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# Regularity theory on A-harmonic system and A-Dirac system

Journal of Inequalities and Applications20142014:443

https://doi.org/10.1186/1029-242X-2014-443

• Accepted: 23 October 2014
• Published:

## Abstract

In this paper, we show the regularity theory on an A-harmonic system and an A-Dirac system. By the method of the removability theorem, we explain how an A-harmonic system arises from an A-Dirac system and establish that an A-harmonic system is in fact the real part of the corresponding A-Dirac system.

## Keywords

• A-harmonic system
• A-Dirac system
• Caccioppoli estimate
• natural growth condition
• removable theorem

## 1 Introduction

In this paper, we consider the regularity theory on an A-Dirac system,
(1.1)
and an A-harmonic system,
(1.2)

Here Ω is a bounded domain in ${R}^{n}$ ($n\ge 2$), $A\left(x,u,\mathrm{\nabla }u\right)$ and $f\left(x,u,\mathrm{\nabla }u\right)$ are measurable functions defined on $\mathrm{\Omega }×{R}^{n}×{R}^{nN}$, N is an integer with $N>1$, $u:\mathrm{\Omega }\to {R}^{n}$ is a vector valued function. Furthermore, $A\left(x,u,\mathrm{\nabla }u\right)$ and $f\left(x,u,\mathrm{\nabla }u\right)$ satisfy the following structural conditions with $m>2$:

(H1) $A\left(x,u,p\right)$ are differentiable functions in p and there exists a constant $C>0$ such that
(H2) $A\left(x,u,p\right)$ are uniformly strongly elliptic, that is, for some $\lambda >0$ we have
$\left(\frac{\partial A\left(x,u,p\right)}{\partial p}{\nu }_{i}^{\alpha }\right){\nu }_{j}^{\beta }\ge \lambda {\left(1+{|p|}^{2}\right)}^{\frac{m-2}{2}}{|\nu |}^{2}.$
(H3) There exist $\beta \in \left(0,1\right)$ and $K:\left[0,\mathrm{\infty }\right)↦\left[0,\mathrm{\infty }\right)$ monotone nondecreasing such that
$|A\left(x,u,p\right)-A\left(\stackrel{˜}{x},\stackrel{˜}{u},p\right)|\le K\left(|u|\right){\left({|x-\stackrel{˜}{x}|}^{m}+{|u-\stackrel{˜}{u}|}^{m}\right)}^{\frac{\beta }{m}}{\left(1+|p|\right)}^{\frac{m}{2}}$

for all $x,\stackrel{˜}{x}\in \mathrm{\Omega }$, $u,\stackrel{˜}{u}\in {R}^{n}$, and $p\in {R}^{nN}$. Without loss of generality, we take $K\ge 1$.

(H4) There exist constants ${C}_{1}$ and ${C}_{2}$ such that
$|f\left(x,u,p\right)|\le {C}_{1}{|p|}^{m}+{C}_{2}.$
(H1) and (H2) imply
$|A\left(x,u,p\right)-A\left(x,u,\xi \right)|\le C{\left(1+{|p|}^{2}+{|\xi |}^{2}\right)}^{\frac{m-2}{2}}|p-\xi |;$
(1.3)
$\left(A\left(x,u,p\right)-A\left(x,u,\xi \right)\right)\left(p-\xi \right)\ge \lambda {\left(1+{|p|}^{2}+{|\xi |}^{2}\right)}^{\frac{m-2}{2}}{|p-\xi |}^{2}$
(1.4)

for all $x\in \mathrm{\Omega }$, $u\in {R}^{n}$ and $p,\xi \in {R}^{nN}$, where $\lambda >0$ is a constant.

Definition 1.1 We say that a function $u\in {W}_{\mathrm{loc}}^{1,m}\left(\mathrm{\Omega }\right)\cap {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ is a weak solution to (1.2), if the equality
${\int }_{\mathrm{\Omega }}A\left(x,u,\mathrm{\nabla }u\right)\mathrm{\nabla }\varphi \phantom{\rule{0.2em}{0ex}}dx={\int }_{\mathrm{\Omega }}f\left(x,u,\mathrm{\nabla }u\right)\varphi \phantom{\rule{0.2em}{0ex}}dx$
(1.5)

holds for all $\varphi \in {W}_{0}^{1,m}\left(\mathrm{\Omega }\right)$ with compact support.

In this paper, we assume that the solutions of the A-harmonic system (1.1) and the A-Dirac system (1.2) exist [1] and establish the regularity result directly. In other words, the main purpose of this paper is to show the regularity theory on an A-harmonic system and the corresponding A-Dirac system. It means that we should know the properties of an A-harmonic operator and an A-Dirac operator. This main context will be stated in Section 2. Further discussion can be found in [210] and the references therein.

In order to prove the main result, we also need a suitable Caccioppoli estimation (see Theorem 3.1). Then by the technique of removable singularities, we can find that solutions to an A-harmonic system satisfying a Lipschitz condition or in the case of a bounded mean oscillation can be extended to Clifford valued solutions to the corresponding A-Dirac system.

The technique of removable singularities was used in [2] to remove singularities for monogenic functions with modulus of continuity $\omega \left(\mathit{r}\right)$, where the sets ${r}^{n}\omega \left(r\right)$ and Hausdorff measure are removable. Kaufman and Wu [11] used the method in the case of Hölder continuous analytic functions. In fact, under a certain geometric condition related to the Minkowski dimension, sets can be removable for A-harmonic functions in Hölder and bounded mean oscillation classes [12]. Even in the case of Hölder continuity, a precise removable sets condition was stated [13]. In [7], the author showed that under a certain oscillation condition, sets satisfying a generalized Minkowski-type inequality were removable for solutions to the A-Dirac system. The general result can be found in [14].

Motivated by these facts, one ask: Does a similar result hold for the more general case of the systems (1.1) and (1.2)? We will answer this question in this paper and obtain the following result.

Theorem 1.2 Let E be a relatively closed subset of Ω. Suppose that $u\in {L}_{\mathrm{loc}}^{m}\left(\mathrm{\Omega }\right)\cap {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ has distributional first derivatives in Ω, u is a solution to the scalar part of A-Dirac system (1.1) under the structure conditions (H1)-(H4) in $\mathrm{\Omega }\setminus E$, and u is of the type of an $m,k$-oscillation in $\mathrm{\Omega }\setminus E$. If for each compact subset K of E
${\int }_{\mathrm{\Omega }\setminus K}d{\left(x,K\right)}^{m\left(k-1\right)-k}<\mathrm{\infty },$
(1.6)

then u extends to a solution of the A-Dirac system in Ω.

## 2 A-Dirac system

In this section, we would introduce the A-Dirac system. Thus the definition of the A-Dirac operator is necessary. We first present the definitions and notations as regards the Clifford algebra at first [7].

We write ${\mathcal{U}}_{n}$ for the real universal Clifford algebra over ${R}^{n}$. The Clifford algebra is generated over R by the basis of the reduced products
$\left\{{e}_{1},{e}_{2},\dots ,{e}_{1}{e}_{2},\dots ,{e}_{1}\cdots {e}_{n}\right\},$
(2.1)

where $\left\{{e}_{1},{e}_{2},\dots ,{e}_{n}\right\}$ is an orthonormal basis of ${R}^{n}$ with the relation ${e}_{i}{e}_{j}+{e}_{j}{e}_{i}=-2{\delta }_{ij}$. We write ${e}_{0}$ for the identity. The dimension of ${\mathcal{U}}_{n}$ is ${2}^{n}$. We have an increasing tower $R\subset C\subset H\subset {\mathcal{U}}_{3}\subset \cdots$ . The Clifford algebra ${\mathcal{U}}_{n}$ is a graded algebra as ${\mathcal{U}}_{n}={⨁}_{l}{\mathcal{U}}_{n}^{l}$, where ${\mathcal{U}}_{n}^{l}$ are those elements whose reduced Clifford products have length l.

For $A\in {\mathcal{U}}_{n}$, $Sc\left(A\right)$ denotes the scalar part of A, that is, the coefficient of the element ${e}_{0}$.

Throughout this paper, $\mathrm{\Omega }\subset {R}^{n}$ is a connected and open set with boundary Ω. A Clifford-valued function $u:\mathrm{\Omega }\to {\mathcal{U}}_{n}$ can be written as $u={\sum }_{\alpha }{u}_{\alpha }{e}_{\alpha }$, where each ${u}_{\alpha }$ is real-valued and ${e}_{\alpha }$ are reduced products. The norm used here is given by $|{\sum }_{\alpha }{u}_{\alpha }{e}_{\alpha }|={\left({\sum }_{\alpha }{u}_{\alpha }^{2}\right)}^{\frac{1}{2}}$, which is sub-multiplicative, $|AB|\le C|A||B|$.

The Dirac operator defined here is
$D=\sum _{j=1}^{n}{e}_{j}\frac{\partial }{\partial {x}_{j}}.$
(2.2)

Also ${D}^{2}=-\mathrm{△}$. Here is Laplace operator.

Throughout, Q is a cube in Ω with volume $|Q|$. We write σQ for the cube with the same center as Q and with side length σ times that of Q. For $q>0$, we write ${L}^{q}\left(\mathrm{\Omega },{\mathcal{U}}_{n}\right)$ for the space of Clifford-valued functions in Ω whose coefficients belong to the usual ${L}^{q}\left(\mathrm{\Omega }\right)$ space. Also, ${W}^{1,m}\left(\mathrm{\Omega },{\mathcal{U}}_{n}\right)$ is the space of Clifford valued functions in Ω whose coefficients as well as their first distributional derivatives are in ${L}^{q}\left(\mathrm{\Omega }\right)$. We also write ${L}_{\mathrm{loc}}^{q}\left(\mathrm{\Omega },{\mathcal{U}}_{n}\right)$ for $\bigcap {L}^{q}\left({\mathrm{\Omega }}^{\prime },{\mathcal{U}}_{n}\right)$, where the intersection is over all ${\mathrm{\Omega }}^{\prime }$ compactly contained in Ω. We similarly write ${W}_{\mathrm{loc}}^{1,m}\left(\mathrm{\Omega },{\mathcal{U}}_{n}\right)$. Moreover, we write ${\mathcal{M}}_{\mathrm{\Omega }}=\left\{u:\mathrm{\Omega }\to {\mathcal{U}}_{n}|Du=0\right\}$ for the space of monogenic functions in Ω.

Furthermore, we define the Dirac Sobolev space
${W}^{D,m}\left(\mathrm{\Omega }\right)=\left\{u\in {\mathcal{U}}_{n}|{\int }_{\mathrm{\Omega }}{|u|}^{m}+{\int }_{\mathrm{\Omega }}{|Du|}^{m}<\mathrm{\infty }\right\}.$
(2.3)

The local space ${W}_{\mathrm{loc}}^{D,m}$ is similarly defined. Notice that if u is monogenic, then $u\in {L}^{m}\left(\mathrm{\Omega }\right)$ if and only if $u\in {W}^{D,m}\left(\mathrm{\Omega }\right)$. Also it is immediate that ${W}^{1,m}\left(\mathrm{\Omega }\right)\subset {W}^{D,m}\left(\mathrm{\Omega }\right)$.

With those definitions and notations and also of the A-Dirac operator, we define the linear isomorphism $\theta :{R}^{n}\to {\mathcal{U}}_{n}^{1}$ by
$\theta \left({\omega }_{1},\dots ,{\omega }_{n}\right)=\sum _{i=1}^{n}{\omega }_{i}{e}_{i}.$
(2.4)
For $x,y\in {R}^{n}$, Du is defined by $\theta \left(\mathrm{\nabla }\varphi \right)=D\varphi$ for a real-valued function ϕ, and we have
$-Sc\left(\theta \left(x\right)\theta \left(y\right)\right)=〈x,y〉,$
(2.5)
$|\theta \left(x\right)|=|x|.$
(2.6)
Here $\stackrel{˜}{A}\left(x,\xi ,\eta \right):\mathrm{\Omega }×{\mathcal{U}}_{1}×{\mathcal{U}}_{n}\to {\mathcal{U}}_{n}$ is defined by
$\stackrel{˜}{A}\left(x,u,\eta \right)=\theta A\left(x,u,{\theta }^{-1}\eta \right),$
(2.7)
which means that (1.5) is equivalent to
$\begin{array}{rl}{\int }_{\mathrm{\Omega }}Sc\left(\theta A\left(x,u,\mathrm{\nabla }u\right)\theta \left(\mathrm{\nabla }\varphi \right)\right)\phantom{\rule{0.2em}{0ex}}dx& ={\int }_{\mathrm{\Omega }}Sc\left(\stackrel{˜}{A}\left(x,u,Du\right)D\varphi \right)\phantom{\rule{0.2em}{0ex}}dx\\ ={\int }_{\mathrm{\Omega }}Sc\left(f\left(x,u,Du\right)\varphi \right)\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
(2.8)

For the Clifford conjugation $\overline{\left({e}_{j1}\cdots {e}_{jl}\right)}={\left(-1\right)}^{l}{e}_{jl}\cdots {e}_{j1}$, we define a Clifford-valued inner product as $\overline{\alpha }\beta$. Moreover, the scalar part of this Clifford inner product $Sc\left(\overline{\alpha }\beta \right)$ is the usual inner product $〈\alpha ,\beta 〉$ in ${R}^{{2}^{n}}$, when α and β are identified as vectors.

For convenience, we replace $\stackrel{˜}{A}$ with A and recast the structure systems above and define the operator:
$A\left(x,\xi ,\eta \right):\mathrm{\Omega }×{\mathcal{U}}_{1}×{\mathcal{U}}_{n}\to {\mathcal{U}}_{n},$
(2.9)

where A preserves the grading of the Clifford algebra, $x\to A\left(x,\xi ,\eta \right)$ is measurable for all ξ, η, and $\xi \to A\left(x,\xi ,\eta \right)$, $\eta \to A\left(x,\xi ,\eta \right)$ are continuous for a.e. $x\in \mathrm{\Omega }$.

Definition 2.1 A Clifford valued function $u\in {W}_{\mathrm{loc}}^{D,m}\left(\mathrm{\Omega },{\mathcal{U}}_{n}^{k}\right)\cap {L}^{\mathrm{\infty }}\left(\mathrm{\Omega },{\mathcal{U}}_{n}^{k}\right)$, for $k=0,1,\dots ,n$, is a weak solution to system (1.1) under conditions (H1)-(H4). If for all $\varphi \in {W}_{0}^{1,m}\left(\mathrm{\Omega },{\mathcal{U}}_{n}^{k}\right)$, then we have
${\int }_{\mathrm{\Omega }}\overline{A\left(x,u,Du\right)}D\varphi \phantom{\rule{0.2em}{0ex}}dx={\int }_{\mathrm{\Omega }}\overline{f\left(x,u,Du\right)}\varphi \phantom{\rule{0.2em}{0ex}}dx.$
(2.10)

## 3 Proof of the main results

In this section, we will establish the main results. At first, a suitable Caccioppoli estimate [7, 15] for solutions to (2.10) is necessary.

Theorem 3.1 Let u be weak solutions to the scalar part of system (1.1) with $\lambda >2{C}_{1}M$ and where (H1)-(H4) are satisfied. Then for every ${x}_{0}\in \mathrm{\Omega }$, ${u}_{0}\in {\mathcal{U}}_{1}^{k}$, ${p}_{0}\in {\mathcal{U}}_{n}^{k}$, and arbitrary $\sigma >1$ we have
$\begin{array}{r}{\int }_{Q}\left[{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}+{|Du-{p}_{0}|}^{m}\right]\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C\left\{\frac{1}{{\left(\sigma |Q|\right)}^{2/n}}{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|u-P|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{{\left(\sigma |Q|\right)}^{m/n}}{\int }_{\sigma Q}{|u-P|}^{m}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}{G}^{2}\phantom{\rule{0.2em}{0ex}}dx\right\},\end{array}$
(3.1)
where $P=u\left(x\right)-{u}_{0}+{p}_{0}\left(x-{x}_{0}\right)$ and
$\begin{array}{rcl}{\int }_{\sigma Q}{G}^{2}\phantom{\rule{0.2em}{0ex}}dx& =& \sigma |Q|\left\{{\left[K\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{\left(\sigma |Q|\right)}^{\frac{2\beta }{n}}\\ +\left({C}_{2}^{2}+{C}_{1}^{2}{|{p}_{0}|}^{2m}\right){\left(\sigma |Q|\right)}^{\frac{2}{n}}\right\}.\end{array}$
(3.2)
Proof Denote $u\left(x\right)-{u}_{0}-{p}_{0}\left(x-{x}_{0}\right)$ by $v\left(x\right)$ and $0<{|Q|}^{\frac{1}{n}}<\sigma {|Q|}^{\frac{1}{n}} for $\sigma >1$, consider a standard cut-off function $\eta \in {C}_{0}^{\mathrm{\infty }}\left(\sigma Q\left({x}_{0}\right)\right)$ satisfying $0\le \eta \le 1$, $|\mathrm{\nabla }\eta |<\frac{1}{{|Q|}^{1/n}}$, $\eta \equiv 1$ on $Q\left({x}_{0}\right)$. Then $\phi ={\eta }^{2}v$ is admissible as a test-function, and we obtain
$\begin{array}{r}{\int }_{\sigma Q}A\left(x,u,Du\right)\cdot \left(Du-{p}_{0}\right){\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}=-2{\int }_{\sigma Q}A\left(x,u,Du\right)\eta v\cdot \mathrm{\nabla }\eta \phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}f\left(x,u,Du\right)\cdot \phi \phantom{\rule{0.2em}{0ex}}dx.\end{array}$
We further have
$\begin{array}{r}-{\int }_{\sigma Q}A\left(x,u,{p}_{0}\right)\cdot \left(Du-{p}_{0}\right){\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}=2{\int }_{\sigma Q}A\left(x,u,{p}_{0}\right)\eta v\cdot \mathrm{\nabla }\eta \phantom{\rule{0.2em}{0ex}}dx-{\int }_{\sigma Q}A\left(x,u,{p}_{0}\right)\cdot D\phi \phantom{\rule{0.2em}{0ex}}dx,\end{array}$
and
${\int }_{\sigma Q}A\left({x}_{0},{u}_{0},{p}_{0}\right)\cdot D\phi \phantom{\rule{0.2em}{0ex}}dx=0.$
$\begin{array}{r}{\int }_{\sigma Q}\left(A\left(x,u,Du\right)-A\left(x,u,{p}_{0}\right)\right)\left(Du-{p}_{0}\right){\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}=-2{\int }_{\sigma Q}\left(A\left(x,u,Du\right)-A\left(x,u,{p}_{0}\right)\right)\left(Du-{p}_{0}\right)\eta v\cdot \mathrm{\nabla }\eta \phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}-{\int }_{\sigma Q}\left(A\left(x,u,{p}_{0}\right)-A\left(x,{u}_{0}+{p}_{0}\left(x-{x}_{0}\right),{p}_{0}\right)\right)\cdot D\phi \phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}-{\int }_{\sigma Q}\left(A\left(x,{u}_{0}+{p}_{0}\left(x-{x}_{0}\right),{p}_{0}\right)-A\left({x}_{0},{u}_{0},{p}_{0}\right)\right)\cdot D\phi \phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}f\left(x,u,Du\right)\cdot \phi \phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le I+II+III+IV+V,\end{array}$
(3.3)
where
$\begin{array}{c}I=2C{\int }_{\sigma Q}{\left(1+{|Du|}^{2}+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}|Du-{p}_{0}|\eta |v||\mathrm{\nabla }\eta |\phantom{\rule{0.2em}{0ex}}dx;\hfill \\ II=K\left(|{u}_{0}|+|{p}_{0}|\right){\int }_{\sigma Q}{|v|}^{\beta }|Du-{p}_{0}|{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx;\hfill \\ III=2K\left(|{u}_{0}|+|{p}_{0}|\right){\int }_{\sigma Q}{|v|}^{\beta +1}|\mathrm{\nabla }\eta |{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}\eta \phantom{\rule{0.2em}{0ex}}dx;\hfill \\ \begin{array}{rl}IV=& K\left(|{u}_{0}|+|{p}_{0}|\right){\int }_{\sigma Q}{\left({|x-{x}_{0}|}^{m}+{|{p}_{0}\left(x-{x}_{0}\right)|}^{m}\right)}^{\frac{\beta }{m}}{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}\\ \cdot \left(2\eta |\mathrm{\nabla }\eta ||v|+{\eta }^{2}|Du-{p}_{0}|\right)\phantom{\rule{0.2em}{0ex}}dx;\end{array}\hfill \\ V={\int }_{\sigma Q}\left({C}_{1}{|Du|}^{m}+{C}_{2}\right)|v|{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx,\hfill \end{array}$

after using (1.3), (H3), (H4).

For positive ε, to be fixed later, using Young’s inequality, we have
$\begin{array}{rl}I\le & 2C{\int }_{\sigma Q}{\left(1+2{|Du-{p}_{0}|}^{2}+3{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}|Du-{p}_{0}|\eta |v||\mathrm{\nabla }\eta |\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\left[{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}|Du-{p}_{0}|\eta |v||\mathrm{\nabla }\eta |\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}{|Du-{p}_{0}|}^{m-1}\eta |v||\mathrm{\nabla }\eta |\phantom{\rule{0.2em}{0ex}}dx\right]\\ \le & C\epsilon {\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+C\frac{1}{\epsilon }{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|v|}^{2}{|\mathrm{\nabla }\eta |}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ +C\epsilon {\int }_{\sigma Q}{|Du-{p}_{0}|}^{m}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+C\left(\epsilon \right){\int }_{\sigma Q}{|v|}^{m}{|\mathrm{\nabla }\eta |}^{m}\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
Using Young’s inequality twice in II, we have
$\begin{array}{rl}II\le & \epsilon {\int }_{\sigma Q}{|Du-{p}_{0}|}^{2}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+\frac{1}{\epsilon }{K}^{2}\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{m}{\int }_{\sigma Q}{|v|}^{2\beta }\phantom{\rule{0.2em}{0ex}}dx\\ \le & \epsilon {\int }_{\sigma Q}{|Du-{p}_{0}|}^{2}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+\frac{1}{\epsilon }{\int }_{\sigma Q}{\left(\frac{1}{{\left(\sigma |Q|\right)}^{1/n}}|v|\right)}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ +\frac{1}{\epsilon }{\left[K\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{\left(\sigma |Q|\right)}^{\left(\frac{2\beta }{1-\beta }+n\right)/n}\\ \le & \epsilon {\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+\frac{1}{\epsilon }{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{\left(\frac{1}{{\left(\sigma |Q|\right)}^{1/n}}\right)}^{2}{|v|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ +\frac{1}{\epsilon }{\left[K\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{\left(\sigma |Q|\right)}^{\left(\frac{2\beta }{1-\beta }+n\right)/n},\end{array}$
and similarly we see
$\begin{array}{rcl}III& \le & \frac{1}{2}{\int }_{\sigma Q}{|v|}^{2}{|\mathrm{\nabla }\eta |}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ +4{K}^{2}\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{m}{\int }_{\sigma Q}{\left(\sigma |Q|\right)}^{\frac{2\beta }{n}}{\left(\frac{|v|}{{\left(\sigma |Q|\right)}^{1/n}}\right)}^{2\beta }{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \le & {\int }_{\sigma Q}{\left(\frac{1}{{\left(\sigma |Q|\right)}^{1/n}}\right)}^{2}{|v|}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\left[4K\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{\left(\sigma |Q|\right)}^{\left(\frac{2\beta }{1-\beta }+n\right)/n}\\ \le & {\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{\int }_{\sigma Q}{\left(\frac{1}{{\left(\sigma |Q|\right)}^{1/n}}\right)}^{2}{|v|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ +{\left[4K\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{\left(\sigma |Q|\right)}^{\left(\frac{2\beta }{1-\beta }+n\right)/n}\end{array}$
and
$\begin{array}{rcl}IV& \le & {\int }_{\sigma Q}K\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}{\left(\sigma |Q|\right)}^{\frac{\beta }{n}}{\left(1+{|{p}_{0}|}^{m}\right)}^{\frac{\beta }{m}}\left(\eta |Du-{p}_{0}|+2\eta |\mathrm{\nabla }\eta ||v|\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le & {\int }_{\sigma Q}K\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}{\left(\sigma |Q|\right)}^{\frac{\beta }{n}}{\left(1+|{p}_{0}|\right)}^{\beta }\left(\eta |Du-{p}_{0}|+2\eta |\mathrm{\nabla }\eta ||v|\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le & \epsilon {\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|\mathrm{\nabla }\eta |}^{2}{|v|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ +\left(4+\frac{1}{\epsilon }\right){K}^{2}\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{2\left(\frac{m}{2}+\beta \right)}{\left(\sigma |Q|\right)}^{\frac{n+2\beta }{n}},\end{array}$
and for positive μ, to be fixed later, this yields
$\begin{array}{rcl}V& =& {\int }_{\sigma Q}{C}_{1}{|Du|}^{m}|u-{u}_{0}-{p}_{0}\left(x-{x}_{0}\right)|{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}\left(\frac{1}{{\left(\sigma |Q|\right)}^{1/n}}|v|\eta \right)\left({C}_{2}{\left(\sigma |Q|\right)}^{\frac{1}{n}}\eta \right)\phantom{\rule{0.2em}{0ex}}dx\\ \le & {\int }_{\sigma Q}{C}_{1}\left[\left(1+\mu \right){|Du-{p}_{0}|}^{m}+\left(1+\frac{1}{\mu }\right){|{p}_{0}|}^{m}\right]|u-{u}_{0}-{p}_{0}\left(x-{x}_{0}\right)|{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\\ +\frac{1}{2}\epsilon {C}_{2}^{2}{\left(\sigma |Q|\right)}^{\frac{n+2}{n}}+\frac{1}{2\epsilon {\left(\sigma |Q|\right)}^{2/n}}{\int }_{\sigma Q}{|v|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \le & {C}_{1}\left(1+\mu \right)\left(2M+{p}_{0}{\left(\sigma |Q|\right)}^{\frac{1}{n}}\right){\int }_{\sigma Q}{|Du-{p}_{0}|}^{m}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+{C}_{1}\left(1+\frac{1}{\mu }\right){|{p}_{0}|}^{m}{\int }_{\sigma Q}|v|{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\\ +\frac{1}{2}\epsilon {C}_{2}^{2}{\left(\sigma |Q|\right)}^{\frac{n+2}{n}}+\frac{1}{2\epsilon }{\int }_{\sigma Q}\frac{1}{{\left(\sigma |Q|\right)}^{2/n}}{|v|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \le & {C}_{1}\left(1+\mu \right)\left(2M+{p}_{0}{\left(\sigma |Q|\right)}^{\frac{1}{n}}\right){\int }_{\sigma Q}{|Du-{p}_{0}|}^{m}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+\frac{1}{\epsilon }{\int }_{\sigma Q}\frac{1}{{\left(\sigma |Q|\right)}^{2/n}}{|v|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ +\frac{\epsilon }{2}\left[{C}_{2}^{2}+{C}_{1}^{2}{\left(1+\frac{1}{\mu }\right)}^{2}{|{p}_{0}|}^{2m}\right]{\left(\sigma |Q|\right)}^{\frac{n+2}{n}}.\end{array}$
By (1.4), we obtain
$\begin{array}{r}{\int }_{\sigma Q}\left(A\left(x,u,Du\right)-A\left(x,u,{p}_{0}\right)\right)\left(Du-{p}_{0}\right){\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\ge \lambda {\int }_{\sigma Q}{\left(1+{|Du|}^{2}+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\ge \lambda {\int }_{\sigma Q}{\left(1+{|Du-{p}_{0}|}^{2}+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\ge \lambda \left\{{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}{|Du-{p}_{0}|}^{m}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\right\}.\end{array}$
Combining these estimates in (3.3) and noting that ${K}^{2}\le {K}^{\frac{2}{1-\beta }}$ (as $K\ge 1$), ${\left(\sigma |Q|\right)}^{\frac{2\beta }{\left(1-\beta \right)n}}\le {\left(\sigma |Q|\right)}^{\frac{2\beta }{n}}$ for $\sigma >1$, ${\left[{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}\ge {\left(1+|{p}_{0}|\right)}^{2\left(\frac{m}{2}+\beta \right)}$, and $4\le {4}^{\frac{2}{1-\beta }}$, we can estimate
$\begin{array}{r}\left[\lambda -2C\epsilon -2\epsilon -{C}_{1}\left(1+\mu \right)\left(2M+{p}_{0}{\left(\sigma |Q|\right)}^{\frac{1}{n}}\right)\right]\\ \phantom{\rule{2em}{0ex}}\cdot \left\{{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}{|Du-{p}_{0}|}^{m}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\right\}\\ \phantom{\rule{1em}{0ex}}\le \left(\frac{C}{\epsilon }+\frac{2}{\epsilon }+C\left(\epsilon \right)+2\right)\left\{\frac{1}{{\left(\sigma |Q|\right)}^{2/n}}{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|u-P|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{{\left(\sigma |Q|\right)}^{m/n}}{\int }_{\sigma Q}{|u-P|}^{m}\phantom{\rule{0.2em}{0ex}}dx\right\}+2\left(\frac{1}{\epsilon }+{4}^{\frac{2}{1-\beta }}\right)\\ \phantom{\rule{2em}{0ex}}\cdot {\left[K\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{\left(\sigma |Q|\right)}^{\frac{n+2\beta }{n}}\\ \phantom{\rule{2em}{0ex}}+\frac{\epsilon }{2}\left[{C}_{2}^{2}+{C}_{1}^{2}{\left(1+\frac{1}{\mu }\right)}^{2}{|{p}_{0}|}^{2m}\right]{\left(\sigma |Q|\right)}^{\frac{n+2}{n}}.\end{array}$
Define $\epsilon =\epsilon \left(\lambda ,m\right)$, $\mu =\mu \left({C}_{1},M,m,\lambda \right)$ small enough, we obtain
$\begin{array}{r}{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}{|Du-{p}_{0}|}^{m}{\eta }^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C\left\{\frac{1}{{\left(\sigma |Q|\right)}^{2/n}}{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|u-P|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{{\left(\sigma |Q|\right)}^{m/n}}{\int }_{\sigma Q}{|u-P|}^{m}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}{G}^{2}\phantom{\rule{0.2em}{0ex}}dx\right\},\end{array}$
where $C=C\left(m,\lambda ,\beta ,M\right)$ and
${\int }_{\sigma Q}{G}^{2}\phantom{\rule{0.2em}{0ex}}dx=\sigma |Q|\left\{{\left[K\left(|{u}_{0}|+|{p}_{0}|\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }}{\left(\sigma |Q|\right)}^{\frac{2\beta }{n}}+\left({C}_{2}^{2}+{C}_{1}^{2}{|{p}_{0}|}^{2m}\right){\left(\sigma |Q|\right)}^{\frac{2}{n}}\right\}.$

Now let the domain of the left-hand side be Q, then we can get the right inequality immediately. □

In order to remove singularity of solutions to A-Dirac system, we also need the fact that real-valued functions satisfying various regularity properties. Thus we have the following.

Definition 3.2 [7]

Assume that $u\in {L}_{\mathrm{loc}}^{1}\left(\mathrm{\Omega },{\mathcal{U}}_{n}\right)$, $q>0$, and that $-\mathrm{\infty }. We say that u is of the type of a $q,k$-oscillation in Ω when
$\underset{2Q\subset \mathrm{\Omega }}{sup}{|Q|}^{-\left(qk+n\right)/qn}\underset{{u}_{Q}\in {\mathcal{M}}_{Q}}{inf}{\left({\int }_{Q}{|u-{u}_{Q}|}^{q}\right)}^{1/q}<\mathrm{\infty }.$
(3.4)

If $q=1$ and $k=0$, then the inequality (3.4) is equivalent to the usual definition of the bounded mean oscillation; when $q=1$ and $0, then the inequality (3.4) is equivalent to the usual local Lipschitz condition [16]. Further discussion of the inequality (3.4) can be found in [8, 17]. In these cases, the supremum is finite if we choose ${u}_{Q}$ to be the average value of the function u over the cube Q.

We remark that it follows from Hölder’s inequality that if $s\le q$ and if u is of the type of an $q,k$-oscillation, then u is of the type of an $s,k$-oscillation.

The following lemma shows that Definition 3.2 is independent of the expansion factor of the sphere.

Lemma 3.3 [7]

Suppose that $F\in {L}_{\mathrm{loc}}^{1}\left(\mathrm{\Omega },R\right)$, $F>0$ a.e., $r\in R$ and ${\sigma }_{1},{\sigma }_{2}>1$. If
$\underset{{\sigma }_{1}Q\subset Q}{sup}{|Q|}^{r}{\int }_{Q}F<\mathrm{\infty },$
then
$\underset{{\sigma }_{2}Q\subset Q}{sup}{|Q|}^{r}{\int }_{Q}F<\mathrm{\infty }.$
(3.5)

Then we proceed to prove the main result, Theorem 1.2.

Proof of Theorem 1.2 Let Q be a cube in the Whitney decomposition of $\mathrm{\Omega }\setminus E$. The decomposition consists of closed dyadic cubes with disjoint interiors which satisfy
1. (a)

$\mathrm{\Omega }\setminus E={\bigcup }_{Q\in \mathcal{W}}Q$,

2. (b)

${|Q|}^{1/n}\le d\left(Q,\partial \mathrm{\Omega }\right)\le 4{|Q|}^{1/n}$,

3. (c)

$\left(1/4\right){|{Q}_{1}|}^{1/n}\le {|{Q}_{2}|}^{1/n}\le 4{|{Q}_{1}|}^{1/n}$ when ${Q}_{1}\cap {Q}_{2}$ is not empty.

Here $d\left(Q,\partial \mathrm{\Omega }\right)$ is the Euclidean distance between Q and the boundary of Ω [18].

If $A\subset {R}^{n}$ and $r>0$, then we define the r-inflation of A as
$A\left(r\right)=\bigcup B\left(x,r\right).$
(3.6)
Let Q be a cube in the Whitney decomposition of $\mathrm{\Omega }\setminus E$. Using the Caccioppoli estimate (3.1), we have
$\begin{array}{r}{\int }_{Q}\left[{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}+{|Du-{p}_{0}|}^{m}\right]\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C\left\{\frac{1}{{\left(\sigma Q\right)}^{2/n}}{\int }_{\sigma Q}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|u-P|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+\frac{1}{{\left(\sigma Q\right)}^{m/n}}{\int }_{\sigma Q}{|u-P|}^{m}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\sigma Q}{G}^{2}\phantom{\rule{0.2em}{0ex}}dx\right\},\end{array}$
with (3.2)
${\int }_{\sigma Q}{G}^{2}\phantom{\rule{0.2em}{0ex}}dx\le C{|\sigma Q|}^{\frac{n+2\beta }{n}}{H}^{2}\left(1+|{u}_{Q}|+|{p}_{0}|\right),$
(3.7)
where
$H\left(t\right)={\left[\stackrel{˜}{K}\left(t\right){\left(1+t\right)}^{\frac{m}{2}}\right]}^{\frac{2}{1-\beta }},\phantom{\rule{1em}{0ex}}\stackrel{˜}{K}\left(t\right)=max\left\{K\left(t\right),{C}_{1},{C}_{2}\right\},$
and choose $|Q|$ small enough such that
${|Q|}^{\frac{\beta }{n}}H\left(1+|{u}_{Q}|+|{p}_{0}|\right)\le 1.$
By the definition of the $q,k$-oscillation condition, we have
$\begin{array}{r}{\int }_{Q}\left[{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}+{|Du-{p}_{0}|}^{m}\right]\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le {C}_{1}{|Q|}^{-\frac{2}{n}}{|Q|}^{\frac{2k+n}{n}}+{C}_{2}{|Q|}^{-\frac{m}{n}}{|Q|}^{\left(mk+n\right)/n}+{C}_{3}|Q|\\ \phantom{\rule{1em}{0ex}}\le C{|Q|}^{a}.\end{array}$
(3.8)
Here $a=\left(n+mk-m\right)/n$. Since the problem is local (use a partition of unity), we show that (2.10) holds whenever $\varphi \in {W}_{0}^{1,m}\left(B\left({x}_{0},r\right)\right)$ with ${x}_{0}\in E$ and $r>0$ sufficiently small. Choose $r=\left(1/5\sqrt{n}\right)min\left\{1,d\left({x}_{0},\partial \mathrm{\Omega }\right)\right\}$ and let $K=E\cap \overline{B}\left({x}_{0},4r\right)$. Then K is a compact subset of E. Also let ${W}_{0}$ be those cubes in the Whitney decomposition of $\mathrm{\Omega }\setminus E$ which meet $B=B\left({x}_{0},r\right)$. Notice that each cube $Q\in {W}_{0}$ lies in $\mathrm{\Omega }\setminus K$. Let $\gamma =m\left(k-1\right)-k$. First, since $\gamma \ge -1$, from [12] we have $m\left(K\right)=m\left(E\right)=0$. Also since $na-n\ge \gamma$, using (1.6) and (3.8), we obtain
$\begin{array}{r}{\int }_{B\left({x}_{0},r\right)}\left[{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}+{|Du-{p}_{0}|}^{m}\right]\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C\sum _{Q\in {W}_{0}}{|Q|}^{a}\le C\sum _{Q\in {W}_{0}}d{\left(Q,K\right)}^{na}\\ \phantom{\rule{1em}{0ex}}\le C\sum _{Q\in {W}_{0}}{\int }_{Q}d{\left(x,K\right)}^{na-n}\phantom{\rule{0.2em}{0ex}}dx\le C{\int }_{K\left(1\right)\setminus K}d{\left(x,K\right)}^{na-n}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\int }_{K\left(1\right)\setminus K}d{\left(x,K\right)}^{\gamma }\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty }.\end{array}$
(3.9)

Hence $u\in {W}_{\mathrm{loc}}^{D,m}\left(\mathrm{\Omega }\right)$.

Next let $B=B\left({x}_{0},r\right)$ and assume that $\psi \in {C}_{0}^{\mathrm{\infty }}\left(B\right)$. Also let ${W}_{j}$, $j=1,2,\dots$ , be those cubes $Q\in {W}_{0}$ with $l\left(Q\right)\le {2}^{-j}$.

Consider the scalar functions
${\varphi }_{j}=max\left\{\left({2}^{-j}-d\left(x,K\right)\right){2}^{j},0\right\}.$
(3.10)
Thus each ${\varphi }_{j}$, $j=1,2,\dots$ , is Lipschitz, equal to 1 on K and as such $\psi \left(1-{\varphi }_{j}\right)\in {W}^{1,m}\left(B\setminus E\right)$ with compact support. Hence
$\begin{array}{r}{\int }_{B}\left[\overline{A\left(x,u,Du\right)}D\psi -\overline{f\left(x,u,Du\right)}\psi \right]\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}={\int }_{B\setminus E}\left[\overline{A\left(x,u,Du\right)}D\left(\psi \left(1-{\varphi }_{j}\right)\right)-\overline{f\left(x,u,Du\right)}\psi \left(1-{\varphi }_{j}\right)\right]\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+{\int }_{B}\left[\overline{A\left(x,u,Du\right)}D\left(\psi {\varphi }_{j}\right)-\overline{f\left(x,u,Du\right)}\psi {\varphi }_{j}\right]\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
(3.11)
Let
$\begin{array}{r}{J}_{1}={\int }_{B\setminus E}\left[\overline{A\left(x,u,Du\right)}D\left(\psi \left(1-{\varphi }_{j}\right)\right)-\overline{f\left(x,u,Du\right)}\psi \left(1-{\varphi }_{j}\right)\right]\phantom{\rule{0.2em}{0ex}}dx,\\ {J}_{2}={\int }_{B}\left[\overline{A\left(x,u,Du\right)}D\left(\psi {\varphi }_{j}\right)-\overline{f\left(x,u,Du\right)}\psi {\varphi }_{j}\right]\phantom{\rule{0.2em}{0ex}}dx.\end{array}$

Since u is a solution in $B\setminus E$, ${J}_{1}=0$.

Next we estimate ${J}_{2}$ as
$\begin{array}{rcl}{J}_{2}& =& {\int }_{B}A\left(x,u,Du\right)\psi D{\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{B}{\varphi }_{j}A\left(x,u,Du\right)D\psi \phantom{\rule{0.2em}{0ex}}dx-{\int }_{B}\overline{f\left(x,u,Du\right)}\psi {\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx\\ =& {J}_{2}^{\prime }+{J}_{2}^{″}+{J}_{2}^{‴}.\end{array}$
(3.12)
Noting that there exists a constant C such that $|\psi |\le C<\mathrm{\infty }$,
$|{J}_{2}^{\prime }|\le C\sum _{Q\in {W}_{j}}{\int }_{B}|A\left(x,u,Du\right)||D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx.$
Recalling that ${|Q|}^{\frac{\beta }{n}}K\left(t\right)\le 1$, we have
$\begin{array}{r}{\int }_{B}|A\left(x,u,Du\right)||D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le {\int }_{B}|A\left(x,u,Du\right)-A\left(x,u,{p}_{0}\right)||D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+{\int }_{B}|A\left(x,u,{p}_{0}\right)-A\left({x}_{0},{u}_{0},{p}_{0}\right)||D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\int }_{B}{\left(1+{|Du|}^{2}+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}|Du-{p}_{0}||D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}K\left(|u|\right){\left({|x-{x}_{0}|}^{m}+{|u-{u}_{0}|}^{m}\right)}^{\frac{\beta }{m}}{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}|D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\int }_{B}\left({\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}+{|Du-{p}_{0}|}^{m-2}\right)|Du-{p}_{0}||D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}K\left(|u|\right)\left({|x-{x}_{0}|}^{\beta }+{|u-{u}_{0}|}^{\beta }\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}|D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\int }_{B}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}|Du-{p}_{0}||D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx+C{\int }_{B}{|Du-{p}_{0}|}^{m-1}|D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}K\left(|u|\right){|x-{x}_{0}|}^{\beta }{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}|D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}K\left(|u|\right){|u-{u}_{0}|}^{\beta }{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}|D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\int }_{B}\left[{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}+{|Du-{p}_{0}|}^{m}\right]\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}\left[{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|D{\varphi }_{j}|}^{2}+{|D{\varphi }_{j}|}^{m}\right]\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}K\left(|u|\right){|Q|}^{\frac{\beta }{n}}{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}|D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}K\left(|u|\right){|Q|}^{\frac{\beta }{n}}{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}|D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{|Q|}^{a}+C{\int }_{B}\left[{|D{\varphi }_{j}|}^{2}+{|D{\varphi }_{j}|}^{m}\right]\phantom{\rule{0.2em}{0ex}}dx+C{\int }_{B}|D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx+C{\int }_{B}|D{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{|Q|}^{a}+C{\int }_{B}\left({2}^{2j}+{2}^{mj}\right)\phantom{\rule{0.2em}{0ex}}dx+C{\int }_{B}{2}^{j}\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
(3.13)
Now for $x\in Q\in {W}_{j}$, $d\left(x,K\right)$ is bounded above and below by a multiple of ${|Q|}^{1/n}$ and for $Q\in {W}_{j}$, ${|Q|}^{1/n}\le {2}^{-j}$. Hence
$\begin{array}{rl}|{J}_{2}^{\prime }|& \le C\sum _{Q\in {W}_{j}}\left({|Q|}^{a}+{|Q|}^{-\frac{m}{n}}|Q|+C{|Q|}^{-\frac{2}{n}}|Q|+{|Q|}^{-\frac{1}{n}}{|Q|}^{n}\right)\\ \le C\sum _{Q\in {W}_{j}}{|Q|}^{a}\le C{\int }_{\bigcup {W}_{j}}d{\left(x,K\right)}^{m\left(k-1\right)-k}.\end{array}$
(3.14)

Since $\bigcup {W}_{j}\subset \mathrm{\Omega }\setminus K$ and $|\bigcup {W}_{j}|\to 0$ as $j\to \mathrm{\infty }$, it follows that ${J}_{2}^{\prime }\to 0$ as $j\to \mathrm{\infty }$.

For
$|{J}_{2}^{″}|\le C\sum _{Q\in {W}_{j}}{\int }_{B}{\varphi }_{j}A\left(x,u,Du\right)D\psi \phantom{\rule{0.2em}{0ex}}dx.$
Similarly, we get
$\begin{array}{r}{\int }_{B}{\varphi }_{j}A\left(x,u,Du\right)D\psi \phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le {\int }_{B}\left(A\left(x,u,Du\right)-A\left(x,u,{p}_{0}\right)\right)D\psi \phantom{\rule{0.2em}{0ex}}dx+{\int }_{B}\left(A\left(x,u,{p}_{0}\right)-A\left({x}_{0},{u}_{0},{p}_{0}\right)\right)D\psi \phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\int }_{B}{\left(1+{|Du|}^{2}+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}|Du-{p}_{0}|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}K\left(|u|\right){\left({|x-{x}_{0}|}^{m}+{|u-{u}_{0}|}^{m}\right)}^{\frac{\beta }{m}}{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}|D\psi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\int }_{B}\left({\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}+{|Du-{p}_{0}|}^{m-2}\right)|Du-{p}_{0}||D\psi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}K\left(|u|\right)\left({|x-{x}_{0}|}^{\beta }+{|u-{u}_{0}|}^{\beta }\right){\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}|D\psi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\int }_{B}{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}|Du-{p}_{0}||D\psi |\phantom{\rule{0.2em}{0ex}}dx+C{\int }_{B}{|Du-{p}_{0}|}^{m-1}|D\psi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}K\left(|u|\right){|x-{x}_{0}|}^{\beta }{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}|D\psi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}K\left(|u|\right){|u-{u}_{0}|}^{\beta }{\left(1+|{p}_{0}|\right)}^{\frac{m}{2}}|D\psi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{\int }_{B}\left[{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|Du-{p}_{0}|}^{2}+{|Du-{p}_{0}|}^{m}\right]\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}\left[{\left(1+{|{p}_{0}|}^{2}\right)}^{\frac{m-2}{2}}{|D\psi |}^{2}+{|D\psi |}^{m}\right]\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+C{\int }_{B}|D\psi |\phantom{\rule{0.2em}{0ex}}dx+C{\int }_{B}K\left(|u|\right){|Q|}^{\frac{\beta }{n}}|D\psi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{|Q|}^{a}+C{\int }_{B}\left({|D\psi |}^{2}+{|D\psi |}^{m}\right)\phantom{\rule{0.2em}{0ex}}dx+C{\int }_{B}|D\psi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C{|Q|}^{a}+C{\int }_{B}\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
Thus,
$|{J}_{2}^{″}|\le C\sum _{Q\in {W}_{j}}\left({|Q|}^{a}+|Q|\right)\le C\sum _{Q\in {W}_{j}}{|Q|}^{a}\le C{\int }_{\bigcup {W}_{j}}d{\left(x,K\right)}^{m\left(k-1\right)-k}.$
(3.15)
Since $u\in {W}_{\mathrm{loc}}^{1,D}\left(\mathrm{\Omega }\right)$ and $|\bigcup {W}_{j}|\to 0$ as $j\to \mathrm{\infty }$, we have ${J}_{2}^{″}\to 0$ as $j\to \mathrm{\infty }$. In order to estimate ${J}_{2}^{‴}$, we should use (H4):
${J}_{2}^{‴}={\int }_{B}\overline{f\left(x,u,Du\right)}\psi {\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx\le C{\int }_{B}{|Du-{p}_{0}|}^{m}\phantom{\rule{0.2em}{0ex}}dx+C{\int }_{B}{|{p}_{0}|}^{m}\phantom{\rule{0.2em}{0ex}}dx={J}_{3}^{\prime }+{J}_{3}^{″}.$
(3.16)
Similar to the estimate of (3.14), using the Caccioppoli inequality (3.1) and the inequality (3.8), we get
$\begin{array}{rl}{J}_{3}^{\prime }& \le C\sum _{Q\in {W}_{j}}{\int }_{Q}{|Du-{p}_{0}|}^{m}\phantom{\rule{0.2em}{0ex}}dx\le C\sum _{Q\in {W}_{j}}{|Q|}^{\frac{\left(n+mk-m\right)}{n}}\\ \le C{\int }_{\bigcup {W}_{j}}d{\left(x,K\right)}^{n+mk-m}\phantom{\rule{0.2em}{0ex}}dx\le C{\int }_{\bigcup {W}_{j}}d{\left(x,K\right)}^{m\left(k-1\right)-k}\phantom{\rule{0.2em}{0ex}}dx.\\ \to 0\phantom{\rule{1em}{0ex}}\left(j\to \mathrm{\infty }\right),\end{array}$
and
$\begin{array}{rl}{J}_{3}^{″}& \le C\sum _{Q\in {W}_{j}}{\int }_{Q}\phantom{\rule{0.2em}{0ex}}dx=C\sum _{Q\in {W}_{j}}|Q|\\ \le C{\int }_{\bigcup {W}_{j}}d{\left(x,K\right)}^{n}\phantom{\rule{0.2em}{0ex}}dx\le C{\int }_{\bigcup {W}_{j}}d{\left(x,K\right)}^{n+mk-m}\phantom{\rule{0.2em}{0ex}}dx\\ \le C{\int }_{\bigcup {W}_{j}}d{\left(x,K\right)}^{m\left(k-1\right)-k}\phantom{\rule{0.2em}{0ex}}dx\\ \to 0\phantom{\rule{1em}{0ex}}\left(j\to \mathrm{\infty }\right).\end{array}$

Hence ${J}_{2}\to 0$.

Combining estimates ${J}_{1}$ and ${J}_{2}$ in (3.11), we prove Theorem 1.2. □

## Declarations

### Acknowledgements

Supported by National Natural Science Foundation of China (No: 11201415); Program for New Century Excellent Talents in Fujian Province University (No: JA14191).

## Authors’ Affiliations

(1)
School of Mathematics and Statistics, Minnan Normal University, Fujian, Zhangzhou, 363000, China

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