,\alpha;D}^{(-\delta_0,\delta_0)}
\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{equation}
In fact, by using a direct computation, we can arrive at
If $z_1>0$, namely, $|y|>1$,
\begin{equation} \label{e3.67}
\begin{gathered}
e^{\delta_0|z_1|}|\hat f(z)|=|y|^{\delta_0}|\hat f(y)|,\\
e^{\delta_0|z_1|}\sum_{k=1}^3|D_z^k\hat f(z)|
\sim \sum_{k=1}^3|y|^{k+\delta_0}|D_y^k\hat f(y)|.
\end{gathered}
\end{equation}
If $z_1<0$, namely, $|y|<1$,
\begin{equation} \label{e3.68}
\begin{gathered}
e^{\delta_0|z_1|}|\hat f(z)|=|y|^{-\delta_0}|\hat f(y)|,\\
e^{\delta_0|z_1|}\sum_{k=1}^3|D_z^k\hat f(z)|
\sim \sum_{k=1}^3|y|^{k-\delta_0}|D_y^k\hat f(y)|.
\end{gathered}
\end{equation}
In addition, for $z,\tilde z\in\tilde\Omega$, we have
\begin{equation} \label{e3.69}
\begin{aligned}
&e^{\delta_0\min\{|z_1|, |\tilde z_1|\}}\frac{|D_z^3\hat
f(z)-D_z^3\hat f(\tilde z)|}{|z-\tilde z|^{\alpha}} \\
&\sim \sup_{y\in D; |y|<1}\sum_{k=1}^3|y|^{k-\delta_0}|D_y^k\hat f(y)|
+\sup_{y\in D;|y|>1} \sum_{k=1}^3|y|^{k+\delta_0}|D_y^k\hat f(y)|\\
&\quad +\sup_{0<d_{y,\tilde y}<1}d_{y,\tilde y}^{3+\alpha-\delta_0}\frac{|D_y^3\hat f(y)-D_y^3
 \hat f(\tilde y)|}{|y-\tilde y|^{\alpha}} \\
&\quad +\sup_{d_{y,\tilde y}>1}d_{y,\tilde y}^{3+\alpha+\delta_0}\frac{|D_y^3\hat f(y)-D_y^3
 \hat f(\tilde y)|}{|y-\tilde y|^{\alpha}},
\end{aligned}
\end{equation}
here $|y|=e^z$, $|\tilde y|=e^{\tilde z}$ and
\[
d_{y,\tilde y}=\begin{cases}
\max(|y|,|\tilde y|)&\text{if }\min(|y|,|\tilde y|)<\min(|y|^{-1},|\tilde y|^{-1})\\
\min(|y|,|\tilde y|)&\text{if }\min(|y|,|\tilde y|)\ge\min(|y|^{-1},|\tilde y|^{-1})
\end{cases}
\]
Therefore, combining \eqref{e3.67}-\eqref{e3.68} with \eqref{e3.69} and
noting $d_{y,\tilde y}\ge r_{y,\tilde y}$ yield \eqref{e3.66}.

Substituting \eqref{e3.66} into \eqref{e3.65}, we obtain
\begin{equation} \label{e3.70}
[D^5w]]_{0,\alpha;D}^{(\delta_0)}\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{equation}
Returning to the coordinate $z=(z_1,z_2,z_3)$ for
$[D^5w]]_{0,\alpha;D}^{(\delta_0)}$, we can derive
\begin{equation} \label{e3.71}
\begin{aligned}
&\sum_{|\beta|\le 5}\sup_{z\in\tilde\Omega}e^{\delta_0z_1}|D^{\beta}w(z)|+
\sum_{|\beta|=5}\sup_{z,\tilde z\in \tilde\Omega; z\neq \tilde z}
e^{\delta_0\min(z_1,\tilde z_1)}\frac{|D_z^{\beta}w(z)-D_z^{\beta}w(\tilde
z)|}{|z-\tilde z|^{\alpha}} \\
&\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{aligned}
\end{equation}
On the other hand, if we introduce the coordinate
transformation:
\begin{equation} \label{e3.72}
\begin{gathered}
\tilde{y}_1=e^{-z_1}\cos(2\pi z_3)\sin(\frac{\pi z_2}{4}+\theta_0),\quad
\tilde{y}_2=e^{-z_1}\sin(2\pi z_3)\sin(\frac{\pi z_2}{4}+\theta_0),\\
\tilde{y}_3=e^{-z_1}\cos(\frac{\pi z_2}{4}+\theta_0),
\end{gathered}
\end{equation}
then by using the same method to deduce \eqref{e3.71}, we can arrive at
\begin{equation} \label{e3.73}
\begin{aligned}
&\sum_{|\beta|\le 5}\sup_{z\in\tilde\Omega}e^{\delta_0z_1}|D^{\beta}w(z)|
+ \sum_{|\beta|=5}\sup_{z,\tilde z\in \tilde\Omega; z\neq \tilde z}
e^{\delta_0\max(z_1,\tilde z_1)}\frac{|D_z^{\beta}w(z)-D_z^{\beta}w(\tilde
z)|}{|z-\tilde z|^{\alpha}} \\
&\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{aligned}
\end{equation}
Combining \eqref{e3.71} with \eqref{e3.73} yields
\begin{gather}\label{e3.74}
[w]_{i,0;\tilde\Omega}^{(\delta_0)}\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}, \quad i=1,2,\dots,5;\\
 \label{e3.75}
\sup_{z_1\tilde z_1>0} e^{\delta_0\min\{|z_1|, |\tilde z_1|\}}
\frac{|D^5w(z)-D^5w(\tilde z)|}{|z-\tilde z|^{\alpha}}
\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{gather}

Next we  consider the case  $z_1\tilde z_1<0$ in \eqref{e3.75}.
Without loss of generality, we assume $z_1<0<\tilde z_1$. Moreover, we
consider the following two cases:
\begin{itemize}
\item[(I)] $\tilde z_1-z_1\ge 1$,

\item[(II)] $\tilde z_1-z_1<1$.
\end{itemize}

For case (I), we have
$$
e^{\delta_0\min\{-z_1,\tilde z_1\}}\frac{|D^5w(z)-D^5w(\tilde z)|}{|z-\tilde z|^{\alpha}}\le
e^{-\delta_0z_1}|D^5w(z)|+e^{\delta_0\tilde z_1}|D^5w(\tilde z)|
\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
$$

For case (II), it follows from \eqref{e3.71} and \eqref{e3.73} that
$$
e^{\delta_0\min\{-z_1,\tilde z_1\}}\frac{|D^5w(z)-D^5w(\tilde z)|}{|z-\tilde z|^{\alpha}}
\le e^{2\delta_0}\cdot e^{-\delta_0z_1}\frac{|D^5w(z)-D^5w(\tilde z)|}{|z-\tilde z|^{\alpha}}
\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
$$
Thus, we have proved that
\begin{equation} \label{e3.76}
[w]_{5,\alpha;\tilde\Omega}^{(\delta_0)}\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{equation}
Namely, by using \eqref{e3.74} and \eqref{e3.76}, the proof is complete.
\end{proof}


Based on Lemmas \ref{lem3.2} and \ref{lem3.3}. we now give the estimate of
$\|u\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}$, for  the solution of \eqref{e3.3}.

\begin{lemma} \label{lem3.4}
Under the assumptions of Lemma \ref{lem3.2}, the
solution $u(z)$ of \eqref{e3.3} satisfies the  estimate
\begin{equation} \label{e3.77}
\|u\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}
\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}).
\end{equation}
\end{lemma}

\begin{proof} To prove \eqref{e3.77}, by using Lemma \ref{lem3.2}, it only
suffices to prove
\begin{equation} \label{e3.78}
\begin{gathered}
|\partial_{z_k}u|_{5,\alpha;\tilde\Omega}^{(\delta_0)}\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}),\quad k=2,3,\\
|\partial_{z_1}^2u|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)} +\sum_{i=1}^2
 |\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}).
\end{gathered}
\end{equation}

Set $w(z)=\partial_{z_2}v$, where $v(z)$ is a solution of \eqref{e3.10}. 
Then it is easy to know that $w(z)$ satisfies the equation \eqref{e3.50} with
$\hat{f}(z)=\partial_{z_2}{f}(z)$. Therefore, by Lemma \ref{lem3.3}, we have
$$
|\partial_{z_2}v|_{5,\alpha;\tilde\Omega}^{(\delta_0)}
\le C|\hat{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}\le C|f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}.
$$
Due to $u(z)=v(z)+h(z)$ with $h(z)=\frac{1}{2}(\tilde g_2(z_1,z_3)
-\tilde g_1(z_1,z_3))z_2^2+\tilde g_1(z_1,z_3)z_2$, then
\begin{equation} \label{e3.79}
|\partial_{z_2}u|_{5,\alpha;\tilde\Omega}^{(\delta_0)}
\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}).
\end{equation}
By using  similar method to the one in Lemma \ref{lem3.3} (compared with the problem
\eqref{e3.50}, $\partial_{z_3}v$ will satisfy the same equation which admits two
Neumann boundary conditions on $y_2=0$ and $y_2$=1 instead of the
Dirichlet boundary conditions of \eqref{e3.50} and the same restrictions in
\eqref{e3.50} as $z_1\to\pm\infty$), we can arrive at
$$
|\partial_{z_3}v|_{5,\alpha;\tilde\Omega}^{(\delta_0)}\le
C|\partial_{z_3}{f}|_{3,\alpha;\tilde\Omega}^{(\delta_0)}\le
C|{f}|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
$$
and further
\begin{equation} \label{e3.80}
|\partial_{z_3}u|_{5,\alpha;\tilde\Omega}^{(\delta_0)}
\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}).
\end{equation}
On the other hand, substituting \eqref{e3.79}-\eqref{e3.80} into the equation
\eqref{e3.3} yields
$$
|\partial_{z_1}^2u|_{4,\alpha;\tilde\Omega}^{(\delta_0)}\le C(|\tilde f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\tilde g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}).
$$
The proof is complete.
\end{proof}

Based on Lemma \ref{lem3.4}, we now derive  uniform estimates on the
solution $\dot u(z)$ to problem \eqref{e3.1}.

\begin{lemma} \label{lem3.5}
Suppose that the assumption \eqref{e3.2} holds,
and $\dot{u}\in C^2(\overline{\tilde{\Omega}})$ is a solution of
\eqref{e3.1}. Then there exists a positive constant $\delta_0$ such that for
any $\dot f\in H_{4,\alpha}^{(\delta_0)}(\tilde{\Omega})$, 
$\dot g_i \in H_{5,\alpha}^{(\delta_0)}(\tilde{\Omega}) (i=1,2)$, we have 
$\dot u\in \mathbb{H}_{6,\alpha}^{(\delta_0)}(\tilde{\Omega})$ with
\begin{equation} \label{e3.81}
\|\dot u\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}
\le C (|\dot f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\dot g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}),
\end{equation}
 where $C>0$ depends only on the constants $\Lambda$ and $\lambda$ in \eqref{e3.2}.
\end{lemma}

\begin{proof} Firstly, we introduce the coordinate transformation
\begin{equation} \label{e3.82}
\tilde{z}_1=k_1z_1, \quad
\tilde{z}_2=k_2z_2, \quad
\tilde{z}_3=k_3z_3
\end{equation}
with $k_1=\frac{1}{\sqrt{c^2(\rho_0)-q_0^2}}$ and
$k_2=k_3=\frac{1}{c(\rho_0)}$.
Under this transformation, the domain $\tilde{\Omega}$ is changed into
the domain $Q\equiv (-\infty,+\infty)\times[0,\frac{1}{c(\rho_0)}]\times(-\infty,+\infty)$,
and the equation \eqref{e3.1} can be rewritten as
\begin{equation} \label{e3.83}
\begin{gathered}
\Delta\dot{u}=\bar f \quad \text{in }  Q,\\
\partial_{\tilde{z}_2}\dot{u}=\tilde{g}_1\quad\text{on } \tilde{z}_2=0,\\
\partial_{\tilde{z}_2}\dot{u}=\tilde{g}_2\quad\text{on } \tilde{z}_2=l,\\
\dot{u}(\tilde{z}_1,\tilde{z}_2,\tilde{z}_3+l)=\dot{u}(\tilde{z}_1,\tilde{z}_2,\tilde{z}_3),\\
\lim_{\tilde{z}_1\to-\infty}\dot{u}=0,\\
\lim_{\tilde{z}_1\to+\infty}\nabla_{\tilde z}\dot{u} \text{ exists}.
\end{gathered}
\end{equation}
where $l=1/c(\rho_0)$, and
\begin{equation} \label{e3.84}
\begin{gathered}
\bar f=\dot{f}+\sum_{i=1}^3(1-k_i^2(c^2(\nabla
v)-\partial_{z_1}^2v))\partial_{\tilde{z}_i}^2\dot{u}-2\sum_{1\le i<j\le3}
k_ik_j\partial_{z_i}v\partial_{z_j}v\partial_{\tilde{z}_i\tilde{z}_j}\dot{u}, \\
\tilde{g}_i=c(\rho_0)\dot{g},\quad i=1,2.
\end{gathered}
\end{equation}

For simplicity and without loss of generality, we assume $l=1$ in
\eqref{e3.83}.
By the assumption $\|v-q_0z_1\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}<\varepsilon$ and
Lemma \ref{lem2.1}, we have
\begin{equation} \label{e3.85}
|\bar f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}\le O(\varepsilon)\|\dot{u}\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}
+|\dot{f}|_{4,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{equation}
On the other hand,  by using Lemma \ref{lem3.4}, one has
\begin{equation} \label{e3.86}
\|\dot{u}\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}\le C(|\bar f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\dot g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}).
\end{equation}
Substituting \eqref{e3.84} into \eqref{e3.85} yields \eqref{e3.81}.

Moreover,  from \eqref{e3.48} and \eqref{e3.49} it follows that
\begin{equation} \label{e3.87}
\begin{gathered}
\lim_{z_1\to+\infty}|\partial_{z_1}\dot{u}|\le C(|\dot{f}|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\dot{g}_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}),\\
\lim_{z_1\to+\infty}\partial_{z_i}\dot{u}=0,\quad
i=2,3.
\end{gathered}
\end{equation}
Therefore, the proof is complete.
\end{proof}

Based on Lemmas \ref{lem3.2} and  \ref{lem3.5},  from the standard
continuity method (see \cite[Theorem 5.2]{11}) we have the 
following result.


\begin{theorem} \label{thm3.6}
There exists a unique solution $\dot u\in \mathbb{H}_{6,\alpha}^{(\delta_0)}(\tilde{\Omega})$
 to problem \eqref{e3.1} for some $\delta_0>0$, which admits the following estimate 
\begin{equation} \label{e3.88}
\|\dot u\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}
\le C \bigl(|\dot f|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^{2}|\dot g_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}\bigr).
\end{equation}
\end{theorem}

\section{Proofs of Theorems \ref{thm1.1}, \ref{thm2.2}, and \ref{thm2.3}}

In this section,  first we use the contraction mapping
principle to show Theorem \ref{thm2.3}. To this end, we define the space
$K=\{\psi(z):\psi(z)-\varphi_0(z)\in \mathbb{H}_{6,\alpha}^{(\delta_0)}(\tilde\Omega),
\psi(z_1,z_2,z_3+1)=\psi(z_1,z_2,z_3),
\|\psi(z)-\varphi_0(z)\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}\le \varepsilon\}$ with
$\varphi_0(z)=q_0z_1$.

Set $\varphi=\dot{\varphi}+\varphi_0$, then $\dot{\varphi}$ satisfies
\begin{equation} \label{e4.1}
\begin{gathered}
L(\psi)\dot{\varphi}=\sum_{i,j=1}^3a_{ij}(z,D\psi)\partial_{z_iz_j}^2\dot{\varphi}
=\dot{f}(z,D\psi,D^2\psi)\quad \text{in } \tilde\Omega,\\
G_1(\psi)\dot\varphi=\partial_{z_2}\dot{\varphi}
 =\dot{g}_1(z,D\psi)\quad\text{on } z_2=0,\\
 G_2(\psi)\dot\varphi=\partial_{z_2}\dot{\varphi}
 =\dot{g}_2(z,D\psi)\quad z_2=1,\\
\dot{\varphi}(z_1,z_2,z_3+1)=\dot{\varphi}(z_1,z_2,z_3),\\
\lim_{z_1\to-\infty}\dot{\varphi}=0,\\
\lim_{z_1\to+\infty}\nabla_z\dot{\varphi}\text{ exists},
\end{gathered}
\end{equation}
where
\begin{gather*}
\begin{aligned}
\dot{f}(z,D\psi,D^2\psi)
&=(L(\varphi_0)\varphi_0-L(\psi)\varphi_0) \\
&\quad +\sum_{i,j=1}^3
(a_{ij}(z,\nabla\psi)-A_{ij}(z,\nabla\psi))\partial_{z_iz_j}\dot{\psi}
-B(z,\nabla\psi)\partial_{z_2}\dot{\psi},
\end{aligned} \\
\dot{g}_i(D\psi)=-b_{i1}(z)\partial_{z_1}\dot{\psi}-b_{i3}(z)\partial_{z_3}\dot{\psi},\quad
i=1,2
\end{gather*}
with $\dot{\psi}=\psi-\varphi_0$.

Define the nonlinear mapping $J$ by $J(\psi)=\varphi$. 

\begin{lemma} \label{lem4.1}
Suppose that $\alpha$ and $\delta_0$ are the
positive constants given in Lemma \ref{lem3.5}. Then there exists an
$\varepsilon_0>0$ such that for any $\varepsilon\in(0,\varepsilon_0)$, $J$ is a mapping
from $K$ to itself.
\end{lemma}

\begin{proof}
 By the definitions of $A_{ij}(z,\nabla\psi)$ and
$B(z,\nabla\psi)$ in \eqref{e2.2}, we arrived at
\begin{gather*}
\begin{aligned}
&|(a_{ij}(z,\nabla\psi)-A_{ij}(z,\nabla\psi))\partial_{z_iz_j}
 \dot{\psi}|_{4,\alpha;\tilde\Omega}^{(\delta_0)} \\
&\le|a_{ij}(z,\nabla\psi)-A_{ij}(z,\nabla\psi)|_{4,\alpha;\tilde\Omega}^{(0)}
\|\dot{\psi}\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}
\le C\varepsilon^2,
\end{aligned}\\
|B(z,\nabla\psi)\partial_{z_2}\dot\psi|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
\le|B(z,\nabla\psi)|_{4,\alpha;\tilde\Omega}^{(0)}
\|\dot\psi\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}\le C\varepsilon^2.
\end{gather*}
In addition, by using $\varphi_0=q_0z_1$, we have
$$
L(\varphi_0)\varphi_0-L(\psi)\varphi_0=0.
$$
Thus, we arrive at
\begin{equation} \label{e4.2}
|\dot{f}|_{4,\alpha;\tilde\Omega}^{(\delta_0)}\le C\varepsilon^2.
\end{equation}
Analogously, one has
\begin{equation} \label{e4.3}
|\dot{g}_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)}\le C\varepsilon^2,\quad i=1,2.
\end{equation}
It follows from Theorem \ref{thm3.6} that
\begin{equation} \label{e4.4}
\|\dot{\varphi}\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}
\le C (|\dot{f}|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
+\sum_{i=1}^2|\dot{g}_i|_{5,\alpha;\tilde\Omega}^{(\delta_0)})
\le C\varepsilon^2,
\end{equation}
where $C>0$  depends only on $\Lambda,\lambda$.

Choose $\varepsilon_0=\frac{1}{2C}$, then for any $0<\sigma<\varepsilon<\varepsilon_0$, by \eqref{e4.4}
we obtain
\begin{equation} \label{e}
\|\dot{\varphi}\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}<\varepsilon.
\end{equation}
This means that the mapping $J$ is from $K$ into itself.
\end{proof}

Next we show that the mapping $J$ defined above is contractible.

\begin{lemma} \label{lem4.2}
Under the assumptions of Lemma \ref{lem4.1}, the
mapping $J$ is a contractible mapping from $K$ to itself.
\end{lemma}

\begin{proof}
 Take $\psi_1, \psi_2\in K$. Let $\varphi_i=J\psi_i$ and 
$\dot{\varphi}_i=\varphi_i-\varphi_0$, then we have
\begin{equation} \label{e4.6}
\begin{gathered}
L(\psi_2)(\varphi_2-\varphi_1)=\dot{f}(z,D\psi_2,D^2\psi_2)-\dot{f}(z,D\psi_1,D^2\psi_1)-
(L(\psi_2)-L(\psi_1))\dot{\varphi}_1\quad \text{in } \tilde\Omega,\\
\partial_{z_2}(\varphi_2-\varphi_1)=\dot{g}_1(z,\psi_2)-\dot{g}_1(z,\psi_1)\quad\text{on }
z_2=0,\\
\partial_{z_2}(\varphi_2-\varphi_1)=\dot{g}_2(z,\psi_2)-\dot{g}_2(z,\psi_1)\quad\text{on }\
z_2=1,\\
(\varphi_2-\varphi_1)(z_1,z_2,z_3+1)=(\varphi_2-\varphi_1)(z_1,z_2,z_3),\\
\lim_{z_1\to-\infty}(\varphi_2-\varphi_1)=0,\\
\lim_{z_1\to+\infty}\nabla_z(\varphi_2-\varphi_1)\text{ exists}.
\end{gathered}
\end{equation}
As in Lemma \ref{lem4.1}, a direct computation yields
\begin{gather*}
|\dot{f}(z,D\psi_2,D^2\psi_2)-\dot{f}(z,D\psi_1,D^2\psi_1)|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
\le C\varepsilon\|\psi_2-\psi_1\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}\,, \\
|\dot{g}_i(z,\psi_2)-\dot{g}_i(z,\psi_1)|_{5,\alpha;\tilde\Omega}^{(\delta_0)}
\le C\varepsilon\|\psi_2-\psi_1\|_{6,\alpha;\tilde\Omega}^{(\delta_0)},\quad i=1,2;\\
|(L(\psi_2)-L(\psi_1))\dot{\varphi}_1|_{4,\alpha;\tilde\Omega}^{(\delta_0)}
\le C\varepsilon \|\psi_2-\psi_1\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}.
\end{gather*}
It follows from Theorem \ref{thm3.6} that
$$
\|\varphi_2-\varphi_1\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}\le
C\varepsilon\|\psi_2-\psi_1\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}.
$$
Choosing appropriately small $\varepsilon_0$ and letting $0<\varepsilon<\varepsilon_0$
yields
$$
\|J(\psi_2)-J(\psi_1)\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}\le
\frac{1}{2}\|\psi_2-\psi_1\|_{6,\alpha;\tilde\Omega}^{(\delta_0)}.
$$
This means that $J$ is a contractible mapping.
\end{proof}

Based on Lemma \ref{lem4.1} and Lemma \ref{lem4.2}, we now show Theorem \ref{thm2.2}.

\begin{proof}[Proof of Theorem \ref{thm2.3}]
 By Lemmas \ref{lem4.1} and \ref{lem4.2}, we know
that the mapping $J\psi=\varphi$ has a unique fixed point in 
$\mathbb{H}_{6,\alpha}^{(\delta_0)}(\tilde\Omega)$.

Next, we show $\lim_{z_1\to +\infty}\nabla_z\varphi(z)$ exists.
Since for $Z_1>Z_2>0$, we have
\begin{align*}
&|\partial_{z_1}\varphi(Z_1,z_2,z_3)-\partial_{z_1}\varphi(Z_2,z_2,z_3)| \\
&=(Z_1-Z_2)| \int_0^1\partial_{z_1}^2\varphi (\theta Z_1+(1-\theta)Z_2, z_2,z_3)d\theta| \\
&\le C(Z_1-Z_2)|\int_0^1 e^{-\delta_0(\theta Z_1+(1-\theta)Z_2)}d\theta|\le Ce^{-\delta_0 Z_2}.
\end{align*}
 This means that there exists a function $q(z_2,z_3)$ such that
$\partial_{z_1}\varphi(z_1,z_2,z_3)$ converges to $q(z_2,z_3)$ uniformly as
$z_1\to +\infty$. On the other hand,
$|\partial_{z_1z_k}^2\varphi(z_1,z_2,z_3)|\le Ce^{-\delta_0z_1}$ for $k=2,3$,
this implies that $\partial_{z_1z_k}^2\varphi(z_1,z_2,z_3)$ converges to $0$
uniformly as $z_1\to +\infty$. Therefore, we can arrive at
$\partial_{z_2}q(z_2,z_3)=\partial_{z_3}q(z_2,z_3)\equiv 0$, namely,
$q(z_2,z_3)\equiv q$, here $q$ is a constant which will be
determined later on. In addition,  $|\partial_{z_k}\varphi(z)|\le
Ce^{-\delta_0|z_1|}$ ($k=2,3$), then $\lim_{z_1\to
\pm\infty}\partial_{z_k}\varphi(z)=0$. From the analysis above, we can also
obtain under the $x$-coordinates,
\begin{equation} \label{e4.7}
\lim_{x_1\to\infty}\partial_{x_1}\varphi=q \quad\text{and}\quad
\lim_{x_1\to\pm\infty}\partial_{x_i}\varphi=0\quad\text{for }
i=2,3.
\end{equation}

We now show that $q=q_0$ holds.
Integrating the mass conservation equation
$\sum_{j=1}^3\partial_{x_j}(\rho(|\nabla\varphi|)\partial_{x_j}\varphi)=0$ in
$\Omega_R=\Omega\cap \{x: -R\le x_1\le R, 0<x_3<1\}$ yields
\begin{equation} \label{e4.8}
0=-\int_{x_1=-R}\rho(\nabla\varphi)\partial_{x_1}\varphi
d\sigma+\int_{x_1=R}\rho(\nabla\varphi)\partial_{x_1}\varphi d\sigma.
\end{equation}
Using \eqref{e4.7} and letting $R\to+\infty$ in \eqref{e4.8}, we arrive at
\begin{equation} \label{e4.9}
\rho(q)q=\rho(q_0)q_0.
\end{equation}
On the other hand, it follows from \eqref{e1.2} that
$$
\rho(q)q=(\frac{\gamma-1}{A\gamma})^{\frac{1}{\gamma-1}}(2C_0-q^2)^{\frac{1}{\gamma-1}}q.
$$
A direct computation yields
\begin{equation} \label{e4.10}
(\rho(q)q)'=(\frac{\gamma-1}{A\gamma})^{\frac{1}{\gamma-1}}(2C_0-q^2)^{-\frac{\gamma+1}{\gamma-1}}
(2C_0-\frac{\gamma+1}{\gamma-1}q^2).
\end{equation}
In addition, by using $q<c(q_0)$ and \eqref{e1.2}, we
have
\begin{equation} \label{e4.11}
C_0=\frac{1}{2}q^2+\frac{c^2(q)}{\gamma-1}>\frac{1}{2}q^2+\frac{1}{\gamma-1}q^2
=\frac{\gamma+1}{2(\gamma-1)}q^2.
\end{equation}
Thus, substituting \eqref{e4.11} into \eqref{e4.10} yields
\begin{equation} \label{e4.12}
(\rho(q)q)'>0.
\end{equation}
Combining \eqref{e4.9} with \eqref{e4.12} implies $q=q_0$.
Thus, we complete the proof.
\end{proof}

Since the proofs of Theorem \ref{thm1.1} and \ref{thm2.2} come directly 
from Theorem \ref{thm2.3}, then we omit them.

\subsection*{Acknowledgments}
Wenxia Chen is supported by the National Natural Science Foundation
of China (No.11501253).
Gang Xu and Qin Xu are supported  by the National
Natural Science Foundations of China (No.11571141).


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\end{document}