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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 63, pp. 1--18.\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-2017/63\hfil 
 H\"older continuity of bounded generalized solutions]
{H\"older continuity of bounded generalized solutions for nonlinear
fourth-order elliptic equations with strengthened coercivity and 
natural growth terms}

\author[M. V. Voitovych \hfil EJDE-2017/63\hfilneg]
{Mykhailo V. Voitovych}

\address{Mykhailo V. Voitovych \newline
Institute of Applied Mathematics and Mechanics,
National Academy of Sciences of Ukraine,
Gen. Batiouk Str. 19, 84116 Sloviansk, Ukraine. \newline
Mariupol State University,
Budivelnykiv Ave. 129a,
87500 Mariupol, Ukraine. \newline
Vasyl Stus Donetsk National University,
600-richya Str. 21,
21021 Vinnytsia, Ukraine.}
\email{voitovichmv76@gmail.com}

\dedicatory{Communicated by Pavel Drabek}

\thanks{Submitted November 11, 2016. Published March 2, 2017.}
\subjclass[2010]{35B45, 35B65, 35J40, 35J62}
\keywords{Nonlinear elliptic equations; strengthened coercivity;
\hfill\break\indent
lower-order term; natural growth; bounded solution, H\"older continuity}

\begin{abstract}
 In this article we extend the author's previous results on the existence
 of bounded generalized solutions of a Dirichlet problem for nonlinear elliptic
 fourth-order equations with the principal part satisfying a strengthened
 coercivity condition, and a lower-order term having a ``natural'' growth
 with respect to the derivatives of the unknown function. Namely, we prove
 the H\"older continuity of bounded generalized solutions of such equations.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded domain of $\mathbb{R}^{n}$, $n\geq 3$.
Then we consider the nonlinear fourth-order elliptic equations of divergent form
\begin{equation}\label{A}
\sum_{|\alpha|=1,2}(-1)^{|\alpha|} D^\alpha A_\alpha
(x,u,\nabla_2u)+B(x,u,\nabla_2u)= 0 \quad \text{in } \Omega,
\end{equation}
where $\alpha=(\alpha_1, \dots, \alpha_n)$ is an $n$-dimensional multiindex with
nonnegative integer components $\alpha_i$, $i=1,\dots, n$,
$|\alpha|=\alpha_1+ \dots + \alpha_n $,
$D^{\alpha}=\partial^{|\alpha|}/\partial x_1^{\alpha_1}\ldots
\partial x_{n}^{\alpha_{n}}$ and $\nabla_2u=\{D^\alpha u: |\alpha|=1,2\}$.

The main structural requirements for the coefficients $A_\alpha$ and $B$ are
the following strengthened coercivity condition:
for a.\,e.\ $x\in \Omega$ and for every $s\in \mathbb{R}$ and
$\xi=\{\xi_{\alpha}\in \mathbb{R}: |\alpha|=1,2\}$,
\begin{equation}\label{strcoercond}
\sum_{|\alpha|=1,2}A_\alpha(x,s,\xi)\xi_{\alpha}
\geq C \Big\{\sum_{|\alpha|=1} |\xi_\alpha|^q +
\sum_{|\alpha|=2} |\xi_\alpha|^{p} \Big\}-f_1(x),
\end{equation}
and the natural growth condition: for a.\,e.\ $x\in \Omega$ and for every
$s\in \mathbb{R}$ and $\xi=\{\xi_{\alpha}\in \mathbb{R}: |\alpha|=1,2\}$
\begin{equation}\label{natgr}
|B(x,s,\xi)|\leq b(|s|)\Big\{\sum_{|\alpha|=1} |\xi_\alpha|^q +
\sum_{|\alpha|=2} |\xi_\alpha|^{p} \Big\}+f_2(x),
\end{equation}
where $C>0$, $p\in(1,n/2)$, $q\in(2p,n)$, $b:\mathbb{R}_{+}\to \mathbb{R}_{+}$
is a continuous nondecreasing function, $f_{1,2}\geq 0$ and
$f_{1,2}\in L^{\tau}(\Omega)$, $\tau>n/q$.

We recall that the strengthened coercivity condition goes back to
Skrypnik \cite{Skr78}, used because of the regularity problem of generalized solutions
from the class $W^{m,p}(\Omega)$ for nonlinear elliptic equations of the divergent
form
\begin{equation}\label{equationA}
\sum_{|\alpha|\leq m}(-1)^{|\alpha|} D^\alpha \mathcal{A}_\alpha
(x,u,\dots,D^{m}u)= 0 \quad \text{in } \Omega.
\end{equation}

In this problem the relation $n=mp$ is important. If $n<mp$, the H\"older
continuity of solutions
is a simple consequence of the embedding theorem
(see, e.g., \cite[Section 7.7]{GlbTr}).
For $n\geq mp$, the embedding theorems do not ensure the boundedness of an
arbitrary solution $u\in W^{m,p}(\Omega)$.

In the case $m=1$ and $n\geq p$, the properties of the boundedness and the
continuity of generalized solutions
are well known (see, e.g.,
\cite{AlbCianSb,AlbFer,FerFus, IwOn,JiangKoskYang},
\cite[Chapter IV,\,\S7, and Chapter IX,\,\S2]{LadUr}).

For $m\geq 2$ and $n=mp$, the boundedness of solutions is established in
 \cite{Frehse,VoitMN,WidmanB}, and the continuity is proved in
\cite[Chapter II, \S\,3]{Skr73}, \cite{Todor,WidmanH}.

Eventually, in the case where $m\geq 2$ and $n>mp$,
there are examples of equations of the form \eqref{equationA} with the smooth
coefficients $\mathcal{A}_\alpha$ and unbounded solutions
(see, e.g., \cite{De Giorgi,Maz'ya}). In this case, Skrypnik \cite{Skr78}
separated a subclass of equations of the form
\eqref{equationA} whose all generalized solutions are bounded and H\"older continuous.
 The separated subclass of equations is characterized by the strengthened
coercivity condition under which the natural energy space is the space
$W^{m,p}(\Omega)\cap W^{1,q}(\Omega)$ with $q>mp$.
In particular, for $m=2$, the structure of this class of equations is
determined by the inequality of the form \eqref{strcoercond}.

Article \cite{Skr78} initiated a research on local properties of
solutions for nonlinear high-order elliptic equations with strengthened
coercivity and sufficiently regular data.
For example, there were established sufficient conditions for regularity of a
boundary point \cite{Skr91} and for the removability of isolated singularities
of solutions \cite{DAsero10}, obtained pointwise estimates for solutions of
some model problems \cite{DAsLar, Skr97}, proved the Harnack inequality
\cite{DAsero06,NicSkr02}.

At the same time, in the cited papers on the equations with the strengthened
coercivity it was assumed that the coefficients $\mathcal{A}_{\alpha}$
satisfy the standard growth conditions for the space
$W^{1,q}_{m,p}(\Omega)=W^{m,p}(\Omega)\cap W^{1,q}(\Omega)$.
In this situation, the solvability of equation \eqref{equationA} is equivalent
to the solvability of the operator equation $\mathcal{A}u=0$ where
the nonlinear operator
$\mathcal{A}: W^{1,q}_{m,p}(\Omega)\to [W^{1,q}_{m,p}(\Omega)]^{\ast}$
 is defined by the equality
$$
\langle\mathcal{A}u,v\rangle=\sum_{|\alpha|\leq
m}\int_{\Omega}\mathcal{A}_{\alpha}(x,u,...,D^{m}u)D^{\alpha}vdx.
$$
So, the standard theory of equations with pseudomonotone operators
(see, e.g., \cite{Lions}) is applicable.
The case where the lower-order term has the natural $(q,p)$-growth like
\eqref{natgr} is beyond the scope of this theory and requires separate consideration.
In this regard, we refer to \cite{Voit11}, \cite{Voit13}, \cite{Voit15} where
the existence and $L^{\infty}$-estimates of solutions of the Dirichlet problem
for nonlinear high-order elliptic equations with the strengthened coercivity
and natural growth terms were established.

Existence and $L^{\infty}$-estimates of bounded solutions of
nonlinear second-order elliptic equations with natural growth
lower-order terms were established for instance in
\cite{BocMurPl92,DrNic}, and the H\"older continuity of the solutions
was proved in \cite[Chapter IX,\,\S2]{LadUr}).

In the present article, we strengthen and supplement our previous results
in \cite{Voit11}, \cite{Voit13}.
Namely, we prove the H\"older continuity in $\Omega$ of every generalized solution
$u\in W^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ of equation \eqref{A} under
the conditions \eqref{strcoercond}, \eqref{natgr}.
For the proof, we use an analogue of Moser's method
(see \cite[Chapter IX]{LadUr}, \cite{Moser}) proposed in \cite{Skr78}.
This method is based on obtaining of uniform $L^{k}$-estimates
(at $k\to +\infty$) for an auxiliary function $\phi(u)$ such that its boundedness
implies the H\"older continuity of the solution $u$. The new point in
the proof (compared to \cite{Skr78}) is an additional requirement on the power
exponent $k$ in the test function in \cite{Skr78}. Namely, the use of
the condition $k\geq k_0$ with a suitable $k_0=k_0(\verb"data")>0$ leads
to the absorption of the lower-order term of natural growth by the coercive
principal part of the equation. In this regard, to complete the proof we
need some further integral estimates of the solution associated with the
use of the Lemma of John-Nirenberg \cite{JohnNir}.

We remark that a theory of existence and properties of solutions
of nonlinear fourth-order elliptic equations with coefficients
satisfying the strengthened coercivity condition and
$L^1$-right-hand sides was developed in
\cite{Kov01,Kov09}, \cite[Part\,I, Section\,2]{KovSkrSh}
(see also \cite{CirmDasLeon} for equations with natural growth terms).

This article is organized as follows.
In Section \ref{main result}, we give the statement of the problem
and present the main result (Theorem \ref{th2.2}).
In the same section, we provide examples of equations that satisfy
all the hypotheses.
In Section \ref{auxresults}, we present some auxiliary results needed
to the proof of Theorem \ref{th2.2} which is set out in Section \ref{proof2}.


\section{Preliminaries and the statement of the main result}\label{main result}

Let $n\in\mathbb{N}$, $n>2$, and let $\Omega$ be a bounded domain
of $\mathbb{R}^n$.

We shall use the following notation:
$\mathbb{R}_{+}=[0,+\infty)$; $\partial S$ is the boundary of the set
 $S\subset \mathbb{R}^n$, $\overline{S}=S\cup\partial S$ is the closure of $S$;
$\Lambda$ is the set of all
$n$-dimensional multi-indices $\alpha$ such that $|\alpha|=1$ or
$|\alpha|=2$; $\mathbb{R}^{n,2}$ is the space of all mappings $\xi:
\Lambda\to\mathbb{R}$; if $ u\in W^{2,1}(\Omega)$, then $\nabla_2
u: \Omega\to\mathbb{R}^{n,2}$, and for every $ x\in\Omega$ and for
every $ \alpha\in\Lambda$, $(\nabla_2u (x))_\alpha = D^\alpha
u(x)$. If $\tau\in[1,+\infty]$, then $\|\cdot\|_{\tau}$ is the norm in
$L^{\tau}(\Omega)$.
For every measurable set
$E\subset\mathbb{R}^n$ we denote by $|E|$ $n$-dimensional
Lebesgue measure of the set $E$.

Let $p\in(1, n/2)$ and $q\in(2p, n)$. We denote by
$W^{1,q}_{2,p}(\Omega)$ the set of all functions in
$W^{1,q}(\Omega)$ that have the second-order generalized
derivatives in $L^p(\Omega)$. The set $W^{1,q}_{2,p}(\Omega)$ is a
Banach space with the norm
$$
\|u\| = \|u\|_{W^{1,q}(\Omega)} + \Big(\sum_{|\alpha|=2}
\int_\Omega |D^\alpha u|^p dx\Big)^{1/p}.
$$
We denote by ${\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)$ the
closure of the set $ C^\infty_0(\Omega)$ in $W^{1,q}_{2,p}(\Omega)$.

We consider the equation
\begin{equation}\label{10}
\sum_{\alpha\in\Lambda}(-1)^{|\alpha|}D^\alpha A_\alpha
(x,u,\nabla_2 u)+B(x,u,\nabla_2 u) = 0 \quad \text{in } \Omega
\end{equation}
under the following assumptions:
\begin{itemize}
\item[(A1)] For every $\alpha\in\Lambda$,
$A_\alpha : \Omega\times\mathbb{R}\times \mathbb{R}^{n,2}\to\mathbb{R}$ and
$B:\Omega\times \mathbb{R}\times \mathbb{R}^{n,2} \to \mathbb{R}$
are Carath\'{e}odory functions, i.e. for every
$(s,\xi)\in\mathbb{R}\times \mathbb{R}^{n,2}$, the functions
$A_\alpha(\cdot,s,\xi)$ and $B(\cdot,s,\xi)$ are measurable on
$\Omega$ and, for almost every $x\in\Omega$, the functions
$A_{\alpha}(x,\cdot,\cdot)$ and $B(x,\cdot,\cdot)$ are continuous in
 $\mathbb{R}\times \mathbb{R}^{n,2}$.

\item[(A2)] For almost every $x\in\Omega$ and for every
$(s,\xi)\in\mathbb{R}\times \mathbb{R}^{n,2}$ the following inequalities hold:
\begin{gather}\label{5}
 \sum_{\alpha\in\Lambda}A_\alpha (x,s,\xi)\xi_\alpha \geq a(|s|)
\Big\{\sum_{|\alpha|=1}|\xi_\alpha|^q +
\sum_{|\alpha|=2}|\xi_\alpha|^{p} \Big\} - g_0(x), \\
\label{3}
 \sum_{|\alpha|=1}|A_\alpha (x,s,\xi)|^{q/(q-1)} \leq a_1(|s|)
\Big\{\sum_{|\alpha|=1}|\xi_\alpha|^q +
\sum_{|\alpha|=2}|\xi_\alpha|^{p} \Big\} + g_1(x), \\
\label{4}
\sum_{|\alpha|=2}|A_\alpha (x,s,\xi)|^{p/(p-1)} \leq a_2(|s|)
\Big\{\sum_{|\alpha|=1}|\xi_\alpha|^q +
\sum_{|\alpha|=2}|\xi_\alpha|^{p} \Big\} + g_2(x), \\
\label{7}
 |B(x,s,\xi)|\leq b(|s|)\Big\{\sum_{|\alpha|=1}|\xi_\alpha|^q +
\sum_{|\alpha|=2}|\xi_\alpha|^{p} \Big\}+ g_3(x),
\end{gather}
where $a:\mathbb{R}_{+}\to (0,+\infty)$ is a continuous nonincreasing function,
$a_1,a_2,b:\mathbb{R}_{+}\to \mathbb{R}_{+}$ are continuous nondecreasing
functions, $g_0, g_1, g_2, g_3$ are nonnegative summable functions on $\Omega$.
\end{itemize}
Assumptions (A1) and (A2) provide the correct setting for the following definition.

\begin{definition} \label{def2.1} \rm
A generalized solution of  \eqref{10} is a
function $u\in W^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ such that
for every function $v\in{\mathaccent"7017
W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$,
\begin{equation}\label{12}
\int_\Omega \Big\{\sum_{\alpha\in\Lambda} A_\alpha(x,u,\nabla_2 u)
D^\alpha v+B(x,u,\nabla_2 u) v\Big\} dx = 0.
\end{equation}
\end{definition}

\begin{remark}\label{rem1}\rm
Note that if in addition to assumptions (A1) and (A2) the functions
$a$, $a_1$, $a_2$ are positive constants, the inequalities
\begin{equation}\label{sgnB}
\begin{gathered}
\sum_{\alpha\in\Lambda}\,[A_{\alpha}(x,s,\xi)-A_{\alpha}(x,s,\xi')]
(\xi_{\alpha}-\xi_{\alpha}')>0, \\
B(x,s,\xi)s\geq -g_4(x), \quad g_4(x)\geq 0
\end{gathered}
\end{equation}
hold for almost every $x\in \Omega$ and any $s\in \mathbb{R}$ and
 $\xi$, $\xi'\in \mathbb{R}^{n,2}$, $\xi\neq\xi'$, and the functions
$g_0$, $g_1$, $g_2$, $g_3$, $g_4$ belong to $L^{\tau}(\Omega)$ with $\tau>n/q$,
then there exists a generalized solution $u\in{\mathaccent"7017
W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ of equation \eqref{10}.
This follows from \cite[Theorem 2.1]{Voit11}. The result remains true
if instead of \eqref{sgnB}, we assume that the function $b$ in \eqref{7}
is bounded and the left-hand side of equation \eqref{10} has an absorption
 term like $c|u|^{q-2}u$, $c>0$ (see \cite[Theorem 2.2]{Voit13}).
\end{remark}

The main result of the present article is a theorem on the local H\"older continuity
of any generalized solution $u\in W^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$
of equation \eqref{10} under assumptions (A1) and (A2).
Following \cite[Chapter 4]{GlbTr}, we recall that a function
$f:\mathcal{D}\to \mathbb{R}$, $\mathcal{D}\subset \mathbb{R}^{n}$
is {\it uniformly H\"older continuous with exponent $\epsilon\in(0,1)$ in
$\mathcal{D}$} if the quantity
$$
[f]_{\epsilon;\mathcal{D}}=
\sup_{x,y\in \mathcal{D}, x\neq y}\frac{|f(x)-f(y)|}{|x-y|^{\epsilon}}
$$
is finite; and {\it locally H\"older continuous with exponent
$\epsilon\in(0,1)$ in the domain $\mathcal{D}$} if $f$ is uniformly
 H\"older continuous with exponent $\epsilon\in(0,1)$ on compact subsets
of $\mathcal{D}$. We denote by $C^{0,\epsilon}(\mathcal{D})$
the set of all functions that are locally H\"older continuous with
exponent $\epsilon\in(0,1)$ in $\mathcal{D}$.

Now let us state the main result of this paper.

\begin{theorem} \label{th2.2}
Assume that conditions {\rm (A1)} and {\rm (A2)} are satisfied with the functions
$g_0$, $g_1$, $g_2$, $g_3$ belonging to $L^{\tau}(\Omega)$, $\tau>n/q$.
Let $u\in W^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ be a generalized
solution of equation \eqref{10} and $\mathrm{M}=\|u\|_{\infty}$.
Then $u\in C^{0,\epsilon}(\Omega)$ and for any domain $\Omega'$ such that
$\overline{\Omega'}\subset\Omega$, we have
$$
[u]_{\epsilon,\overline{\Omega'}}\leq C
$$
where $\epsilon=\epsilon(\verb"data")$ and $C=C(d,\verb"data")$ are positive
constants, $d=\operatorname{dist}(\Omega',\partial\Omega)$ and
$\verb"data"\equiv \big(n, p, q, \tau, |\Omega|,
\mathrm{M}, a(\mathrm{M}), a_1(\mathrm{M}), a_2(\mathrm{M}),
 b(\mathrm{M}), \max_{0\leq i\leq 3}\|g_i\|_{\tau}\big)$.
\end{theorem}

Before proving Theorem \ref{th2.2}, we give several auxiliary results and 
quote some examples where all the hypotheses are verified.

\begin{example}\label{exmpl1}\rm
Let $b$, $\lambda_1$, $\lambda_2: \mathbb{R}_{+}\to \mathbb{R}_{+}$
be continuous nondecreasing functions and let $\mu : \mathbb{R}_{+}\to (0,+\infty)$
be a continuous nonincreasing function. For every $n$-dimensional multiindex $\alpha$,
$|\alpha|=1, 2$, we define the functions
$A_{\alpha}:\Omega\times\mathbb{R}\times\mathbb{R}^{n,2}\to\mathbb{R}$ and
$B:\Omega\times\mathbb{R}\times \mathbb{R}^{n,2} \to \mathbb{R}$ as follows:
\begin{gather*}
A_\alpha(x,s,\xi)
=\begin{cases}
\big(\mu(|s|)+\lambda_1(|s|)\big)\Big(\sum_{|\beta|=1}\xi^2_{\beta}
\Big)^{(q-2)/2}\xi_{\alpha}, \\
\quad\text{if }(x,s,\xi)\in \Omega\times\mathbb{R}\times\mathbb{R}^{n,2},
\; |\alpha|=1, \\[4pt]
\big(\mu(|s|)+\lambda_2(|s|)\big)
\Big(\sum_{|\beta|=2}\xi^2_{\beta}\Big)^{(p-2)/2}\xi_{\alpha}, \\
\quad\text{if } (x,s,\xi)\in \Omega\times\mathbb{R}\times\mathbb{R}^{n,2}, \; |\alpha|=2,
\end{cases}\\
B(x,s,\xi)=b(|s|)\big\{\sum_{|\alpha|=1} |\xi_\alpha|^q +
\sum_{|\alpha|=2} |\xi_\alpha|^{p}\big\}, \quad (x, s, \xi)\in
\Omega\times \mathbb{R}\times \mathbb{R}^{n,2}.
\end{gather*}
These functions satisfy assumptions (A1) and (A2).
Suppose that there exists a generalized solution
$u_0\in W^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ of the equation
\begin{align*}
&\sum_{|\alpha|=2}D^{\alpha}\Big[\big(\mu(|u|)+\lambda_2(|u|)\big)
\Big(\sum_{|\beta|=2}|D^{\beta}u|^2\Big)^{(p-2)/2}D^{\alpha}u\Big]\\
&-\sum_{|\alpha|=1}D^{\alpha}\Big[\big(\mu(|u|)+\lambda_1(|u|)\big)
\Big(\sum_{|\beta|=1}|D^{\beta}u|^2\Big)^{(q-2)/2}D^{\alpha}u\Big]\\
&+b(|u|)\Big(\sum_{|\alpha|=2}|D^{\alpha}u|^{p}+\sum_{|\alpha|=1}
 |D^{\alpha}u|^{q}\Big)=0 \quad \text{in } \Omega.
\end{align*}
Then applying Theorem \ref{th2.2} to the solution $u_0$, we obtain that
$u_0\in C^{0,\epsilon}(\Omega)$ with some $\epsilon$ depending only on
$n$, $p$, $q$, $|\Omega|$, $\mathrm{M}=\|u_0\|_{\infty}$, $\mu(\mathrm{M})$,
$\lambda_1(\mathrm{M})$, $\lambda_2(\mathrm{M})$ and $b(\mathrm{M})$.
\end{example}

\begin{example}\label{exmpl2} \rm
We consider the equation
\begin{equation}\label{exmpl2eq}
\begin{aligned}
&\sum_{|\alpha|=2}D^{\alpha}\Big[
\Big(\sum_{|\beta|=2}|D^{\beta}u|^2\Big)^{(p-2)/2}D^{\alpha}u\Big]
-\sum_{|\alpha|=1}D^{\alpha}\Big[
\Big(\sum_{|\beta|=1}|D^{\beta}u|^2\Big)^{(q-2)/2}D^{\alpha}u\Big]\\
&+u b_1(|u|)\Big(\sum_{|\alpha|=2}|D^{\alpha}u|^{p}
 +\sum_{|\alpha|=1}|D^{\alpha}u|^{q}\Big)=f(x) \quad \text{in } \Omega
\end{aligned}
\end{equation}
where $f\in L^{\tau}(\Omega)$, $\tau> n/q$ and
$b_1:\mathbb{R}_{+} \to \mathbb{R}_{+}$ is a continuous nondecreasing function,
for example, $b_1(s)=s^{\kappa}$, $\kappa>0$, or $b_1(s)=e^{s}$.

The coefficients and the right-hand side of this equation satisfy the
conditions (A1), (A2) and all the assumptions in \cite[Theorem 2.1]{Voit11}
(see Remark \ref{rem1}). Therefore, there exists a generalized solution
$u_1\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ of
equation \eqref{exmpl2eq} such that $\|u_1\|_{\infty}\leq C$, where $C$
is a positive constant depending only on $n$, $p$, $q$, $\tau$, $\|f\|_{\tau}$
and $|\Omega|$. By Theorem \ref{th2.2}, we have that $u_1\in C^{0,\epsilon}(\Omega)$
with some $\epsilon$ depending only on $n$, $p$, $q$, $\tau$, $\|f\|_{\tau}$ and
$|\Omega|$.
\end{example}

\begin{example}\label{exmpl3} \rm
We consider the equation
\begin{align*}
&\sum_{|\alpha|=2}D^{\alpha}\Big[
\Big(\sum_{|\beta|=2}|D^{\beta}u|^2\Big)^{(p-2)/2}D^{\alpha}u\Big]
-\sum_{|\alpha|=1}D^{\alpha}\Big[
\Big(\sum_{|\beta|=1}|D^{\beta}u|^2\Big)^{(q-2)/2}D^{\alpha}u\Big]\\
&+c|u|^{q-2}u+b_2(x)\Big(\sum_{|\alpha|=2}|D^{\alpha}u|^{p}
+\sum_{|\alpha|=1}|D^{\alpha}u|^{q}\Big)=f(x) \quad \text{in} \ \Omega
\end{align*}
where $c>0$, $b_2\in L^{\infty}(\Omega)$, $f\in L^{\tau}(\Omega)$ with
$\tau> n/q$.

By \cite[Theorem 2.2]{Voit13} (see also Remark \ref{rem1}) there exists a
generalized solution $u_2\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)
\cap L^{\infty}(\Omega)$ of this equation such that $\|u_2\|_{\infty}\leq C$,
 where $C$ is a positive constant depending only on $n$, $p$, $q$, $c$,
$\tau$, $\|f\|_{\tau}$, $\|b_2\|_{\infty}$ and $|\Omega|$.
 By Theorem \ref{th2.2}, we have that $u_2\in C^{0,\epsilon}(\Omega)$
with some $\epsilon$ depending only on $n$, $p$, $q$, $c$, $\tau$,
 $\|f\|_{\tau}$, $\|b_2\|_{\infty}$ and $|\Omega|$.
\end{example}

\section{Auxiliary results}\label{auxresults}

The following  is the well-known Sobolev inequality for functions in
${\mathaccent"7017 W}^{1,q}(\Omega)$; see for example
 \cite[Theorem 7.10]{GlbTr}.

\begin{lemma} \label{Sobolev}
Set $q^\ast = nq/(n-q)$. Then ${\mathaccent"7017 W}^{1,q}(\Omega)\subset
L^{q\ast}(\Omega)$. Furthermore, there exists a positive constant $c_{n,q}$
depending only on $n$ and
$q$ such that, for every function $u\in {\mathaccent"7017 W}^{1,q}(\Omega)$,
\begin{equation}\label{2}
\Big(\int_{\Omega} |u|^{q^{\ast}} dx\Big)^{1/q^{\ast}} \leq c_{n,q}\Big(
\sum_{|\alpha|=1} \int_{\Omega} |D^\alpha u|^q dx\Big)^{1/q}.
\end{equation}
\end{lemma}

We denote by $B_{\rho}(y):=\{x\in \mathbb{R}^{n}: |x-y|<\rho\}$
the open ball with center $y$ and radius $\rho>0$; when not important,
or clear from the context, we shall omit denoting the center as follows:
$B_{\rho}\equiv B_{\rho}(y)$.

\begin{lemma} \label{lem4Scr73}
Let $f\in W^{1,q}(B_{\rho})$. Suppose there exists a measurable subset
$G\subset B_{\rho}$ and positive constants $C'$ and $C''$ such that
$$
|G|\geq C'\rho^{n}, \quad \max_{G}|f|\leq C''.
$$
Then
$$
\int_{B_{\rho}}|f|^{q}dx\leq C\rho^{\,q}
\Big(\sum_{|\alpha|=1}\,\int_{B_{\rho}} |D^{\alpha}f|^{q}dx+\rho^{n-q}\Big)
$$
where $C$ is a positive constant depending only on $n$, $q$, $C'$, $C''$.
\end{lemma}

The proof of the above lemma is given in \cite[Chapter 1, \S 2, Lemma 4]{Skr73}.

The following lemma is due to John and Nirenberg \cite{JohnNir}
(see also \cite[Theorem 7.21]{GlbTr}).

\begin{lemma} \label{lemJohnNir}
Let $f\in W^{1,1}(\mathcal{O})$ where $\mathcal{O}$ is a convex domain in
$\mathbb{R}^{n}$. Suppose there exists a positive constant $K$
such that
$$
\sum_{|\alpha|=1}\,\int_{\mathcal{O}\cap B_{\rho}}|D^{\alpha}f|dx
\leq K \rho^{n-1} \quad \text{for all balls $B_{\rho}$}.
$$
Then there exist positive constants $\sigma_0$ and $C$ depending only on
$n$ such that
$$
\int_{\mathcal{O}}\exp\Big(\frac{\sigma}{K}|f-(f)_{\mathcal{O}}|\Big)dx
\leq C(\operatorname{diam}\mathcal{O})^{n}
$$
where $\sigma=\sigma_0|\mathcal{O}|(\operatorname{diam}\mathcal{O})^{-n}$,
$(f)_{\mathcal{O}}=\frac{1}{|\mathcal{O}|}\int_{\mathcal{O}}fdx$.
\end{lemma}

The following result is discussed in \cite[Lemma 8.23]{GlbTr}.

\begin{lemma} \label{lemoscilation}
Let $\omega$ be a non-decreasing function on an interval $(0,R_0]$
satisfying, for all $R\leq R_0$, the inequality
$$
\omega(\vartheta R)\leq \theta\omega(R)+\varphi(R)
$$
where $\varphi$ is also non-decreasing function and $0<\vartheta,\theta<1$.
Then, for any $\delta\in(0,1)$ and $R\leq R_0$, we have
$$
\omega(R)\leq C\Big(\Big(\frac{R}{R_0}\Big)^{\epsilon}\omega(R_0)
+\varphi(R^{\delta}R_0^{1-\delta})\Big)
$$
where $C=C(\vartheta,\theta)$ and $\epsilon=\epsilon(\vartheta,\theta,\delta)$
are positive constants.
\end{lemma}

\section{Proof of Theorem \ref{th2.2}}\label{proof2}

Suppose that conditions \eqref{5}--\eqref{7} hold with the functions
$$
g_0, g_1, g_2, g_3 \in L^{\tau}(\Omega), \quad \tau>n/q.
$$
Let $u\in W^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$ be a generalized solution
of equation \eqref{10}.
We set $\mathrm{M}=\|u\|_{\infty}$, thus
\begin{equation}\label{defMrm}
|u|\leq\mathrm{M}<+\infty \quad \text{on $\Omega$}.
\end{equation}
By $c_i$, $i=0,1,\dots$, we shall denote positive constants
depending only on
$$
\verb"data"\equiv \big(n,\, p,\, q,\, \tau,\, |\Omega|,\,
\mathrm{M},\, a(\mathrm{M}), a_1(\mathrm{M}), a_2(\mathrm{M}),
 b(\mathrm{M}), \max_{0\leq i\leq 3}\|g_i\|_{\tau}\big).
$$

Furthermore, let $\Omega'$ be an arbitrary open subset of $\Omega$ such
that $\overline{\Omega'}\subset\Omega$ and
$d=\operatorname{dist}(\Omega',\partial\Omega)$.
We fix $x_0\in\Omega'$. For every $R\in \big(0,\min\{1,d/4\}\big)$, we set
$$
\mu(R)=\min_{B_{R}(x_0)}u, \quad M(R)=\max_{B_{R}(x_0)}u, \quad
\omega(R)=M(R)-\mu(R).
$$
Here the symbols $\min$ and $\max$ of course stands for
\textit{essential infimum and supremum}.

We fix a positive number $r$ such that
\begin{equation}\label{defr}
r<\min\big\{1-\frac{n}{q\tau},\frac{q-2p}{q-p}\big\}.
\end{equation}
For every $R\in \big(0,\min\{1,d/4\}\big)$, we shall establish the inequality
\begin{equation}\label{relomega}
\omega(R)\leq \theta\omega(2R)+R^{r}
\end{equation}
with a constant $\theta\in(0,1)$ depending only on $\verb"data"$.
This inequality and Lemma \ref{lemoscilation} imply the validity
of Theorem \ref{th2.2}.

To prove \eqref{relomega}, we fix $R$ such that $0<R<\min\{1,d/4\}$ and set
\begin{gather*}
G_1(R)=\Big\{x\in B_{3R/2}(x_0):u(x)\leq \frac{\mu(2R)+M(2R)}{2}\Big\}, \\
G_2(R)=B_{3R/2}(x_0)\setminus G_1(R),
\end{gather*}
and define a function $v_0: B_{2R}(x_0)\to \mathbb{R}$ as follows:
\begin{equation}\label{defV}
v_0(x)=\begin{cases}
 1+\ln\frac{2\omega(2R)}{M(2R)-u(x)+R^{r}}
& \text{if } |G_1(R)|\geq\frac{|B_{3R/2}(x_0)|}{2}, \\[4pt]
 1+\ln\frac{2\omega(2R)}{u(x)-\mu(2R)+R^{r}}
& \text{if } |G_2(R)|\geq\frac{|B_{3R/2}(x_0)|}{2}.
 \end{cases}
\end{equation}
It is easy to see that \eqref{relomega}, and hence Theorem \ref{th2.2},
follows from the estimate
\begin{equation}\label{supestv0}
\|v_0\|_{L^{\infty}(B_{R}(x_0))}\leq c_0,
\end{equation}
For definiteness we assume that the function $v_0$ is defined by the first
line in \eqref{defV}. We can also assume that
\begin{equation}\label{omegageqR}
\omega(2R)\geq \frac{eR^{r}}{2},
\end{equation}
and therefore, $v_0\geq 1$ a.\,e. in $B_{2R}(x_0)$,
otherwise inequality \eqref{relomega} holds.

To derive inequality \eqref{supestv0}, we need some integral estimates of
the solution $u$.
We set
$$
\Phi = \sum_{|\alpha|=1} |D^\alpha u|^q + \sum_{|\alpha|=2} |D^\alpha u|^p.
$$

\begin{lemma} \label{intestphi}
Let $B_{\rho}\subset \Omega$ and let $\zeta \in C^{\infty}_0(\Omega)$ be a
function such that
\begin{equation}\label{zeta01}
 \zeta=0 \text{ in } \Omega\setminus B_{\rho} \quad \text{and} \quad
0\leq \zeta\leq 1.
\end{equation}
Then there exists a positive constant $c_1$
such that
\begin{equation}\label{intestPhiu}
\int_{B_{\rho}}\Phi\, \zeta^{q}dx\leq c_1\rho^{n}
\Big(1+\rho^{-q}+\max_{B_{\rho}}\Big\{\sum_{|\alpha|=1} |D^\alpha \zeta|^q
+ \sum_{|\alpha|=2} |D^\alpha \zeta|^p\Big\}\Big).
\end{equation}
\end{lemma}

\begin{proof}
For every $x\in \Omega$ we set $v_1(x)=e^{\lambda u(x)}\zeta^{q}(x)$ where
\begin{equation}\label{lambdanew}
\lambda=2b(\mathrm{M})/a(\mathrm{M}).
\end{equation}
Simple calculations show that $v_1\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap
L^{\infty}(\Omega)$ and the following assertions hold:
\begin{itemize}
\item[(a)] for every $n$-dimensional multi-index $\alpha$,
$|\alpha|=1$,
$$
D^{\alpha}v_1=\lambda e^{\lambda u}\zeta^{q}D^{\alpha}u
+qe^{\lambda u}\zeta^{q-1}D^{\alpha}\zeta \quad \text{a. e. in } \Omega,
$$

\item[(b)] for every $n$-dimensional multi-index $\alpha$,
$|\alpha|=2$,
\begin{align*}
&|D^{\alpha}v_1-\lambda e^{\lambda u}\zeta^{q}D^{\alpha}u|\\
&\leq \lambda^2e^{\lambda u}\zeta^{q}
\sum_{|\beta|=1}|D^{\beta}u|^2
+2q\lambda e^{\lambda u}\zeta^{q-1}\Big\{\sum_{|\beta|=1}|D^{\beta}u|\Big\}
 \Big\{\sum_{|\beta|=1}|D^{\beta}\zeta|\Big\}\\
&+q(q-1)e^{\lambda u}\zeta^{q-2}\sum_{|\beta|=1}|D^{\beta}\zeta|^2
 +qe^{\lambda u}\zeta^{q-1}|D^{\alpha}\zeta| \quad
\text{a.\,e. in } \Omega.
\end{align*}
\end{itemize}

Since $v_1\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)$,
by \eqref{12}, we have
$$
\int_\Omega \Big\{\sum_{\alpha\in\Lambda} A_\alpha(x,u,\nabla_2 u)
D^\alpha v_1+B(x,u,\nabla_2 u) v_1\Big\} dx = 0.
$$
From this equality, using \eqref{5}, \eqref{7}, \eqref{defMrm}, \eqref{zeta01},
\eqref{lambdanew} and assertions (a) and (b), we deduce that
\begin{equation}\label{absorb1}
b(\mathrm{M})\int_{B_{\rho}}\Phi e^{\lambda u}\zeta^{q}dx
\leq I_1+I_2+I_3+I_4+e^{\lambda \mathrm{M}}\int_{B_{\rho}}(g_3
+\lambda g_0)dx
\end{equation}
where
\begin{gather*}
I_1=q\sum_{\alpha\in\Lambda}\int_{B_{\rho}}|A_\alpha(x,u,\nabla_2 u)|\,
|D^{\alpha}\zeta|\,e^{\lambda u}\zeta^{q-1}dx, \\
I_2=\lambda^2\sum_{|\alpha|=2}\sum_{|\beta|=1}
\int_{B_{\rho}}|A_\alpha(x,u,\nabla_2 u)|\,|D^{\beta}u|^2\,
e^{\lambda u}\zeta^{q}dx, \\
I_3=q^2\sum_{|\alpha|=2}\sum_{|\beta|=1}
 \int_{B_{\rho}}|A_\alpha(x,u,\nabla_2 u)|\,|D^{\beta}\zeta|^2\,
e^{\lambda u}\zeta^{q-2}dx, \\
I_4=2\lambda q\sum_{|\alpha|=2}\sum_{|\beta|=1}\sum_{|\gamma|=1}
 \int_{B_{\rho}}|A_\alpha(x,u,\nabla_2 u)|\,|D^{\beta}u|\,
|D^{\gamma}\zeta|\,
e^{\lambda u}\zeta^{q-1}dx.
\end{gather*}

Let us obtain suitable estimates for the addends in the right-hand side of
\eqref{absorb1}.
\smallskip

\noindent\textit{Estimate for $I_1$.}
 Using the Young's inequality with the exponents $q/(q-1)$ and $q$,
\eqref{3}, \eqref{defMrm} and \eqref{zeta01}, we obtain
\begin{align*}
&q\sum_{|\alpha|=1}\int_{B_{\rho}}|A_\alpha(x,u,\nabla_2 u)|
 \,|D^{\alpha}\zeta|\,e^{\lambda u}\zeta^{q-1}dx\\
&\leq \frac{b(\mathrm{M})}{16}\int_{B_{\rho}}\Phi e^{\lambda u}
 \zeta^{q}dx+c_2\int_{B_{\rho}}g_1dx+c_2\rho^{n}\max_{B_{\rho}}
 \sum_{|\alpha|=1}|D^{\alpha}\zeta|^{q}.
\end{align*}
Using the Young's inequality with the exponents $p/(p-1)$ and $p$, \eqref{4},
\eqref{defMrm} and \eqref{zeta01}, we obtain
\begin{align*}
&q\sum_{|\alpha|=2}\int_{B_{\rho}}|A_\alpha(x,u,\nabla_2 u)|\,
|D^{\alpha}\zeta|\,e^{\lambda u}\zeta^{q-1}dx\\
&\leq \frac{b(\mathrm{M})}{16}\int_{B_{\rho}}
 \Phi e^{\lambda u} \zeta^{q}dx+c_3\int_{B_{\rho}}g_2dx
+c_3\rho^{n}\max_{B_{\rho}}\sum_{|\alpha|=2}|D^{\alpha}\zeta|^{p}.
\end{align*}
We set
$$
\Phi_{\zeta} = \sum_{|\alpha|=1} |D^\alpha \zeta|^q + \sum_{|\alpha|=2}
|D^\alpha \zeta|^p.
$$
From the last two inequalities it follows that
\begin{equation}\label{I1}
I_1\leq \frac{b(\mathrm{M})}{8}\int_{B_{\rho}}\Phi e^{\lambda u} \zeta^{q}dx
+c_4\int_{B_{\rho}}(g_1+g_2)dx+c_4\rho^{n}\max_{B_{\rho}}\Phi_{\zeta}.
\end{equation}
\smallskip

\noindent\textit{Estimates for $I_2$, $I_3$ and $I_4$.}
It is obvious that
\begin{equation}\label{youngpqp}
\frac{p-1}{p}+\frac{2}{q}+\frac{q-2p}{qp}=1, \quad
q-1=(p-1)\frac{q}{p}+(\frac{q}{p}-1).
\end{equation}
Using this equalities, the Young's inequality, \eqref{4}, \eqref{defMrm}
and \eqref{zeta01}, we obtain
\begin{gather}\label{I2}
I_2\leq \frac{b(\mathrm{M})}{8}\int_{B_{\rho}}
\Phi e^{\lambda u} \zeta^{q}dx+c_5\int_{B_{\rho}}g_2dx+c_5\rho^{n}, \\
\label{I3}
I_3\leq \frac{b(\mathrm{M})}{8}\int_{B_{\rho}}\Phi e^{\lambda u}
 \zeta^{q}dx+c_6\int_{B_{\rho}}g_2dx+c_6\rho^{n}\max_{B_{\rho}}\Phi_{\zeta}
 +c_6\rho^{n}, \\
\label{I4}
I_4\leq \frac{b(\mathrm{M})}{8}\int_{B_{\rho}}\Phi e^{\lambda u} \zeta^{q}dx
+c_7\int_{B_{\rho}}g_2dx+c_7\rho^{n}\max_{B_{\rho}}\Phi_{\zeta}+c_7\rho^{n},
\end{gather}

From \eqref{absorb1}, \eqref{I1}, \eqref{I2}--\eqref{I4} it follows that
$$
\frac{b(\mathrm{M})}{2}\int_{B_{\rho}}\Phi e^{\lambda u}\zeta^{q}dx
\leq c_{8}\Big(\int_{B_{\rho}}gdx+\rho^{n}\max_{B_{\rho}}\Phi_{\zeta}+\rho^{n}\Big)
$$
where $g=g_0+g_1+g_2+g_3$.
By H\"older's inequality and the inequality $\tau>n/q$ we have
$$
\int_{B_{\rho}}gdx\leq \|g\|_{\tau}|B_{\rho}|^{(\tau-1)/\tau}
\leq c_9\rho^{n-q}.
$$
The last two inequalities and \eqref{defMrm} imply inequality \eqref{intestPhiu}.
The proof is complete.
\end{proof}

\begin{lemma}\label{intestphi1}
Let $B_{\rho}\subset B_{2R}(x_0)$ and let $\zeta \in C^{\infty}_0(\Omega)$
be a function such that condition \eqref{zeta01} be satisfied.
Then there exists a positive constant $c_{10}$ such that
\begin{equation}\label{intestPhiu1}
\begin{aligned}
&\int_{B_{\rho}}\frac{\Phi\, \zeta^{q}dx}{(M(2R)-u+R^{r})^{q}}\\
&\leq c_{10}\rho^{n} \Big(\rho^{-q}+\max_{B_{\rho}}
 \Big\{\sum_{|\alpha|=1}|D^{\alpha}\zeta|^{q}
 +\sum_{|\alpha|=2}|D^{\alpha}\zeta|^{p}\Big\} \\
&\quad +\rho^{2p-q}\max_{B_{\rho}}\sum_{|\alpha|=2}|D^{\alpha}\zeta|^{p}
+\rho^{-q(q-2p)/(q-p)}\max_{B_{\rho}}\sum_{|\alpha|=1}|D^{\alpha}\zeta|^{qp/(q-p)}
 \Big).
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
For every $x\in B_{2R}(x_0)$, we set $U(x)=M(2R)-u(x)+R^{r}$,
$$
v_2(x)=\begin{cases}
 \zeta^{q}(x)[U(x)]^{1-q} & \text{if }x\in B_{2R}(x_0), \\
 0 & \text{if } x\in \Omega\setminus B_{2R}(x_0).
 \end{cases}
$$
Simple calculations show that
$$
v_2\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)
$$
and the following assertions hold:
\begin{itemize}
\item[(c)] for every $n$-dimensional multi-index $\alpha$,
$|\alpha|=1$,
$$
D^{\alpha}v_2=qU^{1-q}\zeta^{q-1}D^{\alpha}\zeta+(q-1)U^{-q}\zeta^{q}D^{\alpha}u
\quad \text{a. e. in } B_{2R}(x_0),
$$

\item[(d)] for every $n$-dimensional multi-index $\alpha$,
$|\alpha|=2$,
\begin{align*}
&\Big|D^{\alpha}v_2-(q-1)U^{-q}\zeta^{q}D^{\alpha}u\Big|\\
&\leq q\,U^{1-q}\zeta^{q-1}|D^{\alpha}\zeta|+ q(q-1)\,U^{1-q}\zeta^{q-2}
\sum_{|\beta|=1}|D^{\beta}\zeta|^2\\
&\quad +2q(q-1)\,U^{-q}\zeta^{q-1}\Big\{\sum_{|\beta|=1}|D^{\beta}u|\Big\}
 \Big\{\sum_{|\beta|=1}|D^{\beta}\zeta|\Big\}  \\
&\quad +q(q-1)\,U^{-1-q}\zeta^{q}\sum_{|\beta|=1}|D^{\beta}u|^2 \quad
\text{a.\,e. in } B_{2R}(x_0).
\end{align*}
\end{itemize}

Putting the function $v_2$ into \eqref{12} instead of $v$ and using
\eqref{5}, \eqref{7}, \eqref{defMrm}, \eqref{zeta01} and assertions (c) and (d),
we obtain
\begin{equation}\label{puttingv2}
\begin{aligned}
&a(\mathrm{M})\int_{B_{\rho}}\Phi\, U^{-q}\zeta^{q}dx \\
&\leq I'_1+I'_2+I'_3+I'_4+I'_5
+\int_{B_{\rho}}(g_0+(2\mathrm{M}+1)g_3)U^{-q}dx,
\end{aligned}
\end{equation}
where
\begin{gather*}
I'_1=q\sum_{\alpha\in \Lambda}\,\int_{B_{\rho}}|A_{\alpha}(x,u,\nabla_2u)|\,
|D^{\alpha}\zeta|\,U^{1-q}\zeta^{q-1}dx,\\
I'_2=q\sum_{|\alpha|=2}\sum_{|\beta|=1}\,\int_{B_{\rho}}
|A_{\alpha}(x,u,\nabla_2u)|\,|D^{\beta}u|^2U^{-1-q}\zeta^{q}dx,\\
I'_3=q\sum_{|\alpha|=2}\sum_{|\beta|=1}\int_{B_{\rho}}
|A_\alpha(x,u,\nabla_2 u)|\,|D^{\beta}\zeta|^2 U^{1-q}\zeta^{q-2}dx, \\
I'_4=2q\sum_{|\alpha|=2}\sum_{|\beta|=1}\sum_{|\gamma|=1}
 \int_{B_{\rho}}|A_\alpha(x,u,\nabla_2 u)|\,|D^{\beta}u|\,
|D^{\gamma}\zeta| U^{-q}\zeta^{q-1}dx, \\
I'_5=b(\mathrm{M})\int_{B_{\rho}}\Phi\,U^{1-q}\zeta^{q}dx.
\end{gather*}
Next we  obtain suitable estimates for $I'_1$, $I'_2$, $I'_3$, $I'_4$, $I'_5$.
\smallskip

\noindent\textit{Estimate for $I'_1$.}
Using the Young's inequality with the exponents $q/(q-1)$ and $q$, \eqref{3},
\eqref{defMrm} and \eqref{zeta01}, we obtain
\begin{equation}\label{I'11}
\begin{aligned}
&q\sum_{|\alpha|=1}\,\int_{B_{\rho}}|A_{\alpha}(x,u,\nabla_2u)|\,
|D^{\alpha}\zeta|\,U^{1-q}\zeta^{q-1}dx\\
&\leq \frac{a(\mathrm{M})}{20}\int_{B_{\rho}}\Phi\,U^{-q}\zeta^{q}dx
+c_{11}\int_{B_{\rho}}g_1U^{-q}dx
+c_{11}\rho^{n}\max_{B_{\rho}}\sum_{|\alpha|=1}|D^{\alpha}\zeta|^{q}.
\end{aligned}
\end{equation}
We use the Young's inequality with the exponents $p/(p-1)$ and $p$, \eqref{4},
\eqref{defMrm} and \eqref{zeta01} to obtain
\begin{align*}
&q\sum_{|\alpha|=2}\,\int_{B_{\rho}}|A_{\alpha}(x,u,\nabla_2u)|\,
|D^{\alpha}\zeta|\,U^{1-q}\zeta^{q-1}dx\\
&\leq \frac{a(\mathrm{M})}{20}\int_{B_{\rho}}\Phi\,U^{-q}\zeta^{q}dx
+c_{12}\int_{B_{\rho}}g_2U^{-q}dx+c_{12}\sum_{|\alpha|=2}
\int_{B_{\rho}} |D^{\alpha}\zeta|^{p}U^{p-q}\zeta^{q-p}dx,
\end{align*}
whence, taking into account the inequalities $U\geq R^{r}$, $\rho/2<R<1$
and \eqref{defr}, we derive
\begin{equation}\label{I'12}
\begin{aligned}
&q\sum_{|\alpha|=2}\int_{B_{\rho}}|A_{\alpha}(x,u,\nabla_2u)|\,
|D^{\alpha}\zeta| U^{1-q}\zeta^{q-1}dx \\
&\leq \frac{a(\mathrm{M})}{20}\int_{B_{\rho}}\Phi\,U^{-q}\zeta^{q}dx
+c_{12}\int_{B_{\rho}}g_2U^{-q}dx
+ c_{13}\rho^{n-q+2p}\max_{B_{\rho}}\sum_{|\alpha|=2}|D^{\alpha}\zeta|^{p}.
\end{aligned}
\end{equation}
Summing inequalities \eqref{I'11} and \eqref{I'12}, we obtain
\begin{equation}\label{I'1}
\begin{aligned}
I'_1&\leq \frac{a(\mathrm{M})}{10}\int_{B_{\rho}}\Phi U^{-q}\zeta^{q}dx
 +c_{14}\int_{B_{\rho}}(g_1+g_2)U^{-q}dx\\
&\quad +c_{14}\rho^{n}\max_{B_{\rho}}\sum_{|\alpha|=1}
|D^{\alpha}\zeta|^{q}+ c_{14}\rho^{n-q+2p}
\max_{B_{\rho}}\sum_{|\alpha|=2}|D^{\alpha}\zeta|^{p}.
\end{aligned}
\end{equation}
\smallskip

\noindent\textit{Estimate for $I'_2$.}
We use \eqref{defMrm}, the first equality in \eqref{youngpqp},
Young's inequality, \eqref{4} and \eqref{zeta01} to obtain
$$
I'_2 \leq \frac{a(\mathrm{M})}{10}\int_{B_{\rho}}\Phi\,U^{-q}\zeta^{q}dx
+c_{15}\int_{B_{\rho}}g_2U^{-q}dx+c_{15}\int_{B_{\rho}}U^{-q(q-p)/(q-2p)}\zeta^{q}dx.
$$
Estimating the last integral in this inequality by means of the inequalities
$U\geq R^{r}$, $\rho/2<R<1$ and \eqref{defr}, we obtain
\begin{equation}\label{I'2}
I'_2\leq \frac{a(\mathrm{M})}{10}\int_{B_{\rho}}\Phi\,U^{-q}\zeta^{q}dx
+c_{15}\int_{B_{\rho}}g_2U^{-q}dx+c_{16}\rho^{n-q}.
\end{equation}
\smallskip

\noindent\textit{Estimates for $I'_3$ and $I'_4$.}
Using the reasoning similar the proof of \eqref{I'2}, we obtain
\begin{gather}\label{I'3}
\begin{aligned}
I'_3&\leq \frac{a(\mathrm{M})}{10}\int_{B_{\rho}}\Phi\,U^{-q}\zeta^{q}dx
+c_{17}\int_{B_{\rho}}g_2U^{-q}dx\\
&\quad +c_{17}\rho^{n}\max_{B_{\rho}}\sum_{|\alpha|=1}|D^{\alpha}\zeta|^{q}
+c_{17}\rho^{n-q},
\end{aligned} \\
\label{I'4}
\begin{aligned}
I'_4&\leq \frac{a(\mathrm{M})}{10}\int_{B_{\rho}}\Phi\,U^{-q}\zeta^{q}dx
+c_{18}\int_{B_{\rho}}g_2U^{-q}dx\\
&\quad +c_{18}\rho^{n-q(q-2p)/(q-p)}\max_{B_{\rho}}
 \sum_{|\alpha|=1}|D^{\alpha}\zeta|^{qp/(q-p)}.
\end{aligned}
\end{gather}
\smallskip

\noindent\textit{Estimate for $I'_5$.}
We use Young's inequality and Lemma \ref{intestphi}, to obtain
$$
I'_5\leq \frac{a(\mathrm{M})}{10}\int_{B_{\rho}}\Phi\,U^{-q}\zeta^{q}dx
+c_{19}\rho^{n} \Big(\rho^{-q}+\max_{B_{\rho}}
\Big\{\sum_{|\alpha|=1}|D^{\alpha}\zeta|^{q}+\sum_{|\alpha|=2}
|D^{\alpha}\zeta|^{p}\Big\}\Big).
$$
Collecting \eqref{puttingv2}, \eqref{I'1}--\eqref{I'4} and the above inequality,
 we obtain
\begin{align*}
&\frac{a(\mathrm{M})}{2}\int_{B_{\rho}}\Phi\,U^{-q}\zeta^{q}dx \\
&\leq c_{20}\rho^{n-q}+c_{20}\rho^{n}\max_{B_{\rho}}
 \Big\{\sum_{|\alpha|=1}|D^{\alpha}\zeta|^{q}
 +\sum_{|\alpha|=2}|D^{\alpha}\zeta|^{p}\Big\} \\
&\quad +c_{20}\rho^{n-q+2p}\max_{B_{\rho}}\sum_{|\alpha|=2}|D^{\alpha}\zeta|^{p}
 +c_{20}\rho^{n-q(q-2p)/(q-p)}\max_{B_{\rho}}\sum_{|\alpha|=1}
 |D^{\alpha}\zeta|^{qp/(q-p)} \\
&\quad + c_{20}\int_{B_{\rho}}gU^{-q}dx
\end{align*}
where $g=g_0+g_1+g_2+g_3$.
Finally, to obtain \eqref{intestPhiu1}, we estimate the last integral
in this inequality by means of Holder's inequality and the relations
\eqref{defr}, $U\geq R^{r}$ and $\rho/2<R<1$. The proof is complete.
\end{proof}

\begin{lemma}\label{kappa}
For every $\kappa\geq 1$ there is a positive constant
$c=c(\verb"data", \kappa)$
such that $\lim_{k\to +\infty}c(\verb"data", \kappa)=+\infty$ and
\begin{equation}\label{kappasum}
\int_{B_{3R/2}(x_0)}v_0^{\kappa}dx\leq cR^{n}.
\end{equation}
\end{lemma}

\begin{proof}
First, we estimate from above the average integral
$$
(v_0)_{B_{3R/2}(x_0)}=\frac{1}{|B_{3R/2}(x_0)|}\int_{B_{3R/2}(x_0)}v_0dx
$$
by a constant depending only on \verb"data".
For this purpose we choose a function $\zeta_1\in C^{\infty}_0(\Omega)$
such that
\begin{gather*}
0\leq\zeta_1\leq1 \text{ in } \Omega, \quad \zeta_1=1  \text{ in }
B_{3R/2}(x_0), \quad \zeta_1=0  \text{ in }\Omega\setminus B_{2R}(x_0), \\
|D^{\alpha}\zeta_1|\leq K_1R^{-|\alpha|} \quad
 \text{for }|\alpha|=1, 2,
\end{gather*}
where $K_1$ is an absolute constant, not depending on $R$.
Taking into account the facts that $1\leq v_0\leq 1+\ln4$ on $G_1(R)$ and
$|G_1(R)|\geq |B_{3R/2}(x_0)|/2$, and using Holder's inequality,
Lemmas \ref{lem4Scr73} and \ref{intestphi1} and the properties of the
function $\zeta_1$, we obtain
\begin{equation}\label{estintaver}
\begin{aligned}
(v_0)_{B_{3R/2}(x_0)}
&\leq c_{21} R^{-n/q}\Big(\int_{B_{3R/2}(x_0)}v_0^{q}dx\Big)^{1/q}\\
&\leq c_{22} R^{1-n/q}\Big(\int_{B_{3R/2}(x_0)}
 \frac{\Phi \zeta_1^{q}dx}{(M(2R)-u+R^{r})^{q}}+R^{n-q}\Big)^{1/q} \\
&\leq c_{23}.
\end{aligned}
\end{equation}
Next, let $B_{2\rho}\subset B_{2R}(x_0)$, and let
$\zeta_2\in C^{\infty}_0(\Omega)$ be a function such that
\begin{gather*}
0\leq\zeta_2\leq1 \text{ in }\Omega, \quad \zeta_2=1  \text{ in }
B_{\rho}, \quad \zeta_2=0 \text{ in } \Omega\setminus B_{2\rho}, \\
|D^{\alpha}\zeta_2|\leq K_2\rho^{-|\alpha|} \quad
 \text{for } |\alpha|=1, 2,
\end{gather*}
where $K_2$ is an absolute constant, not depending on $\rho$.
Using Holder's inequality, Lemma \ref{intestphi1} and the properties of
the function $\zeta_2$, we derive that
\begin{align*}
\sum_{|\alpha|=1}\,\int_{B_{\rho}}|D^{\alpha}v_0|dx
&\leq c_{24} \rho^{n-n/q}\Big(\sum_{|\alpha|=1}\int_{B_{\rho}}|D^{\alpha}v_0|^{q}dx
\Big)^{1/q}\\
&\leq c_{24} \rho^{n-n/q}\Big(\int_{B_{2\rho}}\frac{\Phi\, \zeta_2^{q}dx}{(M(2R)
-u+R^{r})^{q}}\Big)^{1/q} \\
&\leq c_{25} \rho^{n-1} .
\end{align*}
Hence, by Lemma \ref{lemJohnNir}, we have
\begin{equation}\label{expsumv0}
\int_{B_{3R/2}(x_0)} \exp\Big(c_{26}|\,v_0-(v_0)_{B_{3R/2}(x_0)}|\Big) dx
\leq c_{27} R^{n}.
\end{equation}

Now let $\kappa\geq 1$. Then inequalities \eqref{estintaver} and \eqref{expsumv0}
imply \eqref{kappasum}.
The proof is complete.
\end{proof}

Now we are ready to prove inequality \eqref{supestv0}.

\begin{lemma}\label{lastlemma}
There is a positive constant $c_0$ such that inequality \eqref{supestv0} holds.
\end{lemma}

\begin{proof} We proceed the proof in four steps.
\smallskip

\noindent\textbf{Step 1.} We fix a function
$\psi_0\in C^{\infty}_0(\mathbb{R})$
such that
$$
0\leq\psi_0\leq1  \text{ on } \mathbb{R}, \quad \psi_0=1  \text{ in }
[-1,1], \quad \psi_0=0  \text{ in }\mathbb{R}\setminus (-3/2, 3/2).
$$
For any $x\in\Omega$ we set
$\psi(x)=\psi_0\big(\frac{|x-x_0|}{R}\big)$,
$$
\tilde{v}(x)=\begin{cases}
 [v_0(x)]^{k}\psi^{t}(x)[U(x)]^{1-q} & \text{if } x\in B_{2R}(x_0), \\
 0 & \text{if } x\in \Omega\setminus B_{2R}(x_0),
 \end{cases}
$$
where
$U=M(2R)-u+R^{r}$,
\begin{gather}\label{k0}
k\geq \overline{k}:=\max\{q, (4\mathrm{M}+1)b(\mathrm{M})/ a(\mathrm{M})\}, \\
\label{deft}
 \nu:=\max\{q,2qp/(q-2p)\}< t \leq C_0k,
\end{gather}
and $C_0=C_0(n,p,q,\tau)>1$ is a constant that will be specified below.
Simple calculations show that
$$
\tilde{v}\in{\mathaccent"7017 W}^{1,q}_{2,p}(\Omega)\cap L^{\infty}(\Omega)
$$
and the following assertions hold:
\begin{itemize}
\item[(a1)] for every $\alpha\in\Lambda$, $|\alpha|=1$,
\[
|D^{\alpha}\tilde{v}-(q-1)v_0^{k}\psi^{t}U^{-q}D^{\alpha}u
-kv_0^{k-1}\psi^{t}U^{-q}D^{\alpha}u|
\leq\frac{c_{28}kv_0^{k}\psi^{t-1}}{R\,U^{q-1}} \quad
\text{a.\,e. in } \Omega,
\]

\item[(a2)] for every $\alpha\in\Lambda$, $|\alpha|=2$,
\begin{align*}
&|D^{\alpha}\tilde{v}-(q-1)v_0^{k}\psi^{t}U^{-q}D^{\alpha}u-
kv_0^{k-1}\psi^{t}U^{-q}D^{\alpha}u|\\
&\leq\frac{c_{28}k^2v_0^{k}\psi^{t-2}}{U^{q-1}}
\Big\{\frac{1}{R^2}+\sum_{|\beta|=1}\frac{|D^{\beta}u|^2}{U^2}\Big\} \quad
\text{a.\,e. in } \Omega,
\end{align*}
where $c_{28}>0$ depends only on $C_0$, $\max_{\mathbb{R}}|\psi_0'|$
and $\max_{\mathbb{R}}|\psi_0''|$.
\end{itemize}
Putting the function $\tilde{v}$ in \eqref{12} instead of $v$ and using \eqref{5},
\eqref{7}, and assertions (a1) and (a2),
we obtain
\begin{equation}\label{puttildev}
\begin{aligned}
&(q-1)a(\mathrm{M})\int_{B_{2R}(x_0)}\Phi\,U^{-q} v_0^{k}\psi^{t}dx
+k a(\mathrm{M})\int_{B_{2R}(x_0)}\Phi\,U^{-q} v_0^{k-1}\psi^{t}dx  \\
&\leq b(\mathrm{M})\int_{B_{2R}(x_0)}\Phi\,U^{1-q} v_0^{k}\psi^{t}dx
+\int_{B_{2R}(x_0)}kg_4 v_0^{k}\psi^{t}U^{-q}dx
+\mathcal{I}_1+\mathcal{I}_2,
\end{aligned}
\end{equation}
where $g_4=2g_0+(2M+1)g_3$,
\begin{gather}\label{defI1}
\mathcal{I}_1=\frac{c_{28}k}{R}\sum_{|\alpha|=1}
\int_{B_{2R}(x_0)}|A_{\alpha}(x,u,\nabla_2u)|U^{1-q} v_0^{k}\psi^{t-1}dx, \\
\label{defI2}
\mathcal{I}_2=c_{28}k^2\sum_{|\alpha|=2}\,\int_{B_{2R}(x_0)}
|A_{\alpha}(x,u,\nabla_2u)|U^{1-q} v_0^{k}\psi^{t-2}
\Big\{\frac{1}{R^2}+\sum_{|\beta|=1}\frac{|D^{\beta}u|^2}{U^2}\Big\}dx.
\end{gather}
\smallskip


\noindent\textbf{Step 2.}
We show that the first term in the right-hand side of inequality \eqref{puttildev}
is absorbed by the second term in its left-hand side. For this we need the
inequality
\begin{equation}\label{4M+1}
Uv_0\leq 4 \mathrm{M}+1 \quad \text{a.\,e. in }B_{2R}(x_0).
\end{equation}
To prove it, we consider the function
$$
\chi(s)=(s+R^r)\ln\frac{2\omega(2R)}{s+R^r}, \quad  s\in[0,\omega(2R)].
$$
By \eqref{omegageqR}, we have that
$\chi\geq 0$ in $[0,\omega(2R)]$ and
$\hat{s}:=2e^{-1}\omega(2R)-R^r\in [0,\omega(2R)]$. By standard techniques
of differential calculus we obtain
$$
\max_{s\in [0,\,\omega(2R)]}\chi(s)=\chi(\hat{s})=2e^{-1}\omega(2R)\leq 2\mathrm{M}.
$$
Now inequality \eqref{4M+1} follows from the relations $R\leq 1$ and
$$
Uv_0=M(2R)-u+R^{r}+\chi(M(2R)-u)\leq 4\mathrm{M}+1 \quad \text{a.\,e. in }
B_{2R}(x_0).
$$

Using \eqref{4M+1}, the first term on the right-hand side of inequality
\eqref{puttildev} is estimated in the following way
\begin{equation}\label{estterm}
b(\mathrm{M})\int_{B_{2R}(x_0)}\Phi\,U^{1-q} v_0^{k}\psi^{t}dx
\leq(4\mathrm{M}+1)b(\mathrm{M})\int_{B_{2R}(x_0)}\Phi
U^{-q} v_0^{k-1}\psi^{t}dx.
\end{equation}
Now \eqref{k0}, \eqref{puttildev} and \eqref{estterm} imply the inequality
\begin{equation}\label{absorb}
(q-1)a(\mathrm{M})\int_{B_{2R}(x_0)}\Phi\,U^{-q} v_0^{k}\psi^{t}dx \\
\leq k \int_{B_{2R}(x_0)}g_4 v_0^{k}\psi^{t}U^{-q}dx
+\mathcal{I}_1+\mathcal{I}_2\,.
\end{equation}
\smallskip

\noindent\textbf{Step 3.} Let us estimate from above the quantities
$\mathcal{I}_1$ and $\mathcal{I}_2$, which are defined by \eqref{defI1} and
\eqref{defI2} respectively. We use \eqref{3} and the Young's inequality
$$
|yz|\leq \varepsilon|y|^{q/(q-1)}+\varepsilon^{1-q}|z|^{q},
$$
where
$$
y=|A_\alpha(x,u,\nabla_2 u)|U^{1-q}\psi^{(q-1)t/q}, \quad |\alpha|=1,
\quad z=k\,\psi^{(t-q)/q}/R,
$$
and $\varepsilon$ is an appropriate positive number, to obtain
\begin{equation}\label{estI1}
\begin{aligned}
\mathcal{I}_1
&\leq\frac{(q-1)a(\mathrm{M})}{4} \int_{B_{2R}(x_0)}
 \Phi U^{-q} v_0^{k}\psi^{t}dx\\
&\quad +c_{29}\int_{B_{2R}(x_0)}g_1 v_0^{k}\psi^{t}U^{-q}dx
 +\frac{c_{29}k^{q}}{R^{q}}\int_{B_{2R}(x_0)} v_0^{k}\psi^{t-q}dx,
\end{aligned}
\end{equation}
where $c_{29}>0$ depends only on $c_{28}$, $q$, $a(\mathrm{M})$ and
$a_1(\mathrm{M})$.

Using the first equality in \eqref{youngpqp},  Young's inequality
and \eqref{defr}, we establish that if $\varepsilon>0$, $\alpha$,
$\beta\in\Lambda$, $|\alpha|=2$
and $|\beta|=1$, then
\begin{gather}\label{estA1I2}
\begin{aligned}
&|A_\alpha(x,u,\nabla_2 u)|U^{1-q}\psi^{t-2}\cdot k^2R^{-2} \\
&\leq\varepsilon |A_\alpha(x,u,\nabla_2 u)|^{p/(p-1)}U^{-q}\psi^{t}
 +\varepsilon(1+\varepsilon^{-qp/(q-2p)}) k^{q}R^{-q}\psi^{t-2qp/(q-2p)}
\quad \text{on }\Omega,
\end{aligned}\\
\label{estA2I2}
\begin{aligned}
&|A_\alpha(x,u,\nabla_2 u)|U^{1-q}\psi^{t-2}\cdot k^2|D^{\beta}u|^2U^{-2}\\
&\leq\varepsilon |A_\alpha(x,u,\nabla_2 u)|^{p/(p-1)}U^{-q}\psi^{t}
+\varepsilon|D^{\beta}u|^{q}U^{-q}\psi^{t}\\
&\quad +\varepsilon^{1-qp/(q-2p)} k^{2qp/(q-2p)}R^{-q}\psi^{t-2qp/(q-2p)}
\quad \text{on }\Omega.
\end{aligned}
\end{gather}
From \eqref{4}, \eqref{defI2}, \eqref{estA1I2}, \eqref{estA2I2}, taking into account \eqref{deft} and the suitable choice of $\varepsilon$, we deduce the estimate
\begin{equation}\label{estI2}
\begin{aligned}
&\mathcal{I}_2\leq\frac{(q-1)a(\mathrm{M})}{4}
 \int_{B_{2R}(x_0)}\Phi U^{-q} v_0^{k}\psi^{t}dx\\
&\quad+ c_{30}\int_{B_{2R}(x_0)}g_2 v_0^{k}\psi^{t}U^{-q}dx
 +\frac{c_{30}k^{\nu}}{R^{q}}\int_{B_{2R}(x_0)} v_0^{k}\psi^{t-\nu}dx,
\end{aligned}
\end{equation}
where $c_{30}>0$ depends only on $c_{28}$, $q$, $p$, $a(\mathrm{M})$ and
$a_2(\mathrm{M})$.

From \eqref{absorb}, \eqref{estI1}, \eqref{estI2}, \eqref{deft} and \eqref{defr}
it follows that
\begin{align*}
&\int_{B_{2R}(x_0)}\Phi\,U^{-q} v_0^{k}\psi^{t}dx\\
&\leq \frac{c_{31}k^{\nu}}{R^{q}}\int_{B_{2R}(x_0)} v_0^{k}\psi^{t-\nu}dx
+\frac{c_{31}k^{\nu}}{R^{q-n/\tau}}\int_{B_{2R}(x_0)}
(g_1+g_2+g_4)v_0^{k}\psi^{t-\nu}dx.
\end{align*}
Estimating the last two integrals by H\"older's inequality with the exponents
$\tau$ and $\tau/(\tau-1)$ and taking into account \eqref{k0} and \eqref{deft},
we obtain that for every $k\geq \overline{k}$ and $t\in(\nu,C_0k]$ the
following inequality holds (see also \cite[inequality (39)]{Skr78}):
\begin{equation}\label{baseineq}
\int_{B_{2R}(x_0)}\Phi\,U^{-q} v_0^{k}\psi^{t}dx
\leq \frac{c_{32}k^{\nu}}{R^{q-n/\tau}}\Big(\int_{B_{2R}(x_0)}
 (v_0^{k}\psi^{t-\nu})^{\tau/(\tau-1)}dx\Big)^{(\tau-1)/\tau}.
\end{equation}
\smallskip


\noindent\textbf{Step 4.} We set
\begin{gather*}
J(k,t)=\frac{1}{R^{n}}\int_{B_{2R}(x_0)}v_0^{k}\psi^{t}dx, \quad
k\in\mathbb{R}, \; t>0, \\
\theta=\frac{\tau}{\tau-1}\cdot \frac{q}{q^{\ast}}\,,
 \quad \tilde{\nu}=\frac{(q+\nu)q^{\ast}}{q}.
\end{gather*}
The following assertion holds:
if $k\geq \overline{k}q^{\ast}/q$ and $\tilde{\nu}< t\leq C_0k$, then
\begin{equation}\label{iter1}
J(k,t)\leq c_{33}k^{\tilde{\nu}}[J(k\theta,t\theta-\tilde{\nu})]^{1/\theta}.
\end{equation}
Let $k\geq \overline{k}q^{\ast}/q$ and $\tilde{\nu}< t\leq C_0k$.
Then, applying inequality \eqref{2} to the function
$v^{k/q^{\ast}}_0\psi^{t/q^{\ast}}$, we obtain
\begin{align*}
J(k,t) &\leq \frac{c_{34}k^{q^{\ast}}}{R^{n}}
\Big( \int_{B_{2R}(x_0)} \Phi\, U^{-q} v_0^{kq/q^{\ast}}\psi^{tq/q^{\ast}-q}dx \\
&\quad +\frac{1}{R^{q}}\int_{B_{2R}(x_0)} v_0^{kq/q^{\ast}}\psi^{tq/q^{\ast}-q}dx
 \Big)^{q^{\ast}/q}.
\end{align*}
From this inequality, estimating the first addend in the brackets by means
of \eqref{baseineq} and the second addend by means of H\"older's inequality,
we deduce \eqref{iter1}.

We choose a number $i_0\in\mathbb{N}$ such that 
$\theta^{-i_0}>\overline{k}q^{\ast}/q$ and set $C_0=\tilde{\nu}/(1-\theta)$,
$$
k_{i}=\theta^{-i_0-i}, \quad t_{i}=\frac{\tilde{\nu}(\theta^{-i_0-i}-1)}{1-\theta}, 
\quad J_{i}=J(k_{i},t_{i}), \quad i=0,1,2, \ldots .
$$
Then \eqref{iter1} and the inequality $\theta<1$ imply that for every 
$i=0,1,2, \dots $,
$$
J_{i}^{1/k_{i}}\leq c_{35}J_0^{\theta^{i_0}}.
$$
By Lemma \ref{kappa} we have
$J_0^{\theta^{i_0}}\leq c_{36}$.
From the last two inequalities it follows that
$$
\|v_0\|_{L^{\infty}(B_{R}(x_0))}
=\lim_{i\to\infty}\Big(\frac{1}{R^{n}}\int_{B_{R}(x_0)}v_0^{k_{i}}dx\Big)^{1/k_{i}}
\leq\limsup_{i\to\infty}J_{i}^{1/k_{i}}\leq c_0.
$$
The proof is complete.
\end{proof}

Inequality \eqref{supestv0} implies \eqref{relomega}, and hence according to 
Lemma \ref{lemoscilation}, the assertions of Theorem \ref{th2.2} hold.

\subsection*{Acknowledgements}
This research was supported by the Ministry of Education and Science of
Ukraine (Project No. 0115U000136) and by the State Fund for
Fundamental Research of Ukraine (Project No. 0116U007160).
The author wishes to thank the anonymous referees for the remarks that
allowed us to improve the presentation of this article. 

\begin{thebibliography}{00}

\bibitem{AlbCianSb} A. Alberico, A. Cianchi, C. Sbordone; 
On the modulus of continuity of solutions to the $n$-Laplace equation, 
\emph{Journal of Elliptic and Parabolic Equations}, \textbf{1} (2015),  
No. 1, 1--12.

\bibitem{AlbFer} A. Alberico, V. Ferone;
 Regularity properties of solutions of elliptic equations in $\mathbb{R}^2$ 
in limit cases, \emph{Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend.
Lincei (9) Mat. Appl.}, \textbf{6} (1995), No. 4, 237--250.

\bibitem {BocMurPl92} L. Boccardo, F. Murat, J.-P. Puel;
$L^{\infty}$-estimate for some nonlinear elliptic partial
differential equations and application to an existence result,
\emph{SIAM J. Math. Anal.}, \textbf{23} (1992), No. 2, 326--333.

\bibitem {CirmDasLeon} G. R. Cirmi, S. D'Asero, S. Leonardi; 
Fourth-order nonlinear elliptic equations with lower order
term and natural growth conditions, \emph{Nonlinear Anal.}, \textbf{108} (2014), 
66--86.

\bibitem {DAsLar} S. D'Asero, D. V. Larin; 
On weighted estimates of solutions for nonlinear higher order
elliptic problems, \emph{Appl. Anal.}, \textbf{82} (2003), No. 1, 55--74.

\bibitem {DAsero06} S. D'Asero; 
On Harnack inequality for degenerate nonlinear higher-order
elliptic equations, \emph{Appl. Anal.}, \textbf{85} (2006), No. 8, 971--985.

\bibitem {DAsero10} S. D'Asero; 
On removability of the isolated singularity for solutions of
high-order elliptic equations, \emph{Complex Var. Elliptic Equ.},
 \textbf{55} (2010), No. 5-6, 525--536.


\bibitem {De Giorgi} E. De Giorgi; 
Un esempio di estremali discontinue per un problema variazionale di tipo ellittico,
\emph{Boll. Un. Mat. Ital. (4)}, \textbf{1} (1968), 607--627.

\bibitem {DrNic} P. Dr\'{a}bek, F. Nicolosi; 
Existence of bounded solutions for some degenerated quasilinear elliptic equations,
\emph{Ann. Mat. Pura Appl.}, \textbf{165} (1993), No. 4, 217--238.

\bibitem{FerFus} V. Ferone, N. Fusco; 
Continuity properties of minimizers of integral functionals in a limit case, 
\emph{J. Math. Anal. Appl.}, \textbf{202} (1996), 27--52.

\bibitem{Frehse} J. Frehse; On the boundedness of weak
solutions of higher order nonlinear elliptic partial differential
equations, \emph{Boll. Un. Mat. Ital. (4)}, \textbf{3} (1970),
607--627.

\bibitem {GlbTr} D. Gilbarg, N. S. Trudinger; 
\emph{Elliptic Partial Differential Equations of Second Order}, 
Springer-Verlag, Berlin, 1983.

\bibitem {IwOn} T. Iwanec, J. Onninen; 
Continuity estimates for $n$-harmonic equations, 
\emph{Indiana Univ. Math. J.},  \textbf{56} (2007),  805--824.

\bibitem {JiangKoskYang} R. Jiang, P. Koskela, D. Yang; 
Continuity of solutions to $n$-harmonic equations, \emph{Manuscripta Math.},
\textbf{139} (2012), 237--248.

\bibitem {JohnNir} F. John, L. Nirenberg; 
On functions of bounded mean oscillation, \emph{Comm. Pure Appl. Math.},
\textbf{14} (1961),  415--426.

\bibitem {Kov01} A. A. Kovalevskii; 
Entropy solutions of the Dirichlet problem for a class of fourth-order nonlinear 
elliptic equations with $L^{1}$-right-hand sides, \emph{Izv. Math.},
 \textbf{65} (2001), No. 2, 231--283.

\bibitem {Kov09} A. A. Kovalevsky; 
Nonlinear fourth-order equations with a strengthened ellipticity and $L^{1}$-data. 
On the notions of solution to nonlinear elliptic problems: results and
developments, \emph{Quad. Mat., Dept. Math., Seconda Univ. Napoli,
Caserta}, \textbf{23} (2008), 283--337.

\bibitem{KovSkrSh} A. A. Kovalevsky, I. I. Skrypnik, A. E. Shishkov;
\emph{Singular solutions of nonlinear elliptic and parabolic equations}, 
Walter de Gruyter GmbH, Berlin/Boston, 2016.

\bibitem {LadUr} O. A. Ladyzhenskaya, N. N. Ural'tseva; 
\emph{Linear and Quasilinear Elliptic Equations},
Nauka, Moscow, 1973.

\bibitem {Lions} J.-L. Lions; 
\emph{Quelques M\'ethodes de R\'esolution des Probl\`emes aux Limites 
Non Lin\'eaires}, Dunod, Gauthier-Villars, Paris, 1969.

\bibitem {Maz'ya} V. G. Maz'ya; 
Examples of nonregular solutions of quasilinear elliptic equations with analytic 
coefficients, \emph{Funct. Anal. Appl.}, \textbf{2} (1968), No. 3, 230--234.

\bibitem {Moser} J. Moser; 
A new proof of De Giorgi's theorem concerning the regularity problem for 
elliptic differential equations, \emph{Comm. Pure Appl. Math.},
 \textbf{13} (1960), 457--468.

\bibitem {NicSkr02} F. Nicolosi, I. V. Skrypnik; 
On Harnack type theorems for nonlinear higher order elliptic equations, 
\emph{Nonlinear Anal.}, \textbf{50} (2002), 129--147.

\bibitem{Skr73} I. V. Skrypnik; 
\emph{Nonlinear higher order elliptic equations}, (Russian), 
Naukova dumka, Kiev, 1973.

\bibitem {Skr78} I. V. Skrypnik; 
Higher order quasilinear elliptic
equations with continouos generalized solutions,
\emph{Differential Equations}, \textbf{14} (1978), No. 6, 786--795.

\bibitem {Skr91} I. V. Skrypnik; 
Regularity of a boundary point for a higher-order quasilinear elliptic equation 
(Russian), \emph{Trudy Mat. Inst. Steklov}, \textbf{200} (1991), 310--321;
 translation in Proc. Steklov Inst. Math., \textbf{200} (1993), No. 2, 339--351.

\bibitem {Skr97} I. V. Skrypnik; 
Pointwise estimates of potentials for higher-order capacities, 
\emph{Ukrainian Math. J.}, \textbf{49} (1997), No. 1, 165--180.

\bibitem{Todor} T. G. Todorov; 
The continuity of bounded generalized solutions of
 higher order quasilinear elliptic equations, \emph{Vestnik Leningrad Univ.
 Mat. Mekh. Astronom.} (1975), No. 19, 56--63; English transl. in
 \emph{Vestnik Leningrad Univ. Math.}, \textbf{11} (1978).

\bibitem {Voit11} M. V. Voitovich; 
Existence of bounded solutions for a class of nonlinear fourth-order equations,
\emph{Differ. Equ. Appl.}, \textbf{3} (2011), No. 2, 247--266.

\bibitem {Voit13} M. V. Voitovich; 
Existence of bounded
solutions for nonlinear fourth-order elliptic equations with strengthened 
coercivity and lower-order terms with natural growth, 
\emph{Electron. J. Differential Equations}, \textbf{2013} (2013), No. 102, 1--25.

\bibitem {Voit15} M. V. Voitovich; 
On the existence of bounded generalized solutions of the Dirichlet
problem for a class of nonlinear high-order elliptic equations, 
\emph{J. Math. Sci. (N. Y.)}, \textbf{210} (2015), No. 1, 86--113.

\bibitem {VoitMN} M. V. Voitovich; 
On the boundedness of generalized solutions of higher-order nonlinear 
elliptic equations with data from an Orlicz-Zygmund class,
\emph{Math. Notes}, \textbf{99} (2016), No. 5-6, 840--850.

\bibitem{WidmanB} K.-O. Widman; 
Local bounds for solutions of higher order nonlinear elliptic partial differential
equations, \emph{Math. Z.}, \textbf{121} (1971), No. 1, 81--95.

\bibitem{WidmanH} K.-O. Widman; H\"older continuity
of solutions of elliptic systems, \emph{Manuscripta Math.},
\textbf{5} (1971), No. 4, 299--308.

\end{thebibliography}

\end{document}