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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 79, pp. 1--13.\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-2017/79\hfil
 Harnack's inequality for  $p$-Laplacian equations]
{Harnack's inequality for  $p$-Laplacian equations with Muckenhoupt
 weight degenerating in part of the domain}

\author[Y. A. Alkhutov, S. T. Huseynov \hfil EJDE-2017/79\hfilneg]
{Yuriy A. Alkhutov, Sarvan T. Huseynov}

\address{Yuriy A. Alkhutov \newline
Stoletov Vladimir State University,
Vladimir, Russia}
\email{yuriy-alkhutov@yandex.ru}

\address{Sarvan T. Huseynov \newline
Baku State University, Baku, Azerbaijan}
\email{sarvanhuseynov@rambler.ru}

\dedicatory{Communicated by Ludmila S. Pulkina}

\thanks{Submitted February 5, 2017. Published March 21, 2017.}
\subjclass[2010]{35j92, 35j65, 35j70, 35j62}
\keywords{$p$-Laplasian; Muckenhoupt weight; Harnack's inequality}

\begin{abstract}
 In this article we consider quasi-linear second-order elliptic equations
 of divergence structure with Makenhaupt weight  that degenerates  over
 a small part of the domain. We show that the classical Harnack's
 inequality does not hold in this case, and prove an appropriate Harnack's
 inequality for the considered equation.
\end{abstract}

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

\section{Introduction and statement of results}

On a domain $D\subset \mathbb {R}^n$, $n\ge 2$ we consider
the family of elliptic equations
\begin{equation} \label{e1_1}
L_{\varepsilon } u=\operatorname{div}\big(\omega _{\varepsilon } (x)
|\nabla u|^{p-2} \nabla u)\big)=0,\quad p>1,
\end{equation}
 where $p$ is a constant, $\omega _{\varepsilon } (x)$ is nonnegative
 weight depending on small parameter $\varepsilon $. It is assumed that
the domain $D$ is divided by the  hyperplane
$\Sigma =\{ x:  x_{n} =0\} $ into the parts
$D^{(1)} =D\cap \{ x:  x_{n} >0\} $ and $D^{(2)} =D\cap \{ x:  x_{n} <0\} $, and
\begin{equation} \label{e1_2}
\omega _{\varepsilon } (x)
=\begin{cases}
\varepsilon \omega (x), & x\in D^{(1)} , \\
\omega (x),  & x\in D^{(2)} ,
\end{cases}
\end{equation}
where $\varepsilon \in (0,1]$, $\omega (x)$ is a weight satisfying
 Muckenhoupt's  $A_p $-condition. Note that the weight $\omega (x)$,
defined in the whole space $\mathbb{R}^n $ satisfies to $A_p$-condition
(see \cite{m2}), if
$$
\sup \Big( \frac{1}{|B|} \int_{B} \omega (x)
\,dx\Big) \Big( \frac{1}{|B|} \int _{B}
\omega^{-\frac{1}{p-1}}(x)\,dx\Big)^{p-1} <\infty, \quad 1<p<\infty,
$$
where the supremum is taken over all balls $B\subset {\mathbb R}^n $.

To define a solution of  \eqref{e1_1} we
introduce the class of functions
$$
W_{\rm loc} (D, \omega)=\{u: u \in W_{\rm loc}^{1,1} (D), | \nabla
u|^p\omega \in L_{\rm loc}^{1}(D) \},
$$
where $W_{\rm loc}^{1,1} (D)$ is a classical Sobolev's space of the
functions which are local summable in the domain $D$ together with
all generalized partial derivatives of the first order. As a
solution of the equation  \eqref{e1_1} we take the function
$u\in W_{\rm loc} (D,\omega )$ for which the integral identity
\begin{equation} \label{e1_3}
\int_{D} \omega_\varepsilon (x) | \nabla u|^{p-2}\nabla u\cdot\nabla \xi\,dx=0
\end{equation}
is satisfied on the finite test functions $\xi \in W_{\rm loc}
(D,\omega _{\varepsilon } )$.

The object of the work is the problem of  Harnack's inequality for
the nonnegative solutions of the equation \eqref{e1_1}. A
large number of works is devoted to this problem for the degenerate
equations. The most investigated the case is when the weight
function $\omega (x)$ satisfies the Muckenhoupt's $A_p- $
condition and $\varepsilon =1$.

Below $|E|$ is $n$-dimensional Lebesque measure of the
measurable set $E\subset {\mathbb R}^n $,
$$
d\mu =\omega\,dx,\quad \omega (E)=\int_{E}\omega (x)\,dx,\quad
\int_E  f\,d\mu =\frac{1}{\omega (E)}\int_{E}f\omega\,dx.
$$
The important consequences of the Muckenhoupt's  $A_p $ condition
are the doubling conditions \cite{c1,m2}
\begin{equation} \label{e1_4}
\omega (B_{2r} )\le c\omega (B_r ),
\end{equation}
inversion of the Holder inequality  \cite{c1}
\begin{equation} \label{e1_5}
\Big(\frac{1}{|B_r |} \int_{B_r } \omega ^{1+\delta }
(x)\,dx\Big)^{1/(1+\delta )}
\le C\frac{1}{|B_r |} \int _{B_r } \omega (x)\,dx,
\end{equation}
Friedrichs inequality \cite{f1,h1}
\begin{equation} \label{e1_6}
\begin{gathered}
\int_{\Omega } |\varphi |^p\,d\mu \le c(n,\nu ,p)r^p
\int_{\Omega } |\nabla \varphi |^p\,d\mu , \\
\varphi \in  C^{\infty } (\overline{\Omega }),\quad
\varphi |_{E} =0,\quad |E|\ge \nu |\Omega |,\; \nu >0,
\end{gathered}
\end{equation}
where $\Omega \subset B_r $ is  Lipschitz domain, and Sobolev's
inequality \cite{f1,h1}
\begin{equation} \label{e1_7}
\Big(\hbox{--}\hskip-9pt\int_{B_r } |\varphi |^{pk}\,d\mu \Big)^{1/k}
\le c(n,p)r^p \int_{B_r } |\nabla \varphi |^p\,d\mu ,\quad
 \varphi \in C_0^{\infty } (B_r ),\quad k=\frac{n}{n-1} .
\end{equation}

In \cite{f1,h1} is shown that if $\omega \in A_p $ and $\varepsilon =1$
then solution of the equation \eqref{e1_1} is of Holder
property in $D$ and for all  nonnegative in $B_{4R} \subset D$ solutions
it holds the Harnach inequality
\begin{equation} \label{e1_8}
\inf_{B_{R} } u\ge {\rm const}\cdot \sup_{B_{R} } u.
\end{equation}

For the considered weight $\omega _{\varepsilon } $ the doubling
condition \eqref{e1_4} with a constant independent on
$\varepsilon $, does not hold. This implies that in the center of
the balls on the hyperplane $\Sigma $ the classical Harnack
inequality \eqref{e1_8} does not hold, in which the
constant is independent on $\varepsilon $. This statement is set in
the first section of  \S 2.

In addition to the belonging of the weighting function to the
Muckenhoupt's class $A_p $ it is assumed that in the open balls
$B_{R_0 } $ of small enough radiuses $R_0 $ with the centers on
the hyperplane $\Sigma $ for almost all points $x$from the semiball
$B_{R_0 } \cap \{ x: x_{n} >0\} $ is valid
\begin{equation} \label{e1_9}
\omega (x)\le \gamma \omega (x' ),\; \gamma ={\rm const}>0,
\end{equation}
where $x' $ is a point symmetric to $x$ with respect to the
hyperplane $\Sigma $. In particular to this condition satisfy the
weights $|x|^{\alpha } $, where $-n<\alpha <n(p-1$, and
$|x_{n}|^{\alpha } $, where $-1<\alpha <p-1$. Besides any weight satisfying
to the Muckenhoupt's  $A_p $ condition, that is indeed even with
respect to the hyperplane $\Sigma $ is suitable for this case.

The main aim of this work to formulate and prove the uniform Harnack inequality
over the parameter $\varepsilon $  that corresponds to
the considered equation.  Since the classical Harnack's inequality
\eqref{e1_8} does not hold in the balls with the center on the
hyperplane $\Sigma $, in the formulation of the problem takes part
of such balls and is assumed that
\begin{equation} \label{e1_10}
B_{R}^{-} =B_{R} \cap \{ x:-R<x_{n} <-R/2\} .
\end{equation}



\begin{theorem} \label{thm1}
If the weight  $\omega (x)$ satisfies to the
Muckenhoupt's $A_p $ condition, and the conditions
\eqref{e1_2}, \eqref{e1_9} are satisfied, then for
the nonnegative part in the $B_{4R} \subset D$, with the center on
$\Sigma$, the solution of the equation \eqref{e1_1} satisfies
the inequality
\begin{equation} \label{e1_11}
\inf_{B_{R} } u\ge c_0 \sup_{B_{R}^{-} } u,
\end{equation}
In which the positive constant $c_0 <1$ does not depend on $u$, $R$ and
$\varepsilon $.
\end{theorem}


When $p=2$ and  $\omega \equiv 1$, Theorem \ref{thm1} first was proved in \cite{a2}.
In the present work for the case $\varepsilon =1$ and $p=2$
the inequality \eqref{e1_11} is proved for the non-negative
solutions of the equation with a particular  Muckenhoupt weight
$\omega $, that in general does not satisfy Muckenhoupt's $A_2$-condition.

The end of this work shows that from Theorem \ref{thm1} it follows the Holder
continuity of the solutions at the points $\Sigma \cap D$, and in
consequence Holder continuity of the solutions in the domain of $D$.

Let us consider the family $\{ u^{\varepsilon } (x)\} $ of the solutions of
the equations  $L_{\varepsilon } u^{\varepsilon } =0$ bounded in
$L^{\infty } $ uniformly over $\varepsilon $ on the compact subsets $D$.


\begin{theorem} \label{thm2}
If the weight $\omega $ satisfies the  Muckenhoupt's  $A_p $ condition
and  conditions \eqref{e1_2}, \eqref{e1_9} take place,
then there exists a constant $\alpha \in (0,1)$ depending only on  $p$,
dimension of the space  $n$, constant $\gamma $ from
\eqref{e1_9} and the weight $\omega $, such that the family
$\{ u^{\varepsilon } (x)\} $ is compact in $C^{\alpha } (D' )$ in
any subdomain of $D' \Subset D$.
\end{theorem}


Theorem \ref{thm2} was proved by a different method in an earlier work of the
author \cite{h2}[. In the case when $\omega (x)\equiv 1$ this statement is
given in \cite{h1,h2}. Note also the works \cite{a3,a4}, when
$p=2$ and $\varepsilon =1$, the Holder continuity is proved  for the
solutions of the equations with particular Muckenhoupt's weight
$\omega $, of a more general structure.

\section{Harnack's inequality}

\subsection*{Absence of the classical Harnack's inequality}
Here by $B_{R} $ we denote the open ball of radius $R$ with the
center on the hyperplane $\Sigma $ and
$B_{16r}^{(1)} =B_{16r} \cap\{ x_{n} >0\} $,
$B_{16r}^{(2)} =B_{16r} \cap \{ x_{n} <0\} $.
Choose the points $x_0,  y_0 $ by the way that
$B_{5r}^{x_0 } \subset B_{16r}^{(1)} $,
$B_{5r}^{y_0 } \subset B_{16r}^{(2)} $.
 We assume that the points $x_0 $ and $y_0 $ are
symmetric with respect to the hyperplane $\Sigma $.
Let $\Omega=B_{16r} \backslash (\overline{B_{r/4}^{x_0 } }\cup
\overline{B_{r/4}^{y_0 } })$. Consider the problem
\begin{equation} \label{e2_1}
\begin{gathered}
Lu=\operatorname{div}(\omega _{\varepsilon } (x)|\nabla u|^{p-2} \nabla u)=0
\quad\text{in }\Omega \\
  u=0\quad\text{on } \partial B_{16r} \\
 u=\varepsilon ^{-1} \omega ^{-1} (B_r )\quad\text{on }\partial B_{r/4}^{x_0 }, \quad
 u=\omega ^{-1} (B_r )\quad\text{on } \partial B_{r/4}^{y_0 } .
 \end{gathered}
\end{equation}
Show that in the domain $\Omega $  usual Harnach's inequality
with the constant not depending on $\varepsilon $ does not hold.
Solution of the problem \eqref{e2_1} is a
 minimizer of the variational problem for integral functional
\begin{equation} \label{e2_2}
F[v]=\int_{\Omega } \omega _{\varepsilon } (x)|\nabla v|^p\,dx
\end{equation}
over the smooth functions $v$,
on the closure of the domain $D$,
satisfying boundary conditions \eqref{e2_1}.
Let us continue $u$ to inside of the balls $B_{r/4}^{x_0 } $ and $B_{r/4}^{y_0 } $,
taking $u=\varepsilon ^{-1} \omega ^{-1} (B_r )$ in $B_{r/4}^{x_0 } $
and $u=\omega ^{-1} (B_r )$ in$B_{r/4}^{y_0 } $.

For the compact $K$ belonging to the open set $U$, its $p$-volume
with respect to  $U$ is defined as follows
\[
{\rm cap}_p (K,  U)=\inf \int_{U} \omega (x)|\nabla \varphi |^p\,dx,
\]
where the sharp lower bound is taken over the set of functions from
$C_0^{\infty } (U)$ being equal to unit in the neighborhood of
$K$.

Since the weight $\omega $ satisfies the  Muckenhoupt's
$A_p$-condition then (see \cite{f1}) in the domains
$B_{16r}^{(1)} \backslash\overline{B_{r/4}^{x_0 } }$ and
$B_{16r}^{(2)} \backslash \overline{B_{r/4}^{y_0 } }$ is valid the
 classical Harnachk's inequality that we use in the form
\begin{gather} \label{e2_3}
\sup_{B_{4r}^{x_0 } \backslash \overline{B_{r/2}^{x_0 } }} u
\le C\inf_{B_{4r}^{x_0 } \backslash \overline{B_{r/2}^{x_0 } }} u, \\
\label{e2_4}
\sup_{B_{4r}^{y_0 } \backslash \overline{B_{r/2}^{y_0 } }} u
\le C\inf_{B_{4r}^{y_0 } \backslash \overline{B_{r/2}^{y_0 } }} u.
\end{gather}

By \cite{h1} there exists positive constants  $c_1 $, $c_2 $, not
depending on $r$ such that
\begin{gather} \label{e2_5}
c_1 r^{-p} \omega (B_r^{x_0 } )
\le {\rm cap}_p (\overline{B_{r/4}^{x_0 } },B_{2r}^{x_0 } )
\le c_2 r^{-p} \omega (B_r^{x_0 } ), \\
 \label{e2_6}
c_1 r^{-p} \omega (B_r^{y_0 } )
\le {\rm cap}_p (\overline{B_{r/4}^{y_0 } },B_{2r}^{y_0 } )
\le c_2 r^{-p} \omega (B_r^{y_0 } ).
\end{gather}


\subsection*{Lower estimate of the solution}
Put $w=u\cdot \varepsilon \omega (B_r )$. Then
\[
Lw=0\quad\text{in } B_{16r},\quad  w=1\quad\text{in } \partial B_{r/4}^{x_0 } ,
\quad w=\varepsilon \quad\text{in }\partial B_{r/4}^{y_0 } .
\]
By the miximum principle $0<w\le 1$ in $B_{16r} $.

Let $\eta $ be a cutoff function being equal to zero outside of the ball
$B_{2r}^{x_0 } $, and to unit in the ball $B_r^{x_0 } $,
$|\nabla \eta |\le Cr^{-1} $, $0\le \eta \le 1$.
Since $\omega _{\varepsilon } (x)=\varepsilon \omega (x)$ in $B_{2r}^{x_0 } $,
then by the definition of the volume
\begin{equation} \label{e2_7}
\int_{B_{2r}^{x_0 } } \omega _{\varepsilon } (x)|\nabla (w\eta )|^p\,dx
\ge \varepsilon {\rm cap}_p (\overline{B_{r/4}^{x_0 } },B_{2r}^{x_0 } ).
\end{equation}
Then
\[
\int_{B_{2r}^{x_0 } } \omega _{\varepsilon } (x)|\nabla (w\eta )|^p\,dx
\le C\int_{B_{2r}^{x_0 } } \omega _{\varepsilon } (x)|\nabla w|^p\,dx
+C\varepsilon r^{-p} \int_{B_{2r}^{x_0 } \backslash B_{r/4}^{x_0 } }
\omega (x)w^p\,dx.
\]
Using the estimate $0\le w\le 1$, for which $w^p \le w^{p-1} $, we
obtain
\[
\int_{B_{2r}^{x_0 } } \omega _{\varepsilon } (x)|\nabla (w\eta )|^p\,dx
\le C\int_{B_{2r}^{x_0 } } \omega _{\varepsilon } (x)|\nabla w|^p\,dx
+C\varepsilon r^{-p} \int_{B_{2r}^{x_0 } \backslash B_{r/4}^{x_0 } }
\omega (x)w^{p-1}\,dx.
\]
Let us estimate the last integral in the right hand side of the last
inequality through the sharp upper bound of $w$. Applying Harnack's
inequality \eqref{e2_3} to the solution $u$ of
\eqref{e2_1} and doubling condition \eqref{e1_4}
for the weight $\omega $ we arrive to the estimation
\begin{equation} \label{e2_8}
\int_{B_{2r}^{x_0 } } \omega _{\varepsilon } (x)|\nabla
(w\eta )|^p\,dx\le C\int_{B_{2r}^{x_0 } }
\omega _{\varepsilon } (x)|\nabla w|^p\,dx
+C\varepsilon r^{-p} \omega (B_r )( \min_{\partial
B_r^{x_0 } } w)^{p-1} .
\end{equation}

Now we estimate the first integral in the right hand side of
\eqref{e2_8}. Choosing in the integral identity
\begin{equation} \label{e2_9}
\int_{\Omega } \omega _{\varepsilon } (x)|\nabla w|^{p-2}
\nabla w\cdot \nabla \varphi\,dx=0
\end{equation}
The test function as $\varphi =(1-w)\xi ^p $, where
 $\xi \in C_0^{\infty } (B_{3r}^{x_0 } )$,
$\xi =1$ in $B_{2r}^{x_0 } $,
$|\nabla \xi |\le Cr^{-1} $ and $0\le \xi \le 1$, we obtain
\begin{align*}
\int_{B_{2r}^{x_0 } } \omega _{\varepsilon } (x)|\nabla w|^p\,dx
&\le p\int_{B_{3r}^{x_0 } \backslash B_{2r}^{x_0 } } \omega _{\varepsilon } (x)
 |\nabla w|^{p-1} (1-w)|\nabla \xi |\xi ^{p-1}\,dx \\
&\le  p\int_{B_{3r}^{x_0 } \backslash B_{2r}^{x_0 } }
  \omega _{\varepsilon } (x)|\nabla w|^{p-1} |\nabla \xi |\,dx \\
&\le Cr^{-1} \int_{B_{3r}^{x_0 } \backslash B_{2r}^{x_0 } }
 \omega _{\varepsilon } (x)|\nabla w|^{p-1}\,dx.
\end{align*}
From this by the Holder inequality we find that
\begin{equation}\label{e2_10}
\begin{aligned}
&\int_{B^{x_0}_{2r}}\omega_\varepsilon(x)|\nabla w|^p\,dx \\
&\le C\Big(\int_{B^{x_0}_{3r}\setminus
B^{x_0}_{2r}}\omega_\varepsilon(x)w^{-a}|\nabla w|^p\,dx\Big)^{(p-1)/p}
\Big(\int_{B^{x_0}_{3r}\setminus B^{x_0}_{2r}}\omega_\varepsilon(x)w^{a(p-1)}r^{-p}\,dx
\Big)^{1/p},
\end{aligned}
\end{equation}
where $0<a<1$. To estimate the fist integral in the right hand side
of \eqref{e2_10} we choose in the integral identity \eqref{e2_9}
the test function as $\varphi =w^{1-a} \eta ^p $, where
$\eta \in C_0^{\infty } (B_{4r}^{x_0 } \backslash \overline{B_{3r/2}^{x_0 } })$,
 $\eta =1$ in $B_{3r}^{x_0 } \backslash \overline{B_{2r}^{x_0 } }$,
$|\nabla \eta |\le Cr^{-1} $ and $0\le \eta \le 1$. A a result we obtain
\[
(1-a)\int_{B_{4r}^{x_0 } \backslash B_{3r/2}^{x_0 } }
\omega _{\varepsilon } (x)w^{-a} |\nabla w|^p \eta ^p\,dx
\le p\int_{B_{4r}^{x_0 } \backslash B_{3r/2}^{x_0 } }
\omega _{\varepsilon } (x)|\nabla w|^{p-1} |\nabla \eta |\eta ^{p-1}\,dx.
\]
Applying Young's inequality to the integrand of the right-hand side,
and because of the choice of the cutoff function we obtain the estimate
\begin{equation} \label{e2_11}
\int_{B_{3r}^{x_0 } \backslash B_{2r}^{x_0 } } \omega
_{\varepsilon } (x)w^{-a} |\nabla w|^p\,dx\le Cr^{-p}
\int_{B_{4r}^{x_0 } \backslash B_{3r/2}^{x_0 } }
 \omega _{\varepsilon } (x)w^{p-a}\,dx.
\end{equation}
 Thus from \eqref{e2_10} and \eqref{e2_11}) we have
\begin{align*}
&\int_{B^{x_0}_{2r}}\omega_\varepsilon(x)|\nabla w|^p\,dx \\
&\le C\Big( r^{-p}\int_{B^{x_0}_{4r}\setminus
B^{x_0}_{3r/2}}\omega_\varepsilon(x)w^{p-a}\,dx\Big)^{(p-1)/p}
\Big(\int_{B^{x_0}_{3r}\setminus B^{x_0}_{2r}}\omega_\varepsilon(x)w^{a(p-1)}r^{-p}\,dx
 \Big)^{1/p}.
\end{align*}
We estimate the integrals on the right side through
the upper bounds $w$. Since $\omega _{\varepsilon }
=\varepsilon \omega $ in $B_{2r}^{x_0 } $, using Harnack's
inequality \eqref{e2_3} and doubling condition
\eqref{e1_4} we obtain
\begin{equation} \label{e2_12}
\int_{B_{2r}^{x_0 } } \omega _{\varepsilon } (x)|\nabla w|^p\,dx
\le C\varepsilon r^{-p} \omega (B_r )(\min_{\partial B_r^{x_0 } } w)^{p-1}\,.
\end{equation}
Comparing \eqref{e2_12}, \eqref{e2_8}, \eqref{e2_7},
\eqref{e2_5} and using again in \eqref{e2_5} the
doubling condition  \eqref{e1_4} we obtain
\[
\min_{\partial B_r^{x_0 } } w\ge C,
\]
From which due to the explicit form of $w$ follows,
\begin{equation} \label{e2_13}
\min_{\partial B_r^{x_0 } } u\ge C\varepsilon ^{-1} \omega ^{-1} (B_r )
\end{equation}
with the constant  $C$ that does not depend on $\varepsilon $.

\subsection*{Upper bound of the solution}
Let $\eta _1 \in C_0^{\infty } (B_{2r}^{x_0 } )$,
 $\eta _1 =1$ in $B_r^{x_0 } $ and $\eta _2 \in C_0^{\infty } (B_{2r}^{y_0 } )$,
$\eta _2 =1$ in $B_r^{y_0 } $.
Take $K_1 =\varepsilon ^{-1} \omega ^{-1} (B_r )$,
$K_2 =\omega ^{-1} (B_r )$.

Since the solution $u$ of \eqref{e2_1}  minimizes
the functional \eqref{e2_2}, following to the variational principle,
the choice of the cutoff functions
$\eta _1 $, $\eta _2 $ and condition \eqref{e1_2} we have
\[
\int_{B_{2r}^{y_0 } } \omega (x)|\nabla u|^p\,dx
\le \varepsilon \int_{B_{2r}^{x_0 } } \omega (x)|\nabla (\eta _1 K_1 )|^p\,dx
+\int_{B_{2r}^{y_0 } } \omega (x)|\nabla (\eta _2 K_2 )|^p\,dx.
\]
Hence, from the arbitrary choice of $\eta _1 $ and $\eta _2 $ it
follows that
\[
\int_{B_{2r}^{y_0 } } \omega (x)|\nabla u|^p\,dx
\le \varepsilon K_1^p {\rm cap}_p (\overline{B_{r/4}^{x_0 } },B_{2r}^{x_0 } )
+K_2^p {\rm cap}_p (\overline{B_{r/4}^{y_0 } },B_{2r}^{y_0 } ).
\]
Thus by \eqref{e2_5}, \eqref{e2_6} and
doubling condition \eqref{e1_4}, we obtain
\begin{equation} \label{e2_14}
\int_{B_{2r}^{y_0 } } \omega (x)|\nabla u|^p\,dx
\le \varepsilon K_1^p \omega (B_r )r^{-p} +K_2^p \omega
(B_r )r^{-p} .
\end{equation}
Then by the Friedrichs inequality \eqref{e1_6},
\[
\int_{B_{16r}^{(2)} } \omega (x)u^p\,dx
\le Cr^p \int_{B_{16r}^{(2)} } \omega (x)|\nabla u|^p\,dx
\]
and from \eqref{e2_14} we obtain
\[
\int_{B_r^{y_0 } \backslash B_{r/2}^{y_0 } } \omega (x)u^p\,dx
\le \varepsilon K_1^p \omega (B_r )+K_2^p \omega (B_r ).
\]
Now from \eqref{e2_4} and doubling condition
\eqref{e1_4} we have
\[
\max_{\partial B_r^{y_0 } } u\le C\omega ^{-1/p} (B_r )
(\varepsilon ^{1/p} K_1 \omega ^{1/p} (B_r )+K_2 \omega ^{1/p} (B_r )).
\]
or considering the explicit forms of $K_1 $ and $K_2 $
\[
\max_{\partial B_r^{y_0 } } u\le C(\varepsilon ^{1/p-1}
\omega ^{-1} (B_r )+\omega ^{-1} (B_r )).
\]
or
\begin{equation} \label{e2_15}
\max_{\partial B_r^{y_0 } } u\le C\varepsilon ^{1/p-1} \omega ^{-1} (B_r )
\end{equation}
with the constant $C$ that does not depend on  $\varepsilon $.

If we suppose that the classical Harnack's inequality holds uniformly with
respect to $\varepsilon$  in the domain $\Omega $, then
\[
\min_{\partial B_r^{x_0 } } u\le C\max_{\partial B_r^{y_0 } } u,
\]
where $C$ does not depend on $\varepsilon $.
This inequality leads to the contradiction with the estimates
\eqref{e2_13} and \eqref{e2_15}.

\subsection{Estimation of the minimum of thenon-negative solution}
Below $B_{R} \subset D$ stands for the ball
with the centers on  $\Sigma \cap D$, $B_{R}^{(i)} =B_r \cap
D^{(i)} $ for the semiballs, $i=1,2$. Note that Sobolev's inequality
\eqref{e1_7} entails a corresponding inequality for the
semiballs
\begin{equation} \label{e2_16}
\begin{gathered}
\Big( \hbox{--}\hskip-9pt\int_{B_{R}^{(i)} } |\varphi |^{pk}\,d\mu _i\Big)^{1/k}
\le CR^p \hbox{--}\hskip-9pt\int_{B_{R}^{(i)} } |\nabla \varphi |^p\,d\mu_i , \\
 \varphi \in C_0^{\infty } (B_{R}),\quad  k=\frac{n}{n-1} ,\; i=1,2.
\end{gathered}
\end{equation}

Let $u(x)$ be  a non-negative solution of \eqref{e2_1},
$\tilde{u}(x)$ be even continuation of $u(x)$
from $D^{(2)} $ to $D^{(1)} $ relative to  hyperplane $\Sigma $ and
$B_{4R} \subset D$. Below it is assumed that
\begin{equation} \label{e2_17}
v(x)=\begin{cases}
\min (u(x),\tilde{u}(x)),&\text{if } x\in D^{(1)}  \\
u(x), &\text{if } x\in D^{(2)} .
\end{cases}
\end{equation}


\begin{lemma} \label{lem2.1}
If  \eqref{e1_9} is fulfilled then for any $q>0$, we have
\begin{equation} \label{e2_18}
\inf_{B_{R} } u(x)\ge C\Big( \hbox{--}\hskip-9pt\int_{B_{2R} } v^{-q}
(x)\,d\mu \Big)^{-1/q}
\end{equation}
with the constant $C$ that does not depend on  $u$ or $R$.
\end{lemma}

\begin{proof}
 Not loosing generality we assume that
$u(x)$ is positive. Otherwise one should consider the function
$u(x)+\delta $ and then pass to limit at $\delta \to 0$ in the
estimate \eqref{e2_18}. First we show that for any $R\le
\rho <r\le 2R$ and $q_0 >0$, it holds
\begin{equation} \label{e2_19}
\inf_{B_{\rho } } u(x)\ge C\big(\frac{r-\rho }{r}\big)^{a}
\Big(\hbox{--}\hskip-9pt\int_{B_r^{(1)} } v^{-q_0 } (x)\,d\mu \Big)^{-1/q_0 } ,
\end{equation}
in which  $a=a(n,q_0 ,p)>0$, and $C$ does not depend on $r,\rho $,
$u(x)$ or $\varepsilon $.
Choose in \eqref{e1_3} the test function $\xi =u^{\beta } (x)\eta ^p (x)$,
where $\eta (x)\in C_0^{\infty } (B_{3R} )$ is radially symmetric and
$\beta <1-p$. After some simple estimations using Yuong's inequality
we come to the inequality
\begin{equation} \label{e2_20}
\hbox{--}\hskip-9pt\int_{B_{3R} } |\nabla u|^p u^{\beta -1} \eta ^p \omega _{\varepsilon }\,dx
\le C(p) \hbox{--}\hskip-9pt\int_{B_{3R} } u^{\beta +p-1} |\nabla \eta |^p \omega _{\varepsilon }\,dx.
\end{equation}
In particular, by \eqref{e2_2} and \eqref{e2_9}, we have
\begin{equation} \label{e2_21}
\begin{aligned}
&\hbox{--}\hskip-9pt\int_{B_{3R}^{(2)} } |\nabla u|^p u^{\beta -1} \eta ^p\,d\mu\\
&=\frac{1}{\omega (B_{3R}^{(2)} )}  \hbox{--}\hskip-9pt\int_{B_{3R}^{(2)} } |\nabla u|^p u^{\beta -1}
\eta ^p\,d\mu  \\
&\le C(p,\gamma )\Big(\hbox{--}\hskip-9pt\int_{B_{4R}^{(1)} } u^{\beta +p-1} |\nabla \eta
|^p\,d\mu + \hbox{--}\hskip-9pt\int_{B_{3R}^{(2)} }
u^{\beta +p-1} |\nabla \eta |^p\,d\mu \Big)
\end{aligned}
\end{equation}
Following the Sobolev's embedding theorem \eqref{e2_16}  we obtain
\begin{equation} \label{e2_22}
\begin{aligned}
&\Big(\hbox{--}\hskip-9pt\int_{B_{3R}^{(2)}}u^{k(\beta +p-1)}\eta ^{kp}\,d\mu \Big) ^{1/k}\\
&\le C( |\beta |+p-1)^pR^p
 \Big(\hbox{--}\hskip-9pt\int_{B_{3R}^{(1)}}u^{\beta +p-1}|\nabla \eta|^p\,d\mu
+\hbox{--}\hskip-9pt\int_{B_{3R}^{(2)}}u^{\beta +p-1}|\nabla \eta|^p\,d\mu\Big),
\end{aligned}
\end{equation}
 where $C=C(n,p,\gamma )$.

It is not possible to obtain a similar estimate in the ball
$B_{4R}^{(1)} $ by this method. Take
\begin{equation} \label{e2_23}
G_{R} =B_{3R}^{(1)} \cap \{x:u(x)<\tilde{u}(x)\}
\end{equation}
and assuming that $G_{R} \ne \emptyset $, put in \eqref{e2_3} the test function
\[
\xi (x)=\begin{cases}
(u^{\beta } (x)-\tilde{u}^{\beta } (x))\eta ^p (x) &\text{in  } G_{R} , \\
 0,&\text{in }B_{3R} \backslash G_{R} ,
 \end{cases}
 \]
where $\eta $ and $\beta $ have the same sense as above. This test function
is valid by  condition \eqref{e1_9}. We have
\begin{align*}
&|\beta |\int_{G_{R} } |\nabla u|^p u^{\beta -1} \eta ^p\,d\mu \\
&\le |\beta |\int_{G_{R} } |\nabla u|^{p-1}
|\nabla \tilde{u}|\tilde{u}^{\beta -1} \eta ^p\,d\mu
+p\int_{G_{R} } |\nabla u|^{p-1} |\nabla \eta |\tilde{u}^{\beta
} \eta ^{p-1}\,d\mu \\
&\quad +p\int_{G_{R} } |\nabla
u|^{p-1} |\nabla \eta |u^{\beta } \eta ^{p-1}\,d\mu
\end{align*}

From this and using definition of $G_{R} $ and young's inequality we
find that
\begin{align*}
&\int_{G_{R} } |\nabla u|^p u^{\beta -1} \eta ^p\,d\mu \\
&\le C(p)\Big(\int_{G_{R} } |\nabla \tilde{u}|^p \tilde{u}^{\beta
-1} \eta ^p\,d\mu +\int_{G_{R} }
\tilde{u}^{\beta +p-1} |\nabla \eta |^p\,d\mu
+\int_{G_{R} } u^{\beta +p-1} |\nabla \eta |^p
\,d\mu \Big)
\end{align*}
or since $\tilde{u}^{\beta -1} \le u^{\beta -1} $ on the set
$G_{R} $ we have
\begin{equation} \label{e2_24}
\begin{aligned}
&\int_{B^{(1)}_{3R}\setminus G_R}|\nabla \widetilde
u|^p\widetilde u^{\beta-1}\eta^p\, d\mu
+\int_{G_{R}}|\nabla u|^pu^{\beta -1}\eta ^p\,d\mu\\
&\le C\Big(\int_{B^{(1)}_{3R}}|\nabla \widetilde
u|^p\widetilde u^{\beta-1}\eta^p\, d\mu
+\int_{B^{(1)}_{3R}}u^{\beta+p-1}|\nabla \eta|^p\,d\mu \\
&\quad +\int_{B^{(2)}_{3R}} u^{\beta+p-1}|\nabla \eta|^p\,d\mu\Big),
\end{aligned}
\end{equation}
where $C=C(p)$. Since the function $\tilde{u}$ is an even continuation
of the function $u$ from $D^{(2)} $ to $D^{(1)} $ and the function $\eta $
is even with respect to  the hyperplane $\Sigma $, by \eqref{e2_9} we have
\[
\int_{B_{3R}^{(1)} } |\nabla \tilde{u}|^p \tilde{u}^{\beta -1} \eta ^p\,d\mu
 \le \gamma \int_{B_{3R}^{(2)} } |\nabla u|^p u^{\beta -1} \eta ^p\,d\mu
\]
and from  \eqref{e2_20}, \eqref{e1_2} we have
\[
 \int_{B^{(1)}_{3R}}|\nabla \widetilde
u|^p\widetilde u^{\beta-1}\eta^p\, d\mu
\le C(p,\gamma)\Big( \int_{B^{(1)}_{3R}}u^{\beta+p-1}|\nabla \eta|^p\,d\mu+
\int_{B^{(2)}_{3R}} u^{\beta+p-1}|\nabla \eta|^p\,d\mu\Big).
\]

Considering the last relation in \eqref{e2_24} and using
the definition of the function $v$ (see \eqref{e2_17}), we obtain
\[
\int_{B_{3R}^{(1)} } |\nabla v|^p v^{\beta -1} \eta ^p\,d\mu 
\le C(p,\gamma )\Big(\int_{B_{3R}^{(1)} } v^{\beta +p-1} |\nabla \eta |^p\,d\mu
 +\int_{B_{3R}^{(2)} } u^{\beta +p-1} |\nabla \eta |^p\,d\mu \Big).
\]
Applying here Sobolev's embedding theorem \eqref{e2_16}, by
 the doubling condition \eqref{e1_4} we arrive at
the estimate
\begin{equation}\label{e2_25}
\begin{aligned}
&\Big(\hbox{--}\hskip-9pt\int_{B_{3R}^{(1)}}v^{k(\beta +p-1)}\eta ^{pk}\,d\mu\Big) ^{1/k} \\
&\le C( |\beta |+p-1)^pR^p\Big(\hbox{--}\hskip-9pt\int_{B_{3R}^{(1)}}v^{\beta
+p-1}|\nabla \eta |^p\,d\mu+\hbox{--}\hskip-9pt\int_{B_{3R}^{(2)}}u^{\beta
+p-1}|\nabla \eta |^p\,d\mu\Big),
\end{aligned}
\end{equation}
in which $C=C(n,p,\gamma )$. Thus according to
\eqref{e2_22}, \eqref{e2_25} and definition of the
function $v$,
\begin{align*}
&\Big(\hbox{--}\hskip-9pt\int_{B_{3R}^{(1)}}v^{k(\beta +p-1)}\eta ^{pk}\,d\mu
+\hbox{--}\hskip-9pt\int_{B_{3R}^{(2)}}u^{k(\beta +p-1)}\eta ^{pk}\,d\mu\Big)^{1/k} \\
&\le C( |\beta |+p-1)^pR^p\Big(\hbox{--}\hskip-9pt\int_{B_{3R}^{(1)}}v^{\beta +p-1}|\nabla \eta
|^p\,d\mu+\hbox{--}\hskip-9pt\int_{B_{3R}^{(2)}}u^{\beta +p-1}|\nabla \eta
|^p\,d\mu\Big),
\end{align*}
where $C=C(n,p,\gamma )$. Now from \eqref{e2_7} and
doubling condition \eqref{e2_4} follows that
\begin{equation}\label{e2_26}
\begin{aligned}
&\Big(\hbox{--}\hskip-9pt\int_{B_{3R}}v^{k(\beta +p-1)}\eta ^{pk}\,d\mu \Big)^{1/k} \\
&\le C(n,p,\gamma)( |\beta |+p-1)
^pR^p\hbox{--}\hskip-9pt\int_{B_{3R}}v^{\beta +p-1}|\nabla \eta|^p\,d\mu.
\end{aligned}
\end{equation}

Until now we have  assumed that $G_{R} \ne \emptyset $.
If $G_{R} =\emptyset $ then $v(x)=\tilde{u}(x)$ in $B_{3R}^{(1)} $ and
\eqref{e2_26} follows immediately from
\eqref{e2_22} and the condition \eqref{e1_9}.
Choosing in \eqref{e2_26} test function as $\eta =1$ in
$B_r $, $|\nabla \eta |\le Cr(R(r-\rho ))^{-1} $, by
the condition \eqref{e2_4} we obtain
\begin{equation}\label{e2_27}
\begin{aligned}
& \Big(\hbox{--}\hskip-9pt\int_{B_{\rho}}v^{k(\beta +p-1)}\eta
^{pk}\,d\mu\Big)^{1/k}\\
&\le C(n,p,\gamma)( |\beta |+p-1) ^p( \frac{r}{r-\rho }) ^p
\Big(\hbox{--}\hskip-9pt\int_{B_r}v^{\beta +p-1}\,d\mu\Big).
\end{aligned}
\end{equation}

Let us iterate this inequality. Let $j=0,1,\dots $.
 Denote $r_{j} =\rho +2^{-j} (r-\rho )$, $\chi _{j} =-q_0 k^{j} $ and take in
\eqref{e2_13} $r=r_{j}$, $\rho =r_{j+1}$, $\beta =\chi _i+1-p$.
As a result for
$$
\Phi _{j}=\Big(\hbox{--}\hskip-9pt\int_{B_{r_{j}}^{(1)}}v^{\chi _{j}}\,d\mu\Big)^{1/\chi _{j}}
$$
we obtain the following recurrence relation
\[
\Phi _{j} \le C^{1/|\chi _{j} |} (2^{j} (1+|\chi _{j} |))^{p/|\chi _{j} |}
(\frac{r}{r-\rho } )^{p/|\chi _{j} |} \Phi _{j+1} ,
\]
that implies estimate \eqref{e2_19} (see \cite{m1}).
Taking in this estimate $\rho =R$ and $r=2R$, we obtain
\begin{equation}\label{e2_28}
\inf_{B_{R}}u(x)\ge C\Big(\hbox{--}\hskip-9pt\int_{B_{2R}}v^{-q_0}(x)\,d\mu \Big)^{-1/q_0}.
\end{equation}
To prove \eqref{e2_18} we take $s=2(1+\delta )\delta ^{-1}$,
 where $\delta $ is a constant from \eqref{e2_5}, and
apply to the integral
\[
 \hbox{--}\hskip-9pt\int_{B_{2R} } v^{-q_0 } (x)\omega (x)\,dx
= \hbox{--}\hskip-9pt\int_{B_{2R} } v^{-q_0 } (x)\omega ^{1/p} (x)\omega ^{-1/p} (x)\omega (x)\,dx
\]
triplet Holder inequality with orders $p_1 =p_2 =s$, $p_3=(1+\delta )^{-1} $.
As a result considering  condition\eqref{e2_5} we find
\begin{align*}
&\Big(\frac{1}{\omega (B_{2R} )} \int_{B_{2R} } v^{-q_0 } (x)
 \omega (x)\,dx\Big)^{1/q_0 } \\
&\le \Big(\frac{1}{\omega (B_{2R} )} \Big)^{1/q_0 }
\Big( \int_{B_{2R} } \omega ^{-1} (x)\,dx\Big)^{1/pq_0 }
\Big(\int_{B_{2R} } \omega ^{1+\delta } (x)\,dx\Big)^{1/q_0 (1+\delta )}\\
&\quad\times \Big(\int_{B_{2R} } v^{-pq_0 } (x)\omega (x)\,dx\Big)^{1/pq_0 }\\
&\le  C\Big(\hbox{--}\hskip-9pt\int_{B_{2R} } v^{-pq_0 } (x)\,d\mu \Big)^{1/pq_0 } ,
\end{align*}
That \eqref{e2_28} leads to the estimate
\[
\inf_{B_{R} } u(x)\ge C\Big(\hbox{--}\hskip-9pt\int_{B_{2R} } v^{-pq_0 } (x)\,d\mu \Big)^{1/pq_0 } .
\]
Taking  $q_0 =q/p$ we arrive to\eqref{e2_18}.
The proof is complete.
\end{proof}

The statement of the Lemma \ref{lem2.1} becomes true for the nonnegative
super solutions $u(x)$ of the equation \eqref{e1_1} i.e. for such
nonnegative solutions $u$ that
\[
\int_{D} \omega _{\varepsilon } (x)|\nabla u|^{p-2} \nabla u\cdot \nabla \xi\,dx
\ge 0,\quad  \forall \xi \in C_0^{\infty } (D),\;  \xi \ge 0.
\]

\subsection{Harnack's inequality}
Below $u(x)$ stands for the nonnegative solution of  \eqref{e1_1}
and $B_{3R} \subset D$ for the ball with the center on $\Sigma $. To
prove the Harnack's inequality we  need John-Nirenberg's lemma for
the function $v(x)$defined in \eqref{e2_17}.

\begin{lemma} \label{lem2.2}
For an arbitrary ball $B_{2r} \subset B_{3R} $ we have
\begin{equation} \label{e2_29}
\int_{B_r } |\nabla \ln v|^p\,d\mu
\le Cr^{-p} \omega (B_r ),
\end{equation}
in which the constant $C$ does not depend on $u$, $r$, $R$ or
$\varepsilon $.
\end{lemma}

\begin{proof}
As above without loss of generality we
assume the solution is positive and
$B_r^{(i)} =B_r \cap D^{(i)}$, $i=1,2$. Take the cutoff function
$\eta \in C_0^{\infty }(B_{2r} )$ such that $\eta \equiv 1$ in
$B_r,|\nabla \eta |\le Cr^{-1} $. Assuming in \eqref{e1_3} and
 $\xi =u^{1-p} \eta ^p $ as in \eqref{e2_20} we obtain
\[
\int_{B_{2r} } |\nabla \ln u|^p \eta ^p \omega _{\varepsilon }\,dx
\le C(p)r^{-p} \int_{B_{2r} } \omega _{\varepsilon }\,dx.
\]
If $B_{2r} \cap \Sigma =\emptyset $, then from \eqref{e1_2}
and \eqref{e1_4} we arrive to \eqref{e2_29}.
Now let $B_r^{x_0 } $ be arbitrary open ball of radius $r$ with
the center $x^{0} =(x_1^{0} ,x_2^{0} ,\ldots ,x_{n}^{0} )$, such
that $B_{2r}^{x_0 } \subset B_{3R} $ and
$B_{2r}^{x_0 } \cap \Sigma \ne \emptyset $. To prove the statement
it is sufficient to set
\begin{equation} \label{e2_30}
\int_{B_r^{x_0 } } |\nabla \ln v|^p\,d\mu \le Cr^{-p} \omega (B_r^{x_0 } )
\end{equation}
with the constant $C$, not depending on $u$, $r$, $R$ and $\varepsilon $.
Denote by $y_0 $ the point that is symmetric to $x_0 $ with respect to the hyperplane
 $\Sigma $ and take $d=|x_0 -y_0 |$. It is clear that $0<d<4r$.
Consider the cylinder
\[
\mathcal{C}_r =\Big\{ x: \Big(\sum _{i=1}^{n-1} (x_i -x_i^{0} )^{2} \Big)^{1/2}
 <2r,\; |x_{n} |\le d\Big\}
\]
And introduce the symmetric with respect to the hyperplane $\Sigma $ set
\[
Q_r =B_{2r}^{x_0 } \cup B_{2r}^{y_0 } \cup \mathcal{C}_r .
\]
Let $Q_r^{(i)} =Q_r \cap D^{(i)} $, $i=1,2$. It is not difficult
to see that $B_{2r}^{x_0 } \subset Q_r \subset B_{3R} $ and
$B_r^{x_0 } \subset Q_{r/2} $.

Consider the symmetric with respect to the hyperplane $\Sigma $ cut
off function $\eta \in C_0^{\infty } (Q_r )$, by the way that
$\eta =1$ in $Q_{r/2} $ and $|\nabla \eta |\le Cr^{-1} $. Choosing
in the integral identity (1.3) the test function as $\xi =u^{1-p}
\eta ^p $ we obtain
\[
\int_{Q_r } |\nabla \ln u|^p \eta ^p \omega _{\varepsilon }\,dx
\le C(p)r^{-p} \int_{Q_r } \omega _{\varepsilon }\,dx.
\]
Now from \eqref{e1_2} and \eqref{e1_4} it follows that
\begin{equation} \label{e2_31}
\int_{Q_r^{(2)} } |\nabla \ln u|^p \eta ^p
\,d\mu \le C(p)r^{-p} \omega (B_r^{x_0 } ).
\end{equation}

To prove a similar estimate in $Q_r^{(1)} $ first we assume that
the set $G_{R} $ from \eqref{e2_23} is not empty and choose
in \eqref{e1_3} the test function as
\[
\xi (x)=\begin{cases}
(u^{1-p} (x)-\tilde{u}^{1-p} (x))\eta ^p (x)&\text{in  }  G_{R} , \\
0 &\text{in } B_{3R} \backslash G_{R} ,
 \end{cases}
 \]
where $\eta $ has the same sense as above. Then it is easy to see that
(see \eqref{e1_2})
\begin{align*}
&(p-1)\int_{G_{R} } |\nabla \ln u|^p \eta ^p\,d\mu \\
&\le (p-1)\int_{G_{R} } |\nabla u|^{p-1} |\nabla \ln \tilde{u}|\tilde{u}^{1-p}
\eta ^p\,d\mu
+p\int_{G_{R} } |\nabla u|^{p-1} |\nabla \eta |\tilde{u}^{1-p} \eta ^{p-1}\,d\mu \\
&\quad +p\int_{G_{R} } |\nabla u|^{p-1} |\nabla \eta |u^{1-p} \eta ^{p-1}\,d\mu .
\end{align*}
From this considering $u(x)\le \tilde{u}(x)$ on $G_{R} $, by the
help of Young's inequality we find
\[
\int_{G_{R} } |\nabla \ln u|^p \eta ^p\,d\mu
\le C(p)( \int_{G_{R} } |\nabla \ln \tilde{u}|^p \eta ^p\,d\mu
+\int_{G_{R} } |\nabla \eta |^p\,d\mu )
\]
or adding to both sides of this inequality the integral
\[
\int_{Q_r^{(1)} \backslash G_{R} } |\nabla \ln \tilde{u}|^p \eta ^p\,d\mu ,
\]
because of the choice of the cutoff function $\eta $ and doubling
condition \eqref{e1_4}, we have
\begin{equation} \label{e2_32}
\begin{aligned}
&\int_{Q_r^{(1)} \backslash G_{R} } |\nabla \ln \tilde{u}|^p \eta ^p\,d\mu
+\int_{G_{R} } |\nabla \ln u|^p \eta ^p\,d\mu \\
&\le C(p)\Big( \int_{Q_r^{(1)} } |\nabla \ln \tilde{u}|^p \eta ^p\,d\mu
+r^{-p} \omega (B_r^{x_0 } )\Big).
\end{aligned}
\end{equation}

Since the function $\tilde{u}$ is an even continuation of the
function $u$ from $D^{(2)} $ to $D^{(1)} $ and the cutoff function
is even relative to  $\Sigma $, then according to condition
\eqref{e1_9},
\[
\int_{Q_r^{(1)} } |\nabla \ln \tilde{u}|^p \eta ^p\,d\mu
\le \gamma \int_{Q_r^{(2)} } |\nabla \ln u|^p \eta ^p\,d\mu
\]
and from \eqref{e2_31} it follows that
\[
\int_{Q_r^{(1)} } |\nabla \ln \tilde{u}|^p \eta ^p\,d\mu
\le C(p,\gamma )r^{-p} \omega (B_r^{x_0 } ).
\]
Considering the last relation in the right-hand side of
\eqref{e2_32} we have
\begin{equation} \label{e2_33}
\int_{Q_r^{(1)} \backslash G_{R} } |\nabla \ln
\tilde{u}|^p \eta ^p\,d\mu +\int_{G_{R}
} |\nabla \ln u|^p \eta ^p\,d\mu \le C(p,\gamma
)r^{-p} \omega (B_r^{x_0 } ).
\end{equation}
Now from \eqref{e2_31}, \eqref{e2_33} and definition of the function $v$
(see. \eqref{e2_17}) we arrive to the estimate
\[
\int_{Q_r } |\nabla \ln v|^p \eta ^p\,d\mu
\le C(p,\gamma )r^{-p} \omega (B_r^{x_0 } ),
\]
that implies the relation \eqref{e2_30} since $\eta =1$ in
$B_r^{x_0 } $ and $B_r^{x_0 } \subset Q_r $.

If the set $G_{R} $ is empty then  $v(x)=\tilde{u}(x)$ in
$B_{3R}^{(1)} $ and \eqref{e2_30} follows from
\eqref{e2_31} and condition \eqref{e1_9}.
The proof is complete.
\end{proof}


The statement of the Lemma \ref{lem2.2} is true for the nonnegative supersolutions
of the equation  \eqref{e1_1}. The consequence of this lemma is
John-Nirenberg's lemma the proof of which may be found in \cite{h1}.

\begin{corollary} \label{consequence2.1}
 There exist positive constants $q$ and $C$ not depending on  $u$, $R$ or
$\varepsilon $, such that
\begin{equation} \label{e2_34}
\Big( \hbox{--}\hskip-9pt\int_{B_{2R} } v^{-q} (x)\,d\mu _1 \Big)^{-1/q}
\ge C(n,p)\Big(\hbox{--}\hskip-9pt\int_{B_{2R} } v^{q} (x)\,d\mu _1
\Big)^{1/q} .
\end{equation}
\end{corollary}


\begin{proof}[Proof of Theorem \ref{thm1}]
 Let $u(x)$ be nonnegative
solution of the equation \eqref{e1_1} and $B_{R}^{-} $ be a
set defined in \eqref{e1_10}. Using \eqref{e2_18},
\eqref{e2_34} and doubling condition \eqref{e1_4}
we obtain
\[
\inf_{B_{R} } u(x)\ge C\Big( \hbox{--}\hskip-9pt\int_{B_{2R} } v^{q} (x)\,d\mu \Big)^{1/q}
\ge C\inf_{B_{R}^{-} } u(x).
\]
Now \eqref{e1_11} follows from the classical Harnack's
inequality for the solutions of \eqref{e1_1} in the domain
$D^{(2)} $, according which $\inf_{B_{R}^{-} }u(x)\ge c(n,p)\sup_{B_{R}^{-} } u(x)$.
The proof is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm2}]
 From the results in \cite{f1,h1} is known that solution has the Holder property inside
$D^{(1)} $ and $D^{(2)} $. It remains to prove the Holder property
of the solutions on $\Sigma \cap D$, since the holder property
inside of $D$ may be obtained by elementary gluing of the Holder
property on $\Sigma \cap D$ and $D^{(1)} $, $D^{(2)} $.
Let $B_{4R}\subset D$ be a ball with the center on $\Sigma $ and
\[
M_{4R} =\sup_{B_{4R} } u(x),\quad
 m_{4R} =\inf_{B_{4R} } u(x),\quad
 M_{R}^{-} =\sup_{B_{R}^{-} } u(x),\quad m_{R}^{-} =\inf_{B_{R}^{-} } u(x).
\]
Since the functions $M_{4R} -u(x)$ and  $u(x)-m_{4R} $ are
nonnegative solutions in $B_{4R} $, by the Harnack's inequality
\eqref{e1_11},
\[
M_{4R} -M_{R} \ge c_0 (M_{4R} -m_{R}^{-} ),\quad
m_{R} -m_{4R} \ge c_0 (M_{R}^{-} -m_{4R} ).
\]
Summing these relations and using the fact that $c_0 <1$, we obtain
the scattering lemma
\[
M_{R} -m_{R} \ge (1-c_0 )(M_{4R} -m_{4R} ),
\]
that shows the Holder continuity of the solutions on $\Sigma \cap D$.
The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
The first author was supported by the Ministry of Education and Science
of the Russian Federation (task No 1.3270.2017/PP) and the Russian
Foundation for Basic Research (project No 15-01-00471-a).

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\end{document}