\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 16, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/16\hfil Optimal partial regularity]
{Optimal partial regularity for quasilinear elliptic systems with VMO
coefficients based on A-harmonic approximations}

\author[H. Yu, S. Zheng \hfil EJDE-2015/16\hfilneg]
{Haiyan Yu, Shenzhou Zheng}

\address{Haiyan Yu \newline
Department of Mathematics, Beijing Jiaotong University,
 Beijing 100044, China. \newline
College of Mathematics, Inner Mongolia University for the Nationalities,
Tongliao 028043, China}
\email{12118381@bjtu.edu.cn}

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

\thanks{Submitted August 6, 2014. Published January 20, 2015.}
\subjclass[2000]{35J60, 35B65, 35D30}
\keywords{VMO coefficients; controllable growth; A-harmonic approximation}

\begin{abstract}
 In this article, we consider quasi-linear elliptic systems in
 divergence form with discontinuous coefficients under controllable growth.
 We establish an optimal partial regularity of the weak solutions by a
 modification of A-harmonic approximation argument introduced by Duzaar
 and Grotowski.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded smooth domain of $\mathbb{R}^n (n\ge 2)$ and
$u:\Omega\to \mathbb{R}^N$ be a vectorial-valued function in
Sobolev spaces $W^{1,2}(\Omega,\mathbb{R}^N)$.
In this article, we obtain optimal partial regularity in H\"older spaces
to the weak solution of quasi-linear elliptic systems in divergence form
 under the controllable growth as follows:
\begin{equation}\label{eq1.1}
-D_\alpha(A_{ij}^{\alpha\beta}(x,u)D_\beta u^j)=B_i(x,u,Du),\quad
\text{a. e. } x\in \Omega,\; i=1,2,\dots,N;
\end{equation}
where $A(x,u)=(A_{ij}^{\alpha\beta}(x,u))$ is a VMO function in
$x\in \Omega$ uniformly with  respect to
$u\in \mathbb{R}^N$ and continuous in $u$  uniformly  with respect to
$x\in \Omega$, and $B_i(x,u,D u)$ satisfies the controllable
 growth. In the context, we adopt Einstein's convention by summing over
repeated indices with $\alpha,\beta=1,2,\dots,n$ and $i,j=1,2,\dots,N$.
Therefore, a vectorial-valued function  $u\in W_{\rm loc}^{1,2}(\Omega,\mathbb{R}^N)$
is understood as a  weak solution of \eqref{eq1.1} in the following
distributional sense:
\begin{equation}\label{eq1.2}
\int_\Omega A(x,u)Du\cdot D\varphi\,dx=\int_\Omega B(x,u,Du)\varphi\,dx,\quad
\forall \varphi\in C_0^\infty(\Omega,\mathbb{R}^N).
\end{equation}

Before stating our basic assumptions and main result, let us briefly review some
recent studies on the topic.
As we know, the discontinuity of the coefficients is not so crucial for
H\"older continuity of the weak solutions
of the scalar partial differential equations, which is due to the
famous De Giorgi-Moser-Nash iterating technique, see \cite{GT}.
However, for the vectorial-valued case (i.e. $N>1$) some counterexamples
showed that nonlinear elliptic systems, even in the Euclidian metric,
do not possess everywhere regularity conclusion, see Giaquinta's monograph
\cite{Gi}. In addition, to get the regularity of weak solutions of elliptic
systems, one needs to assume the continuity of coefficients in general.
In fact, the system \eqref{eq1.1} arises naturally in many different contexts.
Giaquinta and Modica \cite{GiM,Gi} first studied partial regularity of
weak solutions of the system \eqref{eq1.1} in the Morrey space
and in the Campanato space \cite{Gi, Ne} when each entry of the leading
coefficients $A(x,u)$ is assumed to be continuous
four order tensorial-valued function.

It is an important observation that many stochastic processes with discontinuous
coefficients reappeared in connected with diffusion approximation \cite{Ky04}.
 However, according to the famous counterexample of Nadirashvili there could not
exist theory of solvability of systems with general discontinuous coefficients
even if they are uniformly bounded and elliptic,
and solutions are understood in a very weak sense. This reminds us of the
significance to treat particular cases of discontinuity.
As an important turning point, Sarason \cite{Sa} introduced the function classes
of the so-called Vanishing Mean Oscillations (briefly called VMO),
which is a class of functions that neither contains nor is contained within
$C^0(\Omega)$ and contains discontinuous functions. Moreover, the VMO functions
 own a good property similar to the class of continuous
functions, which is not shared by general bounded measurable functions and BMO
functions. Since then, the Calder\'{o}n-Zygmund's theory of linear and
nonlinear PDEs with VMO coefficients were immensely developed
which naturally originated from the singular integral operators and the
estimates of commutators with a VMO function \cite{ChFL,BC}. In the meantime,
the regularity in Morrey spaces of weak solutions to PDEs with the discontinuous
leading coefficients was also investigated in a similar approach by
Fazio \cite{FaR} and Fan-Lu-Yang \cite{FLY}.
Very recently, it developed some new different arguments to deal with the
divergence or non-divergence elliptic and parabolic PDEs with the VMO leading
 coefficients, for example a few celebrated approaches of Chiarenza-Frasca-Longo
\cite{ChFL}, Syun-Wang \cite{ByW} and Krylov-Dong-Kim \cite{Ky,DK}.
Now we are in the position to recall some assumptions
imposed on $A(x,u)$ and $B(x,u,Du)$.

\begin{itemize}
\item[(H1)] (uniform ellipticity)
 There exist two constants $0<\lambda\le \Lambda $ such that
\begin{equation}\label{elliptic}
\lambda |\xi|^2\le  A^{\alpha\beta}_{ij}(x,u)\xi_{\alpha}^i\xi_{\beta}^j
\le \Lambda|\xi|^2 ,
\quad \forall  x\in\Omega,\ u\in \mathbb{R}^N,\;
\xi\in \mathbb{R}^{nN}.
\end{equation}

\item[(H2)]  ($A(x,u)$ is VMO in $x$ and continuous in $u$)
$A(\cdot,u)$ is VMO in $x$ uniformly with respect to
$u\in \mathbb{R}^N$ and is continuous in $u$ uniformly with respect to
$x\in \Omega$; that is, $\lim_{s\to{0}}M_s(A(\cdot,u_0))=0$, 
where $M_s(A(\cdot,u))$  referred to section 2, and
there exist a constant and a
continuous concave function 
$\omega:\mathbb R^{+}\to\mathbb R^{+}$ with $\omega(0)=0$,
$0\leq\omega\leq 1$ such that
\begin{equation}\label{continu}
|A^{\alpha\beta}_{ij}(x,u)-A^{\alpha\beta}_{ij}(x,v)|\leq
C\omega(|u-v|^2), \quad\forall u, v \in \mathbb{R}^N,\;  x\in\Omega.
\end{equation}
The modulus of continuity may take a continuous concave function by
$\omega(t)=inf\{\lambda(t):\lambda(t) $ concave and continuous
with $\lambda(t)\ge \alpha(t)$ for any modulus of continuity
$\alpha(t)$\}.

\item[(H3)] (controllable growth)  The lower order item
$B(x,u,Du)$ satisfies the following controllable growth with a constant $L>0$:
\begin{equation}\label{growth}
|B_i(x, u, D u)|\le L \big( |D u|^{2(1-\frac 1{\gamma})}+|u|^{\gamma-1}+g_i \big),
\end{equation}
where
$$
\gamma=\begin{cases}
\frac {2n}{n-2},     & \text{if } n>2,\\
\text{any }\gamma>2, & \text{if }  n=2;
\end{cases}
\quad
g_i\in L^q(\Omega),\; q>\frac n2;
$$
for $\alpha=1,2,\dots,n$ and $ i=1,2,\dots,N$.
\end{itemize}
Let us review some studies on the analogous questions.
Gironimo-Esposito-Sgambati in \cite{GiES} obtained the partial
regularity in Morrey spaces to quasi-linear quadratic functionals with leading
coefficient $A(x,u)$ allowing VMO dependence
on $x$ and continuous dependence on $u$. Later, Zheng \cite{Zh} and
Zheng-Feng \cite{ZhF} derived the partial regularity
in Morrey spaces for quasi-linear elliptic systems with VMO leading
coefficients with the controllable growth and the
natural growth by a reverse H\"older inequality and perturbation argument,
respectively. Chen-Tan \cite{ChT} also got an optimal interior partial regularity
for nonlinear elliptic systems under the controllable growth condition by
the A-harmonic approximation, but their principle coefficients $A(x,u)$
are essentially H\"older continuous in $(x,u)$. Here, we would like to study
the above topic by way of an approach called
A-harmonic approximation. As we know, the argument of harmonic approximation
can go back to De Giorgi's work \cite{DM} who
started to use the idea of approximating almost minimizers and the equation
of minimal surfaces by systems with constant
coefficients. Afterwards, the harmonic approximation argument was efficiently
employed to study $\varepsilon$-regularity
of harmonic maps, see \cite{Sim}. Recently, Duzaar-Mingione-Grotowski-Steffen
in \cite{DuS, DuG,Mi,DM} developed
this approach to so-called A-harmonic approximation, $p$-harmonic approximation
and A-caloric approximation in proving the regularity for nonlinear elliptic
systems with continuous or H\"older continuous coefficients, p-harmonic maps and
parabolic settings, respectively.
In particular, Dan\v{e}\v{c}ek-John-Star\'{a} \cite{DaJS} employed so-called modified
A-harmonic approximation approach to prove the regularity in Morrey's space
of weak solutions of Stokes systems with VMO coefficients. Inspired by his work,
in this paper we should like to prove an optimal partial regularity for quasi-linear
elliptic systems with VMO coefficients under the controllable growth by  a
modification of A-harmonic approximation argument, which avoids to use the
reverse H\"older inequality. We state our main results as follows.

\begin{theorem}\label{main result}
In the case of vectorial-valued functions with $N>1$, suppose that
 $u\in W_{\rm loc}^{1,2}(\Omega,\mathbb{R}^N)$ is a locally weak solution
of the system \eqref{eq1.1}, and $A(x,u)$, $B(x,u,Du)$ satisfy the basic
assumptions {\rm (H1)--(H3)}. Then there exists an open subset
$\Omega_0\subset\Omega$ with $ \operatorname{dim}_H(\Omega\setminus\Omega_0)\le n-2$
such that $u\in C^{0,\alpha}_{\rm loc}
(\Omega_0 ,\mathbb{R}^N),\alpha=2-\frac{n}{q} $
 if $\frac n2<q<n$ or $u\in C^{0,\alpha}_{\rm loc}
(\Omega_0 ,\mathbb{R}^N)$ for all
$ \alpha\in (0,1) $  if $q\ge n$, which $\operatorname{dim}_H$ expresses
the Hausdorff's dimension.
\end{theorem}

This article is organized as follows.
In section 2, we recall some notations and facts, and  give the proof
 of modification of so-called A-harmonic approximation,
Caccioppoli inequality. Section 3 is devoted to prove the main conclusions.


\section{Preliminaries}

We adopt the usual convention of denoting by $C$ a general constant,
which may vary from line to line in the same chain of inequalities.
Let us first recall some notation and basic facts \cite{Sa,St}.

\begin{definition}\rm
 A locally integrable function $f$ is said to belong to
$BMO(\Omega)$(the spaces of bounded mean oscillation), if $f \in
L_{\rm loc}(\Omega)$ and for any $0<s<\infty$, we have
$$
M_s(f, \Omega)=\sup_{x\in \Omega, 0<\rho<s}|\Omega(x, \rho)|^{-1}
\int_{\Omega(x, \rho)}|f(y)-f_{x, \rho}|\,dy< +\infty,
$$
where $\Omega(x, \rho)=\Omega\cap B(x, \rho)$ with any open ball
$B(x, \rho)$ in $\mathbb{R}^n$ centered at $x$ of radius $\rho$,
and $f_{x, \rho}:=\fint_{\Omega(x, \rho)} f(y)\,dy
=\frac 1{|\Omega(x, \rho)|}\int_{\Omega(x, \rho)} f(y)\,dy$.
\end{definition}


\begin{definition}\rm
 A function $f\in L_{\rm loc}(\Omega)$ is said to be in
$VMO(\Omega)$(vanishing mean oscillation in $\Omega$), if
$$
M_0(f)=\lim_{s\to 0}M_s(f,  \Omega)=0.
$$
\end{definition}

As we know, Caccioppoli's inequality is usually a very beginning of
studying regularity to elliptic and parabolic PDEs, see
\cite{Gi}. Here, we provide the so-called second Caccioppoli's inequality.

\begin{lemma}\label{caccioppoli}
 Let $u\in W_{\rm loc}^{1,2}(\Omega,\mathbb{R}^N)$ be a weak solution of
 \eqref{eq1.1} and $A(x,u)$, $B(x,u,Du)$ satisfy the assumption {\rm (H1)--(H3)}.
Then for any $B_\rho(x_0)\subset\Omega$, we have
\begin{equation}\label{Caccio}
\int_{B_{\frac{\rho}{2}}(x_0)}|Du|^2\,dx
\leq\frac{C_1}{\rho^2}\int_{B_{\rho}(x_0)}|u-m|^2\,dx
+C_2\Big(\int_{B_{\rho}(x_0)}(|Du|^2+|u|^\gamma+|g|^{\frac{\gamma}{\gamma-1}})dx
\Big)^{2(1-\frac{1}{\gamma})},
\end{equation}
where $m$ is a vectorial-valued constant in $\mathbb{R}^N$.
\end{lemma}

\begin{proof}
For any $x_0\in \Omega, 0<\rho<\operatorname{dist}(x_0,\partial\Omega)$, denoting
$B_\rho:=B_\rho(x_0)$, we take $\eta\in C^\infty_0(B_\rho(x_0))$
as a cut-off function with $0\le\eta\le1$,  $|D\eta|\le\frac{4}{\rho}$ and
$\eta\equiv 1$ on $B_{\frac{\rho}{2}}(x_0)$.
As usual, we can take the function $\varphi=\eta^2(u-m)$ as a test
function with any vectorial-valued constant $m\in \mathbb{R}^N$. By \eqref{eq1.2}, we have
$$
\int_{B_{\rho}}A(x,u)Du\cdot [2\eta D\eta(u-m)+\eta^2Du]dx=\int_{B_{\rho}}B(x,u,Du)\eta^2(u-m)dx,
$$
which implies
$$
\int_{B_{\rho}}\eta^2 A(x,u)Du\cdot Du\,dx=-2\int_{B_{\rho}}A(x,u)Du\cdot (\eta (u-m)D\eta)+\int_{B_{\rho}}B(x,u,Du)\eta^2(u-m)dx.
$$
By the ellipticity (H1) and the controllable growth (H3) we obtain
\begin{equation} \label{perturbation}
\begin{aligned}
&\lambda\int_{B_{\rho}}|\eta Du|^2\,dx \\
&\le 2\Lambda\int_{B_{\rho}}|\eta Du|\cdot|(u-m)D\eta|\,dx
+L\int_{B_{\rho}}\left(|Du|^{2(1-\frac{1}{\gamma})}+|u|^{\gamma-1}+|g|\right)
|\varphi|dx 
:=I+II.
\end{aligned}
\end{equation}
For $I$, by Young's inequality we have
\begin{equation}\label{I}
I\le \varepsilon\int_{B_{\rho}}|\eta Du|^2\,dx
+\frac{C(\varepsilon)}{\rho^2}\int_{B_{\rho}}|u-m|^2\,dx.
\end{equation}
For $II$, by H\"{o}lder's inequality and Sobolev's inequality and Young's
 inequality we have
\begin{align*}
II&\leq  CL\int_{B_{\rho}}(|Du|^2+|u|^\gamma
+|g|^{\frac{\gamma}{\gamma-1}})^{1-\frac{1}{\gamma}} |\varphi|dx\\
&\leq  CL\Big(\int_{B_{\rho}}(|Du|^2+|u|^\gamma+|g|^{\frac{\gamma}{\gamma-1}})dx
 \Big)^{1-\frac{1}{\gamma}}
\Big(\int_{B_{\rho}}|\varphi|^\gamma\,dx\Big)^{\frac{1}{\gamma}}\\
&\leq CL\Big(\int_{B_{\rho}}(|Du|^2+|u|^\gamma+|g|^{\frac{\gamma}{\gamma-1}})dx
 \Big)^{1-\frac{1}{\gamma}}\Big(\int_{B_{\rho}}|D\varphi|^2\,dx\Big)^{1/2}\\
&\leq \varepsilon\int_{B_{\rho}}|D\varphi|^2\,dx+C(\varepsilon)
\Big(\int_{B_{\rho}}(|Du|^2+|u|^\gamma+|g|^{\frac{\gamma}{\gamma-1}})dx\Big)
^{2(1-\frac{1}{\gamma})}.
\end{align*}
Note that  $D\varphi=2\eta D\eta(u-m)+\eta^2Du$. Then
\begin{equation}\label{II}
II\le\varepsilon\int_{B_{\rho}}|\eta Du|^2\,dx
+C(\varepsilon)\int_{B_{\rho}}|D\eta|^2|u-m|^2\,dx
+C(\varepsilon)\Big(\int_{B_{\rho}}(|Du|^2+|u|^\gamma
+|g|^{\frac{\gamma}{\gamma-1}})dx\Big)^{2(1-\frac{1}{\gamma})}.
\end{equation}
Now by combining \eqref{I} and \eqref{II} it yields
\begin{equation}\label{conc}
(\lambda-2\varepsilon)\int_{B_{\rho}}|\eta Du|^2\,dx\le\frac{C(\varepsilon)}{\rho^2}
\int_{B_{\rho}}|u-m|^2\,dx+C(\varepsilon)
\Big(\int_{B_{\rho}}(|Du|^2+|u|^\gamma+|g|^{\frac{\gamma}{\gamma-1}})dx
\Big)^{2(1-\frac{1}{\gamma})}.
\end{equation}
So, we only choose some $\varepsilon<\lambda/2$, it yields the desired result.
\end{proof}

We are in position to introduce a modification of so-called
A-harmonic approximation lemma. Let us first recall the definition
of locally A-harmonic.

\begin{definition}\label{A} \rm
Let $A\in Bil({B_R(x_0)}\times\mathbb{R}^{N},\mathbb{R}^{n^2\times N^2})$
be a bilinear form with constant coefficients, which satisfies the assumptions
of \eqref{elliptic}. We call a map $h\in W^{1,2}({B_R(x_0)},\mathbb{R}^{N})$
 A-harmonic in ${B_R(x_0)}$ if it satisfies
$$
\int_{{B_R(x_0)}}A(Dh,D\varphi)dx=0,\quad  \forall
 \varphi\in C^1_0({B_R(x_0)},\mathbb{R}^{N}).
$$
\end{definition}

Since $A\in Bil({B_R(x_0)}\times\mathbb{R}^{N},\mathbb{R}^{n^2\times N^2})$
is a bilinear form with constant coefficients, it's well known that for any
A-harmonic function $h$ we have the following inequality.

\begin{lemma}[\cite{Gi}] \label{const-system}
Let $h(x)\in W^{1,2}({B_R(x_0)},\mathbb{R}^N)$ be a weak solution of the
following linear system with constant coefficients
$$
D_\alpha(A_{ij}^{\alpha\beta}D_\beta h_j)=0,\quad i=1,\dots, N.
$$
Then there exists a constant $C=C(n,\lambda,\Lambda)$ such that for any
$x_0\in\Omega$, $0<\rho<R\le \operatorname{dist}(x_0, \partial\Omega)$, it holds
\begin{equation}\label{harm}
\int_{B_\rho(x_0)}|Dh|^2\,dx\leq C\big(\frac{\rho}{R}\big)^n\int_{B_R(x_0)}|Dh|^2\,dx.
\end{equation}
\end{lemma}

Now we  give the modified A-harmonic approximation which is based on the
usual A-harmonic lemma originated by Duzaar and Grotowski' works
\cite{DuG, DM}. In the sequel, suppose that there exist two constants
$0<\lambda\le \Lambda<\infty$  such that the bilinear form
$A\in Bil({B_R(x_0)}\times\mathbb{R}^{N},\mathbb{R}^{n^2\times N^2})$ satisfies
\begin{gather}\label{positive-1}
A^{\alpha\beta}_{ij}(x,u)\xi_{\alpha}^i\xi_{\beta}^j\geq \lambda|\xi|^2,\quad  \forall
 \xi \in \mathbb{R}^{nN}, \\
\label{bounded-1}
A^{\alpha\beta}_{ij}(x,u)\xi_{\alpha}^i\bar \xi_{\beta}^j\leq \Lambda|\xi||\bar{\xi}|,
\quad \forall \xi,\bar{\xi} \in \mathbb{R}^{nN};
\end{gather}

\begin{lemma}\label{A-harm-app}
Consider fixed positive constants $\lambda,\Lambda$ and $n, N\in\mathbb{N}$
with $n\ge 2$ as above. Then for any given $\varepsilon>0$, there exists
$\delta=\delta(n,N,\lambda,\Lambda,\varepsilon)\in(0,1]$ with the following property:
for any bilinear form
$A\in Bil({B_R(x_0)}\times\mathbb{R}^{N},\mathbb{R}^{n^2\times N^2})$
with \eqref{positive-1},\eqref{bounded-1}, assume
$g\in W^{1,2}(B_{R}(x_0),\mathbb{R}^N)$ satisfies
\begin{gather}\label{harmonic6}
R^{-n}\int_{B_{R}(x_0)}|Dg|^2\,dx\leq 1,\\
\label{harmonic7}
\big|R^{-n}\int_{B_{R}(x_0)}A(Dg,D\varphi)dx\big|
\leq \delta\sup_{B_{R}(x_0)}|D\varphi|,\quad
 \forall \varphi\in C_0^\infty(B_{R}(x_0),\mathbb{R}^N);
\end{gather}
there exists an A-harmonic function
 $$
 \omega\in H=\big\{h\in W^{1,2}(B_{R}(x_0),\mathbb{R}^N):
R^{-n}\int_{B_{R}(x_0)}|Dh|^2\,dx\leq 1\big\}
 $$
with
\begin{equation}\label{harmonic8}
R^{-n-2}\int_{B_{R}(x_0)}|\omega-g|^2\,dx\leq\varepsilon.
\end{equation}
\end{lemma}

Thanks to the A-harmonic approximation above, we obtain its modified
version by imitating an argument from Stoke system by
Dan\v{e}\v{c}ek-John-Star\'{a} \cite{DaJS}.

\begin{lemma}[Modification of A-harmonic approximation] \label{modified-A-harmonic}
Let $0<\lambda\le \Lambda<\infty$  and $n\ge 2$ as the above lemma.
Then, for any given $\varepsilon>0$ there exists
$k=k(n,N,\lambda,\Lambda,\varepsilon)>0$ with the following property:
for any $A\in Bil(B_R(x_0)\times\mathbb{R}^{N},\mathbb{R}^{n^2\times N^2})$
satisfying \eqref{positive-1},\eqref{bounded-1}
and any $u\in W^{1,2}(B_R(x_0),\mathbb{R}^N)$, there exists an A-harmonic
function $h\in W^{1,2}(B_R(x_0),\mathbb{R}^N)$ such that
\begin{equation}\label{bound-2}
\int_{B_R(x_0)}|Dh|^2\,dx\leq\int_{B_R(x_0)}|Du|^2\,dx;
\end{equation}
moreover, there exists $\varphi\in C_0^\infty(B_R(x_0),\mathbb{R}^N)$ with
\begin{equation}\label{harmonic4}
\|D\varphi\|_{L^\infty(B_R(x_0),\mathbb{R}^N)}\leq\frac{1}{R};
\end{equation}
such that
\begin{equation}\label{harmonic5}
\int_{B_R(x_0)}|u-h|^2\,dx \leq \varepsilon R^2\int_{B_R(x_0)}|Du|^2\,dx
+k(\varepsilon)\Big[R^{4-n}\Big(\int_{B_R(x_0)}ADu\cdot
D\varphi\,dx\Big)^2\Big].
\end{equation}
\end{lemma}

\begin{proof}
First, observe that it is sufficient to prove the lemma for $x_0=0$ and $R=1$
by a standard scaling argument. In the context, we let $B=B_1(0)$. For any
given $\varepsilon>0$, we pick $\delta=\delta(n,N,\lambda,\Lambda,\varepsilon)$
as the above Lemma \ref{A-harm-app}. Consider $u \in W^{1,2}(B,\mathbb{R}^N)$,
we take
$$
g=u\Big(\int_B |Du|^2\,dx\Big)^{-1/2},
$$
therefore, $\int_{B}|Dg|^2\,dx\leq 1$ which implies \eqref{harmonic6}.
Next, we consider the estimates divided into two cases.
\smallskip

\textbf{Case 1.}  If  for $g$ there holds the inequality \eqref{harmonic7}.
 By Lemma \ref{A-harm-app} there exists an A-harmonic function $\omega$ satisfying
$\int_{B_{\rho}(x_0)}|D\omega|^2\,dx\leq 1 $ and
$\int_{B}|\omega-g|^2\,dx\leq\varepsilon.$

Let $h=\big(\int_B |Du|^2\,dx\big)^{1/2}\omega$, which satisfies
\eqref{bound-2}. In fact, we can easily know $h$ is A-harmonic and
    $$
    \int_B |Dh|^2\,dx=\int_B |Du|^2\,dx\int_B |D\omega|^2\,dx\leq\int_B |Du|^2\,dx.
    $$
Moreover, we have
$$
|u-h|^2=\int_B |Du|^2\,dx\cdot|g-\omega|^2,
$$
which implies
$$
\int_B |u-h|^2\,dx\leq\int_B |Du|^2\,dx
\int_B |g-\omega|^2\,dx \leq\varepsilon\int_B |Du|^2\,dx.
$$
Hence, the inequality \eqref{harmonic5} is valid.
\smallskip


\textbf{Case 2.}
If for $g$ the inequality \eqref{harmonic7} is false.
Then there exists a non-constant function $\psi\in C_0^\infty(B,\mathbb{R}^N)$
such that
$$
\big|\int_{B}A(Dg,D\psi)dx\big|>\delta(\varepsilon)\sup_B|D\psi|.
$$
By taking $\varphi=\psi/\sup_B|D\psi|$ it yields
$\|D\varphi\|_{L^\infty}=1$,
which implies
$$
\frac{1}{\delta(\varepsilon)}\big|\int_{B}A(Dg,D\varphi)dx\big|>1.
$$
Now we take $h=\bar{u}$. By Poincar\'{e} inequality and recalling
$Dg=\big(\int_B |Du|^2\big)^{-1/2}\cdot Du$, it follows that
\begin{align*}
\int_B |u-h|^2\,dx
&= \int_B |u-\bar{u}|^2\,dx\leq C\int_B |Du|^2\,dx \\
&\leq \frac{C}{\delta^2(\varepsilon)} \int_B |Du|^2\,dx
\big|\int_{B}A(Dg,D\varphi)dx\big|^2  \\
&\leq \frac{C}{\delta^2(\varepsilon)}\big|\int_{B}A(Du,D\varphi)dx\big|^2.
\end{align*}
By combining Cases 1 and 2, and taking
$k(\varepsilon)=\frac{C}{\delta^2(\varepsilon)}$,
we obtain the inequality \eqref{harmonic5}.
The proof is complete.
\end{proof}

\begin{lemma}[\cite{EvG}] \label{Haus}
 Let $\Omega$ be an open subset of $\mathbb{R}^n$ and
$u\in L_{\rm loc}(\Omega, \mathbb{R}^N)$. Then for $0\le s<n$ and set
\begin{equation}
E_{s}:=\big\{x\in\Omega:\lim\inf_{\rho\to
0}\rho^{-s}\int_{B_\rho(x) }|u|\,dy>0 \big\},
\end{equation}
there holds the  estimate
$H^{s}(E_{s})=0$.
\end{lemma}

\section{Proof of main result}

In the section, we prove our main result by way of the idea from
modification of A-harmonic approximation argument and perturbation approach.

\begin{proof}[Proof of Theorem \ref{main result}]
For any $x_0\in \Omega$ and fixed
$0<R\le \frac 12 \operatorname{dist}(x_0,\partial\Omega)$.
Without loss of generality, we let $x_0=0$ and
for any $0<\rho<R$ write $B_\rho$ in place of $B_\rho(0)$.
 Now letting  $m=u_{0,\rho}=u_\rho$ in Lemma \ref{caccioppoli}, it follows that
\begin{equation}\label{caccio}
\int_{B_{\frac{\rho}{2}}}|Du|^2\,dx
\leq\frac{C_1}{\rho^2}\int_{B_{\rho}}|u-u_{\rho}|^2\,dx
+C_2\Big(\int_{B_{\rho}}
\big(|Du|^2+|u|^\gamma+|g|^{\frac{\gamma}{\gamma-1}}
\big)dx\Big)^{2(1-\frac{1}{\gamma})},
\end{equation}
Let $\bar{A}=A(\cdot,u_{R})_{R}$  be defined by
$$
\bar A:=A(x,u_{R})_{R}=\fint_{B_R}A(x,u_{R})dx.
$$
Thanks to the modification of A-harmonic Lemma \ref{modified-A-harmonic},
there exists an $\bar{A}$-harmonic
function $h\in W^{1,2}(B_R,\mathbb{R}^N)$ such that the inequalities
\eqref{bound-2},\eqref{harmonic4} and
\eqref{harmonic5} are valid. Therefore, from \eqref{caccio} we have
\begin{equation} \label{caccio1}
\begin{aligned}
&\int_{B_{\frac{\rho}{2}}}|Du|^2\,dx \\
&\leq \frac{2C_1}{\rho^2}\Big(\int_{B_{\rho}}|u-u_\rho-(h-h_{\rho})|^2\,dx
+\int_{B_\rho}|h-h_{\rho}|^2\,dx\Big) \\
&\quad +C\Big(\int_{B_{\rho}}\big(|Du|^2+|u|^\gamma+|g|^{\frac{\gamma}{\gamma-1}}
\big)dx\Big)^{2(1-\frac{1}{\gamma})} \\
&:=\frac{C}{\rho^2}(I_1+I_2)
+C\Big(\int_{B_{\rho}}\big(|Du|^2+|u|^\gamma+|g|^{\frac{\gamma}{\gamma-1}}
\big)dx\Big)^{2(1-\frac{1}{\gamma})}\,.
\end{aligned}
\end{equation}

Next we estimate $I_1$ and $I_2$.
For the estimation of $I_1$, by Poincar\'{e} inequality and Lemma \ref{const-system}
on the system with constant coefficients it follows that
$$
I_1=\int_{B_\rho}|h-h_{\rho}|^2\,dx\leq
C\rho^2\int_{B_\rho}|Dh|^2\,dx\leq C\rho^2\big(\frac{\rho}{R}\big)^n
\int_{B_R}|Dh|^2\,dx.
$$
Hence, from \eqref{bound-2} it yields
\begin{equation}\label{I1}
I_1\le C\rho^2\big(\frac{\rho}{R}\big)^n\int_{B_R}|Du|^2\,dx.
\end{equation}
For  $I_2$, by employing Poincar\'{e} inequality again and
\eqref{harmonic5} in Lemma \ref{modified-A-harmonic}, we have
\begin{equation} \label{I2}
\begin{aligned}
I_2
&=\int_{B_{\rho}}|u-u_\rho-(h-h_{\rho})|^2\,dx
 \le2\int_{B_{\rho}}|u-h|^2\,dx\\
&\leq  C\varepsilon \rho^2\int_{B_\rho}|Du|^2\,dx+Ck(\varepsilon)\rho^{4-n}
\Big(\int_{B_\rho}\bar{A}Du\cdot D\varphi\,dx\Big)^2\\
&\leq  C\varepsilon \rho^2\int_{B_R}|Du|^2\,dx+Ck(\varepsilon)
\rho^{4-n}\Big(\int_{B_\rho}\bar{A}Du\cdot D\varphi\,dx\Big)^2,
\end{aligned}
\end{equation}
where $\varphi$ satisfies $\|D\varphi\|_{L^\infty(B_R,\mathbb{R}^N)}\leq\frac{1}{R}$.
Next, we  estimate the term $\int_{B_\rho}\bar{A}Du\cdot D\varphi\,dx$.
Note that $u$ is a weak solution of \eqref{eq1.1}, then
\begin{align*}
\int_{B_\rho}\bar{A}Du\cdot D\varphi\,dx
&=\int_{B_\rho}[\bar{A}-A(x,u_\rho)]Du\cdot D\varphi\,dx
 +\int_{B_\rho}[A(x,u_\rho)-A(x,u)]Du\cdot D\varphi\,dx\\
&\quad +\int_{B_\rho}B(x,u,Du)\varphi\,dx;
\end{align*}
that is,
\begin{equation} \label{A1}
\begin{aligned}
&\Big(\int_{B_\rho}\bar{A}Du\cdot D\varphi\,dx\Big)^2
\le C\Big(\int_{B_\rho}[\bar{A}-A(x,u_\rho)]Du\cdot D\varphi\,dx\Big)^2 \\
&\quad +C\Big(\int_{B_\rho}[A(x,u_\rho)-A(x,u)]Du\cdot D\varphi\,dx\Big)^2
+C\Big(\int_{B_\rho}B(x,u,Du)\varphi\,dx\Big)^2.
\end{aligned}
\end{equation}
Since $\|D\varphi\|_{L^\infty(B_R,\mathbb{R}^N)}\leq\frac{1}{R}$ in
\eqref{harmonic4} and  $A(\cdot,u)\in VMO\bigcap L^{\infty}(\Omega)$
of the assumptions (H1)--(H2), it follows
\begin{equation} \label{VMO}
\begin{aligned}
\Big(\int_{B_\rho}[\bar{A}-A(x,u_\rho)]Du\cdot D\varphi\,dx\Big)^2 
&\leq \frac{1}{\rho^2}\int_{B_\rho}|Du|^2\,dx \int_{B_\rho}|A(x,u_\rho)-\bar{A}|^2\,dx
\\
&\leq \frac{1}{\rho^2}\cdot2\Lambda\alpha_n\rho^{n}\fint_{B_{\rho}}
|A(x,u_\rho)-\bar{A}|dx \int_{B_{\rho}}|Du|^2\,dx \\
&\leq C(n,\Lambda) M_{s}(A(x,u_\rho))\alpha_n\rho^{n-2}\int_{B_{\rho}}|Du|^2\,dx,
\end{aligned}
\end{equation}
where $\alpha_n$ is the volume of unit ball in $\mathbb{R}^n$.
Similarly, in terms of the continuous assumptions of $A(x,\cdot)$ in $u$
uniformly with respect to $x\in\Omega$ we have the following estimates
\begin{align}
&\Big(\int_{B_\rho}[A(x,u_\rho)-A(x,u)]Du\cdot D\varphi\,dx\Big)^2 \nonumber \\
&\leq \frac{1}{\rho^2}\int_{B_\rho}|Du|^2\,dx
 \int_{B_\rho}|A(x,u_\rho)-A(x,u)|^2\,dx  \nonumber\\
&\leq \frac{1}{\rho^2}\cdot2\Lambda\alpha_n\rho^{n}
 \fint_{B_{\rho}}|A(x,u_\rho)-A(x,u)|dx\cdot\int_{B_{\rho}}|Du|^2\,dx  \nonumber\\
&\leq C\frac{1}{\rho^2}\cdot\Lambda\alpha_n\rho^{n}\fint_{B_{\rho}}
 \omega(|u-u_\rho|)dx\int_{B_{\rho}}|Du|^2\,dx \nonumber \\
&\leq C\Lambda\alpha_n\rho^{n-2}\omega\Big(\fint_{B_{\rho}}
 |u-u_\rho|^2\,dx\Big)\int_{B_{\rho}}|Du|^2\,dx \nonumber\\
&\leq C(n,\Lambda)\rho^{n-2}\omega\Big(\rho^2\fint_{B_{\rho}}|Du|^2\,dx\Big)
\int_{B_{\rho}}|Du|^2\,dx, \label{module}
\end{align}
where we use the Jensen's inequality in the fourth step and the Poincar\'e's
 inequality in the last step.
Finally, we consider the controllable growth condition (H3) it yields
\begin{equation} \label{B}
\begin{aligned}
\Big(\int_{B_\rho}B(x,u,Du)\varphi\,dx\Big)^2
&\le\Big(\int_{B_\rho}|B(x,u,Du)|\,dx\Big)^2 \\
&\leq C\Big(\int_{B_\rho}\big(|Du|^{2(1-\frac{1}{\gamma})}+|u|^{\gamma-1}+|g|
\big)dx\Big)^2 \\
&\leq C\Big(\int_{B_\rho}\big(|Du|^2+|u|^{\gamma}+|g|^{\frac{\gamma}{\gamma-1}}
\big)dx\Big)^{2(1-\frac{1}{\gamma})}
\big(\alpha_n\rho^n\big)^\frac{2}{\gamma} \\
&= C(n)\rho^{n-2}\Big(\int_{B_\rho}\big(|Du|^2+|u|^{\gamma}
+|g|^{\frac{\gamma}{\gamma-1}}\big)dx\Big)^{2(1-\frac{1}{\gamma})}
\end{aligned}
\end{equation}

Now,  substitute estimates \eqref{VMO}, \eqref{module} and \eqref{B}  into \eqref{A},
 it yields
\begin{equation} \label{AA}
\begin{aligned}
&\Big(\int_{B_\rho}\bar{A}Du\cdot D\varphi\,dx\Big)^2\\
&\leq  C(n,\Lambda)\rho^{n-2}\Big(M_s(A(x,u_\rho))+
 \omega\big(\rho^2\fint_{B_{\rho}}|Du|^2\,dx\big)\Big)\int_{B_\rho}|Du|^2\,dx \\
&\quad + C(n)\rho^{n-2}\Big(\int_{B_\rho}\Big(|Du|^2+|u|^{\gamma}
 +|g|^{\frac{\gamma}{\gamma-1}}\Big)dx\Big)^{2\Big(1-\frac{1}{\gamma}\Big)}.
\end{aligned}
\end{equation}
Denoting
\begin{equation}\label{sigma}
\sigma(\rho)=M_s(A(x,u_\rho))+\omega\Big(\rho^2\fint_{B_{\rho}}|Du|^2\,dx\Big)
\end{equation}
and inserting  \eqref{AA} into the estimate of $I_2$, we obtain
 $$
 I_2\le C\Big(\varepsilon+\sigma(\rho)\Big)\rho^2\int_{B_R}|Du|^2\,dx+C\rho^2
 \Big(\int_{B_\rho}\Big(|Du|^2+|u|^{\gamma}
+|g|^{\frac{\gamma}{\gamma-1}}\Big)dx\Big)^{2\Big(1-\frac{1}
 {\gamma}\Big)}.
 $$
Substitute the estimates for I and II into \eqref{caccio1}, we obtain
\begin{equation}\label{2}
\begin{aligned}
\int_{B_{\frac{\rho}{2}}}|Du|^2\,dx
&\le C\Big(\Big(\frac{\rho}{R}\Big)^n+\varepsilon+\sigma(\rho)\Big)
\int_{B_R}|Du|^2\,dx\\
&\quad +C\Big(\int_{B_\rho}\Big(|Du|^2+|u|^{\gamma}+|g|^{\frac{\gamma}{\gamma-1}}\Big)dx\Big)^{2\Big(1-\frac{1}
{\gamma}\Big)}.
\end{aligned}
\end{equation}

It remains to estimate the term of the controllable growth.
 Observe that $g_i\in L^q(\Omega)$  with $q>n/2$ and
\[
\gamma=\begin{cases} \frac {2n}{n-2},  & \text{if } n>2,\\
\text{any }\gamma>2, & \text{if } n=2.
\end{cases}
\]
As we know it is trivial if $n=2$. So, we only consider the case of $n>2$
so that $2(1-\frac{1}{\gamma})=(n+2)/n$, by H\"older inequality
it yields
\begin{align*}
\Big(\int_{B_\rho}\Big(|Du|^2+|u|^{\gamma}+|g|^{\frac{\gamma}{\gamma-1}}\Big)
\,dx\Big)^{2\Big(1-\frac{1} {\gamma}\Big)} 
&\leq C\Big(\int_{B_\rho}|Du|^2+|u|^{\gamma}dx\Big)^{1+\frac{2}{n}}
+C\Big(\int_{B_\rho}|g|^{\frac{2n}{n+2}}dx\Big)^{\frac{n+2}{n}} \\
&\leq C\Big(\int_{B_\rho}|Du|^2+|u|^{\gamma}dx
\Big)^{1+\frac{2}{n}}+C\alpha_n^{\frac{(n+2)q-2n}{n q}}
R^{n+2-\frac{2n}{q}}\|g\|_{L^q}^2,
\end{align*}
putting it into \eqref{2},  yields
\begin{equation}\label{3}
\begin{aligned}
\int_{B_{\frac{\rho}{2}}}|Du|^2\,dx
&\le C\Big(\Big(\frac{\rho}{R}\Big)^n+\varepsilon
+\sigma(\rho)
+\Big(\int_{B_\rho}|Du|^2+|u|^{\gamma}dx\Big)^{2/n}\Big)
\int_{B_R}(|Du|^2\\
&\quad +|u|^{\gamma})dx +CR^{n+2-\frac{2n}{q}}\|g\|_{L^q}^2.
\end{aligned}
\end{equation}
On the other hand,  by a direct calculation it follows that
\begin{align*}
\int_{B_{\frac{\rho}{2}}}|u|^\gamma\,dx
&\le C\int_{B_{\frac{\rho}{2}}}|u_{x_0,\rho}|^\gamma\,dx
+C \int_{B_{\frac{\rho}{2}}}|u-u_{x_0,\rho}|^\gamma\,dx  \\
&\leq C(n)\Big(\frac{\rho}{R}\Big)^n\int_{B_R}|u|^\gamma\,dx
+C\Big(\int_{B_R}|Du|^2\,dx\Big)^{\frac{\gamma}{2}-1}
\Big(\int_{B_R}(|Du|^2+|u|^\gamma)dx\Big).
\end{align*}
Now  add the item $\int_{B_{\frac{\rho}{2}}}|u|^\gamma\,dx$ to both sides
of \eqref{3} to obtain
\begin{equation}\label{4}
\int_{B_{\frac{\rho}{2}}}|Du|^2+|u|^\gamma\,dx
\le C\left(\Big(\frac{\rho}{R}\Big)^n+\varepsilon
+\sigma(\rho)+\delta(\rho)\right)\int_{B_R}\Big(|Du|^2
+|u|^\gamma \Big)dx+CR^{n+2-\frac{2}{q}n}\|g\|_{L^q}^2,
\end{equation}
where
\begin{equation}\label{delta}
\delta(\rho)=\Big(\int_{B_\rho}(|Du|^2+|u|^\gamma)\,dx
\Big)^{2/n}+\Big(\int_{B_\rho}|Du|^2\,dx\Big)^{2/(n-2)}.
\end{equation}
Note that $\delta(\rho)\to 0$ as $\rho\to 0$ due to the absolute continuity
of $\int_{B_\rho}(|Du|^2 +|u|^\gamma)\,dx$ on domain of integration, and if we
assume $\rho^2\fint_{B_{\rho(x)}}|Du|^2dy\to 0$
on $x\in \Omega_0\subset\Omega$ as $\rho\to 0$, then it yields
$\sigma(\rho)=M_s(A(x,u_\rho))+\omega\big(\rho^2\fint_{B_{\rho}}
|Du|^2\,dx\big)<\varepsilon $ as $\rho\to 0$  due to the $VMO$ property of
$A(x,u)$ in $x\in \Omega$.
Observe that $n-2<n+2-\frac{2}{q}n<n$ if $\frac n2<q<n$, by the iteration
lemma it follows
\begin{equation}\label{delta2}
\int_{B_{\frac{\rho}{2}}}\Big(|Du|^2+|u|^\gamma\Big)\,dx
\le C\Big(\frac{\rho}{R}\Big)^{n+2-\frac{2}{q}n}\int_{B_R}
\Big(|Du|^2+|u|^\gamma\Big)\,dx+C\rho^{n+2-\frac{2}{q}n}\|g\|_{L^q(B_R)}^2,
\end{equation}
which implies
$Du\in L^{2,\lambda}(\Omega_0)$ with $\lambda=n+2-\frac{2n}{q}$.
If $q\ge n$, also by the iteration lemma for any $\epsilon>0$ we have
\begin{equation}\label{delta3}
\int_{B_{\frac{\rho}{2}}}\Big(|Du|^2+|u|^\gamma\Big)\,dx
\le C\Big(\frac{\rho}{R}\Big)^{n-\epsilon}\int_{B_R}\Big(|Du|^2
+|u|^\gamma\Big)\,dx+C\rho^{n-\epsilon}\|g\|_{L^q(B_R)}^2,
\end{equation}
which implies $Du\in L^{2,\lambda}(\Omega_0)$ with $\lambda=n-\epsilon$.
Summarizing, in terms of the famous Morrey's lemma one concludes that
$u\in C^{0,\alpha}_{\rm loc}(B_\rho,\mathbb{R}^N),\alpha=2-\frac{n}{q} $
if $n/2 <q<n$ or $u\in C^{0,\alpha}_{\rm loc}(B_\rho,\mathbb{R}^N)$ for all
$\alpha\in (0,1)$ if $q\ge n$.

Finally, let us recall a ``small'' hypothesis of the following
so-called an excess quantity
$$
E(\rho)={\rho}^{2-n}\int_{B_{\rho}(x_0)}|Du|^2\,dx,
$$
According to the definition of $\Omega_0$, we attain
$$
\Omega\setminus\Omega_0
=\big\{x\in\Omega:\liminf_{\rho\to 0}\rho^{2-n}\int_{B_{\rho}}|Du|^2\, dx>0 \big\}.
$$
Therefore, by Lemma \ref{Haus},
${\mathcal H}^{n-2}(\Omega\setminus\Omega_0)=0$.
This completes proof.
\end{proof}

\subsection*{Acknowledgements}
This research was supported by the NSFC, grant No. 11371050.

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\end{document}