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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 49, pp. 1--9.\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-2018/49\hfil $L_\infty$-estimate for the Robin problem]
{$L_\infty$-estimate for the Robin problem of a singular variable
$p$-Laplacian equation in a conical domain}

\author[M. Borsuk \hfil EJDE-2018/49\hfilneg]
{Mikhail Borsuk}

\address{Mikhail Borsuk \newline
Department of Mathematics and Computer Science,
University of Warmia and Mazury in Olsztyn,
10-957 Olsztyn-Kortowo, Poland}
\email{borsuk@uwm.edu.pl}

\thanks{Submitted November 6, 2017. Published February 15, 2018.}
\subjclass[2010]{35J20, 35J25, 35J70}
\keywords{$p(x)$-Laplacian; angular and conical points}

\begin{abstract}
 We establish a bound for the modulus of the weak bounded solution
 to the Robin problem for an elliptic quasi-linear second-order equation
 with the variable $p(x)$-Laplacian.
\end{abstract}

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

\section{Introduction}

The aim of our article is to obtain an estimate for the modulus of
weak bounded solutions to the Robin problem
for quasi-linear elliptic second-order equations with the variable
$p(x)$-Laplacian in a neighborhood of an angular or conical boundary
point in a bounded domain. The Robin boundary conditions are related
to Sturm-Liouville problems which are used in many contexts in science
and engineering. For example, in electromagnetic problems, in heat
transfer problems and for convection-diffusion equations
(Fick's law of diffusion); a well as to study of reflected shocks
in transonic flows.

Let $G\subset \mathbb{R}^n$, $n\geq 2$ be a bounded domain with the boundary
$\Gamma$. We suppose that $\Gamma$ is a smooth surface everywhere except
at the origin $\mathcal{O}\in \Gamma$, and near the point $\mathcal{O}$
it is a conical surface whose vertex is $\mathcal{O}$.

We consider the Robin problem
\begin{equation} \label{RQL} %\tag{$RQL$}
\begin{gathered}
 -\triangle_{p(x)} u+ a_0(x)u|u|^{p(x)-1}+b(u,\nabla u) =f(x),\quad x\in G,\\
|\nabla u|^{p(x)-2}\frac{\partial u}{\partial\overrightarrow{n}}
+ \frac{\gamma}{|x|^{p(x)-1}}u|u|^{p(x)-2} =g(x), \quad x\in \Gamma,
\end{gathered}
\end{equation}
where
\begin{equation}\label{op}
\triangle_{p(x)} u\equiv \operatorname{div}\big(|\nabla u|^{p(x)-2}\nabla u\big).
\end{equation}
We require that the following assumptions hold:
\begin{itemize}
\item [(i)]$p(x)\in C^{(0)}(\overline{G})$ and 
 $1<p_-\leq p(x)\leq p_+=p(0)<n$, $a_0(x)\geq a_0$,
 $a_0={\rm const}>0$ for all $x\in \overline{G}$, $\gamma ={\rm const} >0$;

\item [(ii)] the function $b(u, \xi)$ satisfies in
$\mathfrak{M}=\mathbb{R}\times\mathbb{R}^n$ the inequality
 $$
| b(u,\xi) | \leq \mu |u|^{-1}|\xi|^{p(x)}, \quad 0\leq \mu<1, \;
\forall x\in \overline{G};
$$

\item [(iii)]
\begin{gather*}
|f(x)|\leq f_0|x|^{\beta (x)}, \quad
\beta (x)\geq \beta_0 - \frac{n}{s}, \quad s>\frac{n}{p_-}, \quad
f_0\geq 0,\quad \beta_0>0,\quad \forall x\in \overline{G}; \\
|g(x)|\leq g_0 |x|^{1-p(x)}, \quad g_0\geq 0,\quad \forall x\in \Gamma .
 \end{gather*}
\end{itemize}
The $L_\infty$-regularity of weak solutions for quai-linear equations with
$p(x)$-Laplacian was studied as follows:
\begin{itemize}
 \item in \cite{AB} for $b(u,\xi)\equiv 0$ (the Dirichlet problem),

 \item in \cite{AC, AS} for $ b(u,\xi)$ not depending on $\xi$
(the Dirichlet and the Robin problems),

 \item in \cite{FZ} for
\begin{gather*}
|b(u,\xi)|\leq c_1|\xi|^{\alpha (x)} +c_2|u|^{r(x)-1} +c_3, \\
\alpha (x)=\frac{r(x)-1}{r(x)}p(x),\quad  p(x)\leq r(x) <p^*(x),
\end{gather*}
where $p^*(x)$ is the Sobolev embedding exponent of $p(x)$ (the Dirichlet problem).
\end{itemize}
We define the functions class
$$
\mathfrak{N}^{1,p(x)}_{-1,\infty}(G)
=\Big\{u(x)\in L_\infty (G): \int_{G}\langle |x|^{-p(x)}|u|^{p(x)}
+|u|^{-1}|\nabla u|^{p(x)}\}\rangle \,dx<\infty\Big\}.
$$
It is obvious that $\mathfrak{N}^{1,p(x)}_{-1,\infty}(G^)\subset W^{1,p(x)}(G)$.

\begin{remark} \rm
If $p(x)>n$, by the Sobolev imbedding theorem, we have
$u\in C^{1-\frac{n}{p(0)}}(G)$ (see \cite{ER:00}). Therefore we investigate
 only $p(x)\in (1,n)$ (see assumption (i)).
\end{remark}

\begin{definition} \rm
A function $u$ is called a weak bounded solution of problem \eqref{RQL}
provided that $u(x)\in \mathfrak{N}^{1,p(x)}_{-1,\infty}(G)$ and $u$
satisfies the integral identity
\begin{equation}
\begin{aligned}
 Q(u,\eta)
&:\equiv \int_{G}\langle |\nabla u|^{p(x)-2}u_{x_i}\eta_{x_i}
 +a_0(x)u|u|^{p(x)-1}\eta (x) + b(u,\nabla u)\eta (x) \rangle \,dx \\
&\quad + \gamma \int_{\Gamma}r^{1-p(x)}u|u|^{p(x)-2}\eta (x)ds \\
&=\int_{\Gamma}g(x)\eta (x) ds + \int_{G}f(x)\eta (x) \,dx.
\end{aligned} \label{II} %\tag{$II$}
\end{equation}
for all $\eta (x)\in \mathfrak{N}^{1,p(x)}_{-1,\infty}(G)$.
\end{definition}

\begin{remark}\rm
It is easy to verify that the assumptions (i)--(iii) guarantee the existence
 of integrals over $G$ and $\Gamma$. Therefore, $ Q(u,\eta)$ well defined.
\end{remark}

First we formulate well known lemmas.

\begin{lemma}[{see \cite[Lemma 2.1]{CP:95} and \cite[Lemma 1.60]{BK}}] \label{Lem1.3}
Let us consider the function
\begin{equation*}
 \eta(x)= \begin{cases}
 e^{\varkappa x}-1, & x \geq 0,\\
 -e^{-\varkappa x}+1, & x \leq 0,
 \end{cases}
\end{equation*}
where $\varkappa>0$. Let $a, b $ be positive constants, $m>1$.
If $\varkappa >(2b/a) +m$, then we have
\begin{gather}\label{1.6}
 a\eta'(x)-b\eta(x)\ge \frac{a}{2} e^{\varkappa x}, \quad \forall x\ge0, \\
\label{1.7}
 \eta(x)\ge[\eta(\frac{x}{m})]^m, \quad \forall x\ge0.
\end{gather}
Moreover, there exist a $d\ge 0$ and an $M>0$ such that
\begin{gather}
 \eta(x)\le M\big[\eta\big(\frac{x}{m}\big)\big]^m \quad\text{and}\quad
\eta'(x)\le  M[\eta(\frac{x}{m})]^m,  \quad\forall x\ge d;\label{1.8}\\
 |\eta(x)|\ge x,\quad\forall x\in\mathbb{R}.\label{1.9}
 \end{gather}
\end{lemma}

Next we have Stampacchia's Lemma, see \cite[Lemma 3.11]{MS:68} and \cite{S:63}.

\begin{lemma} \label{degiorgi2}
 Let $\varphi : [k_0, \infty) \to \mathbb{R}$
be a non-negative and non-increasing function which satisfies
\begin{equation} \label{idi}
 \varphi(l) \leq \frac C {(l-k)^\alpha} [\varphi(k)]^\beta \quad \text{for }
 l> k > k_0,
\end{equation}
where $C, \alpha, \beta$ are positive constants with $\beta > 1$. Then
$$
 \varphi (k_0+\delta) = 0,\quad\text{where}\quad
 \delta^\alpha = C | \varphi(k_0)| ^{\beta-1} 2^{\alpha \beta / (\beta-1)}.
$$
\end{lemma}

Our main result is the following.


\begin{theorem}\label{MP}
Let $u(x)$ be a weak solution of \eqref{RQL}. If
 assumptions {\rm (i)--(iii)} hold, then
 there exists a constant $M_0>0$ depending only on
$\operatorname{meas}G$, $n$, $p_{\pm}$, $s$, $\mu$, $f_0$, $g_0$, $a_0$,
$\beta_0$, $\gamma$ and  such that $\|u\|_{L_{\infty}(G)}\le M_0$.
\end{theorem}

\begin{proof}
Let us define the set $A(k)=\{ x\in \overline {G}: |u(x)|>k\}$ and let
 $\chi_{ A(k)}$ be the characteristic function of
the set $A(k)$. We observe that $A(k+d)\subseteq A(k)$ for all
$d>0$.


Putting $\eta((|u|-k)_+)\chi_{ A(k)}\operatorname{sign}u$ as the test
function in \eqref{II}, where $\eta$ is defined by Lemma
\ref{Lem1.3} and $k\ge k_0$ (without loss of generality we can
assume $k_0\ge 1$), we obtain the inequality
\begin{equation}  \label{7} %\label{2.5}
\begin{aligned}
&\int_{A(k)} \Big\{|\nabla u|^{p(x)} \eta'((|u|-k)_+)+\langle a_0(x)|u|^{p(x)}\\
& + b(u, \nabla u) \operatorname{sign}u\rangle
\eta((|u|-k)_+)\Big\}\,dx
+ \gamma\int_{\Gamma\cap A(k)} \big(\frac{|u|}{r}\big)^{p(x)-1}\eta((|u|-k)_+)ds \\
& \le \int_{A(k)} |f(x)| \eta((|u|-k)_+)\,dx
+\int_{\Gamma\cap A(k)} |g(x)|\eta((|u|-k)_+)\,ds.
\end{aligned}
\end{equation}
By  assumptions  (i) and (iii), the inequality
\eqref{7} implies that
\begin{equation}
\begin{aligned}
&\int_{A(k)} \Big\{|\nabla u|^{p(x)} \langle\eta'((|u|-k)_+)-\mu
k_0^{-1}\eta((|u|-k)_+)\rangle \\
& +a_0|u|^{p(x)} \eta((|u|-k)_+)\Big\}\,dx \\
& + \int_{\Gamma\cap A(k)} \big(\gamma |u|^{p(x)-1} -g_0\big)r^{1-p(x)}
 \eta((|u|-k)_+)ds  \\
&\le \int_{A(k)} |f(x)|\eta((|u|-k)_+)\,dx.
\end{aligned}\label{8}
\end{equation}
On the other hand, by assumption  (i) and the definition of $A(k)$, we have
\begin{equation}\label{9}
 |u|^{p(x)}\geq k_0^{p_-}.
\end{equation}
Therefore, the inequality \eqref{8} can be rewritten as
\begin{equation}
\begin{aligned}
&\int_{A(k)} \Big\{|\nabla u|^{p(x)} \langle\eta'((|u|-k)_+)-\mu
k_0^{-1}\eta((|u|-k)_+)\rangle \\
&+a_0|u|^{p(x)} \eta((|u|-k)_+)\Big\}\,dx \\
&+ \int_{\Gamma\cap A(k)} \big(\gamma k_0^{p_- -1} -g_0\big)
 r^{1-p(x)}\eta((|u|-k)_+)\,ds \\
&\le \int_{A(k)} |f(x)| \eta((|u|-k)_+)\,dx.
\end{aligned}\label{10}
\end{equation}
We take
\begin{equation}\label{k01}
 k_0\geq\big(\frac{g_0}{\gamma}\big)^{\frac{1}{p_- -1}}
\end{equation}
and obtain
\begin{equation} \label{12}
\begin{aligned}
&\int_{A(k)} \Big\{|\nabla u|^{p(x)} \langle\eta'((|u|-k)_+)-\mu
k_0^{-1}\eta((|u|-k)_+)\rangle \\
&+a_0|u|^{p(x)} \eta((|u|-k)_+)\Big\}\,dx \\
&\le \int_{A(k)} |f(x)|\eta((|u|-k)_+)\,dx.
\end{aligned}
\end{equation}
Additionally, let us define the sets
\begin{equation}\label{13}
\begin{gathered}
A_-(k)  = A(k)\cap\{|\nabla u|\leq 1\}, \\
A_+(k)  = A(k)\cap\{|\nabla u|\geq 1\}.
\end{gathered}
\end{equation}
Then $ A(k)= A_-(k)\cup A_+(k)$.
Also we define the functions
\begin{equation}\label{14}
v_k(x):=\eta\Big(\frac{(|u|-k)_+}{p_-}\Big),\quad
w_k(x):=\eta\Big(\frac{(|u|-k)_+}{p_+}\Big).
\end{equation}
We note that the inequalities
\begin{gather}
|\nabla u|^{p_+}\leq |\nabla u|^{p(x)}\leq |\nabla u|^{p_-}\quad\text{on }
 A_-(k);\label{A-}\\
|\nabla u|^{p_-}\leq |\nabla u|^{p(x)}\leq |\nabla u|^{p_+}\quad\text{on }
 A_+(k)\label{A+}
\end{gather}
hold by (i).

Direct calculations give
\begin{equation}
\begin{gathered}
|\nabla v_k|=\frac{1}{p_-}|\nabla u|\eta'\Big(\frac{(|u|-k)_+}{p_-}\Big)
=\frac{\varkappa}{p_-}|\nabla u|\exp\Big(\varkappa\frac{(|u|-k)_+}{p_-}\Big),
\quad \varkappa>0 \\
\Longrightarrow\;
|\nabla v_k|^{p_-}=\big(\frac{\varkappa}{p_-}\big)^{p_-}|\nabla u|^{p_-}
 e^{\varkappa (|u|-k)_+},
\end{gathered} \label{15}
\end{equation}
where $\eta$ is given in Lemma \ref{Lem1.3}.
Choosing $\varkappa>p_- + \frac{2\mu}{k_0}$ according to \eqref{1.6}, we have
\begin{equation}\label{16}
\eta'((|u|-k)_+)-\mu k_0^{-1}\eta((|u|-k)_+)
\geq \frac{1}{2}e^{\varkappa (|u|-k)_+}.
\end{equation}
From \eqref{15} and \eqref{16} it follows that
\[
|\nabla u|^{p_-} \langle\eta'((|u|-k)_+)-\mu k_0^{-1}
\eta((|u|-k)_+)\rangle\geq\frac{1}{2}\big(\frac{p_-}{\varkappa}\big)^{p_-}
|\nabla v_k|^{p_-}
\]
which by \eqref{A+} implies
\begin{equation}
\begin{aligned}
&\int_{A_+(k)} |\nabla u|^{p(x)} \langle\eta'((|u|-k)_+)
 -\mu k_0^{-1}\eta((|u|-k)_+)\rangle\, dx \\
&\geq \int_{A_+(k)} |\nabla u|^{p_-} \langle \eta'((|u|-k)_+)
 -\mu k_0^{-1}\eta((|u|-k)_+)\rangle\,dx \\
&\geq \frac{1}{2}\big(\frac{p_-}{\varkappa}\big)^{p_-}
 \int_{A_+(k)}|\nabla v_k|^{p_-}\,dx.
\end{aligned} \label{17}
\end{equation}
Similarly, choosing $\varkappa>p_+ + \frac{2\mu}{k_0}$ and taking into
account \eqref{A-}, we obtain
\begin{equation}\label{18}
\begin{aligned}
&\int_{A_-(k)} |\nabla u|^{p(x)} \langle\eta'((|u|-k)_+)
-\mu k_0^{-1}\eta((|u|-k)_+)\rangle\,dx \\
&\geq \frac{1}{2}\big(\frac{p_+}{\varkappa}\big)^{p_+}
 \int_{A_-(k)}|\nabla w_k|^{p_+}\,dx.
\end{aligned}
\end{equation}
Since $p_+ \geq p_-$,  inequalities \eqref{17} and \eqref{18} hold for
$\varkappa>p_+ + \frac{2\mu}{k_0}$. Therefore, adding
 inequalities \eqref{17} and \eqref{18} we obtain
\begin{equation} \label{19}
\begin{aligned}
&\frac{1}{2}\big(\frac{p_-}{\varkappa}\big)^{p_-}
 \int_{A_+(k)}|\nabla v_k|^{p_-}\,dx
 + \frac{1}{2}\big(\frac{p_+}{\varkappa}\big)^{p_+}
 \int_{A_-(k)}|\nabla w_k|^{p_+}\,dx \\
&\leq \int_{A(k)} |\nabla u|^{p(x)} \langle\eta'((|u|-k)_+)
 -\mu k_0^{-1}\eta((|u|-k)_+)\rangle\,dx
\end{aligned}
\end{equation}
by \eqref{13}.
Finally, from \eqref{12} and \eqref{19} we derive
\begin{align*}
&\frac{1}{2}\big(\frac{p_-}{\varkappa}\big)^{p_-}
 \int_{A_+(k)}|\nabla v_k|^{p_-}\,dx
 + \frac{1}{2}\big(\frac{p_+}{\varkappa}\big)^{p_+}
 \int_{A_-(k)}|\nabla w_k|^{p_+}\,dx \\
&+ a_0 \int_{A(k)} |u|^{p(x)}\eta((|u|-k)_+)\,dx \\
&\le \int_{A(k)} |f(x)| \eta((|u|-k)_+)\,dx.
\end{align*}
Since $\int_{A(k)} = \int_{A_+(k)} +\int_{A_-(k)}$, by \eqref{13} we have
\begin{equation} \label{22}
\begin{aligned}
&\frac{1}{2}\big(\frac{p_-}{\varkappa}\big)^{p_-}
\int_{A_+(k)}|\nabla v_k|^{p_-}\,dx + \frac{1}{2}\big(\frac{p_+}{\varkappa}\big)^{p_+}
\int_{A_-(k)}|\nabla w_k|^{p_+}\,dx \\
&+a_0\int_{A_+(k)} |u|^{p(x)}\eta((|u|-k)_+)\,dx
 + a_0\int_{A_-(k)} |u|^{p(x)}\eta((|u|-k)_+)\,dx \\
& \le \int_{A_+(k)} |f(x)| \eta((|u|-k)_+)\,dx
 + \int_{A_-(k)} |f(x)| \eta((|u|-k)_+)\,dx.
\end{aligned}
\end{equation}
Now, by \eqref{1.7}, \eqref{9} and \eqref{14}, we derive
\begin{equation}
\begin{aligned}
&a_0\int_{A_+(k)} |u|^{p(x)}\eta((|u|-k)_+)\,dx
+ a_0\int_{A_-(k)} |u|^{p(x)}\eta((|u|-k)_+)\,dx \\
&\ge a_0k_0^{p_-}\Big(\int_{A_+(k)}
v_k^{p_-}\,dx  + \int_{A_-(k)}w_k^{p_+}\,dx\Big).
\end{aligned}\label{23}
\end{equation}
From \eqref{22} and \eqref{23} it follows that
\begin{equation} \label{24}
\begin{aligned}
&\frac{1}{2}\big(\frac{p_-}{\varkappa}\big)^{p_-}
\int_{A_+(k)}|\nabla v_k|^{p_-}\,dx
+ \frac{1}{2}\big(\frac{p_+}{\varkappa}\big)^{p_+}
\int_{A_-(k)}|\nabla w_k|^{p_+}\,dx \\
&+ a_0k_0^{p_-}\Big(\int_{A_+(k)} v_k^{p_-}\,dx
  + \int_{A_-(k)}w_k^{p_+}\,dx\Big) \\
&\le \int_{A_+(k)} |f(x)| \eta((|u|-k)_+)\,dx
 + \int_{A_-(k)} |f(x)| \eta((|u|-k)_+)\,dx.
\end{aligned}
\end{equation}
Next, we have
\begin{equation} \label{25}
\begin{aligned}
&\int_{A_{\pm}(k)} |f(x)| \eta((|u|-k)_+)\,dx \\
& =\int_{A_{\pm}(k+d)} |f(x)| \eta((|u|-k)_+)\,dx\\
&\quad + \int_{A_{\pm}(k)\setminus A_{\pm}(k+d)} |f(x)| \eta((|u|-k)_+)\,dx,
\quad \forall d>0.
\end{aligned}
\end{equation}

By \eqref{1.8}, we obtain
$$
\eta((|u|-k)_+)\Big|_{A_{\pm}(k+d)}
\leq M\Big[\eta\Big(\frac{(|u| -k)_+}{p_{\mp}}\Big)\Big]^{p_{\mp}}.
$$
Then \eqref{14} implies
\begin{gather}
\int_{A_{+}(k+d)} |f(x)| \eta((|u|-k)_+)\,dx
 \leq M\int_{A_{+}(k+d)} |f(x)| v_k^{p_-}\,dx;\label{26}\\
\int_{A_{-}(k+d)} |f(x)| \eta((|u|-k)_+)\,dx
\leq M\int_{A_{-}(k+d)} |f(x)| w_k^{p_+}\,dx.\label{27}
\end{gather}
Using the definition of $\eta$ from Lemma \ref{Lem1.3}, we arrive to
\[
\eta((|u|-k)_+)\Biggl |_{A_{\pm}(k)\setminus A_{\pm}(k+d)}
\leq e^{\varkappa d},\quad\forall d>0
\]
which implies
\begin{equation}
\int_{A_{\pm}(k)\setminus A_{\pm}(k+d)} |f(x)| \eta((|u|-k)_+)\,dx
\leq e^{\varkappa d}\int_{A_{\pm}(k)\setminus A_{\pm}(k+d)} |f(x)|\,dx,
\label{28}
\end{equation}
for all $d>0$.
Now, we recall \cite[formula (6.3.9)  page 145]{B}:
\begin{equation}
\begin{gathered}
\begin{aligned}
&\int_{A_{+}(k+d)} |f(x)| v_k^{p_-}\,dx \\
&\leq \varepsilon (1-\theta_-)\Big(\int_{A_{+}(k)} v_k^{p_-^\sharp}\,dx
 \Big)^{\frac{p_-}{p_-^\sharp}} + \theta_-\varepsilon^{\frac{\theta_- -1}{\theta_-}}
\|f\|_{L_{s}(G)}^{\frac{1}{\theta_-}}\int_{A_{+}(k)} v_k^{p_-}\,dx,
\end{aligned}\\
\begin{aligned}
&\int_{A_{-}(k+d)} |f(x)|  w_k^{p_+}\,dx \\
&\leq \varepsilon (1-\theta_+)\Big(\int_{A_{-}(k)} w_k^{p_+^\sharp}\,dx
 \Big)^{\frac{p_+}{p_+^\sharp}} + \theta_+\varepsilon^{\frac{\theta_+ -1}{\theta_+}}
 \|f\|_{L_{s}(G)}^{\frac{1}{\theta_+}}\int_{A_{-}(k)} w_k^{p_+}\,dx,
\end{aligned}\\
\forall\varepsilon >0,\; p_{\mp}^\sharp =\frac{np_{\mp}}{n-p_{\mp}},\;
\theta_{\mp}=1-\frac{n}{sp_{\mp}},
 s >\max\{\frac{n}{p_-},\;\frac{n}{p_+}\}=\frac{n}{p_-}>1.
\end{gathered}\label{28a}
\end{equation}
Then applying \eqref{28a} to \eqref{25}--\eqref{28}, we obtain
\begin{equation}\label{29}
\begin{gathered}
\begin{aligned}
&\int_{A_{+}(k)} |f(x)| \eta((|u|-k)_+)\,dx \\
& \leq M\varepsilon (1-\theta_-)\Big(\int_{A_{+}(k)} v_k^{p_-^\sharp}\,dx
 \Big)^{\frac{p_-}{p_-^\sharp}}+e^{\varkappa d}\int_{A_{+}(k)}
 |f(x)|\,dx \\
&\quad + M\theta_-\varepsilon^{\frac{\theta_- -1}{\theta_-}}\|f\|_{L_{s}(G)}
 ^{\frac{1}{\theta_-}}\int_{A_{+}(k)} v_k^{p_-}\,dx
\end{aligned} \\
\begin{aligned}
&\int_{A_{-}(k)} |f(x)| \eta((|u|-k)_+)\,dx \\
& \leq M \varepsilon (1-\theta_+)\Big(\int_{A_{-}(k)} w_k^{p_+^\sharp}\,dx
 \Big)^{\frac{p_+}{p_+^\sharp}}+e^{\varkappa d}\int_{A_{-}(k)} |f(x)|\,dx \\
&\quad + M \theta_+\varepsilon^{\frac{\theta_+ -1}{\theta_+}}
 \|f\|_{L_{s}(G)}^{\frac{1}{\theta_+}}\int_{A_{-}(k)} w_k^{p_+}\,dx
\end{aligned}
\end{gathered}
\end{equation}
By well known the Sobolev embedding theorem and taking into account \eqref{28a},
we obtain
\begin{equation}\label{30}
\begin{gathered}
\Big(\int_{A_{+}(k)} v_k^{p_-^\sharp}\,dx \Big)^{\frac{p_-}{p_-^\sharp}}
 \leq c_-\int_{A_{+}(k)}(v_k^{p_-} + |\nabla v_k|^{p_-})\,dx; \\
\Big(\int_{A_{-}(k)} w_k^{p_+^\sharp}\,dx \Big)^{\frac{p_+}{p_+^\sharp}}
 \leq c_+\int_{A_{-}(k)}(w_k^{p_+} + |\nabla w_k|^{p_+})\,dx,
\end{gathered}
\end{equation}
where $c_{\mp}$ are positive constants.
Finally, \eqref{24}--\eqref{30} imply that
\begin{equation}
\begin{aligned}
&\big[\frac{1}{2}\big(\frac{p_-}{\varkappa}\big)^{p_-}-Mc_-(1-\theta_-)\varepsilon\big]
 \int_{A_+(k)}|\nabla v_k|^{p_-}\,dx \\
&+\big[\frac{1}{2}\big(\frac{p_+}{\varkappa}\big)^{p_+}-Mc_+(1-\theta_+)\varepsilon\big]
\int_{A_-(k)}|\nabla w_k|^{p_+}\,dx \\
&+ \big[a_0k_0^{p_-}-Mc_-(1-\theta_-)\varepsilon -M\theta_-
 \varepsilon^{\frac{\theta_- -1}{\theta_-}}\|f\|_{L_{s}(G)}^{\frac{1}{\theta_-}}\big]
 \int_{A_+(k)} v_k^{p_-}\,dx \\
& + \big[a_0k_0^{p_-}-Mc_+(1-\theta_+)\varepsilon
 - M \theta_+\varepsilon^{\frac{\theta_+ -1}{\theta_+}}
 \|f\|_{L_{s}(G)}^{\frac{1}{\theta_+}}\big]
 \int_{A_-(k)}w_k^{p_+}\,dx\\
&\le e^{\varkappa d}\int_{A(k)} |f(x)|\,dx,\quad \forall\varepsilon >0.
\end{aligned}\label{32}
\end{equation}
Further, at first, we choose
\begin{equation}\label{ep}
\varepsilon = \frac{1}{4M}\min\Big\{\frac{1}{c_-(1-\theta_-)}
\big(\frac{p_-}{\varkappa}\big)^{p_-}, \;
\frac{1}{c_+(1-\theta_+)}\big(\frac{p_+}{\varkappa}\big)^{p_+}\Big\}
\end{equation}
and next
\begin{equation}\label{k0}
k_0\geq\Big(\frac{2MF}{a_0}\Big)^{\frac{1}{p_-}},
\end{equation}
where
\[
 F=\max\Big\{c_-(1-\theta_-)\varepsilon +\theta_-
 \varepsilon^{\frac{\theta_- -1}{\theta_-}}
 \|f\|_{L_{s}(G)}^{\frac{1}{\theta_-}};\;
  c_+(1-\theta_+)\varepsilon + \theta_+\varepsilon^{\frac{\theta_+ -1}{\theta_+}}
 \|f\|_{L_{s}(G)}^{\frac{1}{\theta_+}}\Big\}.
\]
Thus, by the above arguments, we derive
\begin{equation}\label{33}
\int_{A_+(k)}\big(|\nabla v_k|^{p_-}+v_k^{p_-}\big)\,dx
 + \int_{A_-(k)}\big(|\nabla w_k|^{p_+} + w_k^{p_+}\big) \,dx
\leq C\int_{A(k)} |f(x)|\,dx,
\end{equation}
where $C={\rm const}(n, p_-, p_+, a_0, k_0, \mu, s, \|f\|_{L_s(G)}) >0$.
The inequalities \eqref{30} and \eqref{33} give
\begin{equation}\label{36}
\Big(\int_{A_{+}(k)} v_k^{p_-^\#}\,dx \Big)^{\frac{p_-}{p_-^\#}}
 + \Big(\int_{A_{-}(k)} w_k^{p_+^\#}\,dx \Big)^{\frac{p_+}{p_+^\#}}
\leq \max\{c_-,c_+\} C\int_{A(k)} |f(x)|\,dx,
\end{equation}
for all $k\geq k_0$.
At last, by the H\"older inequality, we have
\begin{equation*}
\int_{A(k)} |f(x)|\,dx\le \|f(x)\|_{L_s(G)} \operatorname{meas}^{1-\frac{1}{s}}A(k);\quad
s>\frac{n}{p_-}>1.
\end{equation*}
Then from \eqref{36} it follows that
\begin{equation} \label{37}
\begin{aligned}
&\Big(\int_{A_{+}(k)} v_k^{p_-^\#}\,dx \Big)^{\frac{p_-}{p_-^\#}}
+ \Big(\int_{A_{-}(k)} w_k^{p_+^\#}\,dx \Big)^{\frac{p_+}{p_+^\#}} \\
&\leq\max\{c_-,c_+\}  C\|f(x)\|_{L_s(G)} \operatorname{meas}^{1-\frac{1}{s}}A(k),
\quad s>\frac{n}{p_-}>1,\;\forall k\geq k_0.
\end{aligned}
\end{equation}
Now, let $l>k>k_0$. By \eqref{1.9} and the definition of the functions
$v_k(x),\,w_k(x)$, we have
 $v_k\geq\frac{1}{p_-}(|u|-k)_+$, $w_k\ge\frac{1}{p_+}(|u|-k)_+$. Therefore,
\begin{equation*}
\int_{A_{+}(l)} v_k^{p_-^\#}\,dx
 \ge\big(\frac{l-k}{p_-}\big)^{p_-^\#}\operatorname{meas}A_{+}(l),\quad
\int_{A_{-}(l)} w_k^{p_+^\#}\,dx
\ge\big(\frac{l-k}{p_+}\big)^{p_+^\#}\operatorname{meas}A_{-}(l).
\end{equation*}
Hence, \eqref{37} together with $A_{\pm}(l)\subseteq A_{\pm}(k) $ imply that
\begin{equation} \label{38}
\begin{aligned}
&\operatorname{meas}A(l)
 =\operatorname{meas}\big(A_{+}(l)\cup A_{-}(l)\big)
 \leq \operatorname{meas}A_{+}(l) + \operatorname{meas}A_{-}(l) \\
&\le \big(\frac{p_-}{l-k}\big)^{p_-^\#}\int_{A_{+}(k)} v_k^{p_-^\#}\,dx
+ \big(\frac{p_+}{l-k}\big)^{p_+^\#}\int_{A_{-}(k)} w_k^{p_+^\#}\,dx\\
&\leq  C_-\big(\frac{p_-}{l-k}\big)^{p_-^\#} \|f(x)\|_{L_s(G)}^{\frac{p_-^\#}{p_-}}
\operatorname{meas}^{\frac{p_-^\#}{p_-}(1-\frac{1}{s})}A(k) \\
&\quad + C_+\big(\frac{p_+}{l-k}\big)^{p_+^\#}
 \|f(x)\|_{L_s(G)}^{\frac{p_+^\#}{p_+}} \operatorname{meas}^{\frac{p_+^\#}{p_+}
 (1-\frac{1}{s})}A(k)
\end{aligned}
\end{equation}
for all $l>k\ge k_0$, where $C_{\mp}=\big(C\max\{c_-,c_+\}
\big)^{p_{\mp}^\#/p_{\mp}}$.
Since $\frac{p_-^\#}{p_-}\leq\frac{p_+^\#}{p_+}$ (see \eqref{28a}), we have
$$\operatorname{meas}^{\frac{p_-^\#}{p_-}(1-\frac{1}{s})}A(k)
\geq \operatorname{meas}^{\frac{p_+^\#}{p_+}(1-\frac{1}{s})}A(k),\quad
\text{if }\operatorname{meas}A(k)\leq 1.
$$
Moreover,
$$
\frac{p_+^\#}{p_+}(1-\frac{1}{s})\geq \frac{p_-^\#}{p_-}(1-\frac{1}{s})>1\quad
\text{for } s>\frac{n}{p_-}>1.
$$
Let us introduce $\psi (k)=\text{meas }A(k)$. Then from \eqref{38} it
follows that
\begin{equation*}
\psi (l)\leq 2\widetilde{C}\psi^{\zeta}(k)
\begin{cases}
\frac{1}{(l-k)^{p_-^\#}}\;\text{if}\quad l-k\geq 1;\\
\frac{1}{(l-k)^{p_+^\#}}\;\text{if}\quad 0< l-k< 1,
\end{cases}
\end{equation*}
 for all $l>k\ge k_0$,
where $\zeta = (1-\frac{1}{s})\frac{n}{n-p_-}>1$,
\[
\widetilde{C}= {\rm const }(n,p_-, p_+,a_0, k_0, \mu, s, \|f\|_{L_s(G)})>0.
\]
By the Stampacchia Lemma, we have that
 $\psi (k_0+\delta)=0$
with $\delta$ depending only on the quantities given in
Theorem \ref{MP}. This fact means that $|u(x)|\leq k_0+\delta$ for
almost all $x\in G$. Thus, we derive $M_0=k_0+\delta$, where $k_0$ is defined
 by \eqref{k01}, \eqref{k0} with \eqref{28a} and \eqref{ep}.
Then Theorem \ref{MP} is proved.
\end{proof}

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\end{document}