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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2019 (2019), No. 01, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2019 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2019/01\hfil Very weak solution for nonlinear elliptic system]
{Integrability of very weak solution to the Dirichlet problem of nonlinear
elliptic system}

\author[Y. Tong, S. Liang, S. Zheng \hfil EJDE-2019/01\hfilneg]
{Yuxia Tong, Shuang Liang, Shenzhou Zheng}

\address{Yuxia Tong \newline
Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China.\newline
College of Science,
North China University of Science and Technology,
Hebei Tangshan 063210, China}
\email{tongyuxia@bjtu.edu.cn}

\address{Shuang Liang \newline
Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China}
\email{shuangliang@bjtu.edu.cn}

\address{Shenzhou Zheng (corresponding author) \newline
Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China}
\email{shzhzheng@bjtu.edu.cn}

\thanks{Submitted December 16, 2017. Published January 2, 2019.}
\subjclass[2010]{35D30, 35K10}
\keywords{Integrability; very weak solution; nonlinear elliptic system;
\hfill\break\indent controllable growth}

\begin{abstract}
 This article concerns the higher integrability of a very weak solution
 $u\in \theta+ W_0 ^{1,r}(\Omega)$ for $\max \{1,p-1\}<r<p<n$ to the
 Dirichlet problem of the nonlinear elliptic system
\begin{gather*}
 -D_\alpha\mathbf{A}_i^\alpha(x,Du)= \mathbf{B}_i(x,Du) \quad \text {in }\Omega,\\
 u=\theta \quad \text {on } \partial\Omega,
 \end{gather*}
 where $\mathbf{A}(x,Du)=\big(\mathbf{A}_i^\alpha(x,Du)\big) $ for
 $\alpha=1,\dots,n$ and $i=1,\dots,m$, and each entry of
 $\mathbf{B}(x,Du)=\big(\mathbf{B}_i(x,Du)\big) $ for $i=1,\dots,m$ satisfies
 the monotonicity and controllable growth. If $\theta \in W^{1,q}(\Omega)$
 for $q>r$, then we derive that the very weak solution $u$ of above-mentioned
 problem is integrable with
 \[
 u\in \begin{cases}
 \theta +L_{\rm weak}^{q^*} (\Omega) & \text{for }1\le q<n,\\
 \theta +L^\tau(\Omega) & \text{for $q=n$  and } 1<\tau<\infty,\\
 \theta +L^\infty (\Omega)  & \text{for } q>n,
 \end{cases}
 \]
 provided that $r$ is sufficiently close to $p$, where $q^*=qn/(n-q)$.
\end{abstract}

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

\section{Introduction}

Let $\Omega\subset \mathbb{R}^{n} $ for $n\ge 2$ be a bounded regular domain. 
By regular domain we understand the domain with a finite measure for which 
the Hodge decomposition Lemma \ref{Hodge-decomposition} below is satisfied. 
The domains with Lipschitz and A-type boundary, for example, always are regular. 
The purpose of this present article is to study a global higher integrability 
of very weak solution to the Dirichlet problem of nonlinear elliptic system:
\begin{equation}\label{elliptic-system}
\begin{gathered}
-D_\alpha\mathbf{A}_i^\alpha(x,Du)= \mathbf{B}_i(x,Du)\quad  \text{in } \Omega,\\
u=\theta \quad  \text {on } \partial\Omega,
\end{gathered}
\end{equation}
where $m\geq 2$ and $\theta (x)\in W^{1,q}(\Omega,\mathbb{R}^{m})$ for 
$q>r$ with $r$ determined later. In the context, we let $1<p<n$, and assume 
that $\mathbf{A}(x,Du)=\big(\mathbf{A}_i^\alpha(x,Du)\big) $ with 
$\alpha=1,\dots,n $ and $i=1,\dots,m$ satisfies the following monotonicity 
and controllable growth: there exist positive constants 
$0<\lambda\le \Lambda_1,\Lambda_2$ such that
\begin{equation} \label{uniform-bound-ellipticity}
\begin{gathered}
  |\mathbf{A}_i^\alpha(x,\xi)|\leq \Lambda_1(|\xi|^{p-1}+a(x)),  \\
  \langle \mathbf{A}(x,\xi_1)-\mathbf{A}(x,\xi_2), \xi_1-\xi_2 \rangle 
\geq \lambda|\xi_1-\xi_2|^p \quad \forall\xi_1,\xi_2 \in\mathbb{R}^{n}\setminus\{0\};
  \end{gathered}
\end{equation}
and $\mathbf{B}(x,Du)=\big(\mathbf{B}_i(x,Du)\big)$ for $i=1,\dots,m$ satisfies
\begin{equation}\label{B-growth}
 |\mathbf{B}_i(x,\xi)|\leq \Lambda_2(|\xi|^{p-\delta}+b(x))
\end{equation}
with $1<\delta<p$, $a(x)\in L^{\frac{q}{p-1}}(\Omega,\mathbb{R}^{m})$ and 
$b(x)\in L^{\frac{nq}{q+np-n}}(\Omega,\mathbb{R}^{m})$.

First of all, let us recall the notation of  \emph{very weak solutions} 
to the Dirichlet problem of \eqref{elliptic-system}.
A mapping $u\in \theta +W_0^{1,r}(\Omega,\mathbb{R}^{m})$ with 
$\max \{1, p-1\} <r<p$ is called a very weak solution to the Dirichlet 
problem \eqref{elliptic-system} if
\begin{equation}\label{very-weak-solution}
\int_\Omega \langle \mathbf{A}(x,Du), D \varphi \rangle dx
 =\int_\Omega \mathbf{B}(x,Du)\cdot \varphi dx
\end{equation}
holds  for all $\varphi \in W_0^{1,r/(r-p+1)} (\Omega,\mathbb{R}^{m})$.


On the basis of the above definition, a crucial fact is that the integrable 
exponent $r$ of $u$ can be smaller than the natural index $p$, which is different 
from the usual hypothesis of  classical weak solution 
$u\in \theta +W_0^{1,p}(\Omega,\mathbb{R}^{m})$. Here, we would like 
to recall recent progresses involving the topic of very weak solution. 
Iwaniec \cite{Twan92} first put forward the concept of the so-called very weak 
solutions for $p$-harmonic tensors and weakly quasiregular mappings with 
the integrability of their weak derivatives being below natural exponent. 
Furthermore, Iwaniec-Sbordone \cite{TwS94} and Iwaniec-Scott-Stroffolini \cite{TwSS99} 
 considered a self-improving regularity for weak minima of variational integrals 
and weakly $p$-harmonic type equations with $r$ sufficiently  close to $p$ 
from lower side, respectively; and got that such very weak solution for 
variational integrals and $p$-harmonic type equations is actually a weak 
solution in the classical sense by way of the so-called Hodge decomposition 
argument concerning disturbance vector field. 
On the other hand, Lewis \cite{Lew93} also obtained a self-improving 
integrability for the derivatives of very weak solutions to certain
 nonlinear elliptic systems by way of the technique of harmonic analysis 
which is rather different from Iwaniec's argument.
 Later, Lewis' harmonic technique was extended to the settings of parabolic
 systems of $p$-Laplacian \cite{KiL00,KiL02}, and various elliptic and 
parabolic systems with non-standard growths 
\cite{BoL07,BoL09,BoL14,Li17}, respectively. This essentially is attained 
by a self-improving integrability of the weak derivatives based on the 
validity of the generalized reverse H\"older inequality \cite{Gi83}. 
In the following, we would like to mention that Greco and Iwaniec 
in \cite{GrIS97} dealt with the nonhomogeneous $p$-harmonic equation
\[
-\operatorname{div} (|\nabla u(x)|^{p-2} \nabla u(x))=-\operatorname{div}f,
\]
and obtained an estimate for the operator $\mathcal H$ which carries given 
vector function $f$ into the gradient field $\nabla u$. Later, 
Zheng-Fang \cite{ZhF98} further considered a local very weak solutions 
for nonlinear elliptic systems \eqref{elliptic-system} with that $\mathbf{B}$ 
satisfies \eqref{B-growth}, $\mathbf{A}(x,Du)$ satisfies 
\eqref{uniform-bound-ellipticity} and
\[
\sum_{1\leq i\leq m,\, 1\leq\alpha\leq n} \mathbf{A}_i^\alpha(x,\xi)\xi_i^\alpha
\geq \lambda|\xi|^p\quad \forall\xi\in\mathbb{R}^{n}\setminus\{0\},
\]
and obtained a self-improving integrability for the derivatives of very 
weak solutions on the basis of the so-called Hodge decomposition of 
perturbation vector fields. For more results for very weak solutions, 
see \cite{BuSC16,GrIS97,Lew93,Twan92}.

The problem under consideration in this paper is global integrability 
property in line with the regularity of boundary data, which is important 
among the regularity theories of nonlinear elliptic PDEs and systems. 
In \cite{GaLC16}, Gao-Liang-Cui studied very weak solution to the following 
boundary value problems of $p$-Laplacian
\begin{gather*}
-\operatorname{div} (|\nabla u(x)|^{p-2} \nabla u(x))=0  \quad x\in \Omega, \\
u(x)=\theta(x)  \quad  x\in \partial \Omega,
\end{gather*}
and obtained a global integrability result, which shows that higher 
integrability of the a boundary datum $\theta$ forces the very weak 
solution $u$ to have a higher integrability. For more information on 
this topic, we refer the readers to \cite{LeS12,LeS14}.

To this end, let us recall some related notations and basic facts. 
The \emph{weak $L^t$-spaces} or
\emph{Marcinkiewicz spaces} (see \cite{GaLC16}) for open subset
 $\Omega \subset \mathbb{R}^{n}$  with parameter $t>0$ is the set of 
all measurable functions $f$ by requiring
\[
|\{x\in \Omega: |f(x)|>s\}| \le \frac k {s ^t}
\]
for some positive constant $k=k(f)$ and every $s >0$, where $|E|$ is 
the $n$-dimensional Lebesgue measure of $E$. We can denoted it by the 
weak $L^t$-space or $L_{\rm weak}^t(\Omega)$.
Note that if $f\in L_{\rm weak}^t (\Omega)$ for some $t>1$ and $|\Omega|<\infty$,
then $f\in L^\tau (\Omega)$ for every $1\le \tau <t$.
Now we are ready to state the main result of this paper.

\begin{theorem}\label{main-result} 
Let $\theta \in W^{1,q}(\Omega,\mathbb{R}^{m})$ for $q>r$. 
Suppose that the operator $\mathbf{A}(x,Du)$ and $\mathbf{B}(x,Du)$ 
satisfy the structural conditions \eqref{uniform-bound-ellipticity} 
and \eqref{B-growth}. Then there exists a constant 
$\varepsilon _0=\varepsilon_0 (n,m,p,\Lambda_1,\Lambda_2,\lambda)>0$, 
such that for every very weak solution 
$u\in \theta +W_0^{1,r}(\Omega,\mathbb{R}^{m})$ for $\max\{1,p-1\}<r<p<n$, 
to the boundary value problem \eqref{elliptic-system}, we have
\begin{equation}\label{result}
u\in \begin{cases}
\theta +L_{\rm weak}^{q^*} (\Omega)  & \text{for } 1\le q<n,\\
\theta +L^\tau (\Omega)  & \text{for } q=n \text{ and  } 1\le \tau <\infty,\\
\theta +L^\infty (\Omega)  & \text{for } q>n,
\end{cases}
\end{equation}
provided that $|p-r|<\varepsilon _0$, where $q^*=\frac {qn}{n-q}$.
\end{theorem}

This article proposes a new way to obtain more properties for general 
elliptic problems, that than those in \cite{ZhF98,Zh08}. 
We have restricted ourselves to the case $\max\{1,p-1\}<r<n$, otherwise any 
function in $W^{1,r}(\Omega)$ for $r\ge n$ is in the space $L^t(\Omega)$ 
for any $1\le t<\infty$ by Sobolev embedding theorem.
As above-mentioned, our proof is inspired by Gao et al and Zheng et al
 \cite{GaLC16,GaLW18,ZhF98,Zh08}. Since for very weak solution one cannot 
take a test function by using a usual weak formulation in the boundary 
value problem \eqref{very-weak-solution}. For this, we have to construct
 a suitable test function by the argument of \emph{Hodge decomposition}. 
That is to say, a main key ingredient is based on choosing an appropriate 
test functions by the so-called Hodge decomposition \cite{TwS94,ZhF98}; 
then we attain our aim in line with Stampacchia lemma \cite{GaLW18}.

The rest of the paper is organized as follows. 
In section 2, we are devoted to presenting some useful lemmas. 
In section 3, we focus on proving our main theorem.

\section{Technical tools}

In this section, we introduce some useful lemmas, which will play essential 
roles in proving our main result. 
Let us denote by $c(n,m,p,\lambda,\Lambda_1,\Lambda_2, \dots)$ a universal 
constant depending only on prescribed quantities and possibly varying from 
line to line in the following context. We  first give a technical lemma 
called Hodge decomposition involved  vector fields, see \cite[Lemma 2.2]{ZhF98}.

\begin{lemma}\label{Hodge-decomposition}
Assume $v\in W_0^{1,r}(\Omega,\mathbb{R}^{m})$ with $\max\{1,p-1\}<r<p$. 
Then there exist $\varphi\in W^{1,\frac{r}{r-p+1}}(\Omega,\mathbb{R}^{m})$ 
and  divergence free matrix field 
$h\in L^{\frac{r}{r-p+1}}(\Omega,\mathbb{R}^{n\times m})$ such that
\[
|\nabla v|^{r-p}\nabla v=\nabla\varphi+h;
\]
moreover,
\[
\|h\|_{L^{\frac{r}{r-p+1}}(\Omega)}\leq c|p-r|\|\nabla v\|_{L^r(\Omega)}^{r-p+1},
\]
where $c=c(n,r,\Omega)$.
\end{lemma}

An efficient tool is the well-known Stampacchia Lemma, which is presented 
in the following lemma, see \cite[Lemma 4.1]{Sta66} or \cite{GaLW18}.

\begin{lemma}\label{technical-result}
Let $\alpha,\beta$ be two positive constants. 
Let $\phi: [s_0, +\infty)\rightarrow [0, +\infty)$ be decreasing and such that
$$
\phi(r)\le  \frac c{(r-s)^{\alpha}}[\phi(s)]^{\beta}
$$
with constants $c>0$ and $r>s\ge s_0$. Then, it leads to the following conclusions:
\begin{itemize}
\item[(i)] if $\beta>1$, we have $\phi(s_0 + d) = 0$ with 
\[
d= \Big(c 2^{\frac {\alpha \beta }{\beta -1}} 
(\phi (s_0)) ^{\beta-1}\Big)^{1/\alpha}.
\]

\item[(ii)] if $\beta =1$, for any  $s\geq s_0$ we have
\[
\phi (s) \leq  \phi (s_0) e^{1-(ce)^{-\frac{1}{\alpha}}(s-s_0)}.
\]

\item[(iii)] if $\beta <1$, for any $s\geq s_0>0$ we have
\[
\phi (s) \le 2^{\frac {\alpha}{(1-\beta)^2}} 
\Big(c^{\frac 1 {1-\beta} }+(2s_0)^{\frac {\alpha}{1-\beta}} \phi (s_0)  
\Big)\Big(\frac{1}{s}\Big)^{\frac {\alpha}{1-\beta}}.
\]
\end{itemize}
\end{lemma}

\section{Proof of Theorem \ref{main-result}}

\begin{proof}
For any $L>0$, we take
\begin{equation}\label{v-def}
v=\begin{cases}
u-\theta +L  & \text{for } u-\theta<-L, \\
0  & \text{for } -L\le u-\theta \le L, \\
u-\theta-L  & \text{for } u-\theta >L,
\end{cases}
\end{equation}
such that, by our assumptions we have $v\in W_0^{1,r}(E)$ with 
$E=\{|u-\theta|>L\}$ and
\begin{equation}\label{nabla-v}
\nabla v =(\nabla u -\nabla \theta)\cdot 1 _{\{|u-\theta|>L\}} \quad  \text {in} \  E.
\end{equation}
Now we introduce the Hodge decomposition involving  disturbance vector field 
$|\nabla v|^{p-2} \nabla v \in L^{r/(r-p+1)} (E)$ shown in Lemma 
\ref{Hodge-decomposition}. Accordingly,
\begin{equation}\label{phi-def}
|\nabla v|^{r-p} \nabla v =\nabla \varphi +h
\end{equation}
with $\varphi \in W_0^{1,r/(r-p+1)}(E)$ and  divergence free matrix field 
$h\in L^{r/(r-p+1)}(E,\mathbb{R}^{n\times m})$. Then we have
\begin{gather}\label{phi-integrability}
\|\nabla \varphi\|_{L^{r/(r-p+1)}(E)} \le C(n,p) \|\nabla v\|_{L^{r}(E)}^{r-p+1},\\
\label{h-def}
\|h\|_{L^{r/(r-p+1)}(E)} \le C(n,p) |p-r|\|\nabla v\|_{L^{r}(E)}^{r-p+1}.
\end{gather}
Extending $\varphi$ by zero value from $E$ to $ \overline{\Omega}$, 
then the above-mentioned term  $\varphi \in W_0^{1,r/(r-p+1)}(\Omega)$ 
can be used as a test function for the integral identity \eqref{very-weak-solution}, 
which yields that
\[
\int_{\Omega} \langle \mathbf{A}(x,D u), D \varphi \rangle dx 
=\int_\Omega  \mathbf{B}(x, D u)\cdot \varphi  dx.
\]
By \eqref{uniform-bound-ellipticity} and Hodge decomposition \eqref{phi-def} 
we conclude that
\begin{align*}
& \int_{\Omega} \langle \mathbf{A}(x,D u), D \varphi \rangle dx \\
& = \int_{E} \langle \mathbf{A}(x,D u), |D v|^{r-p} D v-h \rangle dx\\
& = \int_{E} \langle \mathbf{A}(x,D u),  |D u-D \theta|^{r-p} (D u -D \theta) 
 \rangle dx - \int_{E} \langle \mathbf{A}(x,Du), h \rangle dx\\
& = \int_{E} \langle \mathbf{A}(x,Du)-\mathbf{A}(x,D\theta), 
 (D u -D \theta) \rangle |D u-D\theta|^{r-p}dx \\
&\quad +\int_{E} \langle \mathbf{A}(x,D\theta), (Du -D \theta) 
 \rangle |D u-D\theta|^{r-p}dx- \int_{E} \langle \mathbf{A}(x,Du), h \rangle dx\\
& \geq \lambda\int_{E} |D u-D\theta|^r dx +\int_{E} \langle \mathbf{A}(x,D\theta),
  (Du -D \theta) \rangle |D u-D\theta|^{r-p}dx\\
&\quad - \int_{E} \langle \mathbf{A}(x,Du), h \rangle dx,
\end{align*}
which implies 
\begin{equation}\label{u-theta-a}
\begin{aligned}
\int_{E} |D u-D \theta|^r dx 
 & \leq  c\int_{E} |\mathbf{A}(x,D\theta)| |D u-D\theta|^{r-p+1}dx\\
&\quad  +\int_{E} \langle \mathbf{A}(x,Du), h \rangle dx 
 +\int_\Omega  \mathbf{B}(x,Du)\cdot \varphi \, dx\\
&:= c(I_1+I_2+I_3).
\end{aligned}
\end{equation}
Using \eqref{uniform-bound-ellipticity}, \eqref{B-growth}, \eqref{h-def},
 H\"older inequality and Young inequality we deduce that $I_1,I_2,I_3$ 
can be estimated as follows:
\begin{equation}\label{I-1}
\begin{aligned}
  I_1 &\leq \int_{E} |\mathbf{A}(x,D\theta)||D u-D \theta|^{r-p+1} dx \\
 & \leq \Lambda_1\int_{E} \Big(|D\theta|^{p-1}+a(x)\Big) 
 |D u-D \theta|^{r-p+1} dx \\
  & \leq \varepsilon\cdot c\int_{E} |D u-D \theta|^r dx+c(\varepsilon)\int_{E} 
 |D \theta|^r dx+c(\varepsilon)\int_{E} |a(x)|^{\frac{r}{p-1}} dx
\end{aligned}
\end{equation}
with small $\varepsilon>0$ determined later. For the estimate of $I_2$, 
we derive that
\begin{align}
  I_2 &\leq  \int_{E} |\mathbf{A}(x,Du)| |h|dx  \nonumber\\
& \leq \Lambda_1\int_{E} \Big(|D u|^{p-1}+a(x)\Big) |h| dx  \nonumber\\
& \leq 2^{p-2}\Lambda_1\Big(\int_{E} |D u-D \theta|^{p-1} |h| dx
 +\int_{E} |D \theta|^{p-1}|h|dx\Big)+ \Lambda_1\int_{E} a(x)|h| dx \nonumber\\
& \leq c\Big(\int_{E} |D u-D \theta|^rdx\Big)^{\frac{p-1}{r}}
 \Big(\int_{E} |h|^{\frac{r}{r-p+1}}dx\Big)^{\frac{r-p+1}{r}} \nonumber\\
&\quad +c\Big(\int_{E} |D \theta|^rdx\Big)^{\frac{p-1}{r}}
 \Big(\int_{E} |h|^{\frac{r}{r-p+1}}dx\Big)^{\frac{r-p+1}{r}} \nonumber\\
&\quad +c\Big(\int_{E} |a(x)|^{\frac{r}{p-1}}dx\Big)^{\frac{p-1}{r}}
 \Big(\int_{E} |h|^{\frac{r}{r-p+1}}dx\Big)^{\frac{r-p+1}{r}} \nonumber\\
& \leq  c|p-r|\Big(\int_{E} |D u-D \theta|^rdx\Big)^{\frac{p-1}{r}}
 \Big(\int_{E} |D u-D \theta|^{r}dx\Big)^{\frac{r-p+1}{r}} \nonumber\\
&\quad +c|p-r|\Big(\int_{E} |D \theta|^rdx\Big)^{\frac{p-1}{r}}
 \Big(\int_{E} |D u-D \theta|^{r}dx\Big)^{\frac{r-p+1}{r}} \nonumber\\
&\quad +c|p-r|\Big(\int_{E} |a(x)|^{\frac{r}{p-1}}dx\Big)^{\frac{p-1}{r}}
 \Big(\int_{E} |D u-D \theta|^{r}dx\Big)^{\frac{r-p+1}{r}} \nonumber\\
& \leq c(\varepsilon)|p-r|\int_{E} |D u-D \theta|^r dx+c(\varepsilon)|p-r|
 \int_{E} |D \theta|^r dx \nonumber\\
&\quad +c(\varepsilon)|p-r|\int_{E} |a(x)|^{\frac{r}{p-1}} dx, \label{I-2}
\end{align}
where $0<|p-r|<\varepsilon _0$. For the estimate of $I_3$, we have
\begin{align*}
 I_3 &\leq  \int_{E} |\mathbf{B}(x,Du)| |\varphi|dx \\
& \leq \Lambda_2\int_{E} \Big(|D u|^{p-\delta}+b(x)\Big) |\varphi| dx \\
& \leq \Lambda_2\Big(\int_{E}\Big(|D u|^{p-\delta}+b(x)\Big)^{q_0}  
 dx\Big)^{\frac{1}{q_0}}\Big(\int_{E} |\varphi|^{\frac{nr}{nr-r-np+n}}dx
 \Big)^{\frac{nr-r-np+n}{nr}}\\
& \leq  c\Big(\Big(\int_{E}|D u|^{(p-\delta)q_0}dx\Big)^{\frac{1}{q_0}}
 +\Big(\int_{E}|b(x)|^{q_0}  dx\Big)^{\frac{1}{q_0}}\Big)
 \Big(\int_{E} |D \varphi|^{\frac{r}{r-p+1}}dx\Big)^{\frac{r-p+1}{r}}\\
& \leq c\Big(\int_{E}|D u|^{(p-\delta)q_0}dx\Big)^{\frac{1}{q_0}}
 \Big(\int_{E} |D u-D \theta|^{r}dx\Big)^{\frac{r-p+1}{r}}\\
&\quad +c\Big(\int_{E}|b(x)|^{q_0}  dx\Big)^{\frac{1}{q_0}}
\Big(\int_{E} |D u-D \theta|^{r}dx\Big)^{\frac{r-p+1}{r}}\\
 &:= c(J_1+J_2),
\end{align*}
where $q_0=nr/(r+np-n)$. A direct calculation shows that 
$(p-\delta)\frac{nr}{r+np-n}<r$ with $\delta\in (1,p)$, then one gets that
\begin{align*}
 J_1 &\leq  c|E|^{\frac{1}{q_0}-\frac{p-\delta}{r}} 
 \Big(\int_{E}|D u|^{r}dx\Big)^{\frac{p-\delta}{r}}
 \Big(\int_{E} |D u-D \theta|^{r}dx\Big)^{\frac{r-p+1}{r}} \\
& \leq c|E|^{\frac{1}{q_0}-\frac{p-\delta}{r}}
 \bigg(\Big(\int_{E}|D u-D \theta|^{r}dx\Big)^{\frac{p-\delta}{r}}\\
&\quad +\Big(\int_{E}|D \theta|^{r}dx\Big)^{\frac{p-\delta}{r}}\bigg)
 \Big(\int_{E} |D u-D \theta|^{r}dx\Big)^{\frac{r-p+1}{r}} \\
& =  c|E|^{\frac{1}{q_0}-\frac{p-\delta}{r}}
 \bigg(\Big(\int_{E}|D u-D \theta|^{r}dx\Big)^{\frac{r-\delta+1}{r}}\\
&\quad +\Big(\int_{E}|D \theta|^{r}dx\Big)^{\frac{p-\delta}{r}}
 \Big(\int_{E} |D u-D \theta|^{r}dx\Big)^{\frac{r-p+1}{r}}\bigg)\\
& \leq  c\cdot \varepsilon \int_{E} |D u-D \theta|^r dx
 +c(\varepsilon)|E|^{(\frac{1}{q_0}-\frac{p-\delta}{r})\frac{r}{\delta-1}} \\
&\quad +c(\varepsilon)|E|^{(\frac{1}{q_0}-\frac{p-\delta}{r})\frac{r}{p-1}}
 \Big(\int_{E}|D \theta|^{r}dx\Big)^{\frac{p-\delta}{p-1}},
\end{align*}
and
\[
  J_2 \leq \varepsilon\int_{E} |D u-D \theta|^r dx
+c(\varepsilon)\Big(\int_{E}|b(x)|^{q_0}  dx\Big)^{\frac{r}{q_0(p-1)}}.
\]
Putting estimations of $J_1$ and $J_2$ together, we have
\begin{equation}\label{I-3}
\begin{aligned}
  I_3 &\leq  c\cdot \varepsilon\int_{E} |D u-D \theta|^r dx
 +c(\varepsilon)|E|^{(\frac{1}{q_0}-\frac{p-\delta}{r})\frac{r}{\delta-1}}\\
&\quad  +c(\varepsilon)|E|^{(\frac{1}{q_0}-\frac{p-\delta}{r})\frac{r}{p-1}}
 \Big(\int_{E}|D \theta|^{r}dx\Big)^{\frac{p-\delta}{p-1}}+c(\varepsilon)
\Big(\int_{E}|b(x)|^{q_0}  dx\Big)^{\frac{r}{q_0(p-1)}}.
\end{aligned}
\end{equation}
Therefore, by combining \eqref{I-1}, \eqref{I-2} and \eqref{I-3} we obtain
\begin{equation}\label{u-theta-b}
\begin{aligned}
&\int_{E} |D u-D \theta|^r dx \\
& \leq c\cdot(\varepsilon+|p-r|)\int_{E} |D u-D \theta|^r dx
 +c(\varepsilon)(1+|p-r|)\int_{E}|D \theta|^{r}dx\\
&\quad +c(\varepsilon)|E|^{(\frac{1}{q_0}-\frac{p-\delta}{r})\frac{r}{p-1}}
 \Big(\int_{E}|D \theta|^{r}dx\Big)^{\frac{p-\delta}{p-1}}+
 c(\varepsilon)\int_{E} |a(x)|^{\frac{r}{p-1}} dx\\
&\quad +c(\varepsilon)\Big(\int_{E}|b(x)|^{q_0}  dx
 \Big)^{\frac{r}{q_0(p-1)}}+c(\varepsilon)|E|^{(\frac{1}{q_0}
 -\frac{p-\delta}{r})\frac{r}{\delta-1}}.
\end{aligned}
\end{equation}
Since $|p-r|<\varepsilon_0$, we can take the positive constants $\varepsilon>0$ 
and $\varepsilon_0$ sufficiently small such that 
$c\cdot(\varepsilon+|p-r|)\leq \frac{1}{2}$. Then, the first term in the 
right-hand side of \eqref{u-theta-b} can be absorbed by the left-hand side, 
and we obtain
\begin{equation}\label{u-theta-c}
\begin{aligned}
&\int_{E} |D u-D \theta|^r dx \\
& \leq c\int_{E}|D \theta|^{r}dx+c|E|^{(\frac{1}{q_0}
 -\frac{p-\delta}{r})\frac{r}{p-1}}
 \Big(\int_{E}|D \theta|^{r}dx\Big)^{\frac{p-\delta}{p-1}}
 +c\int_{E} |a(x)|^{\frac{r}{p-1}} dx\\
 &\quad +  c\Big(\int_{E}|b(x)|^{q_0}  dx\Big)^{\frac{r}{q_0(p-1)}}
 +c|E|^{(\frac{1}{q_0}-\frac{p-\delta}{r})\frac{r}{\delta-1}}\\
 &:=   c(K_1+K_2+K_3+K_4+K_5).
\end{aligned}
\end{equation}
Note that $\theta\in W^{1,q}(\Omega)$ for $q>r$, then by  
the H\"older inequality to have
\begin{equation}\label{III-1}
  K_1 \leq \Big(\int_{E} |D \theta|^q dx\Big)^{\frac{r}{q}}|E|^{1-\frac{r}{q}}
 \leq \|D \theta\|^r_{L^q(\Omega)}|E|^{1-\frac{r}{q}}
\end{equation}
and
\begin{align*}
  K_2 &\leq |E|^{(\frac{1}{q_0}-\frac{p-\delta}{r})\frac{r}{p-1}} 
 \Big(\int_{E} |D \theta|^q dx\Big)^{\frac{r}{q}\frac{p-\delta}{p-1}}
 |E|^{(1-\frac{r}{q})\frac{p-\delta}{p-1}}\\
& \leq \|D \theta\|^{r\frac{p-\delta}{p-1}}_{L^q(\Omega)}
|E|^{(\frac{1}{q_0}-\frac{p-\delta}{r})
\frac{r}{p-1}+(1-\frac{r}{q})\frac{p-\delta}{p-1}}.
\end{align*}
By considering $q_0=\frac{nr}{r+np-n}$ for $ \delta\in (1,p)$, we get
\begin{align*}
&\Big(\frac{1}{q_0}-\frac{p-\delta}{r}\Big)\frac{r}{p-1}
 +\Big(1-\frac{r}{q}\Big)\frac{p-\delta}{p-1}\\
& = \frac{r}{p-1}\Big(\frac{r+np-n}{nr}-\frac{p-\delta}{r}\Big)
 +1-\frac{r}{q}+\Big(1-\frac{r}{q}\Big)\Big(\frac{p-\delta}{p-1}-1\Big)\\
& = 1-\frac{r}{q}+\frac{1}{p-1}\Big(\frac{r-n+n\delta}{n}
 +\frac{q-r}{q}(1-\delta)\Big)\\
& = 1-\frac{r}{q}+\frac{1}{p-1} \frac{(\delta-1)rn+rq}{nq}>1-\frac{r}{q},
\end{align*}
which implies 
\begin{equation}\label{K-2}
  K_2 \leq \|D \theta\|^{r\frac{p-\delta}{p-1}}_{L^q(\Omega)}|E|^{1-\frac{r}{q}}
|\Omega|^{\frac{(\delta-1)rn+rq}{nq(p-1)}}
\leq \|D \theta\|^{r\frac{p-\delta}{p-1}}_{L^q(\Omega)}|E|^{1-\frac{r}{q}}
(|\Omega|+1)^{\frac{(\delta-1)pn+pq}{nq(p-1)}}.
\end{equation}
Note that $a(x)\in L^{\frac{q}{p-1}}(\Omega)$ and 
$b(x)\in L^{\frac{nq}{q+np-n}}(\Omega)$, we have
\begin{equation}\label{K-3}
  K_3 \leq \Big(\int_{E} |a(x)|^{\frac{q}{p-1}} dx\Big)^{\frac{r}{q}}
|E|^{1-\frac{r}{q}}\leq \|a(x)\|^{r(p-1)}_{L^{\frac{q}{p-1}}
(\Omega)}|E|^{1-\frac{r}{q}}
\end{equation}
and
\begin{equation}\label{K-4}
\begin{aligned}
  K_4 &\leq  \Big(\int_{E} |b(x)|^{\frac{nq}{q+np-n}} dx
 \Big)^{\frac{(q+np-n)r}{nq(p-1)}}|E|^{\frac{r}{(p-1)q_0}-\frac{(q+np-n)r}{nq(p-1)}}\\
&\leq \|b(x)\|^{\frac{r}{p-1}}_{L^{\frac{nq}{q+np-n}}(\Omega)}
 |E|^{\frac{r}{(p-1)q_0}-\frac{(q+np-n)r}{nq(p-1)}}\\
&=\|b(x)\|^{\frac{r}{p-1}}_{L^{\frac{nq}{q+np-n}}(\Omega)}|E|^{1-\frac{r}{q}}
\end{aligned}
\end{equation}
with $\frac{r}{(p-1)q_0}-\frac{(q+np-n)r}{nq(p-1)}=1-\frac{r}{q}$.
 Similarly, thanks to
\[
 \Big(\frac{1}{q_0}-\frac{p-\delta}{r}\Big)\frac{r}{\delta-1}
=\frac{r+(\delta-1)n}{(\delta-1)n}>1>1-\frac{r}{q},
\]
we obtain
\begin{equation}\label{K-5}
  K_5 \leq |E|^{1-\frac{r}{q}}|\Omega|^{\frac{r}{(\delta-1)n}+\frac{r}{q}}
\leq   |E|^{1-\frac{r}{q}}(|\Omega|+1)^{\frac{p}{(\delta-1)n}+\frac{p}{q}}.
\end{equation}
Putting the estimates of $K_1,K_2,K_3,K_4$ and $K_5$ into \eqref{u-theta-c}, 
it follows that
\begin{equation}\label{u-theta-d}
\begin{aligned}
\int_{E} |D u-D \theta|^r dx 
 & \leq c\Big(\|D \theta\|^r_{L^q(\Omega)}
 +\|D \theta\|^{r\frac{p-\delta}{p-1}}_{L^q(\Omega)} \\
&\quad +\|a(x)\|^{r(p-1)}_{L^{\frac{q}{p-1}}(\Omega)}
 + \|b(x)\|^{\frac{r}{p-1}}_{L^{\frac{nq}{q+np-n}}(\Omega)}
 +1\Big)|E|^{1-\frac{r}{q}},
\end{aligned}
\end{equation}
where $c=c(n,m,p,q,\lambda,\Lambda_1,\Lambda_2,\delta)$.

We now turn our attention to the function $v\in W_0^{1,r}(E)$. 
Since $|v|=(|u-\theta|-L)$ in $E$, then by Sobolev embedding theorem 
and \eqref{nabla-v}, we have
\begin{equation}\label{v-embedding}
\begin{aligned}
\Big( \int_{E}(|u-\theta|-L)^{r^*}dx \Big)^{1/r^*}
&=\Big(\int_{E} |v|^{r^*} dx \Big)^{1/r^*} \\
&\leq  C(n,r) \Big(\int_{E} |D v|^rdx  \Big)^{1/r}\\
& =C(n,r) \Big(\int_{E} |D u-D \theta|^r dx\Big)^{1/r}.
\end{aligned}
\end{equation}
Hence,  considering $\tilde L >L$  yields 
\begin{equation}\label{big-L-L}
\begin{aligned}
\Big(\tilde L-L \Big)^{r^*} |\{|u-\theta|>\tilde L\}|
&= \int_{\{|u-\theta|>\tilde L\}} \big(\tilde L -L\big)^{r^*} dx \\
&\le  \int_{\{|u-\theta|>\tilde L\}}
\big(|u-\theta|-L \big)^{r^*} dx\\
& \le \int_{\{|u-\theta|> L\}} \big(|u-\theta|-L \big)^{r^*} dx.
\end{aligned}
\end{equation}
By collecting \eqref{u-theta-d}, \eqref{v-embedding} and \eqref{big-L-L} 
with $E=\{|u-\theta|>L\}$, we deduce that
\begin{align*}
&\Big( (\tilde L-L)^{r^*} |\{|u-\theta|>\tilde L\}|
\Big) ^{1/r^*} \\
&\leq c_* \Big(\|D \theta\|_{L^q(\Omega)}
 +\|D \theta\|^{\frac{p-\delta}{p-1}}_{L^q(\Omega)}
 +\|a(x)\|^{p-1}_{L^{\frac{q}{p-1}}(\Omega)}
 + \|b(x)\|^{\frac{1}{p-1}}_{L^{\frac{nq}{q+np-n}}(\Omega)}+1\Big) \\
&\quad \times  |\{|u-\theta|>L \}|^{\frac {1} {r} -\frac {1} {q}}.
\end{align*}
where $c_*=c_*(n,m,p,q,\lambda,\Lambda_1,\Lambda_2,\delta)$. 
It actually means that
\begin{equation}\label{u-theta-e}
\begin{aligned}
|\{|u-\theta|>\tilde L\}| 
&\leq \frac {1}{(\tilde L-L)^{r^*}} c^{r^*}_{*} 
 \Big(\|D \theta\|_{L^q(\Omega)}+\|D \theta\|^{\frac{p-\delta}{p-1}}_{L^q(\Omega)}
 +\|a(x)\|^{p-1}_{L^{\frac{q}{p-1}}(\Omega)}\\
&\quad +  \|b(x)\|^{\frac{1}{p-1}}_{L^{\frac{nq}{q+np-n}}(\Omega)}+1\Big)^{r^*}
 |\{|u-\theta|>L \}|^{r^*\big(\frac {1}{r}-\frac{1}{q}\big)}.
\end{aligned}
\end{equation}
Let $\phi (s)=| \{|u-\theta|>s\}|$, $\alpha =r^*$,
 $\beta =r^*\big(\frac{1}{r}-\frac{1}{q}\big) $, 
\[
C=c^{r^*}_{*} \Big(\|D \theta\|_{L^q(\Omega)}
 +\|D \theta\|^{\frac{p-\delta}{p-1}}_{L^q(\Omega)} 
 +\|a(x)\|^{p-1}_{L^{\frac{q}{p-1}}(\Omega)} 
 +\|b(x)\|^{\frac{1}{p-1}}_{L^{\frac{nq}{q+np-n}}(\Omega)}
 +1\Big)^{r^*}
\]
 and $s_0>0$. Then, the above estimation \eqref{u-theta-e} becomes
\begin{equation}\label{u-phi-a}
\phi ( \tilde L ) \le \frac {C}{(\tilde L-L)^\alpha }
\phi (L) ^\beta,
\end{equation}
for $\tilde L>L>0$. Now we are in a position to discuss settings in the 
three cases due to Stampacchia Lemma.
\smallskip

\noindent\textbf{Case (i)} 
 If $1\le q<n$, one has $\beta <1$. In this case, if $s\ge 1$, we then get 
from Lemma \ref{technical-result} that
\[
|\{|u-\theta|>s\} | \le C(\alpha, \beta, s_0) s^{-t},
\]
where
$t= \frac {\alpha}{1-\beta}=q^*$.
If $0<s<1$, one has
\[
|\{|u-\theta|>s\} | \le |\Omega| =|\Omega| s^{q^*}s^{-q^*} \le |\Omega| s^{-q^*}.
\]
In summary, we conclude that
$u\in \theta +L_{\rm weak}^{q^*} (\Omega)$.
\smallskip

\noindent\textbf{Case (ii)}
 If $q=n$, one has $\beta =1$. For any $1\le \tau<\infty$, it follows 
from \eqref{u-phi-a} that
\[
\phi \left( \tilde L \right) \le \frac {C}{(\tilde L-L)^\alpha }
\phi (L)=\frac {C}{(\tilde L-L)^\alpha } \phi (L) ^{1-\frac
{\alpha}{\tau}} \phi(L)^{\frac \alpha \tau} \le \frac {C
|\Omega|^{\frac {\alpha}{\tau}}}{(\tilde L-L)^\alpha }
\phi(L)^{1-\frac \alpha \tau}.
\]
As above, by Stampacchia Lemma we derive
$u\in \theta+ L^{\tau} (\Omega)$.
\smallskip

\noindent{Case (iii)}
  If $q>n$, one has $\beta >1$. Lemma \ref{technical-result} implies 
$\phi(d)=0$ for some constant $d$ depending only on 
$\alpha,\beta,s_0,r,\|D \theta\|_{L^q},\|a(x)\|_{L^{\frac{q}{p-1}}}$ and 
$\|b(x)\|_{L^{\frac{nq}{q+np-n}}}$.
Thus $|\{|u-\theta|>d\}|=0$, which means $u-\theta \le d$, a.e.\ $\Omega$. Therefore
$u\in \theta +L^\infty (\Omega)$,
and the proof is complete.
\end{proof}

\subsection*{Acknowledgments}
This work was partially supported by the Fundamental Research Funds for 
the Central Universities grant no. 2018YJS167 and the National Science 
Foundation of China grant no.\ 11371050.

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\end{document}