The main purpose of this article is to study Besov regularities of weak solutions to sub-elliptic equations of the type

${\text{div}}_{H}A\left(x,\mathfrak{X}u\right)=0$(1.1)

and

${\text{div}}_{H}A\left(x,\mathfrak{X}u\right)={\text{div}}_H\left({\left|F\right|}^{p-2}F\right)$(1.2)

in Ω, respectively. Here Ω is an open and bounded sub-domain in the Heisenberg group ${ℍ}^n={ℝ}^{2n+1}$ ( $n\ge 1$). In the homogeneous Equation (1.1) and the non-homogeneous Equation (1.2), the unknown $u\in H{W}_{\text{loc}}^{1,p}\left(\Omega \right)$, where the sub-elliptic Sobolev space $H{W}^{1,p}\left(\Omega \right)$ will be introduced in Section 2. In both equations, the horizontal divergence operator ${\text{div}}_H$ and the horizontal gradient $\mathfrak{X}$ are defined by

${\text{div}}_{H}F=\underset{i=1}{\overset{2n}{{\displaystyle \sum }}}{X}_i{F}_i,$

$\mathfrak{X}u=\left(X_{1}u,X_{2}u,\cdots ,X_{2n-1}u,X_{2n}u\right)$

in the distributional sense. Moreover, $A:\Omega \times {ℝ}^{2n}\to {ℝ}^{2n}$ is assumed to be a Carathéodory vector field with general growth and uniformly elliptic conditions, that is, there exist constants $\nu ,L,k>0$ and $0<\mu <1$ such that

(A1) $\left[A\left(x,\xi \right)-A\left(x,\eta \right)\right]\cdot \left(\xi -\eta \right)\ge \nu {\left({\mu }^2+{\left|\xi \right|}^2+{\left|\eta \right|}^2\right)}^{\frac{p-2}{2}}{\left|\xi -\eta \right|}^2$,

(A2) $\left|A\left(x,\xi \right)-A\left(x,\eta \right)\right|\le L{\left({\mu }^2+{\left|\xi \right|}^2+{\left|\eta \right|}^2\right)}^{\frac{p-2}{2}}\left|\xi -\eta \right|$,

(A3) $\left|A\left(x,\xi \right)\right|\le k{\left({\mu }^2+{\left|\xi \right|}^2\right)}^{\frac{p-1}{2}}$

for every $\xi ,\eta \in {ℝ}^{2n}$ and for almost all $x\in \Omega$. In (1.2), $F:\Omega \to {ℝ}^{2n}$.

The regularity of solutions to elliptic equations in Euclidean spaces ${ℝ}^n$ has been well studied by Iwaniec [1] , DiBenedetto and Manfredi [2] . Then this theory is extended to the case of general elliptic problems, see in relevant papers [3] - [6] . For the nonlinear Calderón-Zygmund estimate in the Heisenberg group, Goldstein and Zatorska-Goldstein [7] deal with the quadratic case $p=2$. Later on the $H{W}^{1,p}$ estimates for sub-elliptic equations on ${ℍ}^n$ are proved by Mingione, Zatorska-Goldstein and Zhong [8] . They consider the equation of the form

${\text{div}}_H\left[b\left(x\right)a\left(\mathfrak{X}u\right)\right]={\text{div}}_H\left({\left|F\right|}^{p-2}F\right)$

with $b\in {\text{VMO}}_{\text{loc}}\left(\Omega \right)$.

At present, the studies are concerned with the regularity estimates of weak solutions in Besov spaces in both ${ℝ}^n$ and ${ℍ}^n$ ( [9] - [11] ). Besov spaces consist of a wide class of functions compared with the classical Sobolev spaces. Baisón [12] deals with nonlinear elliptic equations in divergence form, and obtains a regularity estimate of weak solutions in Besov spaces. Clop [13] and Lyaghfouri [14] extended the result in Besov spaces by establishing a higher integrability of weak solutions.

For the homogeneous case (1.1), we assume that there exists a function $g\in L^{\frac{Q}{\alpha }}\left(\Omega \right)$ ( $0<\alpha <1$) such that

(A4) $\left|A\left(x,\xi \right)-A\left(y,\xi \right)\right|\le {\text{dist}}_{\text{CC}}{\left(x,y\right)}^{\alpha }\left(g\left(x\right)+g\left(y\right)\right){\left({\mu }^2+{\left|\xi \right|}^2\right)}^{\frac{p-1}{2}}$

for almost every $x,y\in \Omega$ and all $\xi \in {ℝ}^{2n}$. Here ${\text{dist}}_{\text{CC}}\left(x,y\right)$ is the CC-distance between two points x and y in ${ℍ}^n$.

While for the non-homogeneous situation (1.2), we assume that there exists a sequence of measurable non-negative functions $g_k\in L^{\frac{Q}{\alpha }}\left(\Omega \right)\left(k\in \mathbb{N},0<\alpha <1\right)$ satisfying that

(A5) $\{\begin{array}{l}\underset{k=1}{\overset{\infty }{{\displaystyle \sum }}}{\Vert g_k\Vert }_{L^{\frac{Q}{\alpha }}\left(\Omega \right)}^q<\infty \left(1\le q<\infty \right)\\ \left|A\left(x,\xi \right)-A\left(y,\xi \right)\right|\le {\text{dist}}_{\text{CC}}{\left(x,y\right)}^{\alpha }\left(g_k\left(x\right)+g_k\left(y\right)\right){\left({\mu }^2+{\left|\xi \right|}^2\right)}^{\frac{p-1}{2}}\end{array}$

for $\xi \in {ℝ}^{2n}$ and almost all $x,y\in \Omega$ such that $2^{-k}\le {\text{dist}}_{\text{CC}}\left(x,y\right)<2^{-k+1}$. According to (A5), we write ${\left\{g_k\right\}}_k\in l^q\left(L^{\frac{Q}{\alpha }}\left(\Omega \right)\right)$ in short.

By introducing an auxiliary function

$V\left(\xi \right)={\left({\mu }^2+{\left|\xi \right|}^2\right)}^{\frac{p-2}{4}}\xi$(1.3)

with $\xi \in {ℝ}^{2n}$, we present the main results of this article.

Theorem 1.1 Let $0<\alpha <1$ and $2\le p<4$. Assume that A satisfies hypotheses (A1)-(A4) with $0<\mu <1$. If $u\in H{W}_{\text{loc}}^{1,p}\left(\Omega \right)$ is a weak solution to (1.1), then $V\left(\mathfrak{X}u\right)\in B_{2,\infty }^{\alpha }\left(\Omega \right)$ locally.

Theorem 1.2 Let $0<\alpha <1$, $2\le p<4$, and $1\le q<\frac{2Q}{Q-2\alpha }$. Assume that the hypotheses (A1)-(A3) and (A5) hold. If $u\in H{W}_{\text{loc}}^{1,p}\left(\Omega \right)$ is a weak solution to (1.2) with $0<\mu <1$ and ${\left|F\right|}^{p-2}F\in B_{2,q}^{\alpha }\left(\Omega \right)$, then $V\left(\mathfrak{X}u\right)\in B_{2,q}^{\alpha }\left(\Omega \right)$ locally.

See Section 2 for the definitions of $H{W}^{1,p}\left(\Omega \right)$ and $B_{2,q}^{\alpha }\left(\Omega \right)$.

The contribution of the main results is to study a wide class of sub-elliptic equations in the Heisenberg group. Our aim is to obtain a Besov regularity estimate of weak solutions. The hypotheses (A1)-(A4) (or (A5)) shall be an extension of the VMO conditions.

This article is organized as follows. In Section 2 we give some definitions and tools such as classical inequalities, and we present two Lemmas relating to the reverse Hölder type inequalities of weak solutions. In Section 3 and Section 4, we present the proofs of Theorem 1.1 and Theorem 1.2, respectively.

In this section, we present the proofs of Theorem 1.1. Inspired by [13] , for the vector field $A\left(x,\xi \right)$ appeared in (1.2), we introduce

$A_B\left(\xi \right)={\displaystyle {\int }_B^{-}A\left(x,\xi \right)\text{d}x}$(3.1)

for $\xi \in {ℝ}^{2n}$ and a CC-ball $B\subset \Omega$. Then we define

$V\left(x,B\right)=\underset{\xi \in {ℝ}^{2n}}{\text{sup}}\frac{\left|A\left(x,\xi \right)-A_B\left(\xi \right)\right|}{{\left({\mu }^2+{\left|\xi \right|}^2\right)}^{\frac{p-1}{2}}},$(3.2)

where $B\subset \Omega$ is a CC-ball and $x\in \Omega$. It follows that if $A:\Omega \times {ℝ}^{2n}\to {ℝ}^{2n}$ be a Carathéodory vector field such that (A1)-(A4) hold, then A is locally uniformly in VMO, that is,

$\underset{R\to 0}{\lim }\underset{r\left(B\right)<R}{\sup }\underset{c\left(B\right)\in K}{\sup }{\displaystyle {\int }_B^{-}V\left(x,B\right)\text{d}x}=0,$(3.3)

where $K\subset \Omega$, $c\left(B\right)$ and $r\left(B\right)$ denote the center and the radius of the CC-ball B, respectively.

In order to prove Theorem 1.1, we mention that there exists a constant $\hat{C}>0$ such that

${\hat{C}}^{-1}{\left({\mu }^2+{\left|\xi \right|}^2+{\left|\eta \right|}^2\right)}^{\frac{p-2}{2}}\le \frac{{\left|V\left(\xi \right)-V\left(\eta \right)\right|}^2}{{\left|\xi -\eta \right|}^2}\le \hat{C}{\left({\mu }^2+{\left|\xi \right|}^2+{\left|\eta \right|}^2\right)}^{\frac{p-2}{2}}$(3.4)

for any $\xi ,\eta \in {ℝ}^{2n}$ and $\left|\xi -\eta \right|\ne 0$.

We are in a position to present the proof.

Proof of Theorem 1.1. We let $B_{3R}⋐\Omega$ and select a test function $\phi ={\text{Δ}}_{-h}\left({\eta }^2{\text{Δ}}_{h}u\right)$ to (1.1), where $\eta \in C_0^{\infty }\left(B_{3R}\right)$ is a cut-off function satisfying that

$0\le \eta \left(x\right)\le 1,\eta \left(x\right)\equiv 1\text{for}x\in B_{\frac{R}{2}},\eta \left(x\right)\equiv 0\text{for}x\in B_{3R}\backslash B_R,\text{and}\left|\mathfrak{X}\eta \right|\le \frac{C}{R}.$

One gets that

$\begin{array}{c}{G}_1={\displaystyle {\int }_{B_{2R}}\left[A\left(x+h,\mathfrak{X}u\left(x+h\right)\right)-A\left(x+h,\mathfrak{X}u\right)\right]\cdot {\eta }^2{\text{Δ}}_h\mathfrak{X}u\text{d}x}\\ ={\displaystyle {\int }_{B_{2R}}\left[A\left(x,\mathfrak{X}u\right)-A\left(x+h,\mathfrak{X}u\right)\right]\cdot {\eta }^2{\text{Δ}}_h\mathfrak{X}u\text{d}x}\\ +{\displaystyle {\int }_{B_{2R}}\left[A\left(x+h,\mathfrak{X}u\right)-A\left(x+h,\mathfrak{X}u\left(x+h\right)\right)\right]\cdot 2\eta \mathfrak{X}\eta {\text{Δ}}_{h}u\text{d}x}\\ +{\displaystyle {\int }_{B_{2R}}\left[A\left(x,\mathfrak{X}u\right)-A\left(x+h,\mathfrak{X}u\right)\right]\cdot 2\eta \mathfrak{X}\eta {\text{Δ}}_{h}u\text{d}x}\\ =G_2+G_3+G_4.\end{array}$(3.5)

We estimate each $G_i$ ( $1\le i\le 4$) in (3.5). By (A1), it is clear that

$G_1\ge \nu {\displaystyle {\int }_{B_{2R}}{\left({\mu }^2+{\left|\mathfrak{X}u\left(x+h\right)\right|}^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-2}{2}}{\left|{\text{Δ}}_h\mathfrak{X}u\right|}^2{\eta }^2\text{d}x}.$(3.6)

For $G_2$, according to (A4) and (2.2), we obtain that

$\begin{array}{c}{G}_2\le {\displaystyle {\int }_{B_{2R}}{\left|h\right|}^{\alpha }\left(g\left(x\right)+g\left(x+h\right)\right){\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-1}{2}}\left|{\text{Δ}}_h\mathfrak{X}u\right|{\eta }^2\text{d}x}\\ \le \epsilon {\displaystyle {\int }_{B_{2R}}{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-2}{2}}{\left|{\text{Δ}}_h\mathfrak{X}u\right|}^2{\eta }^2\text{d}x}\\ +C{\left|h\right|}^{2\alpha }{\displaystyle {\int }_{B_{2R}}{\left(g\left(x\right)+g\left(x+h\right)\right)}^2{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p}{2}}\text{d}x}\\ \le \epsilon {\displaystyle {\int }_{B_{2R}}{\left({\mu }^2+{\left|\mathfrak{X}u\left(x+h\right)\right|}^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-2}{2}}{\left|{\text{Δ}}_h\mathfrak{X}u\right|}^2{\eta }^2\text{d}x}\\ +C{\left|h\right|}^{2\alpha }{\displaystyle {\int }_{B_{2R}}{\left(g\left(x\right)+g\left(x+h\right)\right)}^2{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p}{2}}\text{d}x}.\end{array}$(3.7)

where $\epsilon >0$ will be chosen later. By (A2) and (2.2), one deduces that

$\begin{array}{c}{G}_3\le C{\displaystyle {\int }_{B_{2R}}{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2+{\left|\mathfrak{X}u\left(x+h\right)\right|}^2\right)}^{\frac{p-2}{2}}\left|{\text{Δ}}_h\mathfrak{X}u\right|\eta \left|\mathfrak{X}\eta \right|\left|{\text{Δ}}_{h}u\right|\text{d}x}\\ \le \epsilon {\displaystyle {\int }_{B_{2R}}{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2+{\left|\mathfrak{X}u\left(x+h\right)\right|}^2\right)}^{\frac{p-2}{2}}{\left|{\text{Δ}}_h\mathfrak{X}u\right|}^2{\eta }^2\text{d}x}\\ +C{\displaystyle {\int }_{B_{2R}}{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2+{\left|\mathfrak{X}u\left(x+h\right)\right|}^2\right)}^{\frac{p-2}{2}}{\left|\mathfrak{X}\eta \right|}^2{\left|{\text{Δ}}_{h}u\right|}^2\text{d}x}.\end{array}$

By applying Lagrange Mean Value Theorem, we obtain

$\begin{array}{l}C{\displaystyle {\int }_{B_{2R}}{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2+{\left|\mathfrak{X}u\left(x+h\right)\right|}^2\right)}^{\frac{p-2}{2}}{\left|\mathfrak{X}\eta \right|}^2{\left|{\text{Δ}}_{h}u\right|}^2\text{d}x}\\ \le C{\left|h\right|}^2{\displaystyle {\int }_{B_{2R+\left|h\right|}}{\left({\mu }^2+2{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-2}{2}}{\left|\mathfrak{X}u\right|}^2\text{d}x}\\ \le C{\left|h\right|}^2{\displaystyle {\int }_{B_{2R+\left|h\right|}}{\left(\mu +\left|\mathfrak{X}u\right|\right)}^p\text{d}x}.\end{array}$(3.8)

To estimate $G_4$, the hypothesis (A4) and (2.2) give us that

$\begin{array}{c}{G}_4\le C{\displaystyle {\int }_{B_{2R}}{\left|h\right|}^{\alpha }\left(g\left(x\right)+g\left(x+h\right)\right){\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-1}{2}}\eta \left|\mathfrak{X}\eta \right|\left|{\text{Δ}}_{h}u\right|\text{d}x}\\ \le \epsilon {\displaystyle {\int }_{B_{2R}}{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-2}{2}}{\eta }^2{\left|{\text{Δ}}_{h}u\right|}^2\text{d}x}\\ +C{\left|h\right|}^{2\alpha }{\displaystyle {\int }_{B_{2R}}{\left(g\left(x\right)+g\left(x+h\right)\right)}^2{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p}{2}}\text{d}x}.\end{array}$(3.9)

Here we notice that

$\epsilon {\displaystyle {\int }_{B_{2R}}{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-2}{2}}{\eta }^2{\left|{\text{Δ}}_{h}u\right|}^2\text{d}x}\le C{\left|h\right|}^2{\displaystyle {\int }_{B_{2R+\left|h\right|}}{\left(\mu +\left|\mathfrak{X}u\right|\right)}^p\text{d}x}.$

Combining the estimates of $G_i$ and choosing $\epsilon$ small enough, we obtain that

$\begin{array}{l}{\displaystyle {\int }_{B_{2R}}{\left({\mu }^2+{\left|\mathfrak{X}u\left(x+h\right)\right|}^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-2}{2}}{\left|{\text{Δ}}_h\mathfrak{X}u\right|}^2{\eta }^2\text{d}x}\\ \le C{\left|h\right|}^{2\alpha }{\displaystyle {\int }_{B_{2R}}{\left(g\left(x\right)+g\left(x+h\right)\right)}^2{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p}{2}}\text{d}x}\\ +C{\left|h\right|}^2{\displaystyle {\int }_{B_{2R+\left|h\right|}}{\left(\mu +\left|\mathfrak{X}u\right|\right)}^p\text{d}x}.\end{array}$(3.10)

By the definition of V (1.3) and the property (3.4), one gets

$\begin{array}{c}{\left|{\Delta }_{h}V\left(\mathfrak{X}u\right)\right|}^2={\left|{\left({\mu }^2+{\left|\mathfrak{X}u\left(x+h\right)\right|}^2\right)}^{\frac{p-2}{4}}\mathfrak{X}u\left(x+h\right)-{\left({\mu }^2+{\left|\mathfrak{X}u\left(x\right)\right|}^2\right)}^{\frac{p-2}{4}}\mathfrak{X}u\left(x\right)\right|}^2\\ \le C{\left({\mu }^2+{\left|\mathfrak{X}u\left(x+h\right)\right|}^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-2}{2}}{\left|{\Delta }_h\mathfrak{X}u\right|}^2.\end{array}$(3.11)

We integrate both sides of (3.11) on $B_{\frac{R}{2}}$ and apply the properties of $\eta$ to get

$\begin{array}{c}{\displaystyle {\int }_{B_{\frac{R}{2}}}{\left|{\text{Δ}}_{h}V\left(\mathfrak{X}u\right)\right|}^2\text{d}x}\le C{\displaystyle {\int }_{B_{\frac{R}{2}}}{\left({\mu }^2+{\left|\mathfrak{X}u\left(x+h\right)\right|}^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p-2}{2}}{\left|{\text{Δ}}_h\mathfrak{X}u\right|}^2{\eta }^2\text{d}x}\\ \le C{\left|h\right|}^{2\alpha }{\displaystyle {\int }_{B_{2R}}{\left(g\left(x\right)+g\left(x+h\right)\right)}^2{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p}{2}}\text{d}x}\\ +C{\left|h\right|}^2{\displaystyle {\int }_{B_{2R+\left|h\right|}}{\left(\mu +\left|\mathfrak{X}u\right|\right)}^p\text{d}x}\end{array}$(3.12)

Dividing both sides of (3.12) by ${\left|h\right|}^{2\alpha }$, it follows that

$\begin{array}{c}{\displaystyle {\int }_{B_{\frac{R}{2}}}{\left|\frac{{\text{Δ}}_{h}V\left(\mathfrak{X}u\right)}{{\left|h\right|}^{\alpha }}\right|}^2\text{d}x}\le C{\displaystyle {\int }_{B_{2R}}{\left(g\left(x\right)+g\left(x+h\right)\right)}^2{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p}{2}}\text{d}x}\\ +C{\left|h\right|}^{2-2\alpha }{\displaystyle {\int }_{B_{2R+\left|h\right|}}{\left(\mu +\left|\mathfrak{X}u\right|\right)}^p\text{d}x}\\ =:P_1+P_2.\end{array}$(3.13)

Finally, we shall give the proof that $P_i$ is bounded for each i. In view of Lemma 2.1, ones get ${\left|\mathfrak{X}u\right|}^p\in L^t\left(\Omega \right)$ with $t>1$. In particular, ${\left|\mathfrak{X}u\right|}^p\in L^{\frac{Q}{Q-2\alpha }}\left(\Omega \right)$. By choosing $0<\left|h\right|<\delta <R$ and (A4), we acquire

$\begin{array}{c}{P}_1\le C{\left({\displaystyle {\int }_{B_{2R}}{\left(g\left(x\right)+g\left(x+h\right)\right)}^{\frac{Q}{\alpha }}\text{d}x}\right)}^{\frac{2\alpha }{Q}}{\left({\displaystyle {\int }_{B_{2R}}{\left[{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p}{2}}\right]}^{\frac{Q}{Q-2\alpha }}\text{d}x}\right)}^{\frac{Q-2\alpha }{Q}}\\ \le C{\left({\displaystyle {\int }_{B_{2R+\left|h\right|}}g{\left(x\right)}^{\frac{Q}{\alpha }}\text{d}x}\right)}^{\frac{2\alpha }{Q}}{\left({\displaystyle {\int }_{B_{2R}}{\left[{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{p}{2}}\right]}^{\frac{Q}{Q-2\alpha }}\text{d}x}\right)}^{\frac{n-2\alpha }{Q}}<\infty .\end{array}$

Because $u\in H{W}_{\text{loc}}^{1,p}\left(\Omega \right)$, we get $P_2<\infty$. It follows that $\underset{\left|h\right|<\delta }{\text{sup}}{\displaystyle {\int }_{B_{\frac{R}{2}}}{\left|\frac{{\text{Δ}}_{h}V\left(\mathfrak{X}u\right)}{{\left|h\right|}^{\alpha }}\right|}^2\text{d}x}<\infty$ with $\delta <R$, that is, $V\left(\mathfrak{X}u\right)\in B_{2,\infty }^{\alpha }\left(\Omega \right)$ locally.

For the non-homogeneous case, we need the following lemma.

Lemma 4.1 Let $A:\Omega \times {ℝ}^{2n}\to {ℝ}^{2n}$ be a Carathéodory vector field such that (A1)-(A3) and (A5) hold. Then A is locally uniformly in VMO, that is,

$\underset{R\to 0}{\text{lim}}\underset{r\left(B\right)<R}{\text{sup}}\underset{c\left(B\right)\in K}{\text{sup}}{\displaystyle {\int }_B^{-}V\left(x,B\right)\text{d}x}=0,$(4.1)

where $V\left(x,B\right)$ is given in (3.2), $K\subset \Omega$, $c\left(B\right)$ and $r\left(B\right)$ denote the center and the radius of the CC-ball B, respectively.

Proof. Given a point $x\in \Omega$, we let $A_k\left(x\right)=\left\{y\in \Omega :2^{-k}\le {\text{dist}}_{\text{CC}}\left(x,y\right)<2^{-k+1}\right\}$. Ones get

$\begin{array}{c}{\displaystyle {\int }_B^{-}V\left(x,B\right)\text{d}x}\le {\displaystyle {\int }_B^{-}\underset{\xi \in {ℝ}^{2n}}{\text{sup}}}{\displaystyle {\int }_B^{-}\frac{\left|A\left(x,\xi \right)-A\left(y,\xi \right)\right|}{{\left({\mu }^2+{\left|\xi \right|}^2\right)}^{\frac{p-1}{2}}}\text{d}y\text{d}x}\\ ={\displaystyle {\int }_B^{-}\underset{\xi \in {ℝ}^{2n}}{\text{sup}}}\frac{1}{\left|B\right|}\underset{k}{{\displaystyle \sum }}{\displaystyle {\int }_{B\cap A_k\left(x\right)}\frac{\left|A\left(x,\xi \right)-A\left(y,\xi \right)\right|}{{\left({\mu }^2+{\left|\xi \right|}^2\right)}^{\frac{p-1}{2}}}\text{d}y\text{d}x}\\ \le \frac{1}{{\left|B\right|}^2}{\displaystyle \underset{k}{\sum }{\displaystyle {\int }_B{\displaystyle {\int }_{B\cap A_k\left(x\right)}{\text{dist}}_{\text{CC}}{\left(x,y\right)}^{\alpha }\left(g_k\left(x\right)+g_k\left(y\right)\right)\text{d}y\text{d}x}}}\\ \le {\left(\frac{1}{{\left|B\right|}^2}{\displaystyle \underset{k}{\sum }{\displaystyle {\int }_B{\displaystyle {\int }_{B\cap A_k\left(x\right)}{\text{dist}}_{\text{CC}}{\left(x,y\right)}^{\frac{Q\alpha }{Q-\alpha }}\text{d}y\text{d}x}}}\right)}^{\frac{Q-\alpha }{Q}}\\ \times {\left(\frac{1}{{\left|B\right|}^2}{\displaystyle \underset{k}{\sum }{\displaystyle {\int }_B{\displaystyle {\int }_{B\cap A_k\left(x\right)}{\left(g_k\left(x\right)+g_k\left(y\right)\right)}^{\frac{Q}{\alpha }}\text{d}y\text{d}x}}}\right)}^{\frac{\alpha }{Q}}\\ \le C\left(Q,\alpha \right){\left|B\right|}^{\frac{\alpha }{Q}}{\left(\frac{1}{{\left|B\right|}^2}{\displaystyle \underset{k}{\sum }{\displaystyle {\int }_B{\displaystyle {\int }_{B\cap A_k\left(x\right)}{\left(g_k\left(x\right)+g_k\left(y\right)\right)}^{\frac{Q}{\alpha }}\text{d}y\text{d}x}}}\right)}^{\frac{\alpha }{Q}}.\end{array}$

By Hölder inequality, we acquire that

$\begin{array}{l}{\left(\frac{1}{{\left|B\right|}^2}{\displaystyle \underset{k}{\sum }{\displaystyle {\int }_B{\displaystyle {\int }_{B\cap A_k\left(x\right)}{\left(g_k\left(x\right)+g_k\left(y\right)\right)}^{\frac{Q}{\alpha }}\text{d}y\text{d}x}}}\right)}^{\frac{\alpha }{Q}}\\ \le C{\left(\frac{1}{{\left|B\right|}^2}\underset{k}{{\displaystyle \sum }}\left|B\cap A_k\left(x\right)\right|{\displaystyle {\int }_B{g}_k{\left(x\right)}^{\frac{Q}{\alpha }}\text{d}x}\right)}^{\frac{\alpha }{Q}}\\ \le \frac{C}{{\left|B\right|}^{\frac{2}{q}}}{\left(\underset{k}{{\displaystyle \sum }}{\Vert g_k\Vert }_{L^{\frac{Q}{\alpha }}\left(B\right)}^q\right)}^{\frac{1}{q}}\frac{1}{{\left|B\right|}^{2\left(\frac{\alpha }{Q}-\frac{1}{q}\right)}}{\left(\underset{k}{{\displaystyle \sum }}{\left|B\cap A_k\left(x\right)\right|}^{\frac{\alpha q}{\alpha q-Q}}\right)}^{\frac{\alpha }{Q}\cdot \frac{\alpha q-Q}{\alpha Q}}\\ \le C\left(Q,\alpha ,q\right){\left|B\right|}^{-\frac{\alpha }{Q}}{\left(\underset{k}{{\displaystyle \sum }}{\Vert g_k\Vert }_{L^{\frac{Q}{\alpha }}\left(B\right)}^q\right)}^{\frac{1}{q}}.\end{array}$

We choose $r>0$ small enough and observe that $x\to {\Vert g_k\Vert }_{l^q\left(L^{\frac{Q}{\alpha }}\left(B_r\left(x\right)\right)\right)}^{}$ is continuous on the set $\left\{x\in \Omega :\text{dist}\left(x,\partial \Omega \right)>r\right\}$. Therefore, there is a point $x_r\in K$ for $r>0$ small enough such that

$\underset{x\in K}{\text{sup}}{\Vert g_k\Vert }_{l^q\left(L^{\frac{Q}{\alpha }}\left(B_r\left(x\right)\right)\right)}^{}={\Vert g_k\Vert }_{l^q\left(L^{\frac{Q}{\alpha }}\left(B_r\left(x\right)\right)\right)}^{}.$

We obtain that

$\underset{r\to 0}{\lim }{\Vert g_k\Vert }_{l^q\left(L^{\frac{Q}{\alpha }}\left(B_r\left(x\right)\right)\right)}^{}={\left(\underset{k}{{\displaystyle \sum }}\underset{r\to 0}{\text{lim}}{\left({\displaystyle {\int }_{B_r\left(x_r\right)}{g}_k^{\frac{Q}{\alpha }}}\right)}^{\frac{q\alpha }{Q}}\right)}^{\frac{1}{q}}.$

Each of the limits on the right hand side equals to 0. Hence we complete the proof. □

With the help of preceding lemma, we have the following result.

Proof of Theorem 1.2. We assume that $B_{3R+1}⋐\Omega$, and choose a test function $\phi ={\Delta }_{-h}\left({\eta }^2{\Delta }_{h}u\right)$ to (1.2), where $\eta \in C_0^{\infty }\left(\Omega \right)$ is a cut-off function satisfying that

$0\le \eta \left(x\right)\le 1,\eta \left(x\right)\equiv 1\text{for}x\in B_{\frac{R}{2}},\eta \left(x\right)\equiv 0\text{for}x\in B_{3R+1}\backslash B_R,\text{and}\left|\mathfrak{X}\eta \right|\le \frac{C}{R}.$

According to the definition of weak solution and choice of test function, we obtain

$\begin{array}{c}{G}_1={\displaystyle {\int }_{B_{2R}}\left[A\left(x+h,\mathfrak{X}u\left(x+h\right)\right)-A\left(x+h,\mathfrak{X}u\right)\right]\cdot {\eta }^2{\text{Δ}}_h\mathfrak{X}u\text{d}x}\\ ={\displaystyle {\int }_{B_{2R}}\left[A\left(x,\mathfrak{X}u\right)-A\left(x+h,\mathfrak{X}u\right)\right]\cdot {\eta }^2{\text{Δ}}_h\mathfrak{X}u\text{d}x}\\ +{\displaystyle {\int }_{B_{2R}}\left[A\left(x+h,\mathfrak{X}u\right)-A\left(x+h,\mathfrak{X}u\left(x+h\right)\right)\right]\cdot 2\eta \mathfrak{X}\eta {\text{Δ}}_{h}u\text{d}x}\\ +{\displaystyle {\int }_{B_{2R}}\left[A\left(x,\mathfrak{X}u\right)-A\left(x+h,\mathfrak{X}u\right)\right]\cdot 2\eta \mathfrak{X}\eta {\text{Δ}}_{h}u\text{d}x}\\ +{\displaystyle {\int }_{B_{2R}}{\text{Δ}}_h\left[{\left|F\right|}^{p-2}F\right]\cdot 2\eta \mathfrak{X}\eta {\text{Δ}}_{h}u\text{d}x}+{\displaystyle {\int }_{B_{2R}}{\text{Δ}}_h\left[{\left|F\right|}^{p-2}F\right]\cdot {\eta }^2{\text{Δ}}_h\mathfrak{X}u\text{d}x}\\ =G_2+G_3+G_4+G_5+G_6.\end{array}$(4.2)

We have estimated the terms $G_1$ to $G_4$ in the proof of Theorem 1.1. Thus it remains to estimate $G_5$ and $G_6$. We apply (2.2) to get

$\begin{array}{c}{G}_5\le C{\displaystyle {\int }_{B_{2R}}\left|{\text{Δ}}_h\left[{\left|F\right|}^{p-2}F\right]\right|\left|{\text{Δ}}_{h}u\right|\eta \text{d}x}\\ \le C{\left|h\right|}^{2\alpha }{\displaystyle {\int }_{B_{2R}}{\left|\frac{{\text{Δ}}_h\left[{\left|F\right|}^{p-2}F\right]}{{\left|h\right|}^{\alpha }}\right|}^2\text{d}x}+C{\displaystyle {\int }_{B_{2R}}{\left|{\text{Δ}}_{h}u\right|}^2{\eta }^2\text{d}x}.\end{array}$

By applying the Lagrange Mean Value Theorem, the second term can be controlled by

$\begin{array}{c}C\left(\mu \right){\displaystyle {\int }_{B_{2R}}{\left|{\text{Δ}}_{h}u\right|}^2{\eta }^2\text{d}x}\le C{\displaystyle {\int }_{B_{2R}}\frac{{\mu }^p}{{\mu }^2}{\left|{\text{Δ}}_{h}u\right|}^2{\eta }^2\text{d}x}\\ \le C{\left|h\right|}^2{\displaystyle {\int }_{B_{2R+\left|h\right|}}\frac{{\left[{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{1}{2}}\right]}^p}{{\left[{\left({\mu }^2+{\left|\mathfrak{X}u\right|}^2\right)}^{\frac{1}{2}}\right]}^2}{\left|\mathfrak{X}u\right|}^2\text{d}x}\\ \le C{\left|h\right|}^2{\displaystyle {\int }_{B_{2R+\left|h\right|}}{\left(\mu +\left|\mathfrak{X}u\right|\right)}^p\text{d}x}.\end{array}$

For the estimate of $G_6$, it is apparent that

$\begin{array}{c}{G}_6\le {\displaystyle {\int }_{B_{2R}}\left|{\text{Δ}}_h\left[{\left|F\right|}^{p-2}F\right]\right|\left|{\text{Δ}}_h\mathfrak{X}u\right|{\eta }^2\text{d}x}\\ \le C{\left|h\right|}^{2\alpha }{\displaystyle {\int }_{B_{2R}}{\left|\frac{{\text{Δ}}_h\left[{\left|F\right|}^{p-2}F\right]}{{\left|h\right|}^{\alpha }}\right|}^2\text{d}x}+\epsilon {\displaystyle {\int }_{B_{2R}}{\left|{\text{Δ}}_h\mathfrak{X}u\right|}^2{\eta }^2\text{d}x}.\end{array}$