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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 142, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/142\hfil Irregular oblique derivative problems]
{Irregular oblique derivative problems for second-order nonlinear elliptic
equations on infinite domains}

\author[G. C. Wen\hfil EJDE-2012/142\hfilneg]
{Guo Chun Wen}  % in alphabetical order

\address{Guo Chun Wen \newline
LMAM, School of Mathematical Sciences, Peking University,
Beijing 100871, China}
\email{Wengc@math.pku.edu.cn}

\thanks{Submitted July 20, 2012. Published August 20, 2012.}
\subjclass[2000]{35J65, 35J25, 35J15}
\keywords{Irregular oblique derivative problem; nonlinear elliptic equations; 
 \hfill\break\indent infinite domains}

\begin{abstract}
 In this article, we study irregular oblique derivative
 boundary-value problems for nonlinear elliptic equations of
 second order in an infinite domain. We first provide the formulation
 of the above boundary-value problem and obtain a representation theorem.
 Then we give a priori estimates of solutions by using the reduction
 to absurdity and the  uniqueness of solutions. Finally by the above
 estimates and the Leray-Schauder theorem, the existence of solutions
 is proved.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Formulation of the problem}

Let $D$ be an $(N+1)$-connected domain including the infinite
point with the boundary $\Gamma=\cup^{N}_{j=0} \Gamma_{j}$ in $\mathbb{C}$, where
$\Gamma\in C_{\mu}^{2}\,(0<\mu<1)$. Without loss of generality, we
assume that $D$ is a circular domain in $|z|>1$, where the boundary
consists of $N+1$ circles $\Gamma_0=\Gamma_{n+1}=\{|z|=1\}$,
$\Gamma_j=\{|z-z_j|=r_j\}$, $j=1,\dots,N$ and $z=\infty\in D$. In this
article, the notation is as the same in References \cite{b1,v1,w1,w2,w3,w4}. 
We consider the second-order nonlinear elliptic equation in the complex
form
\begin{equation}
\begin{gathered}
u_{z\bar z}=F(z,u,u_{z},u_{zz}),\quad 
F=\operatorname{Re} [Qu_{zz}+A_{1}u_{z}] +\hat{A}_2u+A_3, \\
Q=Q(z,u,u_{z},u_{zz}),A_{j}=A_{j}(z,u,u_{z}),\quad j=1,2,3,\;
\hat{A}_2=A_2+|u|^\sigma,
\end{gathered} \label{e1.1}
\end{equation}
satisfying the following conditions.

\subsection*{Condition  (C)}
(1) $Q(z,u,w,U),A_{j}(z,u,w)(j=1,2,3)$ are continuous in $u\in\mathbb{R}$,
$w\in\mathbb{C}$ for almost every $z\in D,\,U\in\mathbb{C}$, and
$Q=0$, $A_{j}=0$ $(j=1,2,3)$ for $z\not\in D$, $\sigma$ is a positive
number.

(2) The above functions are measurable in $D$ for all continuous
functions $u(z)$, $w(z)$ in $\overline{D}$, and satisfy
\begin{equation}
L_{p,2}[A_j(z,u,w),\overline{D}]\le k_0,\quad j=1,2,\;
L_{p,2}[A_3(z,u,w),\bar D]\le k_1, \label{e1.2}
\end{equation}
in which $p_0,p\,(2<p_0\le p),k_0,k_1$ are non-negative constants.

(3) Equation \eqref{e1.1} satisfies the uniform ellipticity condition
\begin{equation}
|F(z,u,w,U_{1})-F(z,u,w,U_2)|\le q_{0}|U_1-U_2|,\quad
A_2\ge0,\; \text{in }D, \label{e1.3}
\end{equation}
for almost every point $z\in D$, any number $u\in \mathbb{R}$,
 $w,U_1,U_2\in \mathbb{C}$, where $q_0(<1)$ is a non-negative constant.

\subsection*{Problem (P)}
In the domain $D$, find a solution $u(z)$ of equation \eqref{e1.1}, which is continuous in $\overline{D}$, and satisfies the boundary conditions
\begin{equation}
\begin{gathered}
\frac12\frac{\partial u}{\partial\nu}+c_1(z)u(z)=c_2(z),\quad
z\in{\Gamma},\; u(a_j)=b_j,j=0,1,\dots,K',\;{\rm i.e.}\\
\operatorname{Re}[\overline{\lambda(z)}u_z]+c_1(z)u=c_2(z),\quad
z\in{\Gamma},\;u(a_j)=b_j, j=0,1,\dots,K',
\end{gathered} \label{e1.4}
\end{equation}
where the vector $\nu$ $(\ne0)$ can be arbitrary at every point on ${\Gamma}$,
$K'$ $(=2K-2N+J+1\ge0)$, $J$ are non-negative integers as stated
below, $\lambda(z)=\cos(\nu,x)+i\sin(\nu,x)={\rm e}^{i(\nu,x)}\ne0$,
$(\nu,x)$ is the angle between $\nu$ and the $x$-axis,
$a_j\,(\in\Gamma_j,j=0,1,\dots,K')$ are distinct points on ${\Gamma}$.
Suppose that $\lambda(z),c_1(z), c_2(z),\,b_j(j$ $=0,1,\dots,K')$
satisfy the conditions
\begin{equation}
\begin{gathered}
C_\alpha[\lambda(z),\Gamma]\le k_0,\quad C_\alpha[c_1(z),\Gamma]\le k_0,\quad
C_\alpha[c_2(z),\Gamma]\le k_2,\\
|b_j|\le k_2,\quad j=0,1,\dots,K',\quad c_1(z)\cos(\nu,n)\ge 0\quad
 \text{on } \Gamma,
\end{gathered}
\label{e1.5}
\end{equation}
in which $\alpha$ $(1/2<\alpha<1)$, $k_2$ are non-negative constants. The
boundary $\partial D={\Gamma}$ can be divided into two parts, namely
$E^+\subset \{z\in\partial D,\cos(\nu,n)\ge0,c_1\ge0\}$ and
$E^-\subset\{z\in\partial D,\cos(\nu,n)\le 0,c_1\le0\}$, and
$E^+\cap E^-=\emptyset$, $E^+\cup E^-={\Gamma}$,
$\overline{E^+}\cap \overline{E^-}=E^0$. For every component
$L={\Gamma}_j\,(0\le j\le N)$ of ${\Gamma}$, there are three cases:

1. $L\subset E^+$.

2. $L\subset E^-$.
In these cases, if $\cos(\nu,n)\equiv0$, $c_1(z)\equiv0$ on
${\Gamma}_j$ $(1\le j\le J,\,J\le N+1)$, and the above identical formulas on
${\Gamma}_j$ $(J<j\le N+1)$ do not hold, then we need the conditions
$\int_{{\Gamma}_j}c_2(z){\rm d}s=0$ $(1\le j\le J)$, and
$u(a_j)=b_j$, $j=0,1,\dots, K'$ $(\ge J)$, in which
$a_j, b_j$ $(j=0,1,\dots,K')$ are as stated before, and denote
${\Gamma}'= \cup_{j=1}^{J}{\Gamma}_j$,
${\Gamma}''=\cup_{j=J+1}^{N+1}{\Gamma}_j$.

3. There exists at least a
point on each component of $L^+=E^+\cap L$ and $L^-=E^-\cap L$, such
that $\cos(\nu,n)\not=0$ at the point, and
$E^0\cap L\in \{a_{0},a_1,\dots,a_{K'}\}$, such that every component of
$L^+$, $L^-$ includes its initial point and does not include its terminal point;
and $a_j\in\overline{L^+}\cap L^-\,(0\le j\le K')$, when the
direction of $\nu$ at $a_j$ is equal to the direction of $L$; and
$a_j\in L^+\cap\overline{L^-}$ $(0\le j\le K')$, when the direction
of $\nu$ at $a_j$ is opposite to the direction of $L$; and
$\cos(\nu,n)$ changes the sign once on the two components with the
end point $a_j$ $(0\le j\le K')$; we may assume that
$u(a_j)=b_j$, $j=0,1,\dots,K'$. The number
\begin{equation}
K=\frac12(K_1+\cdots+K_{N+1}),\quad
K_j={\Delta}_{{\Gamma}_j} \arg\lambda(z),\quad
j=1,\dots,N+1, \label{e1.6}
\end{equation}
is called the index of Problem (P). We can choose $K'=2K-2N+J+1$.
In the following, we shall prove the next theorem.
Now we prove the uniqueness of solutions for Problem (P) of
\eqref{e1.1}.

\begin{theorem} \label{thm1.1}
Suppose that \eqref{e1.1}  satisfy Condition {\rm (C)}. Then Problem {\rm (P)}
for equation \eqref{e1.1} with the condition that $A_3=0$ in $D$,
$c_2=0$ on $\Gamma$ and
$b_j=0(j=0,1,\dots,K')$ has only the trivial solution.
\end{theorem}

\begin{proof}
Let $u(z)$ be any solution of Problem (P)
for equation \eqref{e1.1} with $A_3=0$, $c_2=0$ on $\Gamma$ and
$b_j=0(j=0,1,\dots,K')$. From Condition (C), it is easily seen
that $u(z)$ is a solution of the following uniformly elliptic
equation
\begin{equation}
u_{z \bar z}=\operatorname{Re}[Qu_{zz}+A_1u_{z}]+\hat{A}_2u,\quad
|Q|\le q_0<1,\hat{A}_2=A_2 +|u|^\sigma\ge0\quad \text{in }D, \label{e1.7}
\end{equation}
and satisfies the boundary condition
\begin{equation}
\frac{\partial u}{\partial\nu}+2c_1(z)u(z)=0\quad\text{on }
\Gamma^*,\;u(a_j)=0,\; j=0,1,\dots,K'. \label{e1.8}
\end{equation}
Substitute the solution $u(z)$ into the coefficients of equation \eqref{e1.7},
we can find a solution $\Psi(z)$ of \eqref{e1.7} satisfying the condition
$$
\Psi(z)=1\quad\text{on }\Gamma,
$$
thus the function $U(z)=u(z)/\Psi(z)$ is a solution of the equation
\begin{equation}
U_{z\bar z}=\operatorname{Re}[QU_{zz}+A_0U_z],\quad
A_0=-2(\log\Psi)_{\bar z}+2Q(\log\Psi)_z+A_1, \label{e1.9}
\end{equation}
satisfying the boundary conditions
\begin{equation}
\frac{\partial U}{\partial \nu}+c_1^*(z)U(z)=0\quad\text{on }
\Gamma^*,\; U(a_j)=0,\; j=0,1,\dots,K', \label{e1.10}
\end{equation}
where $a_1^*(z)=c_1(z)+(\partial\Psi/\partial\nu)/\Psi(z)$,
$c_1^*(z)\cos(\nu,n) \geq0$ on $\Gamma^*$.

If $M=\max_{\overline D}U(z)>0$ in $D$, then
there exists a point $z^*\in\Gamma$ such that
$M=U(z^*)=\max_{\overline D}U(z)>0$. When $z^*\in\Gamma'$, noting
that $\cos(\nu,n)\equiv0$,
$c_1(z) \equiv0$, $\partial\Psi(z)/{\partial\nu}\equiv0$ on $\Gamma'$, we
have $\partial U/\partial\nu\equiv0$,
$U(z)\equiv M$ on $\Gamma_j(1\le j\le J')$, this contradicts the point
conditions in \eqref{e1.10}. When $z^*\in\Gamma''$, if $\cos(\nu,n)>0$ at $z^*$,
from \cite[Corollary 2.11, Chapter III]{w1},
we have $\partial U/\partial\nu>0$ at $z^*$, this contradicts
\eqref{e1.10} on $\Gamma''$.
If $\cos(\nu,n)=0$ and $c^*_1(z^*)\neq0$ at $z^*$, then
$\partial U/\partial\nu+c_1^*(z)U\ne0$ at $z^*$, it is also impossible. Denote
by $L$ the longest curve of $\Gamma$ including the point $z^*$, such
that $\cos(\nu,n)=0$ and $c^*(z)=0$, thus $u(z)=M$ on $L$, from the
point conditions in \eqref{e1.10}, any point of
$\tilde T=\{z_0,z_1,\dots,z_{K'}\}$ cannot be an end point of $L$, then there
exists a point $z'\in\Gamma''$, such that at $z'$,
$\cos(\nu,n)>0$ $(<0)$,
${\partial U}/{\partial n}>0$, $\cos(\nu,s)>0$ $(<0)$,
${\partial U}/{\partial s}\ge0$, or $\cos(\nu,n)<0$ $(>0)$,
${\partial U}/{\partial n}>0$, $\cos(\nu,s)>0\,(<0)$,
${\partial U}/{\partial s}\le0$, hence
$$
\frac{\partial U}{\partial\nu}=\cos(\nu,n)\frac{\partial U}
{\partial n}+\cos(\nu,s)\frac{\partial U}{\partial s}>0,\quad\text{or }
<0\text{ at }z'
$$
holds, where $s$ is the tangent vector
at $z'\in\Gamma''$, and then
$$
\frac{\partial U}{\partial\nu}+c^*_1U>0,\quad\text{or }
\frac{\partial U}{\partial\nu}+c^*_1U<0\text{ at }z',
$$
it is also impossible. This shows that $u(z)$ cannot attain its
maximum $M$ at a point $z^*\in\Gamma$. Similarly we can prove that
$u(z)$ cannot attain its minimum at a point $z_*\in\Gamma$, hence
$u(z)=0$ on $\Gamma$, thus $u(z)=0$ in $\overline D$.
\end{proof}

By a similar way as stated before, we can prove the uniqueness
theorem of solutions of Problem (P) for equation \eqref{e1.1} with
$\sigma=0$ as follows.

\begin{corollary} \label{coro1.2}
Suppose that\eqref{e1.1}
with $\sigma=0$ satisfies Condition {\rm (C)} and the following condition,
 for any real functions $u_j(z)\in C^1(\overline{D}),V_j(z)\in
L_{p_0,2}(\overline{D})(j=1,2)$, the following  equality holds:
\begin{align*}
& F(z,u_1,u_{1z},V_1)-F(z,u_2,u_{2z},V_2)\\
&=\operatorname{Re}[\tilde Q(V_1-V_2)+\tilde
A_1(u_1-u_2)_z]+\tilde A_2(u_1-u_2)\quad \text{in }D,
\end{align*}
 where $|\tilde Q|\le q_0$ in $D$,
$A_1,\tilde A_2\in L_{p_0,2}(\overline D)$. Then Problem {\rm (P)} for equation
\eqref{e1.1} has at most one solution.
\end{corollary}


\section{A priori estimates}

We consider the nonlinear elliptic equations of second order
\begin{equation}
u_{z\bar z}-\operatorname{Re}[Qu_{zz}+A_{1}u_{z}]-\hat{A}_2u=A_3,\label{e2.1}
\end{equation}
where $\hat{A}_2=A_2+|u|^\sigma$, $\sigma$ is a positive number, and
assume that the above equation satisfies  Condition (C).

\begin{theorem} \label{thm2.1}
Let \eqref{e2.1} satisfy Condition {\rm (C)}. 
Then any solution of Problem {\rm (P)} for \eqref{e2.1}
satisfies the estimates
\begin{equation}
\begin{gathered}
\hat{C}_\beta[u,\overline{D}]=C^1_\beta[|u|^{\sigma+1},\overline{D}]\le M_1,\quad
\|u\|_{W^2_{p_0,2}(D)}\le M_1, \\
\hat{C}_\beta[u,\overline{D}]\le M_2(k_1+k_2),
\end{gathered} \label{e2.2}
\end{equation}
in which $k=(k_0,k_1,k_2)$, $\beta$ $(0<\beta\le\alpha)$,
$M_{1}=M_1(q_0,p_0, \beta,k,D)$,
$M_2=M_2(q_0,p_0,\beta, k_0,p,D)$ are non-negative
constants.
\end{theorem}

\begin{proof} 
Using the reduction to absurdity, we shall
prove that any solution $u(z)$ of Problem (P) satisfies the
estimate 
\begin{equation}
\hat{C}[u,\overline{D}]=C[|u|^{\sigma+1},\overline{D}]+C[u_z,\overline{D}]\le M_3,
\label{e2.3}
\end{equation}
where $M_3=M_3(q_0,p_0,\alpha,k,p,D)$ is a non-negative constant.
Suppose that \eqref{e2.3} is not true, then there exist sequences of
coefficients $\{A^{(m)}_j \}$ $(j=1,2,3)$,
$\{Q^{(m)}\}$, $\{\lambda^{(m)}(z)\},\{c_j^{(m)}\}$ $(j=1,2)$,
$\{b_j^{(m)}\}(j=0,1,\dots,N_0)$, which satisfy the same conditions of
Condition (C) and \eqref{e1.6}--\eqref{e1.8}, such that
$\{A^{(m)}_j\}$ $(j=1,2,3)$, $\{Q^{(m)}\}$, $\{\lambda^{(m)}(z)\}$,
 $\{c_j^{(m)}\}$ $(j=1,2)$ and $\{b_j^{(m)}\}$ $(j=0,1,\dots,N_0)$
in $\overline{D},\Gamma$
weakly converge or uniformly converge to $A^{(0)}_j$ $(j=1,2,3)$,
$Q^{(0)}$, $\lambda^{(0)}(z)$, $c_j^{(0)}(j=1,2)$, $b_j^{(0)}$
$(j=0,1,\dots,N_0)$, and the corresponding boundary-value problem
\begin{equation}
u_{z\bar z}-\operatorname{Re}[Q^{(m)}u_{zz}+A^{(m)}_{1}u_{z}]-\hat{A}^{(m)}_2u=
A^{(m)}_3,\hat{A}_2^{(m)}=A^{(m)}_2+|u|^\sigma,\label{e2.4}
\end{equation}
and
\begin{equation}
\frac12\frac{\partial u}{\partial\nu}+a^{(m)}_1(z)u=c^{(m)}_2(z)\quad
\text{on } \Gamma,\;u(a_j)=b_j,\,j=0,1,\dots,N_0, \label{e2.5}
\end{equation}
have the solutions $\{u^{(m)}(z)\}$, where
$ \hat{C}[u^{(m)}(z),\overline D]\,(m=1,2,\dots)$ are unbounded.
 Hence we can choose a subsequence of
$\{u^{(m)}(z)\}$ denoted by $\{u^{(m)}(z)\}$ again, such that
$h_m=\hat{C}[u^{(m)}(z),\overline{G}]\to\infty$ as $m\to\infty$. We
can assume $h_m\ge\max[k_1,k_2,1]$. It is obvious that
$\tilde u^{(m)}(z)=u^{(m)}(z)/h_m$ $(m=1,2,\dots)$ are solutions of the
boundary-value problems
\begin{equation}
\tilde{u}_{z\bar z}-\operatorname{Re}[Q^{(m)}\tilde{u}_{zz}+A^{(m)}_{1}\tilde{u}_{z}]-\hat{A}^{(m)}_2
\tilde{u}=A^{(m)}_3/h_m,\label{e2.6}
\end{equation}
and
\begin{equation}
\frac12\frac{\partial\tilde{u}}{\partial\nu}+c^{(m)}_1(z)\tilde{u}=c^{(m)}_2(z)/h_m
\quad\text{on }\Gamma,\; \tilde{u}(a_j)=b^{(m)}_j,\; j=0,1,\dots,N_0. \label{e2.7}
\end{equation}
We can see that the functions in the above equation and boundary conditions
satisfy  condition  (C), \eqref{e1.6}--\eqref{e1.8}, and
\begin{equation}
\begin{gathered}
|u|^{\sigma+1}/h_m\le1,\quad L_{p,2}[A^{(m)}_3/h_m,\overline{D}]\le 1,\\
|c_2^{(m)}/h_m|\le1,\quad |b_j^{(m)}/h_m|\le1,\quad
j=0,1,\dots,N_0,
\end{gathered} \label{e2.8}
\end{equation}
hence from \cite[Theorem 4.10, Chapter III]{w1}, we  obtain the
estimate
$$
\hat{C}_{\beta}[\tilde{u}^{(m)}(z),\overline{D}]\le M_4,\|\tilde{u}^{(m)}(z)\|_
{W^2_{p_0,2}(D)}\le M_4,
$$
in which $M_4=M_4(q_0,p_0,\beta,k,D)$ is a non-negative constant.
Thus from the sequence of functions $\{\tilde{u}^{(m)}(z)\}$, we can choose
the subsequence denoted by $\{\tilde{u}^{(m)}(z)\}$, which  converges uniformly
to $\tilde{u}^{(0)}(z)$ in $\overline{D}$, and their partial derivatives
$\tilde{u}^{(m)}_x,\tilde{u}^{(m)}_y$ in
$\overline{D}$ are uniformly convergent and
$\tilde{u}^{(m)}_{xx},\tilde{u}^{(m)}_{yy},\tilde{u}^{(m)}_{xy}$ in
$\overline{D}$ weakly convergent. This
shows $\tilde{u}_0(z)$ is a solution of the boundary-value problem
\begin{equation}
\tilde{u}_{0z\bar z}-\operatorname{Re}[Q^{(0)}\tilde{u}_{0zz}+A^{(0)}_{1}\tilde{u}_{0z}]
-\hat{A}^{(0)}_2\tilde{u}_0=0, \label{e2.9}
\end{equation}
and
\begin{equation}
\frac12\frac{\partial\tilde{u}_0}{\partial\nu}+c^{(0)}_1(z)\tilde{u}_0
=0\quad\text{on }\Gamma, \; u_0(a_j)=0,\; j=0,1,\dots,N_0. \label{e2.10}
\end{equation}
We see that \eqref{e2.9} possesses the condition $A^{(0)}_3=0$ and \eqref{e2.10}
is the homogeneous boundary condition. On the basis of Theorem \ref{thm1.1}, the
solution satisfies $\tilde{u}_0(z)=0$.
However, from $\hat{C}[\tilde{u}^{(m)}(z),\overline{D}]=1$, we can derive
that there exists a point $z^*\in\overline{D}$, such
that $[|\tilde{u}_0(z)|^{\sigma+1}+|\tilde{u}_{0z}|]_{z=z^*}\ne0$, which is
impossible. This shows the first of two estimates in \eqref{e2.2} is true.
It is not difficult to verify the third estimate in \eqref{e2.2}.
\end{proof}

\section{Solvability}

By the above estimates and the Leray-Schauder theorem, we can prove
the existence of solutions of Problem (P) for equation \eqref{e1.1}. We
first introduce the nonlinear elliptic equation of second order
\begin{equation}
\begin{aligned}
u_{z\bar z}
&=f_m(z,u,u_z,u_{zz}),\,f_m(z,u,u_z,u_{zz}) \\
&=\operatorname{Re}[Q_mu_{zz}+A_{1m}u_z]+\hat{A}_{2m}u+A_3\quad\text{in }D,
\end{aligned} \label{e3.1}
\end{equation}
with the coefficients
\begin{gather*}
Q_m=\begin{cases}
Q &\text{in } D_m \\
0 &\text{in } \mathbb{C}\setminus D_m
\end{cases}
\quad
A_{jm}=\begin{cases} A_j & \text{in } D_m \\
0 &\text{in } \mathbb{C}\setminus D_m
\end{cases}
\quad
j=1,3,\\
\hat{A}_{2m}=\begin{cases}\hat{A}_2 &\text{in } D_m\\
0 & \text{in } \mathbb{C}\setminus D_m
\end{cases}
\end{gather*}
where $D_m=\{z\in D: \text{dist}(z,\Gamma\cup\{\infty\})\ge1/m\}$, $m$ is a positive 
integer.

\begin{theorem} \label{thm3.1}
If \eqref{e3.1} satisfies Condition {\rm (C)}, and $u(z)$ is any solution of
 Problem {\rm (P)} for equation \eqref{e3.1}, then $u(z)$ can be expressed 
in the form
$$
u(z)=U(z)+\tilde{v}(z)=U(z)+\hat v(z)+v(z),
$$
where $\tilde{v}(z)=\hat v(z)+v(z)$ is a solution of \eqref{e3.1} with the
homogeneous Dirichlet boundary condition
\begin{equation}
\tilde{v}(z)=0\quad\text{on }\,\partial D_0=\{|z|=1\}.
\end{equation}
Here
$$
v(z)=Hf_m=\frac2\pi\int\int_{D_0}\,\frac{f_m(1/\zeta)}{|\zeta|^4}
\ln\big|\frac{1 -\zeta z}\zeta\big|d\sigma_\zeta,
$$
in which $D_0$ is the image  under the mapping $z=1/\zeta$,
$U(z)$ is a solution of the Dirichlet boundary-value problem
for $U_{z\bar z}=0$ in $D$, and $U(z)$ and $\tilde{v}(z)$ satisfy the estimates
\begin{equation}
\hat{C}^1_{\beta}[U,\overline{D}]+\|U|_{W^2_{p_0,2}(D)}\le M_5,\quad
\hat{C}^1_{\beta} [\tilde{v},\overline{D_0}]+\|\tilde{v}\|_{W^2_{p_0,2}(D_0)}\le
M_6, \label{e3.2}
\end{equation}
 where $\beta(>0),\,M_j=M_j(q_0,p_0,\beta,k,D_m)$ $(j=5,6)$ are non-negative
constants.
\end{theorem}

\begin{proof} It is clear that the solution $u(z)$ can be
expressed as before. On the basis of Theorem \ref{thm2.1}, it is easy to see
that $\tilde{v}$ satisfies the second estimate in \eqref{e3.2}, and then we
know that $U(z)$ satisfies the first estimate of \eqref{e3.2}.
\end{proof}

\begin{theorem} \label{thm3.2}
If \eqref{e1.1} satisfies Condition {\rm (C)}, then Problem {\rm (P)}
 for  equation \eqref{e1.1} has
a solution.
\end{theorem}

\begin{proof} 
To prove the existence of solutions
of Problem (P) for \eqref{e3.1} by using the Leray-Schauder theorem, we
introduce the equation with the parameter $t\in[0,1]$:
\begin{equation}
V_{z\bar z}=tf_m(z,u,u_z,(U+V)_{zz})\quad\text{in }D.\label{e3.3}
\end{equation}
Denote by $B_{M}$ a bounded open set in the Banach space
 $B=\hat W^{2}_{p_0,2}(D_0)=\hat{C}^1_{\beta}(\overline{D_0})\cap
W_{p_0,2}^2(D_0)(0<\beta\le\alpha)$, the elements of which are real
functions $V(z)$ satisfying the inequalities
\begin{equation}
\hat{C}_\beta^{1}[V(z),\overline{D_0}]+\|V\|_{W^2_{p_0,2}(D_0)} <M_7=M_6+1, \label{e3.4}
\end{equation}
in which $M_6$ is a non-negative constants as
stated in \eqref{e3.2}. We choose any function $V(z)\in\overline{B_M}$ and make
an integral $v(z)=H\rho$ as follows:
\begin{equation}
v(z)=H\rho=\frac2{\pi}\int\int_{D_0}\frac{\rho(1/\zeta)}{|\zeta|^4}\log\bigg|\frac{1
-\zeta z}\zeta\bigg|d\sigma_\zeta,  \label{e3.5}
\end{equation}
where $\rho(z)=V_{z\bar z}$. Next we find a solution $\hat v(z)$ of the
 boundary-value problem in $D_0$:
\begin{gather}
\hat v_{z\bar z}=0\quad\text{in } D_0, \label{e3.6}\\
\hat v(z)=-v(z)\quad\text{on } \partial D_0. \label{e3.7}
\end{gather}
Denote $\tilde{v}(z)=\hat v(z)+v(z)$. Moreover we find a solution $U(z)$
of the boundary-value problem in $D$:
\begin{gather}
U_{z\bar z}=0\quad\text{in }D, \label{e3.8}\\
\frac12\frac{\partial U}{\partial\nu}+c_1(z)U=c_2(z)-\frac{\partial\tilde{v}}
{\partial\nu}-c_1(z)\tilde{v}\quad\text{on }\Gamma.
\label{e3.9}
\end{gather}
Now we discuss the equation
\begin{equation}
\tilde{V}_{z\bar z}=tf_m(z,u,u_z,U_{zz}+\tilde{v}_{zz}),\quad
0\le t\le 1, \label{e3.10}
\end{equation}
where $u(z)=U(z)+\tilde{v}(z)$. By Condition (C), the principle of
contracting mapping and the results in Subsection 3.2, Problem
(D) for the equation \eqref{e3.10} in $D_0$ has a unique solution
$\tilde{V}(z)$ with the boundary condition
$$
\tilde{V}(z)=0\quad\text{on }\partial D_0.
$$
Denote by $\tilde{V}=S(V,t)$ $(0\le t\le1)$ the mapping from $V$ onto
$\tilde{V}$. Furthermore, if $u(z)$ is a solution of Problem (P) in $D$
for the equation
\begin{equation}
u_{z\bar z}=tf_m(z,u,u_z,u_{zz}),\quad 0\le t\le 1, \label{e3.11}
\end{equation}
then from Theorem \ref{thm2.1}, the solution $u(z)$ of Problem (P) for
\eqref{e3.11} satisfies \eqref{e2.2}, consequently
$\tilde{V}(z)=u(z)-U(z)\in B_M$.
Set $B_0=B_M\times [0,1]$. In the following, we shall verify that
the mapping $\tilde{V}=S(V,t)$ satisfies the following three conditions of
Leray-Schauder theorem:

1. For every $t\in[0,1]$, $\tilde{V}=S(V,t)$ continuously maps the Banach
space $B$ into itself, and is completely continuous in $B_M$.
Besides, for every function $V(z)\in\overline{B_M}$, $S(V,t)$ is uniformly
continuous with respect to $t\in[0,1]$.

In fact, we arbitrarily choose $V_n(z)\in\overline{B_M}$, $n=1,2,\dots$.
 It is clear that from $\{V_n(z)\}$ there exists a subsequence
$\{V_{n_k}(z)\}$, such that $\{V_{n_k}(z)\}$, $\{V_{n_kz}(z)\}$ and
corresponding functions $\{U_{n_k}(z)\}$, $\{U_{n_kz}(z)\}$
uniformly converge to $V_0(z)$, $V_{0z}(z)$, $U_0(z)$, $U_{0z}(z)$
in $\overline{D}$ respectively. We can find a solution $\tilde{V}_0(z)$ of
Problem (D) for the equation
$$
\tilde{V}_{0z\bar z}=tf_m(z,u_0,u_{0z},U_{0zz}+\tilde{v}_{0zz}),\quad
0\le t\le 1.
$$
Noting that $u_{n_kz\bar z}=U_{n_kz\bar z}+\tilde{v}_{n_kz\bar z}$, from
$\tilde{V}_{n_k}=S(V_{n_k},t)$ and $\tilde{V}_0=S(V_0,t)$, we have
\begin{align*}
(\tilde{V}_{n_k}-\tilde{V}_0)_{z\bar z}
&=t[f_m(z,u_{n_k},u_{n_kz},U_{n_kzz} +\tilde{v}_{n_kzz}) \\
&\quad -f_m(z,u_{n_k},u_{n_kz},U_{n_kzz}+\tilde{v}_{0zz})+C_{n_k}(z)], \quad
0\le t\le 1,
\end{align*}
where
\[ 
C_{n_k}=f_m(z,u_{n_k},u_{n_kz},U_{n_kzz}+\tilde{v}_{0zz}) 
-f_m(z,u_0,u_0,U_{0zz}+\tilde{v}_{0zz}),\,z\in D_0.
\]
Similarly to \cite[(2.4.18), Chapter 2]{w4}, we obtain
$$
L_{p_0,2}[C_{n_k},\overline{D_0}]\to0 \quad\text{as }k\to\infty.
$$
Similarly to \eqref{e2.2}--\eqref{e2.10}, we obtain
\begin{equation}
\|\tilde{V}_{n_k}-\tilde{V}_0\|_{\hat W^2_{p_0,2}(D_0)}
\le L_{p_0,2}[C_{n_k},\overline{D_0}]/[1-q_0],  \label{e3.12}
\end{equation}
where $q_0<1$. It is easy to show that
$\|\tilde{V}_{n_k}-\tilde{V}_0\|_{\hat W^2_{p_0,2}(D)}\to 0$ as
$k\to\infty$. Moreover, from Theorem \ref{thm2.1}, we can verify that
from $\{\tilde{V}_{n_k}(z)-\tilde{V}_0(z)\}$, there exists a subsequence,
 denoted by $\{\tilde{V}_{n_k}(z)-\tilde{V}_0(z)\}$ again, such that
$C^1_\beta[\tilde{V}_{n_k}-\tilde{V}_0,\overline{D_0}]\to0$ as $k\to\infty$.
This shows that the complete
continuity of $\tilde{V}=S(V,t) (0\le t\le1)$ in $\overline{B_M}$. By using a
similar method, we can prove that $\tilde{V}=S(V,t)(0\le t\le1)$
continuously maps $\overline{B_M}$ into $B$, and $\tilde{V}=S(V,t)$ is
uniformly continuous with respect to $t\in[0,1]$ for $V\in\overline{B_M}$.

2. For $t=0$, from Theorem \ref{thm2.1} and \eqref{e3.4}. It is clear that
 $\tilde{V}(z)=S(V,0)\in B_M$.

3. From Theorem \ref{thm2.1} and \eqref{e3.4}, we see that
 $\tilde{V}=S(V,t)(0\le t\le1)$ does not have a solution 
$\tilde{V}(z)$ on the boundary $\partial B_M=\overline{B_M}\setminus B_M$.

Hence by the Leray-Schauder theorem, we know that Problem (P)
for the equation \eqref{e3.3} with $t=1$, namely \eqref{e3.1} has a solution
$u(z)=U(z)+\tilde{v}(z)=U(z)+\hat v(z)+v(z)\in B_M$.
\end{proof}

\begin{theorem} \label{thm3.3}
Under the conditions in Theorem  \ref{thm3.1}, Problem {\rm (P)} 
for  equation  \eqref{e1.1} has a solution.
\end{theorem}

\begin{proof}
 By Theorems \ref{thm2.1} and \ref{thm3.2}, Problem (P) for
the equation \eqref{e3.1} possesses a solution $u_m(z)$, and the solution
$u_m(z)$ of Problem (P) for \eqref{e3.1} satisfies the estimate \eqref{e2.2},
where $m=1,2,\dots$. Thus, we can choose a subsequence
$\{u_{m_k}(z)\}$, such that $\{u_{m_k}(z)\}$, $\{u_{m_kz}(z)\}$ in
$\overline{D}$ uniformly converge to $u_0(z)$, $u_{0z}(z)$ respectively.
Obviously, $u_0(z)$ satisfies the boundary conditions of Problem
(P) for equation \eqref{e1.1}.
\end{proof}

We can choose $K'=2K-2N+J+1$. By using the similar method as Section
1-3, we can prove the following theorem.

\begin{theorem} \label{thm3.4}
Under the above conditions, Problem {\rm (P)} for the equation \eqref{e1.1} has a solution.
Moreover we have the solvability result of Problem {\rm (P)} for
\eqref{e1.1} with the boundary condition
$$
\frac12\frac{\partial u}{\partial\nu}+c_1(z)u(z)=c_2(z),\quad z\in{\Gamma}.
$$
 When $K\ge N-1/2$, the general solution includes $K'+1=2K-2N+2+J$ arbitrary real
constants.
\end{theorem}

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\end{thebibliography}

\end{document}