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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 143, pp. 1--7.\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-2018/143\hfil Harnack inequality for a $(p,q)$-Laplacian]
{Harnack inequality for  $(p,q)$-Laplacian equations
 uniformly degenerated in a part of domain}

\author[S. T. Huseynov \hfil EJDE-2018/143\hfilneg]
{Sarvan T. Huseynov}

\address{Sarvan T. Huseynov \newline
Baku State University, Baku AZ1148, Azerbaijan}
\email{sarvanhuseynov@rambler.ru}


\dedicatory{Communicated by Ludmila S. Pulkina}

\thanks{Submitted January 5, 2018. Published July 17, 2018.}
\subjclass[2010]{35J62, 35J65, 35J70, 35J92}
\keywords{$p(x)$-Laplacian; elliptic equation; Harnack inequality}

\begin{abstract}
 We consider a $(p,q)$-Laplace equation with the exponent values $p,q$
 depending on the boundary which is divided into two parts by a hyperplane.
 Assuming that the equation is uniformly degenerate with respect to
 a small parameter in the part of domain where $q<p$,
 a special Harnack inequality is proved for non-negative solutions.
\end{abstract}

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


\section{Statement of main result}

We consider the elliptic equation
\begin{equation} \label{e1.1}
L_\varepsilon u= \operatorname{div}(\omega_\varepsilon(x)
|\nabla u|^{p(x)-2}\nabla u)=0
\end{equation}
in a domain $D\subset \mathbb{R}^n$, $n\ge 2$,
with a positive weight $\omega_\varepsilon(x)$, and an exponent
to be defined below. Assume that the domain is divided by the
 hyperplane $\Sigma=\{x:x_n=0\}$  into two parts
$D^{(1)}=D\cap\{x:x_n>0\}$, $D^{(2)}=D\cap\{x:x_n<0\}$,  and that
\begin{gather} \label{e1.2}
  \omega_{\varepsilon}(x)= \begin{cases}
  \varepsilon,& \text{if }x\in D^{(1)} \\
  1,&\text{if } x\in D^{(2)},
  \end{cases}\quad \varepsilon\in (0,1], \\
 \label{e1.3}
p(x)= \begin{cases}
  q, &\text{if }x\in D^{(1)} \\
  p, &\text{if }x\in D^{(2)},
  \end{cases}\quad 1<q<p.
\end{gather}
To define the solution of  \eqref{e1.1}, we introduce a class
of functions related to the exponent $p(x)$:
$$
W_{\rm loc}(D)=\{u:u\in W^{1,1}_{\rm loc}(D), |\nabla u|^{p(x)}\in
L^1_{\rm loc}(D)\}\,.
$$
This set   is a Sobolev space of functions locally
summable in $D$ together with their first order generalized derivatives.

 By a solution of to \eqref{e1.1}, we mean a function
$u\in W_{\rm loc}(D)$, which satisfies the integral identity
\begin{equation} \label{e1.4}
 \int_D\omega_\varepsilon(x)|\nabla u|^{p(x)-2}\nabla
u\cdot\nabla\varphi\,dx=0
\end{equation}
for the test functions  $\varphi\in C^\infty_0(D)$.

For the exponent $p(\cdot)$,  given by \eqref{e1.3}, the smooth functions are dense
in $W_{\rm loc}(D)$ (see \cite{z1}). Therefore, finite functions from $W_{\rm loc}(D)$
can be considered as test functions in \eqref{e1.4}.

$p$-Laplace type equations with a variable nonlinearity exponent, $p(x)$,
 and variational problems with integrants satisfying non-standard coerciveness
and growth conditions occur in the modeling of composite materials and
electrorheological fluids whose characteristics depend on the considered
electromagnetic field. In this work, we consider a model for the case of
plane boundary between two different phases. Note that the problem is
complicated by the degeneration, uniform in $\varepsilon$, in the domain $D^{(1)}$.

In each of the domains $D^{(i)}$,  $i=1,2$,  the regularity of the solution has
been well studied before (see \cite{l1}). It was proved in \cite{a1} that for $p$
defined by \eqref{e1.3} and for every fixed $\varepsilon\in (0,1]$,
every solution of the equation \eqref{e1.1} in the arbitrary subdomain
$D' \Subset D$  belongs to the space $C^\alpha(D')$  of H\"older
functions in $D'$. The independence of the Hölder exponent $\alpha$  on
$\varepsilon$ in case $p=q$  has been established in \cite{a2,h3},
and for our equation it was proved in \cite{h2}.
         	
Harnack inequality  plays an important role in the qualitative theory
of differential equations (see \cite{s1}):
if $p(x)\equiv p$,  then the following inequality holds for the solution $u$
 of the equation \eqref{e1.1} which is non-negative in the ball $B_{4R}\subset D$:
\begin{equation} \label{e1.5}
\inf_{B_R}u\ge \gamma(n,p) \sup_{B_R} u.
 \end{equation}
In \cite{a5}, it was shown that the classical inequality \eqref{e1.5} does not
hold for the solutions of the equation \eqref{e1.1} if  $\varepsilon=1$ and
$q<p$. This inequality is not satisfied in the balls $B_R$  centered on the
hyperplane  $\Sigma$. To state the result obtained in \cite{a5},
denote by $B_R^{-}$
the set  $\{x\in B_R: x_n<-R/2\}$. It was established in \cite{a5} that if $u$
is a non-negative solution of the equation \eqref{e1.1} in the ball
$B_{8R}\subset D$  centered on the hyperplane $\Sigma$, then the following
inequality holds in the concentric ball $B_R$   of radius $R$:
\begin{equation} \label{e1.6}
\inf_{B_R}u+R \ge C(n,p,q) \sup_{B_R^{-}}u.
\end{equation}
Along with the invalidity of classical Harnack inequality  \eqref{e1.5};
it was proved in \cite{a5} that for large values of  $R$ the term  $R$ in \eqref{e1.6}
cannot be replaced by $R^\nu$ when $\nu<(p-q)/(p-1)$.
Note that in case $p=q=2$ the Harnack inequality of the form  \eqref{e1.6}
with no  $R$ has been first obtained in \cite{a3}, and  in \cite{h1}
  in case  $q=p\ne 2$.

In this work, we establish the Harnack inequality of the form \eqref{e1.6}
with a constant $C$  independent of $\varepsilon$. Our main result
is the following theorem.

\begin{theorem} \label{thm1.1}
If \eqref{e1.2} and \eqref{e1.3} hold, and  $u$ is a non-negative
solution of \eqref{e1.1} in the ball  $B_{8R}\subset D$ centered on
the hyperplane  $\Sigma$, then the inequality \eqref{e1.6} holds in
the concentric ball $B_R$  of radius $R$  with the constant  $C$
depending only on $n$, $p$  $q$.
\end{theorem}

The proof is based on the modified technique of Mozer \cite{m1},
developed in \cite{a4,a6}, where the domains $D^{(1)}$ and $D^{(2)}$ play
different roles.

The assertion of Theorem \ref{thm1.1} Also holds for the equation
$$
\operatorname{div}\Big( \omega_{\varepsilon}(x) |\nabla u|^{p(x)-2}a\nabla u \Big)=0,
$$
 where $\alpha$  is a measurable uniformly positive definite matrix.
Besides, the constant in \eqref{e1.6}, it will additionally depend on ellipticity
constants of this matrix.

\section{Proof of main result}

Below $B_R$  will denote an open ball centered on $\Sigma\cap D$,  so that
$B_{4R}\subset D$,  $B^{(i)}_R=D^{(i)}\cap B_R$, $i=1,2$, $u$ is a non-negative
solution of the equation \eqref{e1.1} and $w=u+R$.
Here $|E|$  is  $n$-dimensional Lebesgue measure of the measurable set
$E\subset \mathbb{R}^n$ , and
$$
-\hspace{-0.38cm}\int_{E} f\,dx=\frac{1}{|E|}\int _{E} f\,dx.
$$

Let us first establish auxiliary estimates for the solutions.
Taking  $\varphi=w^\beta\eta^p$, as a test function in the integral
identity \eqref{e1.4} with   $\beta<1-p$, $\eta\in
C^\infty_0(B_{3R})$ and $0\le\eta\le 1$ by \eqref{e1.2} we have
\begin{equation} \label{e2.1}
\int_{B^{(2)}_{3R}}|\nabla
w|^{p}w^{\beta-1}\eta^p\,dx\le  C\Bigl(\int_{B^{(2)}_{3R}}w^{\beta+p-1}
|\nabla\eta|^p\,dx+
\int_{B^{(1)}_{3R}}w^{\beta+q-1}|\nabla\eta|^q\,dx\Bigr).
\end{equation}
Below  $\tilde f$  will denote a continuation of a function from
$D^{(2)}$ to $D^{(1)}$ even with respect to the hyperplane  $\Sigma$. Let
\begin{equation} \label{e2.2}
 G_R=B^{(1)}_{3R}\cap\{x:w(x)<\tilde w(x)\}
\end{equation}
and, assuming $G_R\ne\emptyset$,
$$
  \varphi (x)=\begin{cases}
  (w^{\gamma}(x)-\tilde w^{\gamma}(x))\eta^q(x) &\text{in } G_R \\
  0 &\text{in } B_{3R}\setminus G_R,
  \end{cases}
$$
as a test function in \eqref{e1.4}, with the constant $\gamma<1-q$
to be defined later. Then we obtain (see \eqref{e1.2})
\begin{equation} \label{e2.3}
\begin{aligned}
&|\gamma|\int_{G_R}|\nabla w|^qu^{\gamma-1}\eta^q\,dx \\
&\le |\gamma|\int_{G_R}|\nabla w|^{q-1}|\nabla\tilde w|\tilde w^{\gamma-1}\eta^q\,dx
 +q\int_{G_R}|\nabla w|^{q-1}|\nabla\eta|\tilde w^\gamma\eta^{q-1}\,dx \\
&\quad +q\int_{G_R}|\nabla w|^{q-1}|\nabla\eta|w^\gamma\eta^{q-1}\,dx.
\end{aligned}
\end{equation}
Let us estimate the integrands on the right-hand side of \eqref{e2.3}
by using Young's inequality, definition of $G_R$  and relation  $\gamma<0$.
We have
\begin{gather} \label{e2.4}
\begin{aligned}
|\nabla w|^{q-1}|\nabla\tilde w|\tilde u^{\gamma-1}\eta^q
&\le \varepsilon_1|\nabla w|^{q}\tilde w^{\gamma-1}\eta^q+C(\varepsilon_1,q)
 |\nabla\tilde w|^q\tilde w^{\gamma-1}\eta^q\\
&\le \varepsilon_1|\nabla
w|^{q}u^{\gamma-1}\eta^q+C(\varepsilon_1,q)|\nabla\tilde w|^q\tilde
w^{\gamma-1}\eta^q,
\end{aligned} \\
\label{e2.5}
\begin{aligned}
|\nabla w|^{q-1}|\nabla\eta|\tilde w^\gamma\eta^{q-1}
&\le |\nabla w|^{q-1}|\nabla\eta|w^\gamma\eta^{q-1}\\
&\le\varepsilon_2|\nabla
w|^{q}w^{\gamma-1}\eta^q+C(\varepsilon_2,q)w^{\gamma+q-1}|\nabla\eta|^q,
\end{aligned} \\
\label{e2.6}
|\nabla w|^{q-1}|\nabla\eta|w^\gamma\eta^{q-1}
\le\varepsilon_3|\nabla
w|^{q}w^{\gamma-1}\eta^q+C(\varepsilon_3,q)w^{\gamma+q-1}|\nabla\eta|^q.
\end{gather}
Considering the relations \eqref{e2.4}--\eqref{e2.6} in \eqref{e2.3},
 by a proper choice of $\varepsilon_1$, $\varepsilon_2$ and $\varepsilon_3$,
we have
\begin{equation} \label{e2.7}
 \int_{G_R}|\nabla w|^qw^{\gamma-1}\eta^q\,dx
 \le C(q)\Bigl(\int_{G_R}|\nabla \tilde w|^q\tilde
w^{\gamma-1}\eta^q\,dx+\int_{G_R}w^{\gamma+q-1}|\nabla\eta|^q\,dx\Bigr).
\end{equation}

Introduce the constant $\gamma$ as
\begin{equation} \label{e2.8}
\gamma=\beta+p-q.
\end{equation}
Then
$$
|\nabla \tilde w|^q\tilde w^{\gamma-1}\eta^q=|\nabla \tilde
w|^q\tilde w^{(\beta-1)q/p}\tilde w^{(\beta-1)(p-q)/p+p-q}\eta^q,
$$
and, by Young's inequality,
\begin{equation} \label{e2.9}
|\nabla \tilde w|^q\tilde w^{\gamma-1}\eta^q\le R^{p-q}|\nabla
\tilde w|^p\tilde w^{\beta-1}\eta^p+R^{-q}\tilde w^{\beta+p-1}.
\end{equation}
Now we can rewrite the inequality \eqref{e2.7} as
\begin{equation} \label{e2.10}
\begin{aligned}
\int_{G_R}|\nabla w|^qu^{\gamma-1}\eta^q\,dx
&\le C(q)\Bigl (R^{p-q}\int_{G_R}|\nabla \tilde
w|^p\tilde w^{\beta-1}\eta^p\,dx \\
&\quad +R^{-q}\int_{G_R}\tilde w^{\beta+p-1}\,dx
 +\int_{G_R}w^{\beta+p-1}|\nabla\eta|^q\,dx\Bigr ).
\end{aligned}
\end{equation}
Let
$$
 v(x)= \begin{cases}
  w(x), &\text{if } x\in D^{(2)} \\
\min{(w(x),\tilde w(x))}, &\text{if } x\in D^{(1)}\,.
  \end{cases}
$$
Note that \eqref{e2.10} implies
\begin{equation} \label{e2.11}
\begin{aligned}
&\int_{B^{(1)}_{3R}}|\nabla v|^qv^{\gamma-1}\eta^q\,dx\\
&\le C(q)\Bigl(R^{p-q}\int_{B^{(1)}_{3R}}|\nabla \tilde w|^p\tilde
w^{\beta-1}\eta^p\,dx+R^{-q}\int_{B^{(1)}_{3R}}\tilde w^{\beta+p-1}\,dx \\
&\quad +\int_{B^{(1)}_{3R}}v^{\beta+p-1}|\nabla\eta|^q\,dx\Bigr).
\end{aligned}
\end{equation}
To prove the theorem, it suffices to add the integral
$$
\int_{B^{(1)}_{3R}\setminus G_R}|\nabla \tilde w|^q\tilde
w^{\gamma-1}\eta^q\,dx
$$
to both sides of the inequality \eqref{e2.10} and then use
\eqref{e2.9} on the right-hand side.

Using the definition of the function $v$,  we rewrite  \eqref{e2.1} as follows:
\begin{equation} \label{e2.12}
\int_{B^{(2)}_{3R}}|\nabla
w|^{p}w^{\beta-1}\eta^p\,dx\le C\Bigl(\int_{B^{(2)}_{3R}}
w^{\beta+p-1}|\nabla\eta|^p\,dx+
\int_{B^{(1)}_{3R}}v^{\beta+q-1}|\nabla\eta|^q\,dx\Bigr).
\end{equation}
Hence, from \eqref{e2.11} and the properties of even continuation of a
function we obtain
\begin{equation} \label{e2.13}
\begin{aligned}
\int_{B^{(1)}_{3R}}|\nabla v|^qv^{\gamma-1}\eta^q\,dx
&\le C(q)\Bigl (\int_{B^{(2)}_{3R}}
w^{\beta+p-1}(R^{-q}+R^{p-q}|\nabla\eta|^p)\,dx \\
&\quad +\int_{B^{(1)}_{3R}}(v^{\beta+p-1}+v^{\beta+q-1}R^{p-q})
 |\nabla\eta|^q\,dx\Bigr).
\end{aligned}
\end{equation}

Let us estimate from below the integrand on the left-hand side of \eqref{e2.13}
using the inequality \eqref{e2.9} with  $\tilde w$ replaced by  $w$.
 Also, note that  $v^q\le R^{q-p}v^p$ as  $w\ge R$. Taking into account
this relation, we can rewrite \eqref{e2.12} and \eqref{e2.13} as
\begin{equation} \label{e2.14}
\begin{aligned}
&\int_{B^{(2)}_{3R}}|\nabla w|^{q}w^{\gamma-1}\eta^q\,dx \\
&\le C\Bigl(R^{p-q}\int_{B^{(2)}_{3R}}w^{\beta+p-1}|\nabla\eta|^p\,dx
 +\int_{B^{(1)}_{3R}}v^{\beta+p-1}|\nabla\eta|^q\,dx\Bigr)
\end{aligned}
\end{equation}
and
\begin{equation} \label{e2.15}
\begin{aligned}
&\int_{B^{(1)}_{3R}}|\nabla v|^qv^{\gamma-1}\eta^q\,dx \\
&\le C(q)\Bigl(\int_{B^{(2)}_{3R}}
w^{\beta+p-1}(R^{-q}+R^{p-q}|\nabla\eta|^p)\,dx+
\int_{B^{(1)}_{3R}}v^{\beta+p-1}|\nabla\eta|^q\,dx\Bigr)
\end{aligned}
\end{equation}
respectively. Summing both sides of the inequalities \eqref{e2.14} and
\eqref{e2.15}, and using again the definition of the function $v$, we obtain
\begin{align*}
&\int_{B_{3R}}|\nabla v|^{q}v^{\gamma-1}\eta^q\,dx \\
&\le  C(q)\Bigl(\int_{B_{3R}}v^{\beta+p-1}(R^{-q}+R^{p-q}|\nabla\eta|^p)\,dx+
\int_{B_{3R}}v^{\beta+p-1}|\nabla\eta|^q\,dx\Bigr)
\end{align*}
Hence, from the choice of  $\gamma$ (see \eqref{e2.8} and by
the Sobolev embedding theorem, we conclude that
\begin{equation} \label{e2.16}
\begin{aligned}
&\Bigl(-\hspace{-0.38cm}\int_{B_{3R}}v^{k(\beta+p-1)}\eta^k\,dx\Bigr)^{1/k}\\
&\le C(n,p,q)|\beta|^q\Bigl(-\hspace{-0.38cm}\int_{B_{3R}}v^{\beta+p-1}(1+R^{p}
|\nabla\eta|^p+R^q|\nabla\eta|^q)\,dx\Bigl),
\end{aligned}
\end{equation}
where  $k=n/(n-1)$. Iterating the relation  \eqref{e2.16} ) by Mozer method,
we arrive at the following conclusion.

\begin{lemma} \label{lem2.1}
 For every  $q_0>0$, we have
\begin{equation} \label{e2.17}
\inf_{B_R}v(x)\ge C(n,p,q,q_0) \Bigl (-\hspace{-0.38cm}\int_{B_{2R}}
v^{-q_0}(x)\,dx \Bigr)^{-1/q_0}.
\end{equation}
\end{lemma}

As $w\ge v$,  \eqref{e2.17} implies
\begin{equation} \label{e2.18}
\inf_{B_R}w(x)\ge C(n,p,q,q_0) \Bigl (-\hspace{-0.38cm}\int_{B_{2R}}
v^{-q_0}(x)\,dx \Bigr)^{-1/q_0}.
\end{equation}

\begin{lemma} \label{lem2.2}
For every ball  $B_{2r}\subset B_{3R}$ centered in
$B_{3R}$, it holds
\begin{equation} \label{e2.19}
\int _{B_r}|\nabla\ln{v}|^q\,dx\le Cr^{n-q},
\end{equation}
where the constant $C$ does not depend on $u$, $R$  and $r$.
\end{lemma}

\begin{proof}
 As before, it is assumed below that $B^{(i)}_r=D^{(i)}\cap B_r$ $i=1,2$.
Consider a cutting function  $\eta\in C^\infty_0(B_{3R})$, such that
$\eta\equiv 1$ in $B_r$, $|\nabla\eta|\le Cr^{-1}$.
Assuming  $\varphi=w^{1-p}\eta^p$ in the integral identity  \eqref{e1.4},
 by simple calculation with the consideration of \eqref{e1.2} we obtain
$$
\int _{B^{(2)}_{2r}}|\nabla\ln{w}|^p\eta^p\,dx\le C\Bigl
(\int _{B^{(2)}_{2r}}|\nabla\eta|^p\,dx
+\int _{B^{(1)}_{2r}}w^{q-p}|\nabla\eta|^q\,dx\Bigr).
$$

Or, from  $w^{q-p}\le R^{q-p}$,
$$
\int _{B^{(2)}_{2r}}|\nabla\ln{w}|^p\eta^p\,dx\le C\bigl
(r^{n-p}+R^{q-p}r^{n-q}\bigr )\le Cr^{n-p}.
$$
Thus,
\begin{equation} \label{e2.20}
 \int _{B^{(2)}_{2r}}|\nabla\ln{w}|^q\eta^q\,dx\le
Cr^{n-q}
\end{equation}
which proves  \eqref{e2.19} in the case $B_r\subset D^{(2)}$.

Now let  $B_r\cap D^{(1)}\neq\emptyset$. To prove the similar estimate
in $B^{(1)}_r$ we first assume that the set  $G_R$ defined by  \eqref{e2.2}
is not empty and consider
$$
  \varphi (x)=\begin{cases}
  (w^{1-q}(x)-\tilde w^{1-q}(x))\eta^q(x) &\text{in } G_R \\
  0 & \text{in } B_{3R}\setminus G_R.
  \end{cases}
$$
as a test function in \eqref{e1.4}. Then it is not difficult to see that
\begin{align*}
&\int _{G_R}|\nabla\ln{w}|^q\eta^q\,dx \\
&\le  \int_{G_R}|\nabla w|^{q-1}|\nabla\ln{\tilde w}|\tilde w^{1-q} \eta^q\,dx \\
&\quad  +q\int _{G_R}|\nabla w|^{q-1}|\nabla \eta|\tilde w^{1-q}  \eta^{q-1}\,dx
 +q\int _{G_R}|\nabla w|^{q-1}|\nabla\eta|w^{1-q}\eta^{q-1} \,dx.
\end{align*}
As  $w(x)\le\tilde w(x)$ in $G_R$,  then, by Young's inequality, we obtain
\begin{equation} \label{e2.21}
 \int _{G_R}|\nabla \ln{w}|^q\eta^q\,dx
\le C \Bigl(\int _{G_R}|\nabla \ln{\tilde w}|^q\eta^q\,dx+ \int _{G_R}|\nabla
\eta|^q\,dx\Bigr).
\end{equation}

First consider the case where the center of the ball $B_r$  is located
in $\overline D^{(2)}$.  Then, by \eqref{e2.20},
\begin{equation} \label{e2.22}
 \int _{B^{(1)}_{2r}}|\nabla\ln{\tilde w}|^q\eta^q\,dx
\le Cr^{n-q}.
\end{equation}
Summing \eqref{e2.21} and \eqref{e2.22}, we have
\begin{equation} \label{e2.23}
 \int _{B^{(1)}_{2r}}|\nabla\ln{v}|^q\eta^q\,dx
\le C \Bigl (\int _{B^{(1)}_{2r}}|\nabla \ln{\tilde w}|^q \eta^q\,dx +
r^{n-q}\Bigr )
\le C r^{n-q}.
\end{equation}
Hence, by \eqref{e2.20}, we obtain the inequality
\begin{equation} \label{e2.24}
 \int _{B_{2r}}|\nabla\ln{v}|^q\eta^q\,dx\le C r^{n-q},
\end{equation}
which implies \eqref{e2.19}.

If the set  $G_R$ is empty, then  $v(x)=\tilde w(x)$ in $B^{(1)}_{3R}$,
and \eqref{e2.19}follows from \eqref{e2.20}. Now consider the case where
$G_R\ne\emptyset$  and the center of the ball $B_r$  is located in $D^{(1)}$.
 Denote by $\hat B_r$  the image of the ball  $B_r$ under mirror reflection
with respect to the hyperplane $\Sigma$.  By inequality \eqref{e2.20},
for the ball  $\hat B_{2r}$ we have
\begin{equation} \label{e2.25}
 \int _{\hat B^{(2)}_{2r}}|\nabla\ln{
w}|^q\eta^q\,dx=\int _{B^{(1)}_{2r}}|\nabla\ln{\tilde
w}|^q\eta^q\,dx\le Cr^{n-q}.
\end{equation}

As above, summing \eqref{e2.21} and \eqref{e2.25}, we obtain again \eqref{e2.23},
which, combined with \eqref{e2.20}, leads to the inequality \eqref{e2.24},
which in turn implies \eqref{e2.19}. If the set  $G_R$ is empty,
then \eqref{e2.19} follows from \eqref{e2.20} and \eqref{e2.25}.
The proof is complete.
\end{proof}

John-Nirenberg lemma is a corollary of \eqref{e2.19}:
there exist the positive constants   $q_0$ and $C$,  independent of
 $u$ and  $R$ such that
\begin{equation} \label{e2.26}
 \Bigl(-\hspace{-0.38cm}\int _{B_{2R}} v^{-q_0}(x)\,dx
\Bigr)^{-1/q_0}\ge C \Bigl(-\hspace{-0.38cm}\int _{B_{2R}} v^{q_0}(x)\,dx
\Bigr)^{1/q_0}.
\end{equation}


\begin{proof}[Proof of Theorem \ref{thm1.1}]
 As before, let  $w=u+R$.  Using \eqref{e2.18} and  \eqref{e2.26}, we obtain
$$
\inf_{B_R}u(x)\ge C \Bigl(-\hspace{-0.38cm}\int _{B_{2R}} v^{q}(x)\,dx
\Bigr)^{1/q}\ge C\inf_{B^-_{R}}w(x).
$$
Then \eqref{e1.6} follows from the classical Harnack inequality for the
solutions of the equation \eqref{e1.1} in the domain $D^{(2)}$,  which states
 $\inf_{B^-_{R}}w(x)\ge c\sup_{B^-_{R}}w(x)$.
Theorem 1 is proved.
\end{proof}

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