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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 69, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/69\hfil Oblique problem for elliptic semi-linear equations]
{Oblique derivative problem for elliptic second-order semi-linear
equations in a domain with a conical boundary point}

\author[M. Bodzioch, M. Borsuk \hfil EJDE-2018/69\hfilneg]
{Mariusz Bodzioch, Mikhail Borsuk}

\address{Mariusz Bodzioch \newline
Faculty of Mathematics and Computer Science,
University of Warmia and Mazury in Olsztyn,
Sloneczna 54, 10-710 Olsztyn, Poland}
\email{mariusz.bodzioch@matman.uwm.edu.pl}

\address{Mikhail Borsuk \newline
Faculty of Mathematics and Computer Science,
University of Warmia and Mazury in Olsztyn,
Sloneczna 54, 10-710 Olsztyn, Poland}
\email{borsuk@uwm.edu.pl}

\thanks{Submitted March 2, 2018. Published March 14, 2018}
\subjclass[2010]{35J20, 35J25, 35J61}
\keywords{Elliptic equations; oblique problem; conical points.}

\begin{abstract}
 This article concerns the oblique boundary value problem for elliptic
 semi-linear equations in a domain with a conical point on the boundary.
 We investigate the asymptotic behavior of strong solutions near a boundary
 conical point. New regularity theorems are established under the
 least possible assumptions on the equation coefficients.
 The investigation of asymptotic properties of solutions can be used to
 obtain new solvability theorems. The results obtained in this paper
 are extensions of our previous results to a wider class of elliptic equations.
\end{abstract}

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

\section{Introduction}

Problems in which the boundary value condition has the form
$\mathcal{B}(x,u,Du)=0$, where $\mathcal{B}$ depends on the gradient $Du$
of the unknown function $u$ in a suitable way, are called
\emph{oblique derivative problems}. The two-dimensional basic theory
of linear oblique derivative problems is quite old. For two-dimensional
domains Talenti \cite{t67} established $W^{2,2}$-solvability under the
assumption that $a^{ij}$ are measurable functions only. For higher-dimensional
case the $W^{2,2}$-regularity and invertibility properties for some linear
oblique derivative problems are obtained if $a^{ij} \in  W^{1,n}(\Omega)$
(see Miranda \cite{m63}, Chicco \cite{c77}) or if $a^{ij}$ are measurable
functions satisfying the Cordes condition (Chicco \cite{c77}).
Agmon, Douglis and Nirenberg in \cite{adn59}, using explicit representations
for solutions to derive suitable $L^p$-estimates, established that the
condition $a^{ij}\in C^0(\overline{\Omega})$ is sufficient to
 $W^{2,p}$-regularity of solutions for all values of $p\in(0,\infty)$.

It should be noted that investigations in the aforementioned works refer to
linear boundary value problems in sufficiently smooth domains.
 However, many problems of physics and technology lead to boundary-value problems
in domains with a non-smooth boundary, in particular, in domains which have
 a finite number of angular ($n=2$) or conical ($n\geq 3$) points on the boundary.
The theory of linear boundary value problems in non-smooth domains was described
in well-known survey of Kondrat'ev and Oleinik \cite{ko83} and in the work of
Kufner and S\"{a}ndig \cite{ks87}, as well as in the monograph of Kozlov et al.\
 \cite{kmr97}.

In non-smooth domains, linear oblique derivative problems were studied earlier
 by Faierman \cite{f87}, Garroni, Solonnikov and Vivaldi \cite{gsv96},
Grisvard \cite{g85}, Lieberman \cite{l87a,l85a,l01}, Reisman \cite{r95},
and others. Lieberman considered the oblique derivative problem in Lipschitz
domains. His results concern elliptic equations with H\"{o}lder-continuous
coefficients. The local and global maximum principle for general second-order
linear and quasi-linear elliptic oblique derivative problems were established
by him. Grisvard in his work investigated the properties of the second weak
derivatives of the oblique problem for the Laplace operator in a plane domain
with a polygonal boundary. Solonnikov et al. proved the uniqueness of solutions
and obtained \emph{a priori} estimates for weak solutions of the Laplace
operator in the Sobolev-Kondrat'ev weighted spaces.

Some properties of solutions of the semi-linear problem in a smooth domain and
in a neighbourhood of an isolated singular point were studied by Kondrat'ev et al.,
see e.g. \cite{kl88a,kn93a}. Other problems for elliptic semi-linear equations
were considered by Veron et al., see e.g.\ \cite{bvggv99,fv96,pv00}.

The oblique derivative problem plays a major role in the study of reflected shocks
in transonic flow \cite{ckl00}. Another important application of this theory
is the capillary problem (see e.g. \cite{f86}). In geodesy, the most fundamental
problems of the gravity field determination from boundary observations are
translated into exterior boundary value problems for the Laplace or Poisson
equations, see e.g. \cite{ddo06,s97}.

The aim of this paper is to describe the asymptotic behavior of strong solutions
to the oblique problem for general semi-linear second-order elliptic equations
near the boundary conical point, i.e. we obtain the estimation of the type
$|u(x)|=O(|x|^{\alpha})$ with the sharp exponent $\alpha$. In our previous
papers (\cite{b12,bb14,bb15,bb15a,b13,bk06}) we obtained similar
results for linear and quasi-linear oblique problems. The result presented
in this paper extends our previous results to a wider class of elliptic
equations. New regularity theorems were established. Our results refer to
general equations of second-order. It should be pointed out that assumptions
concerning of the equation coefficients are the least restrictive possible,
i.e.\ the leading coefficients of the equation must be Dini-continuous at
the conical point and the lower coefficients can grow in a particular way.

This paper is organized as follows. At first, we introduce notations for
a domain with a conical boundary point and introduce function spaces that
are used in the following sections. Next, we formulate the boundary-value
problem with oblique derivative for semi-linear elliptic equations in a
domain with a conical boundary point. The problem assumptions are also
formulated. In section \ref{sec:main-thm-form} we describe the main results,
i.e.\ Theorems \ref{sec:slmainthm}--\ref{sec:slmainthm3}.
In Theorem \ref{sec:slmainthm} we assume that the leading coefficients
are Dini-continuous at zero. Then, we generalize the result assuming that
the condition of Dini-continuity is not satisfied. It gives less accurate
regularity of solutions. In Theorem \ref{sec:slmainthm3}, we obtain estimates
for a particular function describing the growth of the leading coefficients.
In the next two sections, we derive global and local weighted estimates
that are used in the last section to prove the main theorems.

\section{Preliminaries}

Let $\mathcal{K}$ be an open cone $\{(r,\omega): 0<r<\infty,  \omega\in\Omega\}$
with the vertex at $\mathcal{O}$ with boundary
$\partial\mathcal{K}=\{(r,\omega): 0<r<\infty, \omega_1
=\frac{\omega_0}{2}\in (0, \pi), \omega_i\in\Omega, i\geq 2\}$.
Let $G\subset \mathbb{R}^n$ be a bounded domain.

Let us introduce the following notations for a domain $G$ which has a conical
point at $\mathcal{O}\in\partial G$.
\begin{itemize}
\item $\Omega$: is a subdomain of the unit sphere $S^{n-1}$;
\item $\partial\Omega$: the boundary of $\Omega$;
\item $G_a^b:=G\cap\{(r,\omega):0\leq a<r<b,\omega \in \Omega\}$:
 a layer in $\mathbb{R}^n$;
\item $\Gamma_a^b:=\partial G\cap\{(r,\omega):0\leq a<r<b,\omega \in \partial\Omega\}$:
 the lateral surface of the layer $G_a^b$;
\item $G_d:=G\backslash G_0^d$,  $\Gamma_d:=\partial G\backslash \Gamma_0^d$,
 $\Omega_{\varrho}:=\overline{G_0^d}\cap\partial B_{\varrho}(0)$, $\varrho\leq d$;
\item $G^{(k)} := G_{2^{-(k+1)}d}^{2^{-k}d}$, $k=0,1,2,\ldots$.
\end{itemize}

Without loss of generality we assume that there exists $0<b\ll 1$ such that
$G_0^{b}$ is a \emph{rotational cone} with the vertex at $\mathcal{O}$
and the aperture $\omega_0\in(0,\pi)$ (see Figure \ref{sec:fig1}), thus
\begin{equation*}
\Gamma_0^{b}=\big\{(r,\omega) x_1^2=\cot^2\frac{\omega_0}{2}
\sum_{i=2}^n x_1^2,\ r\in(0,b),\ \omega_1
=\frac{\omega_0}{2},\ \omega_0\in(0, \pi)\big\}.
\end{equation*}

\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{fig1}
\caption{$n$-dimensional bounded domain with a boundary conical point}
\label{sec:fig1}
\end{figure}

We use standard function spaces: $C^k(\overline{G})$, $C_0^k(G)$ with the
norm $|u|_{k,G}$; the Lebesgue space $L^p(G)$, $p\geq 1$, with the norm
$\|u\|_{L^p(G)}=\big(\int_G |u|^pdx\big)^{1/p}$;
the Sobolev space $W^{k,p}(G)$ for integer $k\geq0$, $1\leq p<\infty$,
which is a set of all functions $u\in L^p(G)$ such that for every multi-index
$\beta$ with $|\beta|\leq k$   weak partial derivatives $D^{\beta}u$ belong
to $L^p(G)$, equipped with the finite norm
\[
\|u\|_{W^{k,p}(G)}=\Big(\int_G\sum_{|\beta|\leq k}|D^{\beta}u|^pdx\Big)^{1/p};
\]
 the weighted Sobolev space $V_{p,\alpha}^k(G)$ for integer $k\geq 0$,
$1<p<\infty$ and $\alpha\in \mathbb{R}$ is the space of distributions
$u\in \mathcal{D}'(G)$ with the finite norm
\[
\|u\|_{V_{p,\alpha}^k(G)}= \Big(\int_G\sum_{|\beta|\leq k} r^{\alpha+p(|\beta|-k)}
| D^{\beta}u|^pdx\Big)^{1/p}
\]
 and $V_{p,\alpha}^{k-\frac{1}{p}}(\Gamma)$ that is the space of functions $g$,
given on $\partial G$, with the norm
\[
\|g\|_{V_{p,\alpha}^{k-\frac{1}{p}}(\partial G)}
=\inf \|\mathcal{G}\|_{V_{p,\alpha}^k(G)},
\]
 where the infimum is taken over
all functions $g$ such that $\mathcal{G}\big|_{\partial G}=g$ in the sense of traces.

 For $p=2$ we use the notation
$$
W^k(G)= W^{k,2}(G), \quad \mathring{W}^k_{\alpha}(G)=V_{2,\alpha}^k(G),\quad
\mathring{W}_{\alpha}^{k-\frac{1}{2}}(\Gamma)=V_{2,\alpha}^{k-\frac{1}{2}}(\Gamma).
$$

\section{Setting of the problem}

Let $G\subset \mathbb{R}^n$ be a bounded domain with the boundary $\partial G$
that is a smooth surface everywhere except at the origin $\mathcal{O}\in \partial G$
 and near $\mathcal{O}$ it is a conical surface. We consider the
semi-linear problem
\begin{equation}\label{sec:sl}
\begin{gathered}
a^{ij}(x)u_{x_ix_j}+a^i(x)u_{x_i}+a(x)u(x)=h(u)+f(x), \quad
 x\in G, \\
h(u)=a_0(x)u(x)|u(x)|^{q-1},\quad  q\in (0,1), \\
\frac{\partial u}{\partial \vec{n}} +\chi(\omega)\frac{\partial u}{\partial r}
+\frac{1}{|x|}\gamma(\omega)u(x)=g(x), \quad
 x\in \partial G\backslash\mathcal{O}.
\end{gathered}
\end{equation}
where $\vec{n}$ denotes the unite exterior normal vector to
$\partial G\backslash \mathcal{O}$; $(r,\omega)$ are spherical coordinates
in $\mathbb{R}^n$ with the pole $\mathcal{O}$.

\begin{remark} \label{rmk1} \rm
For $q=1$, problem \eqref{sec:sl} takes the form of linear problem with
$a(x)\longmapsto a(x)-a_0(x)$, which was considered in \cite{bb14}.
\end{remark}

\begin{definition} \label{def1} \rm
A function $u$ is called a strong solution of problem \eqref{sec:sl}
 provided that for any $\epsilon>0$ function
$u\in W_{\rm loc}^{2,n}(G)\cap W^2\left(G_{\epsilon}\right)
\cap C^0(\overline{G})$ and satisfies the equation of \eqref{sec:sl}
for almost all $x\in G_{\epsilon}$ as well as the boundary condition
in the sense of traces on $\Gamma_{\epsilon}$.
\end{definition}

Regarding the problem we assume that the following conditions are satisfied:
\begin{itemize}
\item[(A1)]
the uniform ellipticity condition
\begin{equation} \label{itm:semi-as-a}
\nu|\xi|^2\leq \sum_{i,j=1}^n a^{ij}(x)\xi_i\xi_j
\leq \mu|\xi|^2, \quad \forall \xi\in\mathbb{R}^n, \;
 x\in\overline{G}
\end{equation}
with the ellipticity constants $\nu,\mu>0$;
 $a^{ij}=a^{ji}$; $a^{ij}(0)=\delta_i^j$;

\item[(A2)]
$a^{ij}\in C^0(\overline{G})$,
$a^i\in L^p(G)$, $p>n$;
$a\in L^n(G)\cap {\mathring{W}_{4-n}^0(G)}$,
$a(x)\leq 0$,  $a_0\in {L^{\frac{n}{1-q}}(G)\cap V^0_{\frac{2}{1-q},
\frac{4}{1-q}-n}(G)}$; there exists a monotonically increasing nonnegative
continuous function $\mathcal{A}$, $\mathcal{A}(0)=0$, such that for
$x,y\in\overline{G}$
\begin{equation} \label{itm:semi-as-b}
\Big(\sum_{i,j=1}^n|a^{ij}(x)-\delta_i^j|^2\Big)^{1/2}
+|x|\Big(\sum_{i=1}^n|a^i(x)|^2\Big)^{1/2}
 +|x|^{2}|a(x)|\leq \mathcal{A}(|x|);
\end{equation}

\item[(A3)] % \label{itm:semi-as-c}
$\gamma(\omega),\chi(\omega)\in C^1(\overline{\Omega})$ and there exist numbers
 $\chi_0\geq 0$,  $\gamma_0>0$, such that $\gamma(\omega)\geq \gamma_0>0$,
$\chi_0\geq \chi(\omega)>0$;

\item[(A4)]
$f\in L^n(G)\cap \mathring{W}_{4-n}^0(G)$,
$g \in {{V_{n,0}^{1-\frac{1}{n}}(\partial G)}
\cap \mathring{W}_{4-n}^{1\slash 2}(\partial G)}$,
there exist nonnegative numbers $f_1,\ g_0,\ g_1,\ a_1$ and $s>0$, such that
\begin{equation}  \label{itm:semi-as-d}
\begin{gathered}
|f(x)|\leq f_1|x|^{s-2}, \quad |g(x)|\leq g_0|x|^{s-1}, \quad
 |\nabla g|\leq g_1|x|^{s-2}, \\
\|a_0\|^{\frac{1}{1-q}}_{\frac{2}{1-q},\frac{4}{1-q}-n(G_0^{\varrho})}
\leq a_1\varrho^{s},\ \varrho\in(0,1);
\end{gathered}
\end{equation}

\item[(A5)] %\label{itm:semi-as-e}
$a_0\in V^0_{\frac{n}{1-q},\frac{2qn}{1-q}}(G)$.
\end{itemize}

\section{Main result}\label{sec:main-thm-form}

Let us consider the following eigenvalue problem for the Laplace-Beltrami
operator $\Delta_{\omega}$ on the unit sphere
\begin{equation} \label{sec:evp}
\begin{gathered}
\Delta_{\omega}\psi+\lambda(\lambda+n-2)\psi(\omega)=0,
\quad \omega\in\Omega, \\
\frac{\partial \psi}{\partial \vec{\nu}}
+[\lambda\chi(\omega)+\gamma(\omega)]\psi(\omega)=0, \quad
 \omega\in \partial\Omega,
\end{gathered}
\end{equation}
which consists of the determination of all values $\lambda$ (eigenvalues),
for which \eqref{sec:evp} has non-zero weak solutions $\psi(\omega)$
(eigenfunctions). Here $\vec{\nu}$ denotes the unite exterior normal vector
to $\partial \mathcal{K}$ at the points of $\partial \Omega$ and functions
$\chi(\omega)\geq0$, $\gamma(\omega)>0$ are $C^0(\partial \Omega)$-functions
 (see \cite{b12,bb14}). Let us define the number
\begin{equation}\label{sec:semiks}
k_s=\sqrt{g_0^2+a_1^2+\frac{1}{2s}(f_1^2+g_1^2)}.
\end{equation}

Our main results are the following statements, whose proofs are
Section \ref{sec:semilinpowermod}.

\begin{theorem}\label{sec:slmainthm}
Let $u$ be a strong solution of problem \eqref{sec:sl} and $\lambda>1$
be the smallest positive eigenvalue of problem \eqref{sec:evp}.
Suppose that assumptions {\rm (A1)--(A5)} with $\mathcal{A}(r)$ being
Dini-continuous at zero are satisfied. Suppose in addition that there
 exists a nonnegative constant $k_0$, such that
\begin{equation}\label{sec:sla0}
\|a_0\|_{L^{\frac{n}{1-q}}(G_{\varrho\slash 4}^{2\varrho})}
\leq k_0\varrho^{1-2q}\psi^q(\varrho),
\end{equation}
where
\begin{equation}\label{sec:slmpsi}
\psi(\varrho)=\begin{cases}
\varrho^{\lambda} & \text{if }  s>\lambda, \\
\varrho^{\lambda}\ln\frac{1}{\varrho} & \text{if }
  s=\lambda, \\
\varrho^s & \text{if }  s<\lambda,
\end{cases}
\end{equation}
with  $0<\varrho<b$.
Then there are positive constants $d\in(0,b)$ and $c_1$, $c_2$, which
depend only on $\nu$, $\mu$, $s$, $b$, $\lambda$, $\gamma_0$,
 $\chi_0$, $k_0$, $f_1$, $g_0$, $g_1$, $a_1$, $\|\gamma\|_{C^1(\partial \Omega)}$,
 $\|\chi\|_{C^1(\partial \Omega)}$, $\operatorname{diam} G$,
 $\operatorname{meas} G$, on the modulus of continuity of the leading
coefficients and on the quantity $\int_0^1\frac{\mathcal{A}(r)}{r}dr$,
and do not depend on $u$, such that for all $x\in G_0^d$:
\begin{itemize}
\item if $0<q<1-\frac{2}{\lambda}$ and $\lambda>s$, then
\begin{equation}\label{sec:slmainb}
|u(x)|\leq c_1 |x|^{\frac{2}{1-q}};
\end{equation}

\item if $1-\frac{2}{\lambda}\leq q\leq 1$, then
\begin{equation}\label{sec:slmaina}
|u(x)|\leq c_2\begin{cases} |x|^{\lambda} & \text{if }  \lambda <s, \\
 |x|^{\lambda}\ln\frac{1}{|x|} & \text{if }  \lambda=s, \\
 |x|^s & \text{if }  \lambda>s.
\end{cases}
\end{equation}
\end{itemize}
\end{theorem}

\begin{theorem}\label{sec:slmainthm2}
Let $u$ be a strong solution of problem \eqref{sec:sl} and $\lambda>1$ be
the smallest positive eigenvalue of problem \eqref{sec:evp}.
Suppose that assumptions {\rm (A1)--(A5)} are satisfied with $\mathcal{A}(r)$,
 which is a continuous at zero function, but \emph{not Dini} continuous.
Suppose in addition that there exists a nonnegative constant $k_0$,
such that \eqref{sec:sla0} holds with
\begin{equation}\label{sec:slmpsi2}
\psi(\varrho)=\begin{cases}
\varrho^{\lambda-\epsilon} & \text{if }  s>\lambda, \\
\varrho^{s-\epsilon} & \text{if }  s\leq\lambda,
\end{cases}
\end{equation}
and $0<\varrho<b$.
Then, for any $\epsilon >0$, there are positive constants $d\in(0,b)$ and
$c_1$, $c_2$, which depend only on $\nu$, $\mu$, $s$, $b$, $\lambda$,
$\gamma_0$, $\chi_0$, $k_0$, $f_1$, $g_0$, $g_1$, $a_1$,
$\|\gamma\|_{C^1(\partial \Omega)}$, $\|\chi\|_{C^1(\partial \Omega)}$,
$\operatorname{diam} G$, $\operatorname{meas} G$, on the modulus of continuity
of the leading coefficients and do not depend on $u$, such that for all
$x\in G_0^d$
\begin{itemize}
\item if $0<q<1-\frac{2}{\lambda}$ and $\lambda>s$, then
\begin{equation*}
|u(x)|\leq c_1 |x|^{\frac{2}{1-q}+\epsilon};
\end{equation*}

\item if $1-\frac{2}{\lambda}\leq q\leq 1$, then
\begin{equation*}
|u(x)|\leq c_2\begin{cases} |x|^{\lambda-\epsilon}
& \text{if }  \lambda \leq s, \\
|x|^{s-\epsilon} & \text{if }  \lambda>s.
\end{cases}
\end{equation*}
\end{itemize}
\end{theorem}

\begin{theorem}\label{sec:slmainthm3}
Let $u$ be a strong solution of problem \eqref{sec:sl} and $\lambda>1$
be the smallest positive eigenvalue of problem \eqref{sec:evp}.
Suppose that assumptions {\rm (A1)--(A5)} are satisfied with
 $\mathcal{A}(r)\sim \frac{1}{\ln\frac{1}{r}}$. Suppose in addition that there
exists a nonnegative constant $k_0$, such that \eqref{sec:sla0} holds with
\begin{equation}\label{sec:slmpsi3}
\psi(\varrho)=\ln^{c_s(\lambda)}\big(\frac{1}{\varrho}\big)
\begin{cases} \varrho^{\lambda} & \text{if }  s>\lambda, \\
\varrho^s & \text{if }  s\leq\lambda,
\end{cases}
\end{equation}
and $ 0<\varrho<b$.
Then there are positive constants $d\in(0,b)$ and $c_1$, $c_2$, $c_s$,
which depend only on $\nu$, $\mu$, $s$, $b$, $\lambda$, $\gamma_0$,
$\chi_0$, $k_0$, $f_1$, $g_0$, $g_1$, $a_1$,
$\|\gamma\|_{C^1(\partial \Omega)}$, $\|\chi\|_{C^1(\partial \Omega)}$,
$\operatorname{diam} G$, $\operatorname{meas} G$, on the modulus of continuity
 of the leading coefficients and do not depend on $u$, such that for all
$x\in G_0^d$
\begin{itemize}
\item if $0<q<1-\frac{2}{\lambda}$ and $\lambda>s$, then
\begin{equation*}
|u(x)|\leq c_1 \ln^{c_s(\lambda)}\big(\frac{1}{|x|}\big) |x|^{\frac{2}{1-q}};
\end{equation*}

\item if $1-\frac{2}{\lambda}\leq q\leq 1$, then
\begin{equation*}
|u(x)|\leq c_2\ln^{c_s(\lambda)}\big(\frac{1}{|x|}\big)
\begin{cases} |x|^{\lambda} & \text{if }  \lambda \leq s, \\
|x|^{s} & \text{if }  \lambda>s.
  \end{cases}
\end{equation*}
\end{itemize}
\end{theorem}

\section{Global integral weighted estimate}

Let us introduce the function
\begin{equation}\label{sec:sl1}
M(\epsilon)=\max_{x\in\overline{\Omega}_{\epsilon}} |u(x)|
\end{equation}
then because of $u\in C^0(\overline{G})$,
\begin{equation}\label{sec:sl2}
\lim_{\epsilon \to 0^+} M(\epsilon)=|u(0)|.
\end{equation}

\begin{lemma}\label{sec:sllemma}
Let $u$ be a strong solution of problem \eqref{sec:sl} and assumptions
{\rm (A1)--(A3)} be satisfied. Then there exists a positive constant
$c_0$ depending only on $\nu$, $\mu$, $G$, $\max_{x,y\in G}\mathcal{A}(|x-y|)$,
$\|\chi\|_{C^1(\partial \Omega)}$, $\|\gamma\|_{C^1(\partial \Omega)}$, such that
\begin{equation}\label{sec:sl3}
\lim_{\epsilon\to 0^+} \epsilon^{2-n}\Big|\int_{\Omega_{\epsilon}}u
\frac{\partial u}{\partial r}d\Omega_{\epsilon}\Big| \leq c_0 |u(0)|^2.
\end{equation}
\end{lemma}

\begin{proof}
Let us consider the set $G_{\epsilon}^{2\epsilon}$,
$\Omega_{\epsilon}\subset \partial G_{\epsilon}^{2\epsilon}$.
By \cite[Lemma 6.36]{l96} we have
$$
\int_{\Omega_{\epsilon}}|w|d\Omega_{\epsilon}
\leq c\int_{G_{\epsilon}^{2\epsilon}}\left(|w|+|\nabla w|\right)dx,
$$
where $c$ depends only on the domain $G$. Setting
$w=u\frac{\partial u}{\partial r}$ and using the Cauchy inequality, we obtain
\begin{equation}\label{sec:g13}
\int_{\Omega_{\epsilon}}\big|u\frac{\partial u}{\partial r}\Big|d\Omega_{\epsilon}
\leq c\int_{G_{\epsilon}^{2\epsilon}}\left(r^2u_{xx}^2
+|\nabla u|^2+r^{-2}u^2\right)dx.
\end{equation}
Let us consider two sets $G_{\epsilon\slash 2}^{5\epsilon\slash 2}$ and
$G_{\epsilon}^{2\epsilon}\subset G_{\epsilon\slash 2}^{5\epsilon\slash 2}$,
 and a new variable $x'$ defined by $x=\epsilon x'$. Thus, the function
$w(x')=u(\epsilon x')$ satisfies in $G_{1\slash 2}^{5\slash 2}$
the  problem
\begin{equation}\label{sec:sl5}
\begin{gathered}
\begin{aligned}
&a^{ij}(\epsilon x')w_{x_i'x_j'}+\epsilon a^i(\epsilon x')w_{x_i'}
+\epsilon^2 a(\epsilon x')w(x')\\
&=\epsilon^2 h(w(x'))+\epsilon^2 f(\epsilon x'), \quad
 x'\in G_{1\slash 2}^{5\slash 2},
\end{aligned} \\
 h(w(x'))=a_0(\epsilon x')w(x')|w(x')|^{q-1},\quad q\in (0,1), \\
 \frac{\partial w}{\partial \vec{n}'} +\chi(\omega)\frac{\partial w}{\partial r'}
+\frac{1}{|x'|}\gamma(\omega)w(x')=\epsilon g(\epsilon x'), \quad
 x'\in \Gamma_{1\slash 2}^{5\slash 2}.
\end{gathered}
\end{equation}
We apply $L^2$-estimate for the solution of problem \eqref{sec:sl5}
(see \cite[Theorem 15.3]{adn59}). As a result we obtain the following
estimation
\begin{align*}
&\int_{G_1^2}\left(w_{x'x'}^2+|\nabla' w|^2+w^2\right)dx'\\
&\leq c_1\int_{G_{1\slash 2}^{5\slash 2}}[\epsilon^4 f^2
 +\epsilon^4 h^2(w)+w^2]dx'
+c_2\epsilon^2 \inf\int_{G_{1\slash 2}^{5\slash 2}}
\left(|\nabla' \mathcal{G}|^2+|\mathcal{G}|^2\right)dx',
\end{align*}
here infimum is taken over all
$\mathcal{G}\in W^1\big(G_{1\slash2}^{5\slash 2}\big)$, such that
$\mathcal{G}\big|_{\Gamma_{1\slash2}^{5\slash 2}}=g$ and constants
$c_1$ and $c_2$ are positive and depend only on $\nu$, $\mu$, $G$,
$\max_{x,y\in G_{1\slash2}^{5\slash2}}\mathcal{A}(|x-y|)$,
$\|\chi\|_{C^1(\Gamma_{1\slash2}^{5\slash2})}$,
$\|\gamma\|_{C^1(\Gamma_{1\slash2}^{5\slash2})}$.
Returning to the variable $x$ we obtain
\begin{equation} \label{sec:sl6}
\begin{aligned}
&\int_{G_{\epsilon}^{2\epsilon}}
 \left(r^{4-n}u_{xx}^2 + r^{2-n}|\nabla u|^2 + r^{-n}u^2\right)dx \\
&\leq c_3 \int_{G_{\epsilon\slash 2}^{5\epsilon\slash 2}}
 \big(r^{4-n} f^2+ r^{4-n}h^2 + r^{-n}u^2\big)dx  \\
&\quad + c_4\inf \int_{G_{\epsilon\slash 2}^{5\epsilon\slash 2}}
 \left(r^{4-n}|\nabla \mathcal{G}|^2+r^{2-n}|\mathcal{G}|^2\right)dx.
\end{aligned}
\end{equation}
Now, by the mean value theorem with regard to $u\in C^0(\overline{G})$
and \eqref{sec:sl1}, we obtain
\begin{equation} \label{sec:g20c}
\begin{aligned}
\int_{G_{\epsilon\slash 2}^{5\epsilon\slash 2}}r^{-n}u^2dx
&=\int_{\epsilon\slash 2}^{5\epsilon\slash 2}r^{-1}\int_{\Omega}u^2(r,\omega)
 d\Omega dr\leq 2\epsilon(\theta_1\epsilon)^{-1}
 \int_{\Omega}u^2(\theta_1\epsilon,\omega)d\Omega\\
&\leq 2{\theta_1}^{-1} M^2(\theta_1\epsilon)\operatorname{meas} \Omega
\end{aligned}
\end{equation}
for some $\frac{1}{2}<\theta_1<\frac{5}{2}$.
By \eqref{sec:g13}, \eqref{sec:sl6} and \eqref{sec:g20c}, it follows that
\begin{equation} \label{sec:sl8}
\begin{aligned}
&\epsilon^{2-n}\big|\int_{\Omega_{\epsilon}}u
\frac{\partial u}{\partial r}d\Omega_{\epsilon}\big|\\
&\leq c_5M^2(\theta_1\epsilon)
 + c_6\int_{G_{\epsilon\slash 2}^{5\epsilon\slash 2}}
  (r^{4-n}h^2+r^{4-n}f^2)dx + c_7\|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}
 (\Gamma_{\epsilon\slash 2}^{5\epsilon \slash 2})}.
\end{aligned}
\end{equation}
Taking into account that $q<1$, by the Young inequality with $\frac{1}{q}$
 and $\frac{1}{1-q}$, we deduce
\begin{equation}  \label{sec:sl9}
\begin{aligned}
r^{4-n}h^2&=r^{4-n}a_0^2|u|^{2q}
 = \left(r^{-nq}|u|^{2q}\right)\left(r^{4-n+nq}a_0^2\right) \\
&\leq \delta r^{-n}u^2 + c(\delta,q)r^{\frac{4}{1-q}-n}a_0^{\frac{2}{1-q}}, \quad
\forall \delta>0.
\end{aligned}
\end{equation}
By assumptions about functions $f$, $g$ and $a_0$, from \eqref{sec:g20c}
and \eqref{sec:sl9}, we obtain that
\begin{equation}\label{sec:sl10}
\lim_{\epsilon\to 0^+}  \Big\{ \int_{G_{\epsilon\slash 2}^{5\epsilon\slash 2}}
\left[r^{4-n}h^2+r^{4-n}f^2\right]dx
+ \|g\|^2_{\mathring{W}^{1\slash 2}
_{4-n}(\Gamma_{\epsilon\slash 2}^{5\epsilon \slash 2})} \Big\} =0
\end{equation}
and therefore, by \eqref{sec:sl2}, \eqref{sec:sl8} and \eqref{sec:sl10},
 we finally obtain \eqref{sec:sl3}.
\end{proof}

\begin{theorem} \label{sec:slgthm}
Let $u$ be a strong solution of problem \eqref{sec:sl} and assumptions
{\rm (A1)--(A3)} be satisfied. Then $u\in{\mathring{W}^2_{4-n}(G)}$ and
\begin{equation} \label{sec:slg}
\begin{aligned}
&\|u\|_{\mathring{W}^2_{4-n}(G)}
+\Big(\int_G a_0r^{2-n}|u|^{1+q}dx\Big)^{1/2}
+ \Big(\int_{\partial G} r^{1-n}\gamma(\omega)u^2ds\Big)^{1/2} \\
&\leq C\Big(|u|_{2,G}+\|a_0\|^{\frac{1}{1-q}}_{V^0_{\frac{2}{1-q},
 \frac{4}{1-q}-n}(G)}+\|f\|_{\mathring{W}^0_{4-n}(G)}
 + \|g\|_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}\Big),
\end{aligned}
\end{equation}
where $C>0$ depends on $\nu$, $\mu$, $b$, $q$, $n$,
$\operatorname{meas} G$, $\|\chi\|_{C^1(\partial \Omega)}$,
$\|\gamma\|_{C^1(\partial \Omega)}$ and on modulus of continuity of the leading
coefficients.
\end{theorem}

\begin{proof}
Let us rewrite the equation of \eqref{sec:sl} in the  form
\begin{equation}\label{sec:slg1}
\Delta u= h(u)+f(x)-\Big[\big(a^{ij}(x)-\delta_i^j\big)u_{x_ix_j}
+a^i(x)u_{x_i}+a(x)u(x)\Big],
\end{equation}
multiply both sides by $r^{2-n}u(x)$ and integrate over $G_{\epsilon}$.
As a result we obtain
\begin{equation} \label{sec:slg2}
\begin{aligned}
\int_{G_{\epsilon}}r^{2-n}u\Delta u dx
& = \int_{G_{\epsilon}} r^{2-n}u\left(h+f\right)dx\\
&\quad - \int_{G_{\epsilon}} r^{2-n}u\big[\big(a^{ij}-\delta_i^j\big)u_{x_ix_j}
 +a^iu_{x_i}+au\big]dx.
\end{aligned}
\end{equation}
Calculating the integral from the left side by parts, using the boundary
condition, the representation
$\partial G_{\epsilon}=\Gamma_{\epsilon}\cup \Omega_{\epsilon}$,
$d\Omega_{\epsilon}=\epsilon^{n-1}d\Omega$, and the fact that
$$
x_i\cos(\vec{n},x_i)\Big|_{\Omega_{\epsilon}}=\epsilon, \quad
 x_i\cos\left(\vec{n},x_i\right)\Big|_{\Gamma_0^d}=0,
$$
(see \cite[Lemma 1.10]{bk06}), we obtain
\begin{equation} \label{sec:slg7}
\begin{aligned}
&\int_{G_{\epsilon}}r^{2-n}|\nabla u|^2 dx
  +\int_{\Gamma_{\epsilon}}\gamma(\omega)r^{1-n}u^2ds
  + \frac{n-2}{2}\int_{\Omega} u^2 d\Omega \\
&+\int_{G_{\epsilon}}r^{2-n}a_0(x)|u|^{q+1}dx\\
&= \int_{\Gamma_{\epsilon}} r^{2-n} ug\,ds
 - \int_{\Gamma_{\epsilon}}\chi(\omega)r^{2-n}u\frac{\partial u}{\partial r} ds
 - \epsilon^{2-n}\int_{\Omega_{\epsilon}} u\frac{\partial u}{\partial r}
 d\Omega_{\epsilon} \\
&\quad +\frac{n-2}{2}\int_{\Gamma_d}r^{-n}u^2x_i\cos(\vec{n},x_i)ds
  - \int_{G_{\epsilon}}r^{2-n}uf\,dx  \\
&\quad + \int_{G_{\epsilon}} r^{2-n}u\big[\big(a^{ij}-\delta_i^j\big)u_{x_ix_j}
 +a^iu_{x_i}+au\big]dx
 \end{aligned}
\end{equation}
(see \cite[formulas (13)-(17)]{bb14}).
Repeating verbatim estimations of formulas (26)-(33) of the proof of
\cite[Theorem 3.1]{bb14}, taking $\delta_1=1$, identity \eqref{sec:slg7}
takes the form
\begin{equation}\label{sec:slg8}
\begin{split}
&\int_{G_{\epsilon}}r^{2-n}|\nabla u|^2 dx
  +\int_{\Gamma_{\epsilon}}\gamma(\omega)r^{1-n}u^2ds
 + \int_{G_{\epsilon}}r^{2-n}a_0(x)|u|^{q+1}dx\\
&\leq \epsilon^{2-n}\Big|\int_{\Omega_{\epsilon}} u
 \frac{\partial u}{\partial r}d\Omega_{\epsilon}\Big|
 +\frac{1}{2}\int_{\partial \Omega}\chi\left(\frac{\omega_0}{2},
 \omega'\right)u^2\left(\epsilon,\frac{\omega_0}{2},\omega'\right)d\omega' \\
&\quad +2\int_{G_{\epsilon}}\mathcal{A}(r)\left(r^{4-n}u^2_{xx}
 +r^{2-n}|\nabla u|^2+r^{-n}u^2\right)dx
 + \frac{\delta}{2}\int_{G_{\epsilon}} r^{-n}u^2dx \\
&\quad +C(\chi_0,d)\int_{G_d}\left(u^2_{xx}+|\nabla u|^2 +u^2\right)dx
  + \frac{1}{2\delta}\|f\|^2_{\mathring{W}^0_{4-n}(G)}\\
&\quad +\frac{c}{\gamma_0}\|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)},
\end{split}
\end{equation}
for all $\delta>0$. By assumption \eqref{itm:semi-as-b}, function
$\mathcal{A}(r)$ is continuous at zero and $\mathcal{A}(0)=0$. Therefore
for all $\delta>0$ there exists $d>0$ such that
$\mathcal{A}(r)<\delta$ for all $ 0<r<d<b$.
Hence, by \eqref{sec:sl6}, \eqref{sec:g20c} and \eqref{sec:sl9},
assuming that $2\epsilon<d$, we obtain for all $\delta>0$
\begin{align*}%\label{sec:slg9}
&\int_{G_{\epsilon}}\mathcal{A}(r)\left(r^{4-n}u^2_{xx}+r^{2-n}|\nabla u|^2
 +r^{-n}u^2\right) dx\\
&= \int_{G_{\epsilon}^{2\epsilon}}\mathcal{A}(r)
 \left(r^{4-n}u^2_{xx}+r^{2-n}|\nabla u|^2+r^{-n}u^2\right)dx \\
&\quad + \int_{G_{2\epsilon}^d}\mathcal{A}(r)
 \left(r^{4-n}u^2_{xx}+r^{2-n}|\nabla u|^2+r^{-n}u^2\right)dx \\
&\quad + \int_{G_d}\mathcal{A}(r)\Big(r^{4-n}u^2_{xx}+r^{2-n}|\nabla u|^2
 +r^{-n}u^2\Big)dx \\
&= C\mathcal{A}(2\epsilon)\Big( M^2(\epsilon)
 +\int_{G_{\epsilon\slash 2}^{5\epsilon\slash 2}}
 r^{\frac{4}{1-q}-n}a_0^{\frac{2}{1-q}}(x)dx
 + \|f\|^2_{\mathring{W}^0_{4-n}(G_{\epsilon\slash 2}^{5\epsilon\slash 2})}\\
&\quad +\|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}
\left(\Gamma_{\epsilon\slash 2}^{5\epsilon\slash 2}\right)}\Big)
+\delta\int_{G_{2\epsilon}^d}\left(r^{4-n}u^2_{xx}+r^{2-n}|\nabla u|^2
 +r^{-n}u^2\right)dx \\
&\quad + C_1(d,\text{diam}\, G)\int_{G_d}\left(u^2_{xx}+|\nabla u|^2+u^2\right)dx.
\end{align*}
Substituting $\epsilon=2^{-k-1}d$ to \eqref{sec:sl6}, we obtain
\begin{align*}
&\int_{G^{(k)}}\left(r^{4-n}u^2_{xx}+r^{2-n}|\nabla u|^2+r^{-n}u^2\right)dx\\
&\leq C_2\int_{G^{(k-1)}\cup G^{(k)}\cup G^{(k+1)}}
 \left(r^{-n}u^2+r^{4-n}f^2+r^{4-n}h^2\right)dx \\
&\quad +C_3\inf\int_{G^{(k-1)}\cup G^{(k)}\cup G^{(k+1)}}
 \left(r^{4-n}|\nabla\mathcal{G}|^2+r^{2-n}|\mathcal{G}|^2\right)dx,
\end{align*}
here infimum is taken over all $\mathcal{G}\in {\mathring{W}^1_{4-n}(G)}$
such that $\mathcal{G}\big|_{\partial G}=g$. Summing these inequalities over
$k=0,1,\dots ,\lfloor\log_2\left(\frac{d}{4\epsilon}\right)\rfloor$, for any
$\epsilon\in\left(0,\frac{d}{2}\right)$, we obtain

\begin{equation} \label{sec:slg10}
\begin{aligned}
&\int_{G_{2\epsilon}^d}\left(r^{4-n}u^2_{xx}+r^{2-n}|\nabla u|^2+r^{-n}u^2\right)dx \\
&\leq C_2 \int_{G_{\epsilon}^{2d}}\left(r^{-n}u^2+r^{4-n}f^2+r^{4-n}h^2\right)dx
+ C_3\|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}\left(\Gamma_{\epsilon}^{2d}\right)}.
\end{aligned}
\end{equation}
By \eqref{sec:slg8} and \eqref{sec:slg10}, we have
\begin{equation} \label{sec:slg11}
\begin{aligned}
&\int_{G_{\epsilon}}r^{2-n}|\nabla u|^2 dx
 +\int_{\Gamma_{\epsilon}}r^{1-n}\gamma(\omega)u^2ds
 + \int_{G_{\epsilon}}r^{2-n}a_0(x)|u|^{q+1}dx\\
&\leq \epsilon^{2-n}\big|\int_{\Omega_{\epsilon}} u\frac{\partial u}{\partial r}
 d\Omega_{\epsilon}\big| +\int_{\partial \Omega}\chi
 \left(\frac{\omega_0}{2},\omega'\right)u^2
 \left(\epsilon,\frac{\omega_0}{2},\omega'\right)d\omega' \\
&\quad +\delta\int_{G_{\epsilon}}r^{\frac{4}{1-q}-n}a_0^{\frac{2}{1-q}}(x)dx
 +\mathcal{A}(2\epsilon)\Big(M^2(\epsilon)
 +\int_{G_{\epsilon\slash 2}^{5\epsilon\slash 2}}r^{\frac{4}{1-q}-n}
 a_0^{\frac{2}{1-q}}(x)dx\\
&\quad +\|f\|^2_{\mathring{W}^0_{4-n}
 \left(G_{\epsilon\slash 2}^{5\epsilon\slash 2}\right)}
 +\|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}
 \left(\Gamma_{\epsilon\slash 2}^{5\epsilon\slash 2}\right)}\Big)
 + \delta\int_{G_{\epsilon}}r^{-n}u^2dx \\
&\quad + C(\chi_0,d,\text{diam}\, G)\int_{G_d}
 \left(u^2_{xx}+|\nabla u|^2 +u^2\right)dx \\
&\quad + C_4\Big(\|f\|^2_{\mathring{W}^0_{4-n}(G)}
 + \|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}\Big), \quad
 \forall\ \delta>0,
\end{aligned}
\end{equation}
where $C_4>0$ does not depend on $\epsilon$. By \cite[Lemma 2.5]{bb14},
taking into account that $\chi(\omega)\leq \chi_0$ and
$0<\gamma_0\leq \gamma(\omega)$, we find
\begin{equation} \label{sec:slg12}
\begin{aligned}
\int_{G_{\epsilon}}r^{-n}u^2dx
&\leq \frac{1}{\lambda(\lambda+n-2)}\int_{G_{\epsilon}}r^{2-n}|\nabla u|^2dx\\
&\quad +\frac{1}{\lambda(\lambda+n-2)}\Big(1+ \frac{\lambda\chi_0}{\gamma_0}\Big)
 \int_{\Gamma_{\epsilon}}r^{1-n}\gamma(\omega)u^2ds \\
&\quad + d^{-n}\int_G u^2 dx.
\end{aligned}
\end{equation}
We apply $L^2$-estimate for the solution of problem \eqref{sec:sl}
(see \cite[Theorem 15.3]{adn59}). As a result we obtain the  estimate
\begin{equation} \label{sec:slg13}
\begin{aligned}
&\int_{G_d}\left(u^2_{xx}+|\nabla u|^2 +u^2\right)dx \\
&\leq C_5\int_{G_{d\slash 2}}\left(f^2+h^2+u^2\right)dx
 + C_6\|g\|^2_{W^{1\slash 2}(\Gamma_{d\slash 2})}\\
&\leq C_7\Big(\int_{G_{d\slash 2}}r^{\frac{4}{1-q}-n}a_0^{\frac{2}{1-q}}dx
 + |u|^2_{2,G} + \|f\|^2_{\mathring{W}^{0}_{4-n}(G)}
 + \|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}\Big),
\end{aligned}
\end{equation}
where the constant $C_7>0$ depends only on $\nu$, $\mu$, $d$, $G$,
$\max_{x,y\in G}\mathcal{A}(|x-y|)$, $\|\chi\|_{C^1(\partial G)}$,
$\|\gamma\|_{C^1(\partial G)}$. Now, let us choose
\[
\delta \leq \min\big\{\frac{\gamma_0(\lambda+n-2)}
 {4(\gamma_0+\lambda\chi_0)},\frac{\lambda(\lambda+n-2)}{2}\big\}.
\]
Thus, by \eqref{sec:slg11}--\eqref{sec:slg13}, we obtain
\begin{equation} \label{sec:slg14}
\begin{aligned}
&\int_{G_{\epsilon}}r^{2-n}|\nabla u|^2 dx +\int_{\Gamma_{\epsilon}}r^{1-n}
 \gamma(\omega)u^2ds + \int_{G_{\epsilon}}r^{2-n}a_0|u|^{q+1}dx\\
&\leq \epsilon^{2-n}\big|\int_{\Omega_{\epsilon}} u\frac{\partial u}{\partial r}
 d\Omega_{\epsilon}\big| +\int_{\partial \Omega}
 \chi\left(\frac{\omega_0}{2},\omega'\right)u^2\left(\epsilon,\frac{\omega_0}{2},
 \omega'\right)d\omega' \\
&\quad +C_8\int_{G}r^{\frac{4}{1-q}-n}a_0^{\frac{2}{1-q}}(x)dx
 +\mathcal{A}(2\epsilon)\Big(M^2(\epsilon) \\
&\quad +\int_{G_{\epsilon\slash 2}^{5\epsilon\slash 2}}r^{\frac{4}{1-q}-n}a_0^{\frac{2}{1-q}}dx +\|f\|^2_{\mathring{W}^0_{4-n}\left(G_{\epsilon\slash 2}^{5\epsilon\slash 2}\right)}+\|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}\left(\Gamma_{\epsilon\slash 2}^{5\epsilon\slash 2}\right)}\Big) \\
&\quad + C_9\Big(|u|^2_{2,G}+\|f\|^2_{\mathring{W}^0_{4-n}(G)}
 + \|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}\Big).
\end{aligned}
\end{equation}
By  Lemma \ref{sec:sllemma}, \eqref{sec:slg12}, as well as
$u\in C^0(\overline{G})$, using the Fatou's lemma, we can pass
 in \eqref{sec:slg14} to the limit $\epsilon\to0^+$. As a result we obtain
\begin{equation} \label{sec:slg15}
\begin{aligned}
&\int_{G}\left(r^{2-n}|\nabla u|^2 + r^{-n}u^2\right) dx
 +\int_{\partial G}r^{1-n}\gamma(\omega)u^2ds + \int_{G}r^{2-n}a_0|u|^{q+1}dx\\
&\leq C_8\int_{G}r^{\frac{4}{1-q}-n}a_0^{\frac{2}{1-q}}dx
+ C_{10}\Big(|u|^2_{2,G}+\|f\|^2_{\mathring{W}^0_{4-n}(G)}
 + \|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}\Big).
\end{aligned}
\end{equation}
Passing to the limit $\epsilon\to 0^+$ in \eqref{sec:slg10} and taking into
account \eqref{sec:slg15}, we obtain
\begin{equation} \label{sec:slg16}
\begin{aligned}
\int_{G_0^d} r^{4-n}u^2_{xx}dx
&\leq C_{11}\int_{G} r^{-n}u^2dx
+ C_2\int_G r^{\frac{4}{1-q}-n}a_0^{\frac{2}{1-q}}dx \\
&\quad+ C_2 \|f\|^2_{\mathring{W}^{0}_{4-n}(G)} 
C_3\|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}.
\end{aligned}
\end{equation}
Finally, by \eqref{sec:slg15} and \eqref{sec:slg16}, we obtain the required
estimation \eqref{sec:slg}.
\end{proof}


\begin{theorem}\label{sec:slgthm2}
Let $u$ be a strong solution of problem \eqref{sec:sl} and assumptions 
{\rm (A1)--(A3)} be satisfied. Then $(u-u(0))\in{\mathring{W}^2_{4-n}(G)}$ and
\begin{equation} \label{sec:slga}
\begin{aligned}
&\|u-u(0)\|_{\mathring{W}^2_{4-n}(G)}
 +\Big(\int_G  a_0r^{2-n}|u-u(0)|^{1+q}dx\Big)^{1/2} \\
&+ \Big(\int_{\partial G} r^{1-n}\gamma(\omega)|u-u(0)|^2ds\Big)^{1/2} \\
&\leq C\Big(|u(0)|\cdot\|a\|_{\mathring{W}^0_{4-n}(G)}+|u|_{2,G}
 +\|a_0\|^{\frac{1}{1-p}}_{V^0_{\frac{2}{1-q},\frac{4}{1-q}-n}(G)}\\
&\quad + \|f\|_{\mathring{W}^0_{4-n}(G)}
+ \|g\|_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}\Big)
\end{aligned}
\end{equation}
where $C>0$ depends on $\nu$, $\mu$, $b$, $q$, $n$, $\operatorname{diam} G$,
$\|\chi\|_{C^1(\partial \Omega)}$, $\|\gamma\|_{C^1(\partial \Omega)}$
and on modulus of continuity of the leading coefficients.
\end{theorem}

\begin{proof}
Setting $v(x)=u(x)-u(0)$, we have $v(x)\in {C^0(\overline{G})}$,
 $v(0)=0$ and $v$ is a strong solution of the  problem
\begin{gather*}
\begin{aligned} 
&a^{ij}(x)v_{x_ix_j}+a^i(x)v_{x_i}+a(x)v(x) \\
&=h(u)+f(x)-a(x)u(0) 
\equiv h(u)+f_0(x), \quad x\in G,
\end{aligned} \\ 
h(v)=a_0(x)\left(v+u(0)\right)\left|v+u(0)\right|^{q-1},\quad  q\in (0,1), \\ 
\frac{\partial v}{\partial \vec{n}} +\chi(\omega)\frac{\partial v}{\partial r}
+\frac{1}{|x|}\gamma(\omega)v(x)=g(x)-\frac{1}{|x|}\gamma(\omega)u(0)
\equiv g_0(x), \quad  x\in \partial G\backslash\mathcal{O}.
\end{gather*}
Without loss of generality we can suppose that $u(0)\geq 0$. 
Then $g_0(x)\leq g(x)$, since $\gamma(\omega)>0$. We have that 
$f_0(x)\in {\mathring{W}^0_{4-n}(G)}$. Proceeding step by step the arguments 
of the proof of Theorem \ref{sec:slgthm} for the function $v$ we obtain 
the required estimation \eqref{sec:slga}.
\end{proof}

\begin{corollary}
Let $u$ be a strong solution of problem \eqref{sec:sl} and assumptions 
{\rm (A1)--(A3)} be satisfied. Then $u(0)=0$.
\end{corollary}

\begin{proof}
By the Cauchy inequality we have
$$
\frac{1}{2}|u(0)|^2\leq |u(x)|^2 + |u(x)-u(0)|^2.$$
Thus
\begin{equation}\label{sec:slg17}
\frac{1}{2}|u(0)|^2\int_{G_0^d}r^{-n} dx 
\leq \int_{G_0^d} r^{-n}|u(x)|^2dx + \int_{G_0^d}r^{-n}|u(x)-u(0)|^2 dx.
\end{equation}
By Theorems \ref{sec:slgthm} and \ref{sec:slgthm2} both integrals from 
the right side are finite, i.e. the right side of \eqref{sec:slg17} is finite. 
On the other hand, because of 
$\int_{G_0^d}r^{-n}dx \sim \int_0^d \frac{dr}{r} = \infty$, 
the left side of this inequality is infinite if $u(0)\not = 0$. 
It leads to a contradiction. Thus, it must be $u(0)=0$.
\end{proof}

\section{Local integral weighted estimates}

\begin{theorem} \label{sec:sllthm}
Let $u$ be a strong solution of problem \eqref{sec:sl} and assumptions 
{\rm (A1)--(A4)} be satisfied with $\mathcal{A}(r)$ that is Dini-continuous at zero.
Then there are $d\in(0,b)$ and a constant $c>0$ depending only on
 $\nu$, $\mu$, $b$, $s$, $\lambda$, $q$, $\gamma_0$, $\chi_0$, 
$\operatorname{meas} G$, $\operatorname{diam} G$, $\|\chi\|_{C^1(\partial \Omega)}$, 
$\|\gamma\|_{C^1(\partial \Omega)}$ and on the quantity 
$\int_0^d \frac{\mathcal{A}(\tau)}{\tau}d\tau$, such that for all 
$\varrho\in (0,d)$,
\begin{equation} \label{sec:sll}
\begin{aligned}
\|u\|_{\mathring{W}^2_{4-n}(G_0^{\varrho})}
&\leq   c\Big(\|u\|_{L^2(G)}+\|a_0\|_{V^0_{\frac{2}{1-q},\frac{4}{1-q}-n}(G)}
+\|f\|_{\mathring{W}^0_{4-n}(G)} \\
&\quad + \|g\|_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}+k_s\Big)
 \begin{cases}
 \varrho^{\lambda} & \text{if }  s>\lambda, \\
 \varrho^{\lambda}\ln\frac{1}{\varrho} & \text{if }  s=\lambda, \\
 \varrho^s & \text{if }  s<\lambda,
\end{cases}
\end{aligned}
\end{equation}
where $k_s$ is defined by \eqref{sec:semiks}.
\end{theorem}

\begin{proof}
By Theorem \ref{sec:slgthm} we have that $u\in{\mathring{W}^2_{4-n}(G)}$. 
Let us now introduce the  function
$$
\widetilde{U}(\varrho)=\int_{G_0^{\varrho}}r^{2-n}|\nabla u|^2dx
+\int_{\Gamma_0^{\varrho}}r^{1-n}\gamma(\omega)u^2 ds, \quad 0<\varrho<d<b.
$$
Multiplying both sides of \eqref{sec:slg1} by $r^{2-n}u(x)$ and integrating 
it over the domain $G_0^{\varrho}$, $0<\varrho <d$, we obtain
\begin{equation} \label{sec:sll0}
\begin{aligned}
\int_{G_0^{\varrho}}r^{2-n}u\Delta u dx
&= \int_{G_0^{\varrho}} r^{2-n}u(h+f)dx\\
&\quad - \int_{G_0^{\varrho}} r^{2-n}u[(a^{ij}-\delta_i^j)u_{x_ix_j}+a^iu_{x_i}
 +au]dx.
\end{aligned}
\end{equation}
Calculating the integral from the left side by parts, similarly
to \eqref{sec:slg7}, we obtain
\begin{equation} \label{sec:sll0a}
\begin{aligned}
&\int_{G_0^{\varrho}}r^{2-n}|\nabla u|^2dx
 +\int_{\Gamma_0^{\varrho}}\chi(\omega)r^{2-n}u\frac{\partial u}{\partial r}ds
 +\int_{\Gamma_0^{\varrho}}\gamma(\omega)r^{1-n}u^2ds\\
& + \int_{G_0^{\varrho}} r^{2-n}uh\,dx\\
&=\int_{\Omega}\Big(\varrho u\frac{\partial u}{\partial r}
 +\frac{n-2}{2}u^2\Big)d\Omega+ \int_{\Gamma_0^{\varrho}}r^{2-n}ug\,ds
-\int_{G_0^{\varrho}}r^{2-n}ufdx\\
&\quad +\int_{G_0^{\varrho}}r^{2-n}u[(a^{ij}-\delta_i^j)u_{x_ix_j} +a^iu_{x_i} +au]dx.
\end{aligned}
\end{equation}
Similarly to \eqref{sec:slg10}, we have
\begin{equation*}
\int_{G_0^{\varrho}}r^{4-n}u_{xx}^2 dx
\leq C_1\int_{G_0^{2\varrho}}\left(r^{-n}u^2+r^{4-n}h^2+r^{4-n}f^2\right)dx\\
+C_2\|g\|^2_{\mathring{W}_{4-n}^{1\slash 2}(\Gamma_0^{2\varrho})}.
\end{equation*}
Repeating verbatim estimations of formulas (52)--(55) of the proof of
\cite[Theorem 4.1]{bb14}, identity \eqref{sec:sll0a} together with the above
inequality, takes the form
\begin{align*}
&[1-(\mathcal{A}(\varrho)+\delta)]\widetilde{U}(\varrho)
+\int_{G_0^{\varrho}}r^{2-n}a_0|u|^{1+q}dx \\
&\leq \frac{\varrho}{2\lambda}\widetilde{U}'(\varrho)
 + \mathcal{A}(\varrho)\widetilde{U}(2\varrho)
 + c_1\delta^{-1} \Big(\|f\|^2_{\mathring{W}^0_{4-n}\left(G_0^{2\varrho}\right)} \\
&\quad + \|a_0\|^{\frac{2}{1-q}}_{V^0_{\frac{2}{1-q},\frac{4}{1-q}-n}
 \left(G_0^{2\varrho}\right)}
  +\|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}(\Gamma_0^{2\varrho})}\Big), \quad
  \forall \delta>0,
\end{align*}
where constant $c_1>0$ depends on $\gamma_0$, $\chi_0$, $\lambda$.
We used inequalities \eqref{sec:sl9} for $\delta =1$ and \eqref{sec:slg10}
for $\epsilon =0$. Now, using assumption (A4), we finally obtain
\begin{equation}\label{sec:sll2}
[1-(\mathcal{A}(\varrho)+\delta)]\widetilde{U}(\varrho)
\leq \frac{\varrho}{2\lambda}\widetilde{U}'(\varrho)
+ \mathcal{A}(\varrho)\widetilde{U}(2\varrho)
+c_2k_s^2\delta^{-1}\varrho^{2s}, \quad \forall  \delta>0.
\end{equation}
Moreover, by Theorem \ref{sec:slgthm}, we have the initial condition
\begin{equation}\label{sec:sll3}
\begin{aligned}
\widetilde{U}(d)
&\leq C\Big(\|u\|^2_{L^2(G)}+ \|f\|^2_{\mathring{W}^0_{4-n}(G)}
 + \|a_0\|^{\frac{2}{1-q}}_{V^0_{\frac{2}{1-q},\frac{4}{1-q}-n}(G)} \\
&\quad +\|g\|^2_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}\Big)\equiv U_0.
\end{aligned}
\end{equation}
The differential inequality \eqref{sec:sll2} with the initial condition
\eqref{sec:sll3} is the Cauchy problem of \cite[Theorem 1.57]{bk06} and 
it is the same type as (57) and (58) in \cite{bb14}.
 Repeating verbatim investigations for $s>\lambda$, $s=\lambda$ and 
$s<\lambda$ in the proof of \cite[Theorem 4.1]{bb14}, we obtain
\begin{equation}\label{sec:sll4}
\widetilde{U}(\varrho)\leq c\left(U_0+k_s^2\right)
\begin{cases} \varrho^{2\lambda} & \text{if }  s>\lambda, \\
 \varrho^{2\lambda}\ln\frac{1}{\varrho} & \text{if }  s=\lambda, \\
 \varrho^{2s} & \text{if }  s<\lambda,
\end{cases}
\end{equation}
here constant $c>0$ depends only on $\lambda$, $d$, $s$ and on
$\int_0^d \frac{\mathcal{A}(\varsigma)}{\varsigma}d\varsigma$.
Finally, taking into account \eqref{sec:sl9}, \eqref{sec:slg10},
\eqref{sec:slg12}, \eqref{sec:sll3} and \eqref{sec:sll4},
we obtain the required estimate \eqref{sec:sll}.
\end{proof}

\begin{theorem} \label{sec:sllthm2}
Let $u$ be a strong solution of problem \eqref{sec:sl} and assumptions
 {\rm (A1)--(A4)} be satisfied with function $\mathcal{A}(r)$ continuous 
at zero, but \emph{not Dini} continuous. Then for all
 $\epsilon>0$ there are $d\in(0,b)$ and a constant $c_{\epsilon}>0$ depending 
only on $\nu$, $\mu$, $b$, $s$, $\lambda$, $q$, $\epsilon$, $\gamma_0$, $\chi_0$, 
$\operatorname{meas} G$, $\operatorname{diam} G$, 
$\|\chi\|_{C^1(\partial \Omega)}$, $\|\gamma\|_{C^1(\partial \Omega)}$, 
such that for all $\varrho\in(0,d)$
\begin{equation} \label{sec:sll5}
\begin{aligned}
\|u\|_{\mathring{W}^2_{4-n}(G_0^{\varrho})}
& \leq   c\Big(\|u\|_{L^2(G)}+\|a_0\|_{V^0_{\frac{2}{1-q},\frac{4}{1-q}-n}(G)}
+\|f\|_{\mathring{W}^0_{4-n}(G)} \\
&\quad+ \|g\|_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}+k_s\Big)
 \begin{cases} 
\varrho^{\lambda-\epsilon} & \text{if }  s>\lambda, \\
\varrho^{s-\epsilon} & \text{if }  s\leq\lambda,
\end{cases}
\end{aligned}
\end{equation}
where $k_s$ is defined by \eqref{sec:semiks}.
\end{theorem}

\begin{proof}
Similarly to the proof of Theorem \ref{sec:sllthm}, we obtain the Cauchy
 problem \eqref{sec:sll2}-\eqref{sec:sll3}. Repeating verbatim the proof 
of \cite[Theorem 4.2]{bb14} and taking into account Theorem \ref{sec:slgthm} 
we obtain estimation \eqref{sec:sll5}.
\end{proof}

\begin{theorem}\label{sec:sllthm3}
Let $u$ be a strong solution of problem \eqref{sec:sl} and assumptions 
{\rm (A1)--(A4)} be satisfied with function
 $\mathcal{A}(r)\sim\frac{1}{\ln\frac{1}{r}}$. Then there are $d\in(0,b)$ 
and constants $c>0$ and $c_s>0$ depending only on $\nu$, $\mu$, $b$, $s$, 
$\lambda$, $q$, $\epsilon$, $\gamma_0$, $\chi_0$, $\operatorname{meas} G$, 
$\operatorname{diam} G$, $\|\chi\|_{C^1(\partial \Omega)}$,
 $\|\gamma\|_{C^1(\partial \Omega)}$, such that for all $\varrho\in(0,d)$,
\begin{equation} \label{sec:sll6}
\begin{aligned}
\|u\|_{\mathring{W}^2_{4-n}(G_0^{\varrho})}
&\leq   c\Big(\|u\|_{L^2(G)}+\|a_0\|_{V^0_{\frac{2}{1-q},\frac{4}{1-q}-n}(G)}
+\|f\|_{\mathring{W}^0_{4-n}(G)} \\
&\quad + \|g\|_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}+k_s\Big)
 \ln^{c_s(\lambda)}\left(\frac{1}{\varrho}\right)
 \begin{cases} \varrho^{\lambda} & \text{if }  s>\lambda, \\
 \varrho^{s} & \text{if }  s\leq\lambda,
\end{cases}
\end{aligned}
\end{equation}
where $k_s$ is defined by \eqref{sec:semiks}.
\end{theorem}

\begin{proof}
Similarly to the proof of Theorem \ref{sec:sllthm}, we obtain the Cauchy 
problem \eqref{sec:sll2}--\eqref{sec:sll3}. Repeating verbatim the proof 
of \cite[Theorem 4.3]{bb14} and taking into account Theorem \ref{sec:slgthm} 
we obtain estimation \eqref{sec:sll6}.
\end{proof}

\section{Power modulus of continuity} \label{sec:semilinpowermod}

\begin{theorem} \label{sec:semithad} 
Let $u\in W^{2,n}(G)$ be a strong solution of \eqref{sec:sl}. 
Suppose that assumptions  {\rm (A1)--(A5)} are satisfied. Then there is a 
positive constant $c$, such that
\begin{equation*}%\label{sec:sll7}
\|u\|_{V_{n,0}^2(G)}\leq c\Big(\|f\|_{L^n(G)}+\|a_0\|^{\frac{1}{1-q}}
_{V^0_{\frac{n}{1-q},\frac{2qn}{1-q}}(G)}
+\|g\|_{V_{n,0}^{1-\frac{1}{n}}(\partial G)} + \|u\|_{L^n(G')}\Big),
\end{equation*}
for an arbitrary nonempty open set $G'\subset\subset G$.
\end{theorem}

\begin{proof}
By \cite[Theorem 1.4.1]{kmr01}, for a solution $u\in V_{n,0}^2(G)$ of the problem
\begin{gather*}
 a^{ij}(x)u_{x_ix_j}+a^i(x)u_{x_i}+a(x)u(x)=F(x), \quad x\in G, \\ 
\frac{\partial u}{\partial \vec{n}} +\chi(\omega)\frac{\partial u}{\partial r}
+\frac{1}{|x|}\gamma(\omega)u(x)=g(x), \quad x\in \partial G\backslash\mathcal{O},
\end{gather*}
the estimate
\begin{equation*}%\label{sec:sl18}
\|u\|_{V_{n,0}^2(G)}
\leq c_0 \Big( \|F\|_{L^n(G)} + \|g\|_{V_{n,0}^{1-\frac{1}{n}}(\partial G)}
+\|u\|_{L^n(G')}\Big)
\end{equation*}
holds for any nonempty open set $G'\subset\subset G$, provided that 
$\lambda>1$ and $F\in L^n(G)$, where constant $c_0$ depends only on $\nu$, 
$\mu$, $n$, $\chi_0$, $\gamma_0$, $\max_{x\in \overline{G}}\mathcal{A}(|x|)$,
 $\|a^i\|_{p,G}$, $\|a\|_{\frac{p}{2},G}$, $p>n$, and the domain $G$. 
Thus, using the Jensen inequality with $F(x)=f(x)+a_0(x)u|u|^{q-1}$, we obtain
\begin{equation} \label{sec:sl19}
\begin{aligned}
&\int_G \left(r^{-2n}|u|^n+r^{-n}|\nabla u|^n + |u_{xx}|^n \right)dx\\
&\leq C\Big\{ \int_G \left(|a_0|^n |u|^{qn} + |f|^n\right)dx
 + \int_{G'}|u|^ndx+ \|g\|^n_{V_{n,0}^{1-\frac{1}{n}}(\partial G)}\Big\}.
\end{aligned}
\end{equation}
Using the Young inequality and taking into account $q\in(0,1)$, we deduce
\begin{equation}\label{sec:sl20}
|a_0|^n|u|^{qn} = \left(r^{-2qn}|u|^{qn}\right)\left(r^{2qn}|a_0(x)|^n\right)
\leq \epsilon^{\frac{qn}{q-1}}r^{\frac{2qn}{1-q}}|a_0|^{\frac{n}{1-q}}
+ \epsilon^n r^{-2n}|u|^n,
\end{equation}
for all $\epsilon>0$. Choosing $\epsilon= 2^{-n}$, from \eqref{sec:sl19}
and \eqref{sec:sl20}, we obtain the required estimation.
\end{proof}

\begin{remark} \rm
Choosing in Theorem \ref{sec:semithad} the domain $G'$ such that 
$(\operatorname{diam} G')^{2n}<1/2$, we have
$$
\int_{G'}|u|^ndx
\leq \int_{G'}r^{2n}r^{-2n}|u|^ndx
\leq (\operatorname{diam} G')^{2n}\int_{G'}r^{-2n}|u|^ndx
\leq \frac{1}{2}\int_{G'}r^{-2n}|u|^ndx.
$$
Thus, formula \eqref{sec:sl19} takes the form
\begin{align*}
&\int_G \left(r^{-2n}|u|^n+r^{-n}|\nabla u|^n + |u_{xx}|^n \right)dx\\
&\leq C_1\Big\{ \int_G \left(|a_0|^n |u|^{qn} + |f|^n\right)dx 
 + \|g\|^n_{V_{n,0}^{1-\frac{1}{n}}(\partial G)}\Big\}.
\end{align*}
and the statement of Theorem \ref{sec:semithad} takes the form
$$
\|u\|_{V_{n,0}^2(G)}
\leq c_1\Big(\|f\|_{L^n(G)}+\|a_0\|^{\frac{1}{1-q}}_{V^0_{\frac{n}{1-q},
\frac{2qn}{1-q}}(G)}+\|g\|_{V_{n,0}^{1-\frac{1}{n}}(\partial G)}\Big).
$$
\end{remark}

Now we will prove the main results.

\begin{proof}[Proof of Theorem \ref{sec:slmainthm}]
We consider two sets $G_{\varrho\slash 4}^{2\varrho}$ and 
$G_{\varrho\slash 2}^{\varrho}\subset G_{\varrho\slash 4}^{2\varrho}$, 
$\varrho>0$. We make the transformation: $x=\varrho x'$, 
$u(\varrho x')=\psi(\varrho)v(x')$, where function $\psi(\varrho)$ is defined 
by \eqref{sec:slmpsi}. The function $v(x')$ satisfies the problem
\begin{gather*}
\begin{aligned}
&a^{ij}(\varrho x')v_{x_i'x_j'}+\varrho a^i(\varrho x')v_{x_i'}
 +\varrho^2a(\varrho x')v \\
&=\frac{\varrho^2}{\psi(\varrho)}f(\varrho x')+a_0(\varrho x')
  \varrho^2 \psi^{q-1}(\varrho)v|v|^{q-1}, \quad x'\in G_{1\slash 4}^2,
\end{aligned} \\ 
\frac{\partial v}{\partial \vec{n'}} +\chi(\omega)\frac{\partial v}{\partial r'}
+\frac{1}{|x'|}\gamma(\omega)v=\frac{\varrho}{\psi(\varrho)}g(\varrho x'), 
\quad  x'\in \Gamma_{1\slash 4}^2.
\end{gather*}
Based on the local maximum principle (see \cite[Theorem 3.3]{l87a},
\cite[Theorem 4.3]{l01} and \cite[Corollary 7.34]{l96}) we conclude that
\begin{equation} \label{sec:slc1}
\begin{aligned}
&\sup_{x'\in G_{1\slash 2}^1}|v(x')| \\
&\leq C\Big[\Big(\int_{G_{1\slash 4}^2}v^2dx'\Big)^{1/2}
 +\frac{\varrho^2}{\psi(\varrho)}\|f\|_{L^n(G_{1\slash 4}^2)} \\
&\quad +\varrho^2\psi^{q-1}(\varrho)
 \Big(\int_{G_{1\slash 4}^2}|a_0(\varrho x')|^n|v|^{qn}dx'\Big)^{1/n}
 + \frac{\varrho}{\psi(\varrho)}\sup_{x'\in G_{1\slash 4}^2}|g(\varrho x')|\Big],
\end{aligned}
\end{equation}
where constant $C>0$ depends only on $\nu$, $\mu$, $b$, $\gamma_0$, $\chi_0$,
$n$, $G$, $\omega_0$, $\max_{\omega\in \partial \Omega} \gamma(\omega)$,
$g_1$, $\|a\|_{L^n(G_{1\slash 4}^4)}$,
$\|(\sum_{i=1}^n |a^i|^2)^{1/2}\|_{L^p(G_{1\slash 2}^2)}$, $p>n$.
Now, we use the Young inequality. Taking into account that $q\in(0,1)$, we deduce
\begin{equation} \label{sec:slc2}
\begin{aligned}
|a_0(\varrho x')|^n|v|^{qn}
&= \left(|x'|^{-2qn}|v|^{qn}\right)\left(|x'|^{2qn}|a_0(\varrho x')|^n\right)\\
&\leq \epsilon^{\frac{qn}{q-1}}|x'|^{\frac{2qn}{1-q}}
 |a_0(\varrho x')|^{\frac{n}{1-q}} + \epsilon^n |x'|^{-2n}|v|^n,
\quad \forall \epsilon>0.
\end{aligned}
\end{equation}
By \eqref{sec:slc1} and \eqref{sec:slc2}, we obtain
\begin{equation} \label{sec:slc3}
\begin{aligned}
&\sup_{x'\in G_{1\slash 2}^1}|v(x')|\\
& \leq C\Big(\int_{G_{1\slash 4}^2}v^2dx'\Big)^{1/2}
 +C\frac{\varrho^2}{\psi(\varrho)}\|f\|_{L^n\left(G_{1\slash 4}^2\right)}
 +  C\frac{\varrho}{\psi(\varrho)}\sup_{x'\in G_{1\slash 4}^2}|g(\varrho x')|\\
&\quad +C\varrho^2\psi^{q-1}(\varrho)
 \Big[\epsilon\Big(\int_{G_{1\slash 4}^2}|x'|^{-2n}|v|^n dx'\Big)^{1/n} \\
&\quad + \epsilon^{\frac{q}{q-1}}\Big( \int_{G_{1\slash 4}^2}|x'|^{\frac{2qn}{1-q}}
 |a_0(\varrho x')|^{\frac{n}{1-q}}dx'\Big)^{1/n}\Big],
\end{aligned}
\end{equation}
for all $\epsilon>0$. Returning to the variable $x$ and to the function $u$, we have
\begin{equation} \label{sec:slc4}
\begin{aligned}
&\Big(\int_{G_{1\slash 4}^2} v^2 dx'\Big)^{1/2} \\
&= \Big(\frac{1}{\psi^2(\varrho)}\int_{G_{1\slash 4}^2}u^2(\varrho x')dx'\Big)^{1/2}
\leq \frac{2^{\frac{n}{2}}}{\psi(\varrho)}
 \Big(\int_{G_{\varrho\slash 4}^{2\varrho}}r^{-n}u^2(x) dx\Big)^{1/2}\\
&\leq c\Big(\|u\|_{L^2(G)}+\|a_0\|_{V^0_{\frac{2}{1-q},\frac{4}{1-q}-n}(G)}
 +\|f\|_{\mathring{W}^0_{4-n}(G)} \\
&\quad + \|g\|_{\mathring{W}^{1\slash 2}_{4-n}(\partial G)}+k_s\Big)
 = \text{const.},
\end{aligned}
\end{equation}
by the definition of the function $\psi(\varrho)$ and Theorem \ref{sec:sllthm}.
By assumption (A4),
\begin{align*}
\frac{\varrho^2}{\psi(\varrho)}\|f\|_{L^n\left(G_{1\slash 4}^2\right)}
&= \frac{\varrho^2}{\psi(\varrho)}
\Big(\int_{G_{1\slash 4}^2}|f(\varrho x')|^ndx'\Big)^{1/n} 
\leq \frac{\varrho}{\psi(\varrho)}
 \Big(\int_{G_{\varrho\slash 4}^{2\varrho}}|f(x)|^ndx\Big)^{1/n} \\
&\leq f_1 \frac{\varrho}{\psi(\varrho)}
 \Big(\int_{\varrho \slash 4}^{2\varrho}r^{n(s-2)}r^{n-1}dr
\operatorname{meas}\Omega\Big)^{1/n} \\
&\leq c_1f_1\frac{\varrho^s}{\psi(\varrho)}
 = c_1f_1
\begin{cases} \varrho^{s-\lambda}<1 & \text{if }  s>\lambda, \\
\frac{1}{\ln\frac{1}{\varrho}}<1 & \text{if }  s=\lambda, \\
1 & \text{if }  s<\lambda.
\end{cases}
\end{align*}
Hence
\begin{equation}\label{sec:slc5}
\frac{\varrho^2}{\psi(\varrho)}\|f\|_{L^n\left(G_{1\slash 4}^2\right)}
\leq c_1f_1=\text{const.}
\end{equation}
Similarly
\begin{equation*}
\frac{\varrho}{\psi(\varrho)}\sup_{x'\in G_{1\slash 4}^{2}} |g(\varrho x')|
\leq \frac{\varrho}{\psi(\varrho)}g_1\varrho^{s-1}
= g_1 \begin{cases} \varrho^{s-\lambda}<1 & \text{if }  s>\lambda, \\
\frac{1}{\ln\frac{1}{\varrho}}<1 & \text{if }  s=\lambda, \\
1 & \text{if }  s<\lambda.
\end{cases}
\end{equation*}
Thus
\begin{equation}\label{sec:slc6}
\frac{\varrho}{\psi(\varrho)}\sup_{x'\in G_{1\slash 4}^{2}} |g(\varrho x')|
 \leq g_1 = \text{const.}
\end{equation}
We calculate
\begin{equation}\label{sec:slc7}
\begin{aligned}
\Big(\int_{G_{1\slash 4}^2}|x'|^{-2n}|v|^ndx'\Big)^{1/n}
&= \varrho \Big(\int_{G_{\varrho\slash 4}^{2\varrho}}r^{-2n}|v|^ndx\Big)^{1/n}\\
&=\frac{\varrho}{\psi(\varrho)}
 \Big(\int_{G_{\varrho\slash 4}^{2\varrho}} r^{-2n}|u|^ndx\Big)^{1/n}
\end{aligned}
\end{equation}
and
\begin{equation}\label{sec:slc8}
\Big(\int_{G_{1\slash 4}^{2}}|x'|^{\frac{2qn}{1-q}}
|a_0(\varrho x')|^{\frac{n}{1-q}}dx'\Big)^{1/n}
\leq c(q,n)\varrho^{-1}\Big(\int_{G_{\varrho\slash 4}^{2\varrho}}
 |a_0|^{\frac{n}{1-q}}dx\Big)^{1/n}.
\end{equation}
Choosing $\epsilon = \psi(\varrho)/\varrho$ in \eqref{sec:slc3},
because of \eqref{sec:sl19} and \eqref{sec:sl20}, by \eqref{sec:slc7}
and \eqref{sec:slc8}, we obtain
\begin{equation} \label{sec:slc9}
\begin{aligned}
&\epsilon\Big(\int_{G_{1\slash 4}^2}|x'|^{-2n}|v|^ndx'\Big)^{1/n} \\
&\leq c(n,q,b) \Big[\int_{G}\left(|a_0(x)|^{\frac{n}{1-q}}
  + |f(x)|^n\right)dx\Big]^{1/n}
  +c(n,q,b)\|g\|^n_{V_{n,0}^{1-\frac{1}{n}}(\Gamma)} \\
&\leq c(n,q,b,s,f_1,k_0)=\text{const.}
\end{aligned}
\end{equation}
and
\begin{equation} \label{sec:slc10}
\begin{aligned}
&\epsilon^{\frac{q}{q-1}}\Big(\int_{G_{1\slash 4}^{2}}|x'|^{\frac{2qn}{1-q}}
|a_0(\varrho x')|^{\frac{n}{1-q}}dx'\Big)^{1/n} \\
&\leq c_2 \Big(\frac{\varrho}{\psi(\varrho)}\Big)^{\frac{q}{1-q}} \frac{1}{\varrho}
\Big(\int_{G}|a_0|^{\frac{n}{1-q}}dx\Big)^{1/n}
 \leq c_2 k_0=\text{const.},
\end{aligned}
\end{equation}
by \eqref{sec:sla0} and assumption (A4). From \eqref{sec:slc4}--\eqref{sec:slc6},
\eqref{sec:slc9} and \eqref{sec:slc10}, with regard to \eqref{sec:slc3}, we have
\begin{equation}\label{sec:slc11}
\sup_{x'\in G_{1\slash 2}^1}|v(x')|\leq c_3(1+\varrho^2\psi^{q-1}(\varrho)).
\end{equation}
We need to show that for all $\varrho>0$,
\begin{equation}\label{sec:slc12}
\varrho^2\psi^{q-1}(\varrho) < \infty.
\end{equation}
Let us assume $0<q<1-\frac{2}{\lambda}$ and $\lambda > s$.
In this case we have that  if $s\leq \frac{2}{1-q}$, then
$$
\varrho^2\psi^{q-1}(\varrho) = \varrho^{s(q-1)+2}<\infty
$$
holds for all $\varrho>0$. Choosing the best exponent
$s=\frac{2}{1-q}<\lambda$, we obtain the required estimation \eqref{sec:slmainb}.
In fact, by \eqref{sec:slc11} and \eqref{sec:slc12}, it follows
$$
|v(x')|\leq M_0'=\text{const.}
$$
for all $x'\in G_{1\slash 2}^1.$ Returning to the variable $x$, we obtain
$$
|u(x)|\leq M_0'\psi(\varrho)=M_0'\varrho^{\frac{2}{1-q}}
$$
for all $x\in G_{\varrho\slash 2}^{\varrho}$, $0<\varrho < b$.
Setting $|x|=2\varrho/3$, we obtain \eqref{sec:slmainb}.

Let us assume that $1-\frac{2}{\lambda}\leq q\leq 1$. Thus for all $\varrho>0$
$$
\varrho^2\psi^{q-1}(\varrho) 
= \begin{cases} \varrho^{2+\lambda(q-1)}<\infty & \text{if }  s>\lambda, \\ 
 \varrho^{2+\lambda(q-1)}\ln^{\frac{3}{2}(q-1)}\frac{1}{\varrho}<\infty 
 & \text{if }  s=\lambda, \\ 
\varrho^{2+s(q-1)}\leq \varrho^{2q}\varrho^{(\lambda-2)(q-1)}<\infty 
& \text{if }  s<\lambda. 
\end{cases}
$$
Repeating verbatim the proof of \eqref{sec:slmainb}, we obtain the 
estimate \eqref{sec:slmaina}. The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{sec:slmainthm2}] 
Repeating verbatim the proof of Theorem \ref{sec:slmainthm},
taking into account \eqref{sec:slmpsi2} and applying Theorem \ref{sec:sllthm2}, 
we obtain the desired result.
\end{proof}

\begin{proof}[Proof of Theorem \ref{sec:slmainthm3}]
 Repeating verbatim the proof of Theorem \ref{sec:slmainthm}, 
taking into account \eqref{sec:slmpsi3} and applying Theorem \ref{sec:sllthm3}, 
we obtain the desired result.
\end{proof}

\begin{thebibliography}{00}

\bibitem{adn59} S.~Agmon, A.~Douglis, L.~Nirenberg;
\newblock {Estimates near the boundaryfor solutions of elliptic partial
  differential equations satisfying general boundary conditions, I}.
\newblock {\em Comm. Pure Appl. Math.}, 12: 623--727, 1959.

\bibitem{bvggv99} M.~F. Bidaut-Veron, V.~Galaktionov, Ph. Grillot, L.~Veron;
\newblock {Singularities in a semilinear elliptic equation with a non-Lipschitz
  nonlinearity}.
\newblock {\em J. Diff. Eqn.}, 154: 318--338, 1999.

\bibitem{b12} M.~Bodzioch;
\newblock {Oblique derivative problem for linear second-order elliptic
  equations with the degeneration in a 3-dimensional bounded domain with the
  boundary conical point}.
\newblock {\em Electron. J. Differential Equ.}, 2012 (228): 1--28, 2012.

\bibitem{bb14} M.~Bodzioch, M.~Borsuk;
\newblock {On the degenerate oblique derivative problem for elliptic
  second-order equation in a domain with boundary conical point}.
\newblock {\em Complex Variables and Elliptic Equations}, 59(3): 324--354, 2014.

\bibitem{bb15} M.~Bodzioch, M.~Borsuk;
\newblock {Behavior of strong solutions to the degenerate oblique derivative
  problem for elliptic quasi-linear equations in a neighborhood of a boundary
  conical point}.
\newblock {\em Complex Variables and Elliptic Equations}, 60(4): 510--528, 2015.

\bibitem{bb15a} M.~Bodzioch, M.~Borsuk;
\newblock {The degenerate second-order elliptic oblique derivative problem in a
  domain with conical boundary point}.
\newblock {\em Current Trends in Analysis and Its Applications, Trends in
  Mathematics}, 2015: 11--18, 2015.

\bibitem{b13} M.~Borsuk;
\newblock {The behavior near the boundary corner point of solutions to the
  degenerate oblique derivative problem for elliptic second-order equations in
  a plane domain}.
\newblock {\em J. Diff. Equat.}, 254: 1601--1625, 2013.

\bibitem{bk06} M.~Borsuk, V.~Kondrat'ev;
\newblock {\em {Elliptic Boundary Value Problems of Second Order in Piecewise
  Smooth Domains}}. \newblock Elsevier, 2006.

\bibitem{ckl00} S.~Canic, B.~L. Keyftiz, G.~M. Lieberman;
\newblock {A proof of existence of perturbed steady transonic shocks via a free
  boundary problem}.
\newblock {\em Comm. Pure Appl. Math.}, 53: 484--511, 2000.

\bibitem{c77} M.~Chicco;
\newblock {Terzo problema al contorno per una classe di equazioni ellittiche
  del secondo ordine a coefficienti discontinui}.
\newblock {\em Ann. Mat. Pura Appl.}, 112(4): 241--259, 1977.

\bibitem{ddo06} G.~D\'{i}az, J. I. D\'{i}az, J.~Otero;
\newblock {On an oblique boundary value problem related to the Backus problem
  in Geodesy}.
\newblock {\em Nonlinear Analysis: Real World Applications}, 7: 147--166, 2006.

\bibitem{fv96} J.~Fabbri, L.~Veron;
\newblock {Singular boundary value problems for nonlinear elliptic equations in
  nonsmooth domains}.
\newblock {\em Adv. Diff. Eqn.}, 1(6): 1075--1098, 1996.

\bibitem{f87} M.~Faierman;
\newblock {Regularity of solutions of an elliptic boundary value problem in a
  rectangle}.
\newblock {\em Comm. in PDE}, 12: 285--305, 1987.

\bibitem{f86} R.~Finn;
\newblock {\em {Equilibrium Capilary Surfaces}}, volume 284.
\newblock Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1986.

\bibitem{gsv96} M.~G. Garroni, V.~A. Solonnikov, M.~A. Vivaldi;
\newblock {On the oblique derivative problem in an infinite angle}.
\newblock {\em Topological methods in nonlinear analysis}, 7(2): 299--325, 1996.

\bibitem{g85} P.~Grisvard;
\newblock {\em {Elliptic problems in nonsmooth domains}}.
\newblock Pitman Advanded Publishing Program, Boston - London - Melbourne,
  1985.

\bibitem{kl88a} V.~A. Kondrat'ev, E.~M. Landis;
\newblock {On qualitative properties of solutions of a nonlinear equation of
  second order [Russian]}.
\newblock {\em Mat. Sb.}, 135(3):1023--1042, 1988.
\newblock {English Translation in Math. USSR Sb, 63(2): 337-351, 1989}.

\bibitem{kn93a} V.~A. Kondrat'ev, V.~A. Nikishkin;
\newblock {On isolated singularities of solutions of the Emden - Fowler type
  equations [Russian]}.
\newblock {\em Differ. uravn.}, 29(6): 1025--1038, 1993.

\bibitem{ko83} V.~A. Kondrat'ev, O.~A. Oleinik;
\newblock {Boundary value problems for partial differential equations in
  nonsmooth domains}.
\newblock {\em Russian Math. Surveys}, 38(2): 1--86, 1983.

\bibitem{kmr97} V.~A. Kozlov, V.~G. Maz'ya, J.~Rossmann.
\newblock {\em {Elliptic Boundary Value Problems in Domains with Point
  Singularities}}, volume~52.
\newblock AMS. Mathematical surveys and monographs, 1997.

\bibitem{kmr01} V.~A. Kozlov, V.~G. Maz'ya, J.~Rossmann;
\newblock {\em {Spectral Problems Associated with Corner Singularities of
  Solutions to Elliptic Equations}}, volume~85.
\newblock AMS. Mathematical surveys and monographs, 2001.

\bibitem{ks87} A.~Kufner, A.~M. S\"{a}ndig;
\newblock {Some applications of weighted Sobolev spaces}.
\newblock {\em Teubner-Texte Math.}, 100: 1--268, 1987.

\bibitem{l85a} G.~M. Lieberman;
\newblock {The Perron process applied to oblique derivative problems}.
\newblock {\em Adv. in Math.}, 55: 161--172, 1985.

\bibitem{l87a} G.~M. Lieberman;
\newblock {Local estimates for subsolutions and supersolutions of oblique
  derivative problems for general second order elliptic equations}.
\newblock {\em Trans. of AMS}, 304: 343--353, 1987.

\bibitem{l96} G.~M. Lieberman;
\newblock {\em {Second order parabolic diffeential equations}}.
\newblock World Scientific, Singapore - New Jersey - London - Hong Kong, 1996.

\bibitem{l01} G.~M. Lieberman;
\newblock {Pointwise estimate for oblique derivative problems in nonsmooth
  domains}.
\newblock {\em J. Diff. Eqn.}, 173(1): 178--211, 2001.

\bibitem{m63} C.~Miranda;
\newblock {Sulle equazioni ellittiche del secondo ordine di tipo non
  variazionale, a coefficienti discontinui}.
\newblock {\em Ann. Mat. Pura Appl.}, 63: 353--386, 1963.

\bibitem{pv00} S.~I. Pohozaev, L.~Veron;
\newblock {Nonexistence results of solutions of semilinear differential
  inequalities on the Heisenberg group}.
\newblock {\em Manuscripta Mathematica}, 102: 85--99, 2000.

\bibitem{r95} H.~Reisman;
\newblock {Second order elliptic boundary value problem in a domain with  edges}.
\newblock {\em Comm. in PDE}, 6: 1023--1042, 1995.

\bibitem{s97} F.~Sans\`{o};
\newblock {The hierarchy of Geodetic BVP's}.
\newblock {\em Lecture Notes in Earth Sciences}, 65, 1997.

\bibitem{t67} G.~Talenti;
\newblock {Problemi di derivata obliqua per equazioni ellittiche in due  variabili}.
\newblock {\em Boll. Un. Mat. Ital.}, 22(3): 505--526, 1967.

\end{thebibliography}

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