Global higher integrability for very weak solutions to nonlinear subelliptic equations

Abstract

In this paper we consider the following nonlinear subelliptic Dirichlet problem:

$$ \textstyle\begin{cases} X^{*}A(x,u,Xu)+ B(x,u,Xu)=0,& x\in\Omega,\\ u-u_{0}\in W_{X,0}^{1,r}(\Omega), \end{cases} $$

where $X=\{X_{1},\ldots,X_{m}\}$ is a system of smooth vector fields defined in $\mathbf{R}^{n}$ with globally Lipschitz coefficients satisfying Hörmander’s condition, and we prove the global higher integrability for the very weak solutions.

1 Introduction and main result

The theory of very weak solutions was introduced in the work of Iwaniec and Sbordone [1]. Iwaniec and Sbordone realized that the usual Sobolev assumption for weak solutions to p-harmonic equation can be relaxed to a slightly weaker Sobolev space and proved that very weak solutions are actually classical weak solutions by using the nonlinear Hodge decomposition to construct suitable test functions. Based on Whitney’s extension theorem and theory of $A_{p}$ weights, Lewis [2] showed a completely different proof and obtained the same result to certain elliptic systems. After [1] and [2], many authors have devoted their energy to the study of the regularity of such solutions; see for example [3–5] and the references therein. We mention here that Xie and Fang [5] obtained the global higher integrability for very weak solutions to a class of nonlinear elliptic systems with Lipschitz boundary condition by using Hodge decomposition to construct a suitable test function. Recently the authors in [6] proved the global regularity result for a second-order degenerate elliptic systems of p-Laplacian type in the Euclidean setting.

In 2005, Zatorska-Goldstein [7] showed the local higher integrability of very weak solutions to the nonlinear subelliptic equations

$$ X^{*}A(x,u,Xu)+ B(x,u,Xu)=0, \quad x\in\Omega, $$

(1.1)

where $\Omega\subset\mathbf{R}^{n}$ is a bounded domain and $X=\{ X_{1},\ldots,X_{m}\}$ ($m\leq n$) is a system of smooth vector fields in $\mathbf{R}^{n}$ with globally Lipschitz coefficients satisfying the Hörmander’s condition and $X^{*}=(X_{1}^{*},\ldots,X_{m}^{*})$ is a family of operators formal adjoint to $X_{j}$ in $L^{2}$.

In this work we are concerned with the boundary value problem for (1.1) with the boundary condition $u-u_{0}\in W_{X,0}^{1,r}(\Omega)$, i.e.,

$$ \textstyle\begin{cases} X^{*}A(x,u,Xu)+ B(x,u,Xu)=0,& x\in\Omega,\\ u-u_{0}\in W_{X,0}^{1,r}(\Omega), \end{cases} $$

(1.2)

and establish the global higher integrability for very weak solutions. We assume that the functions $A=(A_{1},\ldots,A_{m}):\mathbf {R}^{n}\times\mathbf{R}\times\mathbf{R}^{m}\rightarrow\mathbf {R}^{m}$ and $B:\mathbf{R}^{n}\times\mathbf{R}\times\mathbf{R}^{m}\rightarrow \mathbf{R}$ are both Carathéodory functions satisfying

$$\begin{aligned} &\bigl\vert A(x,u,\xi) \bigr\vert \leq\alpha\bigl(\vert u \vert ^{p-1}+\vert \xi \vert ^{p-1}\bigr) , \end{aligned}$$

(1.3)

$$\begin{aligned} &\bigl\vert B(x,u,\xi) \bigr\vert \leq\alpha\bigl(\vert u \vert ^{p-1}+\vert \xi \vert ^{p-1}\bigr) , \end{aligned}$$

(1.4)

$$\begin{aligned} &\bigl\langle A(x,u,\xi)-A(x,v,\zeta),\xi-\zeta\bigr\rangle \geq \beta \vert \xi-\zeta \vert ^{2}\bigl(\vert \xi \vert +\vert \zeta \vert \bigr)^{p-2} , \end{aligned}$$

(1.5)

for a.e. $x\in\mathbf{R}^{n}$, $u\in\mathbf{R}$ and $\xi\in\mathbf {R}^{m}$. Here $p\geq2$, α, β are positive constants.

A function $u\in W_{X}^{1,r}(\Omega)$ ($r< p$) is called a very weak solution to (1.1) if

$$ \int_{\Omega}A(x,u,Xu)\cdot X\varphi\,dx+ \int_{\Omega}B(x,u,Xu)\varphi\,dx=0 $$

(1.6)

holds for all $\varphi\in C_{0}^{\infty}(\Omega)$.

In the above definition, the very weak means the integrable exponent is strictly lower than the natural exponent p and if $r=p$, this is the classical definition of weak solution to (1.1).

To get our result, some regularity assumption introduced in [8] should be imposed on Ω. Let us first recall the notion of uniform $(X,p)$-fatness which can be found in [9]: A set $E\subset\mathbf{R}^{n}$ is called uniformly $(X,p)$ -fat if there exist constants $C_{0},R_{0}>0$ such that

$$ \operatorname{cap}_{p}\bigl(E\cap\bar{B}(x,R),B(x,2R)\bigr)\geq C_{0} \operatorname{cap}_{p}\bigl(\bar{B}(x,R),B(x,2R) \bigr) $$

for all $x\in\partial E$ and $0< R< R_{0}$, where $\operatorname{cap}_{p}$ is the variational p-capacity defined in Section 2.

We consider the following hypotheses on Ω:

($H_{1}$):

there exists a constant $C_{1}\geq1$ such that, for all $x\in\Omega$,

$$ \vert B_{\rho(x)} \vert \leq C_{1}\bigl\vert B_{\rho(x)}\cap\bigl( \mathbf{R}^{n}\setminus\Omega\bigr) \bigr\vert , $$

where $\rho(x)=2\operatorname{dist}(x,\mathbf{R}^{n}\setminus\Omega)$;

($H_{2}$):

the complement $\mathbf{R}^{n}\setminus\Omega$ of Ω is uniformly $(X,p)$-fat.

Under the hypotheses stated above, we prove the following.

Theorem 1.1

Assume that $u_{0}\in W_{X}^{1,s}(\Omega)$, $s>p$. Then there exists a $\delta>0$ such that if $u\in W_{X}^{1,p-\delta}(\Omega)$ is a very weak solution to the Dirichlet problem (1.2), we have $u\in W_{X}^{1,p+\tilde{\delta}}(\Omega)$ for some $\tilde{\delta}>0$.

The key technical tool in proving Theorem 1.1 is a Sobolev type inequality with a capacity term. With it we can prove a reverse Hölder inequality for the generalized gradient Xu of a very weak solution, which allows us to get the global higher integrability of Xu. This paper is organized as follows. In Section 2 we collect some known results on Carnot-Carathéodory spaces and prove a Sobolev type inequality characterized by capacity. Section 3 is devoted to the proof of Theorem 1.1.

2 Some known results and a Sobolev type inequality

Let $\{X_{1},\ldots,X_{m}\}$ be a system of $C^{\infty}$-smooth vector fields in $\mathbf{R}^{n} (n\geq3)$ satisfying Hörmander’s condition (see [10]):

$$ \operatorname{rank} \bigl( \operatorname{Lie}\{X_{1},\ldots ,X_{m}\} \bigr) =n. $$

The generalized gradient is denoted by $Xu=(X_{1}u,\ldots,X_{m}u)$ and its length is given by

$$ \bigl\vert Xu(x) \bigr\vert = \Biggl( \sum_{j=1}^{m} \bigl\vert X_{j} u(x) \bigr\vert ^{2} \Biggr) ^{\frac{1}{2}}. $$

An absolutely continuous curve $\gamma:[a,b]\rightarrow\mathbf {R}^{n}$ is said to be admissible with respect to the system $\{X_{1},\ldots,X_{m}\}$, if there exist functions $c_{i}(t), a\leq t\leq b$, satisfying

$$ \sum_{i=1}^{m} c_{i}(t)^{2} \leq1 \quad\mbox{and}\quad\gamma'(t)=\sum _{i=1}^{m} c_{i}(t)X_{i}\bigl( \gamma(t)\bigr). $$

The Carnot-Carathéodory distance $d(x,y)$ generated by $\{ X_{1},\ldots,X_{m}\}$ is defined as the infimum of those $T>0$ for which there exists an admissible path $\gamma:[0, T]\rightarrow\mathbf{R}^{n}$ with $\gamma(0)=x$, $\gamma(T)=y$.

By the accessibility theorem of Chow [11], the distance d is a metric and therefore $(\mathbf{R}^{n},d)$ is a metric space which is called the Carnot-Carathéodory space associated with the system $\{ X_{1},\ldots,X_{m}\}$. The ball is denoted by

$$ B(x_{0},R)=\bigl\{ x\in\mathbf{R}^{n}:d(x,x_{0})< R \bigr\} . $$

For $\sigma>0$ and $B=B(x_{0}, R)$, we will write σB to indicate $B(x_{0},\sigma R)$ and diamΩ the diameter of Ω with respect to d.

It was proved in [12] that the identity map is a homeomorphism of $(\mathbf{R}^{n},d)$ into $\mathbf{R}^{n}$ with the usual Euclidean metric, and every set which is bounded with respect to the Euclidean metric is also bounded with respect to d. Moreover, by a result of Garofalo and Nhieu [13], Proposition 2.11, if the given vector fields have globally Lipschitz coefficients in addition, then a subset of $\mathbf{R}^{n}$ is bounded with respect to d if and only if it is bounded with respect to the Euclidean metric.

Hereafter we assume that the vector fields $X_{1},\ldots,X_{m}$ satisfy the Hörmander condition and have globally Lipschitz coefficients.

Lemma 2.1

[12, 14]

For every bounded open set $\Omega\subset\mathbf{R}^{n}$ there exists $C_{d}\geq1$ such that

$$\begin{aligned} \bigl\vert B(x,2R) \bigr\vert \leq C_{d}\bigl\vert B(x,R) \bigr\vert \end{aligned}$$

(2.1)

for any $x\in\Omega$ and $0< R\leq5\operatorname{diam}\Omega$.

Here, $\vert B(x,R) \vert$ denotes the Lebesgue measure of $B(x,R)$. The best constant $C_{d}$ in (2.1) is called the doubling constant, the measure such that (2.1) holds is called a doubling measure and the homogeneous dimension relative to Ω is $Q=\log_{2}C_{d}$.

Given $1\leq p<\infty$, we define the Sobolev space $W_{X}^{1,p}(\Omega)$ by

$$ W_{X}^{1,p}(\Omega)= \bigl\{ u\in L^{p}(\Omega ):X_{j}u\in L^{p}(\Omega), j=1,2,\ldots,m \bigr\} , $$

endowed with the norm

$$ \Vert u \Vert _{W_{X}^{1,p}(\Omega)}=\Vert u \Vert _{L^{p}(\Omega)}+\Vert Xu \Vert _{L^{p}(\Omega)}. $$

Here, $X_{j}u$ is the distributional derivative of $u\in L_{\operatorname{loc}}^{1}(\Omega)$ given by the identity

$$ \langle X_{j}u,\varphi\rangle= \int_{\Omega}u X_{j}^{*}\varphi\,dx, \quad \varphi \in C_{0}^{\infty}(\Omega). $$

The space $W_{X}^{1,p}(\Omega)$ is a Banach space which admits $C^{\infty}(\Omega)\cap W_{X}^{1,p}(\Omega)$ as its dense subset. The completion of $C_{0}^{\infty}(\Omega)$ under the norm $\Vert \cdot \Vert _{W_{X}^{1,p}(\Omega)}$ is denoted by $W_{X,0}^{1,p}(\Omega )$. The following Sobolev-Poincaré inequalities can be found in [14] and [15]:

Lemma 2.2

Let Q be the homogeneous dimension relative to Ω, $B=B(x_{0},R)\subset\Omega, 0< R<\operatorname {diam}\Omega, 1\leq p<\infty$. There exists a constant $C>0$ such that, for every $u\in W^{1,p}_{X}(B)$,

$${({⨍}_B|u-u_B{|}^{\kappa p}dx)}^{\frac{1}{\kappa p}}\le CR{({⨍}_B|Xu{|}^{p}dx)}^{\frac{1}{p}},$$

where $u_B={⨍}_{B}udx=\frac{1}{|B|}{\int }_{B}udx$, and $1\leq\kappa\leq{Q/(Q-p)}$, if $1\leq p< Q$; $1\leq \kappa<\infty$, if $p\geq Q$. Moreover, for any $u\in W^{1,p}_{X,0}(B)$,

$${({⨍}_B|u{|}^{\kappa p}dx)}^{\frac{1}{\kappa p}}\le CR{({⨍}_B|Xu{|}^{p}dx)}^{\frac{1}{p}}.$$

Next we recall a Gehring lemma on the metric measure space $(Y,d,\mu )$, where d is a metric and μ is a doubling measure.

Lemma 2.3

[7]

Let $q\in[q_{0},2Q]$, $q_{0}>1$ is fixed. Assume that functions f, g are nonnegative and $g\in L_{\operatorname{loc}}^{q}(Y,\mu)$, $f\in L_{\operatorname {loc}}^{r_{0}}(Y,\mu)$, for some $r_{0}>q$. If there exist constants $b>1$ and θ such that for every ball $B\subset\sigma B\subset Y$ the following inequality holds:

$${⨍}_B{g}^{q}d\mu \le b[{\left({⨍}_{\sigma B}gd\mu \right)}^q+{⨍}_{\sigma B}{f}^{q}d\mu ]+\theta {⨍}_{\sigma B}{g}^{q}d\mu ,$$

then there exist nonnegative constants $\theta_{0}=\theta _{0}(q_{0},Q,C_{d},\sigma)$ and $\varepsilon_{0}=\varepsilon _{0}(b,q_{0},Q,C_{d},\sigma)$ such that if $0<\theta<\theta_{0}$ then $g\in L_{\operatorname{loc}}^{p}(Y,\mu)$ for $p\in [q,q+\varepsilon_{0})$.

For the Hardy-Littlewood maximal functions

$$ M f(x)=\sup_{R>0}\frac{1}{\vert B(x,R) \vert } \int_{B(x,R)}\bigl\vert f(y) \bigr\vert \,dy $$

and

$$ M_{\Omega}f(x)=\sup_{R>0}\frac{1}{\vert B(x,R) \vert } \int_{B(x,R)\cap\Omega}\bigl\vert f(y) \bigr\vert \,dy, $$

we will use the following properties proved in [14] and [15].

Lemma 2.4

If $f\in L^{p}(\Omega)$, $1< p\leq\infty$, then $M_{\Omega}f\in L^{p}(\Omega)$ and there exists a constant $C=C(C_{d}, p)>0$ such that

$$ \Vert M_{\Omega}f \Vert _{L^{p}(\Omega)}\leq C\Vert f \Vert _{L^{p}(\Omega).} $$

Lemma 2.5

If $u\in W^{1,p}_{X,\rm loc}(\Omega)$, $1< p <\infty$, then there exists $C>0$ such that, for a.e. $x, y\in \Omega$,

$$ \bigl\vert u(x)-u(y) \bigr\vert \leq Cd(x,y) \bigl( M_{\Omega }\vert Xu \vert (x)+M_{\Omega} \vert Xu \vert (y) \bigr) . $$

Moreover, for any $B=B(x_{0},R)\subset\Omega$ and $u\in W^{1,p}_{X}(B)$, we have

$$ \bigl\vert u(x)-u_{B} \bigr\vert \leq CRM_{B}\vert Xu \vert (x),\quad\textit{a.e.}~x\in B. $$

(2.2)

It is worth noting that from Lemma 2.5 and Lemma 2.2 we can infer that, for a.e. $x\in B$ and $u\in W^{1,p}_{X,0}(B)$,

$$ \bigl\vert u(x) \bigr\vert \leq CRM_{B}\vert Xu \vert (x). $$

(2.3)

Let $\omega(x)\geq0$ be a locally integrable function, we say that $\omega\in A_{p}$, $1< p <\infty$, if there exists some positive constant A such that

$$\underset{B\subset {\mathbf{R}}^n}{sup}\left({⨍}_B\omega dx\right){\left({⨍}_B{\omega }^{\frac{1}{1-p}}dx\right)}^{p-1}\le A<\mathrm{\infty }.$$

Lemma 2.6

Assume $\omega\in L_{\operatorname {loc}}^{1}(\mathbf{R}^{n})$ is nonnegative and $1< p <\infty$. Then $\omega\in A_{p}$ if and only if there exists a constant $C>0$ such that

$$ \int_{\mathbf{R}^{n}}\vert Mf \vert ^{p}\omega\,dx\leq C \int_{\mathbf{R}^{n}}\vert f \vert ^{p}\omega\,dx, $$

for all $f\in L^{p}(\omega(x)\,dx)$.

The $(X,p)$ -capacity of a compact set $K\subset\Omega$ in Ω is defined by

$$ \operatorname{cap}_{p} (K,\Omega)=\inf\biggl\{ \int_{\Omega} \vert Xu \vert ^{p}\,dx:u\in C_{0}^{\infty}(\Omega), u=1 \mbox{ on } K \biggr\} $$

and for an arbitrary set $E\subset\Omega$, the $(X,p)$-capacity of E is

$$ \operatorname{cap}_{p} (E,\Omega)=\inf_{\substack{G\subset\Omega\operatorname{open}\\ E\subset G}}\sup _{\substack{K\subset G\\{K~\mathrm{compact}}}} \operatorname{cap}_{p}(K,\Omega). $$

We will use the following two-sided estimate of $(X,p)$-capacity in [16]: For $x\in\Omega$ and $0< R<\operatorname{diam}\Omega$, there exist $C_{1}, C_{2}>0$ such that

$$ C_{1}\frac{\vert B(x,R) \vert }{R^{p}}\leq {\operatorname{cap}}_{p} \bigl(\bar{B}(x,R),B(x,2R)\bigr)\leq C_{2}\frac{\vert B(x,R) \vert }{R^{p}}. $$

(2.4)

Lemma 2.7

[9]

If $\mathbf {R}^{n}\backslash\Omega$ is uniformly $(X,p)$-fat, then there exists $1< q< p$ such that $\mathbf {R}^{n}\backslash\Omega$ is also uniformly $(X,q)$-fat.

The uniform $(X,q)$-fatness also implies uniform $(X,p)$-fatness for all $p\geq q$, which is a simple consequence of Hölder’s and Young’s inequality.

At the end of this section we prove a Sobolev type inequality characterized by capacity. A similar inequality in the Euclidean setting can be found in [8].

Lemma 2.8

Let $\Omega\subset\mathbf {R}^{n}$ be a bounded open set with the homogeneous dimension Q, $1< q<\infty$ and $0< R<\operatorname{diam}\Omega$. For any $x\in \Omega$, denote $B=B(x,R)$ and $N(\varphi)=\{y\in\bar{B}:\varphi (y)=0\}$. Then there exists a constant $C=C(Q,q)>0$ such that, for all $\varphi\in C^{\infty}(2B)\cap W_{X}^{1,q}(2B)$,

$${({⨍}_{2B}|\phi {|}^{\kappa q}dx)}^{\frac{1}{\kappa q}}\le C{(\frac{1}{{cap}_q(N(\phi ),2B)}{\int }_{2B}|X\phi {|}^{q}dx)}^{\frac{1}{q}},$$

(2.5)

where $1\leq\kappa\leq{Q/(Q-q)}$ if $1\leq q< Q$ and $1\leq\kappa <\infty$ if $q\geq Q$.

Proof

We always assume $\varphi_{2B}\neq0$; otherwise, (2.5) follows immediately from Lemma 2.2 and (2.4). Let $\eta\in C_{0}^{\infty}(2B), 0\leq\eta\leq1$ such that $\eta =1$ on B̄ and $\vert X\eta \vert \leq\frac{c}{R}$. Denoting $v=\eta(\varphi_{2B}-\varphi)/\varphi_{2B}$, then $v\in C_{0}^{\infty}(2B)$ and $v=1$ in $N(\varphi)$. It follows from Lemma 2.2 that

$$\begin{aligned} {\operatorname{cap}}_{q}\bigl(N(\varphi),2B\bigr)&\leq \int_{2B}\vert Xv \vert ^{q}\,dx \\ &\leq \vert \varphi_{2B} \vert ^{-q} \int_{2B}\vert X\eta \vert ^{q}\vert \varphi -\varphi_{2B} \vert ^{q}\,dx+\vert \varphi _{2B} \vert ^{-q} \int_{2B}\vert X\varphi \vert ^{q}\,dx \\ &\leq C\vert \varphi_{2B} \vert ^{-q} \int_{2B}\vert X\varphi \vert ^{q}\,dx, \end{aligned}$$

and then

$$ \vert \varphi_{2B} \vert \leq C \biggl( \frac{1}{\operatorname{cap}_{q}(N(\varphi),2B)} \int_{2B}\vert X\varphi \vert ^{q}\,dx \biggr) ^{\frac{1}{q}}. $$

(2.6)

Then Lemma 2.2 and (2.6) lead to

$$\begin{array}{rl}{({⨍}_{2B}|\phi {|}^{\kappa q}dx)}^{\frac{1}{\kappa q}}& \le {({⨍}_{2B}|\phi -{\phi }_{2B}{|}^{\kappa q}dx)}^{\frac{1}{\kappa q}}+|{\phi }_{2B}|\\ & \le CR{({⨍}_{2B}|X\phi {|}^{q}dx)}^{\frac{1}{q}}+C{(\frac{1}{{cap}_q(N(\phi ),2B)}{\int }_{2B}|X\phi {|}^{q}dx)}^{\frac{1}{q}}\\ & \le C{(\frac{1}{{cap}_q(N(\phi ),2B)}{\int }_{2B}|X\phi {|}^{q}dx)}^{\frac{1}{q}},\end{array}$$

where in the last step we used the estimate

$$ {\operatorname{cap}}_{q}\bigl(N(\varphi),2B\bigr)\leq{ \operatorname{cap}}_{q}(\bar{B},2B)\leq C\vert B \vert R^{-q}. $$

The proof is complete. □

3 Proof of Theorem 1.1

Assume that the function $u\in W_{X}^{1,p-\delta}(\Omega)(\delta <\frac{1}{2})$ is a very weak solution to the Dirichlet problem (1.2). Choose a ball $B_{0}$ such that $\overline{\Omega}\subset\frac {1}{2}B_{0}$ and let B be a ball of radius R with $3B\subset B_{0}$ for fixed $0< R<1$. There are two cases: (i) $3B\subset\Omega$ or (ii) $3B\backslash\Omega\neq\emptyset$. In the case (i), the following estimate has been proved in [7]:

$${⨍}_{\frac{B}{2}}|Xu{|}^{p-\delta }dx\le \theta {⨍}_{3B}|Xu{|}^{p-\delta }dx+b[{⨍}_{3B}|u{|}^{p-\delta }+{({⨍}_{3B}|Xu{|}^{t}dx)}^{\frac{p-\delta }{t}}],$$

(3.1)

where θ small enough, $b>1$, $\max \{ 1,(p-\delta)_{*} \} < t< p-\delta$.

When $3B\backslash\Omega\neq\emptyset$, a similar inequality (see (3.31) below) will be achieved.

Step 1. Let η be a smooth cut-off function on 2B, i.e. $\eta\in C_{0}^{\infty}(2B)$ such that

$$ 0\leq\eta\leq1, \qquad\eta=1\quad \mbox{on } B \quad\mbox{and} \quad \vert X \eta \vert \leq c/R. $$

Define $\hat{u}=\eta(u-u_{0})$ and

$$ E_{\mu}=\bigl\{ x\in\mathbf{R}^{n}:M\vert X \hat{u} \vert (x)\leq\mu\bigr\} , \quad\operatorname{for } \mu>0. $$

(3.2)

We conclude from Lemma 2.5 and the assumption $(H_{1})$ that û is Lipschitz continuous on $E_{\mu}\cup(\mathbf {R}^{n}\setminus\Omega)$.

Indeed, if $x,y\in E_{\mu}\cap\Omega$, then Lemma 2.5 implies $\vert \hat{u}(x)-\hat{u}(y) \vert \leq c\mu d(x,y)$; if $x,y\in\mathbf{R}^{n}\setminus\Omega$, then $\hat{u}(x)=\hat {u}(y)=0$. We set $B_{\rho_{x}}=B(x,\rho_{x})$ with $\rho _{x}=2\operatorname{dist}(x,\mathbf{R}^{n}\setminus\Omega)$ for the case $x\in E_{\mu}\cap\Omega$ and $y\in\mathbf{R}^{n}\setminus\Omega$. Since û is zero on $\mathbf{R}^{n}\setminus\Omega$, it follows that

$$\begin{aligned} \int_{{B_{{\rho_{x}}}} \cap({\mathbf{{R}}^{n}}\setminus\Omega )}\vert \hat{u} - {\hat{u}_{{B_{{\rho_{x}}}}}} \vert \,dz & =\int_{{B_{{\rho_{x}}}} \cap({\mathbf{{R}}^{n}}\setminus\Omega)} \vert {\hat{u}_{{B_{{\rho_{x}}}}}} \vert \,dz \\ &=\bigl\vert {B_{{\rho_{x}}}} \cap\bigl({\mathbf{{R}}^{n}} \setminus\Omega\bigr) \bigr\vert \vert {\hat{u}_{{B_{{\rho _{x}}}}}} \vert \end{aligned}$$

and then, from assumption $(H_{1})$ and Lemma 2.2,

$$\begin{array}{rl}|{\hat{u}}_{B_{{\rho }_x}}|& \le C_1\frac{|B_{{\rho }_x}\cap ({\mathbf{R}}^n\setminus \mathrm{\Omega })|}{|B_{{\rho }_x}|}|{\hat{u}}_{B_{{\rho }_x}}|\\ & =\frac{C_1}{|B_{{\rho }_x}|}{\int }_{B_{{\rho }_x}\cap ({\mathbf{R}}^n\setminus \mathrm{\Omega })}|\hat{u}-{\hat{u}}_{B_{{\rho }_x}}|dz\\ & \le C_1{⨍}_{B_{{\rho }_x}}|\hat{u}-{\hat{u}}_{B_{{\rho }_x}}|dz\le c{C}_1{\rho }_x{⨍}_{B_{{\rho }_x}}|X\hat{u}|dz\\ & \le c{C}_1{\rho }_{x}M|X\hat{u}|(x)\le c{C}_1\mu {\rho }_x.\end{array}$$

Therefore, we have by (2.2) and (3.2)

$$\begin{aligned} \bigl\vert \hat{u}(x)-\hat{u}(y) \bigr\vert &=\bigl\vert \hat{u}(x) \bigr\vert \\ &\leq\bigl\vert \hat{u}(x)-\hat{u}_{B_{\rho_{x}}} \bigr\vert +\vert \hat{u}_{B_{\rho_{x}}} \vert \\ &\leq c{\rho_{x}}M\vert X\hat{u} \vert (x)+c C_{1}\mu {\rho_{x}} \\ &\leq c C_{1}\mu{\rho_{x}} \\ &\leq cC_{1}\mu d(x,y). \end{aligned}$$

It follows that û is a Lipschitz function on $E_{\mu}\cup (\mathbf{R}^{n}\setminus\Omega)$ with the Lipschitz constant $cC_{1}\mu$.

As in [7], we can use the Kirszbraun theorem (see e.g. [17]) to extend û to a Lipschitz function $v_{\mu}$ defined on $\mathbf{R}^{n}$ with the same Lipschitz constant. Moreover, there exists $\mu_{0}$ such that, for every $\mu\geq\mu_{0}$, $\operatorname{supp}{v_{\mu}}\subset3B\cap\Omega$.

In fact, let $D=2B\cap\Omega$ and $x\in\mathbf{R}^{n}\backslash (3B\cap\Omega)$, we have by Lemma 2.1 that

$$M|X\hat{u}|(x)=\underset{B^{\prime }\ni x,B^{\prime }\cap 2B\ne \mathrm{\varnothing }}{sup}{⨍}_{B^{\prime }}|X\hat{u}|(y)dy\le \frac{C_d}{|2B|}{\int }_D|X\hat{u}|(y)dy,$$

where $\vert B' \vert >\vert B \vert$, $C_{d}$ is the doubling constant. Setting

$$ \mu_{0}=\frac{C_{d}}{\vert 2B \vert } \int_{D}\vert X\hat{u} \vert (y)\,dy, $$

then $M\vert X\hat{u} \vert (x)\leq\mu, \mu\geq\mu_{0}$, which implies $v_{\mu}(x)=\hat{u}(x)=0$ for $x\in\mathbf {R}^{n}\backslash(3B\cap\Omega)$. So we can take the function $v_{\mu}$ as a test function in (1.6).

Let $\mu\geq\mu_{0}$ and take $v_{\mu}$ as a test function in (1.6) to have

$$\begin{aligned} \int_{3B \cap\Omega}A(x,u,Xu) \cdot X{v_{\mu}}\,dx + \int_{3B \cap\Omega} B(x,u,Xu){v_{\mu}}\,dx=0. \end{aligned}$$

Noting that ${v_{\mu}}=\hat{u}$ on $(3B\cap\Omega)\cap{E_{\mu}}$ and that $\operatorname{supp}\hat{u}\subset D$, we have by the structure conditions on $A(x,u,\xi)$ and $B(x,u,\xi)$

$$\begin{aligned} & \int_{D\cap E_{\mu}} A(x,u,Xu)\cdot X\hat{u}\,dx+ \int_{D\cap E_{\mu}} B(x,u,Xu)\hat{u}\,dx \\ &\quad\leq \int_{(3B\cap\Omega)\backslash E_{\mu}} \bigl\vert A(x,u,Xu) \bigr\vert \vert Xv_{\mu} \vert \,dx+ \int_{(3B\cap\Omega)\backslash E_{\mu}} \bigl\vert B(x,u,Xu) \bigr\vert \vert v_{\mu} \vert \,dx \\ &\quad\leq c\mu \int_{(3B\cap\Omega)\backslash E_{\mu}} \bigl( \vert u \vert ^{p-1}+\vert Xu \vert ^{p-1} \bigr)\,dx, \end{aligned}$$

(3.3)

where in the last inequality we use the fact that $\vert Xv_{\mu} \vert \leq c\mu$, $\vert v_{\mu} \vert \leq cR\mu$ (see [7]).

Multiplying both sides of (3.3) by $\mu^{-(1+\delta)}$ and integrating over $(\mu_{0},\infty)$, we get

$$\begin{aligned} L&:= \int_{\mu_{0}}^{\infty} \int_{D\cap E_{\mu}}{\mu}^{-(1+\delta)} \bigl( A(x,u,Xu)\cdot X \hat{u}+B(x,u,Xu)\hat{u} \bigr) \,dx\,d\mu \\ &\leq c \int_{\mu_{0}}^{\infty} \int_{(3B\cap\Omega)\backslash E_{\mu}}\mu^{-\delta} \bigl( \vert u \vert ^{p-1}+\vert Xu \vert ^{p-1} \bigr) \,dx\,d\mu:=P. \end{aligned}$$

(3.4)

Interchanging the order of integration and applying (3.2), we have

$$\begin{aligned} P &=c \int_{3B} \int_{{\mu_{0}}}^{M\vert X\hat{u} \vert } {{\mu^{ - \delta}} \bigl( { \vert u{ \vert ^{p - 1}} + \vert Xu{ \vert ^{p - 1}}} \bigr)\, d\mu\,dx} \\ &\leq\frac{c}{1-\delta} \int_{(3B\cap\Omega)\backslash E_{\mu_{0}}}\bigl(M\vert X\hat{u} \vert \bigr)^{1-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu \vert ^{p-1}\bigr)\,dx \\ &\leq c \int_{3B\cap\Omega} \bigl( \vert u \vert ^{p-\delta}+\vert Xu \vert ^{p-\delta} \bigr)\,dx+c \int_{3B\cap\Omega}\bigl(M\vert X\hat{u} \vert \bigr)^{p-\delta}\,dx. \end{aligned}$$

(3.5)

Using Lemma 2.4 and Lemma 2.8, we have

$$\begin{aligned} & c \int_{3B\cap\Omega}\bigl(M\vert X\hat{u} \vert \bigr)^{p-\delta}\,dx \\ &\quad \leq c \int_{D}\vert X\hat{u} \vert ^{p-\delta}\,dx \\ &\quad \leq c \int_{D}\vert Xu-Xu_{0} \vert ^{p-\delta}\,dx+\frac{c}{R^{p-\delta}} \int_{2B}\vert u-u_{0} \vert ^{p-\delta}\,dx \\ &\quad \leq c \int_{D}\vert Xu-Xu_{0} \vert ^{p-\delta}\,dx +\frac{c\vert 2B \vert }{R^{p-\delta}} \biggl( \frac{1}{\operatorname{cap}_{p-\delta}(N(u-u_{0}),2B)} \int_{2B}\vert Xu-Xu_{0} \vert ^{p-\delta}\,dx \biggr) , \end{aligned}$$

where $N(u-u_{0})= \{ x\in\bar{B}:u(x)=u_{0}(x) \}$. Since $u-u_{0}$ vanishes outside Ω, we have $\mathbf{R}^{n}\setminus\Omega \subset\{u-u_{0}=0\}$. On the other hand, by Lemma 2.7 and assumption $(H_{2})$, there exists $\delta_{0}$ such that if $0<\delta<\delta _{0}$, $\mathbf{R}^{n}\setminus\Omega$ is uniformly $(X,p-\delta )$-fat, and hence

$$\begin{aligned} {\operatorname{cap}}_{p-\delta}\bigl(N(u-u_{0}),2B \bigr)&\geq{\operatorname{cap}}_{p-\delta}\bigl(\bar{B}\cap\bigl( \mathbf{R}^{n}\setminus\Omega\bigr),2B\bigr) \\ &\geq c \operatorname{cap}_{p-\delta}(\bar{B},2B)\geq c\vert B \vert R^{-(p-\delta)}. \end{aligned}$$

(3.6)

From (3.6) and the doubling condition, we derive

$$\begin{aligned} c \int_{3B\cap\Omega}\bigl(M\vert X\hat{u} \vert \bigr)^{p-\delta}\,dx&\leq c \int_{D}\vert X\hat{u} \vert ^{p-\delta}\,dx \\ &\leq c \int_{D}\vert Xu \vert ^{p-\delta}\,dx+c \int_{D}\vert Xu_{0} \vert ^{p-\delta}\,dx, \end{aligned}$$

(3.7)

and then (3.5) becomes

$$ P\leq c \int_{3B\cap\Omega} \vert u \vert ^{p-\delta}\,dx+c \int_{3B\cap\Omega} \vert Xu_{0} \vert ^{p-\delta}\,dx+c \int_{3B\cap\Omega} \vert Xu \vert ^{p-\delta}\,dx. $$

(3.8)

As regards the estimation of L, by changing the order of integration, we have

$$\begin{aligned} L&= \int_{\mu_{0}}^{\infty} \int_{D}\mu^{-(1 +\delta)}\bigl(A(x,u,Xu)\cdot X\hat{u} + B(x,u,Xu)\hat{u}\bigr)\chi_{\{M\vert X\hat{u} \vert (x)\leq\mu\} }\,dx\,d\mu \\ &= \int_{D\backslash{E_{\mu_{0}}}} \int_{M\vert X\hat{u} \vert }^{\infty}{\mu^{-(1 + \delta )}}\bigl(A(x,u,Xu) \cdot X\hat{u} + B(x,u,Xu)\hat{u}\bigr)\,dx\,d\mu \\ & \quad {}+ \int_{D \cap{E_{{\mu_{0}}}}} \int_{{\mu_{0}}}^{\infty}{\mu^{-(1 +\delta)}}\bigl(A(x,u,Xu) \cdot X\hat{u} + B(x,u,Xu)\hat{u}\bigr)\,dx\,d\mu \\ &=\frac{1}{\delta} \int_{D\backslash E_{\mu_{0}}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(A(x,u,Xu)\cdot X\hat{u}+B(x,u,Xu)\hat{u}\bigr)\,dx \\ &\quad{}+\frac{1}{\delta} \int_{D\cap E_{\mu_{0}}}\mu_{0}^{-\delta}\bigl(A(x,u,Xu) \cdot X\hat{u}+B(x,u,Xu)\hat{u}\bigr)\,dx. \end{aligned}$$

Since $D\setminus E_{\mu_{0}}=D\setminus(D\cap E_{\mu_{0}})$, (1.3) and (1.4) imply

$$\begin{aligned} L&=\frac{1}{\delta} \int_{D}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}A(x,u,Xu)\cdot X\hat{u}\,dx-\frac{1}{\delta} \int_{D\cap E_{\mu_{0}}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}A(x,u,Xu)\cdot X\hat{u}\,dx \\ &\quad{}+\frac{1}{\delta} \int_{D\backslash E_{\mu_{0}}} \bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}B(x,u,Xu)\hat{u}\,dx \\ &\quad{}+\frac{1}{\delta} \int_{D\cap E_{\mu_{0}}}\mu_{0}^{-\delta}\bigl(A(x,u,Xu) \cdot X\hat{u}+B(x,u,Xu)\hat{u}\bigr)\,dx \\ &\geq\frac{1}{\delta} \int_{D}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}A(x,u,Xu)\cdot X\hat{u}\,dx \\ &\quad{}-\frac{2\alpha}{\delta} \int_{D\cap E_{\mu_{0}}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu \vert ^{p-1}\bigr)\vert X\hat{u} \vert \,dx \\ &\quad{}-\frac{\alpha}{\delta} \int_{D}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu \vert ^{p-1}\bigr)\vert \hat{u} \vert \,dx \\ &:=\frac{1}{\delta}(I_{1}-2\alpha I_{2}-\alpha I_{3}). \end{aligned}$$

(3.9)

Step 2. Next, we will estimate $I_{i}$ ($i=1,2,3$) one by one.

Now for estimation of $I_{1}$. To this end, define the sets

$$\begin{aligned}& {D_{1}} = \bigl\{ x \in D\setminus B:M\vert X\hat{u} \vert \le \delta\bigl({M_{D}}\vert Xu-Xu_{0} \vert \bigr)\bigr\} , \\& {D_{2}}= \bigl\{ x \in D\setminus B:M\vert X\hat{u} \vert > \delta \bigl({M_{D}}\vert Xu-Xu_{0} \vert \bigr)\bigr\} \end{aligned}$$

and $B_{\Omega}=B\cap\Omega$. Thus

$$\begin{aligned} I_{1} &= \int_{B_{\Omega}\cup D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(A(x,u,Xu)-A(x,u_{0},Xu_{0})\bigr) \cdot\eta X(u-u_{0})\,dx \\ &\quad{} + \int_{B_{\Omega}\cup D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}A(x,u_{0},Xu_{0})\cdot\eta (Xu-Xu_{0})\,dx \\ &\quad{} + \int_{D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}A(x,u,Xu)\cdot X\eta(u-u_{0})\,dx \\ &\quad{} + \int_{D_{1}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}A(x,u,Xu)\cdot X\hat{u}\,dx. \end{aligned}$$

Since $( M\vert X\hat{u} \vert ) ^{-\delta}\le \vert X\hat{u} \vert ^{-\delta}$ a.e., it follows from (1.5) and (1.3) that

$$\begin{aligned} I_{1}&\geq\beta \int_{B_{\Omega}}\bigl( M\vert X\hat{u} \vert \bigr)^{-\delta} \vert Xu-Xu_{0} \vert ^{p}\,dx \\ &\quad{} -\alpha\biggl( \int_{B_{\Omega}} \vert X\hat{u} \vert ^{-\delta} \bigl( \vert u_{0} \vert ^{p-1}+ \vert Xu_{0} \vert ^{p-1} \bigr) \\ &\quad{} \times \vert Xu-Xu_{0} \vert \,dx + \int_{D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(\vert u_{0} \vert ^{p-1}+\vert Xu_{0} \vert ^{p-1}\bigr)\vert Xu-Xu_{0} \vert \,dx \\ &\quad{} + \int_{D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu \vert ^{p-1}\bigr)\bigl\vert X\eta(u-u_{0}) \bigr\vert \,dx \\ &\quad{} + \int_{D_{1}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu \vert ^{p-1}\bigr)\vert X\hat{u} \vert \,dx \biggr) \\ &:=I_{11}-\alpha(I_{12}+I_{13}+I_{14}+I_{15}). \end{aligned}$$

(3.10)

Since the function $(M\vert X\hat{u} \vert )^{-\delta}$ is an $A_{p}$-weight, we obtain from Lemma 2.6 that

$$ I_{11}\geq c\beta \int_{B_{\Omega}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(M_{B_{\Omega}} \vert Xu-Xu_{0} \vert \bigr)^{p}\,dx. $$

By the doubling condition and Lemma 2.8 we see that, for $x\in\frac{B}{2}\cap\Omega$,

$$\begin{array}{rl}M|X\hat{u}|(x)& \le \underset{B^{\prime }\ni x,B^{\prime }\subset B}{sup}{⨍}_{B^{\prime }}|X\hat{u}|dy+\underset{B^{\prime }\ni x,B^{\prime }\cap \partial B\ne \mathrm{\varnothing }}{sup}{⨍}_{B^{\prime }}|X\hat{u}|dy\\ & \le M_{B_{\mathrm{\Omega }}}|X(u-u_0)|+\frac{c}{R}{({⨍}_{2B}|u-u_0{|}^{s^{\prime }}dx)}^{\frac{1}{s^{\prime }}}+c{(\frac{1}{|2B|}{\int }_D|Xu-X{u}_0{|}^{s^{\prime }}dx)}^{\frac{1}{s^{\prime }}}\\ & \le M_{B_{\mathrm{\Omega }}}|X(u-u_0)|+\frac{c}{R}{\left(\frac{1}{{cap}_{s^{\prime }}(N(u-u_0),2B)}{\int }_{2B}\right|X(u-u_0){|}^{s^{\prime }}dx)}^{\frac{1}{s^{\prime }}}\\ & +c{(\frac{1}{|2B|}{\int }_D|Xu-X{u}_0{|}^{s^{\prime }}dx)}^{\frac{1}{s^{\prime }}}\\ & \le M_{B_{\mathrm{\Omega }}}|X(u-u_0)|+c{(\frac{1}{|2B|}{\int }_D|Xu-X{u}_0{|}^{s^{\prime }}dx)}^{\frac{1}{s^{\prime }}},\end{array}$$

(3.11)

where $\max \{ 1,(p-\delta)_{*} \} < s'< p-\delta$ is such that $\mathbf{R}^{n}\setminus\Omega$ is uniformly $(X,s')$-fat and the last inequality comes from an argument similar to (3.6).

To continue, we define

$$ G= \biggl\{ x\in\frac{B}{2}\cap\Omega:M_{B_{\Omega}}\bigl\vert X(u-u_{0}) \bigr\vert \geq c \biggl( \frac{1}{\vert 2B \vert } \int_{D}\vert Xu-Xu_{0} \vert ^{s'}\,dx \biggr) ^{\frac{1}{s'}} \biggr\} . $$

So from (3.11) we see that $M\vert X\hat{u} \vert \leq cM_{B_{\Omega}} \vert X(u-u_{0}) \vert$ on G, and then

$$\begin{aligned} I_{11}&\geq c \int_{G}\bigl(M_{B_{\Omega}} \vert Xu-Xu_{0} \vert \bigr)^{-\delta}\bigl(M_{B_{\Omega}} \vert Xu-Xu_{0} \vert \bigr)^{p}\,dx \\ &\geq c \int_{\frac{B}{2}\cap\Omega} \vert Xu-Xu_{0} \vert ^{p-\delta}\,dx-c\vert B \vert \biggl( \frac{1}{\vert 2B \vert } \int_{D}\vert Xu-Xu_{0} \vert ^{s'}\,dx \biggr) ^{\frac{p-\delta}{s'}} \\ &\geq c \int_{\frac{B}{2}\cap\Omega} \vert Xu \vert ^{p-\delta}\,dx-c \int_{D}\vert Xu_{0} \vert ^{p-\delta}\,dx-c\vert B \vert \biggl( \frac{1}{\vert 2B \vert } \int_{D}\vert Xu \vert ^{s'}\,dx \biggr) ^{\frac{p-\delta}{s'}}. \end{aligned}$$

(3.12)

Using the fact $X\hat{u}=X(u-u_{0})$ on B and Young’s inequality, we have

$$\begin{aligned} I_{12} &\leq c \int_{D} \bigl( \vert u_{0} \vert ^{p-\delta}+\vert Xu_{0} \vert ^{p-\delta} \bigr)\,dx +c \varepsilon \int_{D}\vert Xu-Xu_{0} \vert ^{p-\delta}\,dx \\ &\leq c \int_{D}\vert u_{0} \vert ^{p-\delta}\,dx+c \int_{D}\vert Xu_{0} \vert ^{p-\delta}\,dx+c\varepsilon \int_{D}\vert Xu \vert ^{p-\delta}\,dx. \end{aligned}$$

(3.13)

Next from the definition of $D_{2}$ and Lemma 2.4, we see

$$\begin{aligned} I_{13} &\leq c\delta^{-\delta} \int_{D}\bigl(M_{D}\vert Xu-Xu_{0} \vert \bigr)^{1-\delta}\bigl(\vert u_{0} \vert ^{p-1}+\vert Xu_{0} \vert ^{p-1}\bigr)\,dx \\ &\leq c \int_{D} \bigl( \vert u_{0} \vert ^{p-\delta}+\vert Xu_{0} \vert ^{p-\delta} \bigr)\,dx +c \varepsilon \int_{D}\vert Xu-Xu_{0} \vert ^{p-\delta}\,dx \\ &\leq c \int_{D}\vert u_{0} \vert ^{p-\delta}\,dx+c \int_{D}\vert Xu_{0} \vert ^{p-\delta}\,dx+c\varepsilon \int_{D}\vert Xu \vert ^{p-\delta}\,dx. \end{aligned}$$

(3.14)

For $I_{14}$, we have by using $\vert X\eta(u-u_{0}) \vert \leq \vert X\hat{u} \vert +\vert Xu-Xu_{0} \vert$

$$\begin{aligned} I_{14} &\leq c \int_{D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu-Xu_{0} \vert ^{p-1}+\vert Xu_{0} \vert ^{p-1}\bigr)\bigl\vert X\eta(u-u_{0}) \bigr\vert \,dx \\ &\leq c \int_{D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu_{0} \vert ^{p-1}\bigr) \bigl(\vert X\hat{u} \vert + \vert Xu-Xu_{0} \vert \bigr)\,dx \\ & \quad {} +c \int_{D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta} \vert Xu-Xu_{0} \vert ^{p-1}\vert X \eta \vert \vert u-u_{0} \vert \,dx \\ &\leq c \int_{D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu_{0} \vert ^{p-1}\bigr)\vert X\hat{u} \vert \,dx \\ & \quad {} +c \int_{D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu_{0} \vert ^{p-1}\bigr)\vert Xu-Xu_{0} \vert \,dx \\ & \quad {} +\frac{c}{R} \int_{D_{2}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta} \vert Xu-Xu_{0} \vert ^{p-1}\vert u-u_{0} \vert \,dx \\ &:=K_{1}+K_{2}+K_{3}. \end{aligned}$$

Using Young’s inequality and (3.7), we get

$$\begin{aligned} K_{1}&\leq c \int_{D}\bigl(\vert u \vert ^{p-1}+\vert Xu_{0} \vert ^{p-1}\bigr)\vert X\hat{u} \vert ^{1-\delta}\,dx \\ &\leq c \int_{D}\vert u \vert ^{p-\delta}\,dx+c \int_{D}\vert Xu_{0} \vert ^{p-\delta}\,dx+c\varepsilon \int_{D}\vert X{u} \vert ^{p-\delta}\,dx. \end{aligned}$$

(3.15)

By the definition of $D_{2}$ and noting that $\vert X(u-u_{0}) \vert \le M_{D}\vert X(u-u_{0}) \vert$ a.e. D,

$$\begin{aligned} K_{2} &\leq c\delta^{-\delta} \int_{D}\vert Xu-Xu_{0} \vert ^{1-\delta} \bigl(\vert u \vert ^{p-1}+\vert Xu_{0} \vert ^{p-1}\bigr)\,dx \\ &\leq c \int_{D}\vert u \vert ^{p-\delta}\,dx+c \int_{D}\vert Xu_{0} \vert ^{p-\delta}\,dx+c\varepsilon \int_{D}\vert Xu \vert ^{p-\delta}\,dx. \end{aligned}$$

(3.16)

Finally, by Young’s inequality,

$$\begin{aligned} K_{3} &\leq\frac{c\delta^{-\delta}}{R} \int_{D_{2}}\bigl(M_{D}\vert Xu-Xu_{0} \vert \bigr)^{-\delta} \vert Xu-Xu_{0} \vert ^{p-1} \vert u-u_{0} \vert \,dx \\ &\leq\frac{c\delta^{-\delta}}{R} \int_{D}\vert Xu-Xu_{0} \vert ^{p-1-\delta} \vert u-u_{0} \vert \,dx \\ &\leq c\varepsilon \int_{{D}}\vert Xu-Xu_{0} \vert ^{p -\delta}\,dx + c \int_{{D}}{\biggl\vert {\frac{{u-{u_{0}}}}{R}} \biggr\vert ^{p-\delta}}\,dx. \end{aligned}$$

(3.17)

In order to estimate the second component of the right-hand side, we let $s''=(p-\delta)(1-\vartheta)$, where $0<\vartheta<\frac{p-\delta }{p-\delta+Q}$ if $p-\delta\leq Q$ and $0<\vartheta<\min \{ \frac {p-\delta-Q}{p-\delta},\frac{1}{2} \}$ if $p-\delta> Q$. Denote

$$ \kappa= \textstyle\begin{cases} \frac{Q}{Q-s''},&s''< Q,\\ 2,&s''>Q, \end{cases} $$

then $\kappa s''\geq p-\delta$. Using Lemma 2.7 and Lemma 2.8, we derive

$$\begin{array}{rl}{\left({⨍}_{2B}\right|\frac{u-u_0}{R}{|}^{p-\delta }dx)}^{\frac{1}{p-\delta }}& \le c{R}^{-1}{({⨍}_{2B}|u-u_0{|}^{\kappa s^{″}}dx)}^{\frac{1}{\kappa s^{″}}}\\ & \le c{R}^{-1}{\left(\frac{1}{{cap}_{s^{″}}(N(u-u_0),2B)}{\int }_{2B}\right|X(u-u_0){|}^{s^{″}}dx)}^{\frac{1}{s^{″}}}\\ & \le c{(\frac{1}{|2B|}{\int }_D|Xu-X{u}_0{|}^{s^{″}})}^{\frac{1}{s^{″}}},\end{array}$$

where the proof of the last inequality is similar to (3.6). Therefore,

$$ c \int_{2B}\biggl\vert \frac{u-u_{0}}{R}\biggr\vert ^{p-\delta}\,dx \leq c\vert 2B \vert \biggl( \frac{1}{\vert 2B \vert } \int_{D}\vert Xu-Xu_{0} \vert ^{s''}\,dx \biggr) ^{\frac{{p-\delta}}{s''}}. $$

(3.18)

Inserting (3.18) into (3.17), we have

$$\begin{aligned} K_{3}&\leq c\varepsilon \int_{D} \vert Xu \vert ^{p -\delta}\,dx + c \int_{D} \vert Xu_{0} \vert ^{p - \delta}\,dx \\ &\quad {}+ c\vert 2B \vert \biggl( \frac{1}{\vert 2B \vert } \int_{D} \vert Xu \vert ^{s''}\,dx \biggr) ^{\frac{{p-\delta}}{s''}}. \end{aligned}$$

(3.19)

A combination of (3.15), (3.16) and (3.19) implies

$$\begin{aligned} I_{14}&\leq c \int_{D}\bigl(\vert u \vert ^{p-\delta}+\vert Xu_{0} \vert ^{p-\delta}\bigr)\,dx \\ &\quad{} +c\varepsilon \int_{D}\vert Xu \vert ^{p-\delta}\,dx + c\vert 2B \vert \biggl( \frac{1}{\vert 2B \vert } \int_{D} \vert Xu \vert ^{s''}\,dx \biggr) ^{\frac{{p-\delta}}{s''}}. \end{aligned}$$

(3.20)

The definition of $D_{1}$ and Lemma 2.4 give

$$\begin{aligned} I_{15} &\leq c \int_{D_{1}}\bigl(M\vert X\hat{u} \vert \bigr)^{1-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu \vert ^{p-1}\bigr)\,dx \\ &\leq c\delta^{1-\delta} \int_{D}\bigl(M_{D}\vert Xu-Xu_{0} \vert \bigr)^{1-\delta}\bigl(\vert u \vert ^{p-1}+\vert Xu \vert ^{p-1}\bigr)\,dx \\ &\leq c\delta^{1-\delta} \biggl[ \int_{D}\vert Xu-Xu_{0} \vert ^{p-\delta}\,dx + \int_{D} \bigl( \vert u \vert ^{p-\delta}+\vert Xu \vert ^{p-\delta} \bigr)\,dx \biggr] \\ &\leq c \int_{D} \bigl( \vert u \vert ^{p-\delta}+\vert Xu_{0} \vert ^{p-\delta} \bigr)\,dx+c\delta \int_{D}\vert Xu \vert ^{p-\delta}\,dx. \end{aligned}$$

(3.21)

The previous estimates show that

$$\begin{aligned} I_{1}&\geq c \int_{\frac{B}{2}\cap\Omega} \vert Xu \vert ^{p-\delta}\,dx -c \int_{D} \bigl( \vert u \vert ^{p-\delta}+\vert u_{0} \vert ^{p-\delta}+\vert Xu_{0} \vert ^{p-\delta} \bigr)\,dx \\ & \quad{} - c(\varepsilon+\delta) \int_{D}\vert Xu \vert ^{p-\delta}\,dx-c\vert 2B \vert \biggl( \frac{1}{\vert 2B \vert } \int_{D}\vert Xu \vert ^{t}\,dx \biggr) ^{\frac{p-\delta}{t}}, \end{aligned}$$

(3.22)

where $t=\max\{s',s''\}< p-\delta$.

Now we address the estimation of $I_{2}$. Using (3.7), we have

$$\begin{aligned} I_{2}&\leq \int_{D}\vert u \vert ^{p-1}\vert X\hat{u} \vert ^{1-\delta}\,dx + \int_{D\cap{E_{\mu_{0}}}}\bigl(M\vert X\hat{u} \vert \bigr)^{-\delta} \vert Xu \vert ^{p-1}\vert X\hat{u} \vert \,dx \\ &\leq c \int_{D}\vert u \vert ^{p-\delta}\,dx+c\varepsilon \int_{D}\vert X\hat{u} \vert ^{p-\delta}\,dx + \int_{D\cap E_{\mu_{0}}}\vert Xu \vert ^{p-1}\bigl(M\vert X \hat{u} \vert \bigr)^{1-\delta}\,dx \\ &\leq c \int_{D} \bigl( \vert u \vert ^{p-\delta}+\vert Xu_{0} \vert ^{p-\delta} \bigr)\,dx \\ &\quad{} + c\varepsilon \int_{D}\vert Xu \vert ^{p-\delta}\,dx + \int_{D\cap E_{\mu_{0}}}\vert Xu \vert ^{p-1}\bigl(M\vert X \hat{u} \vert \bigr)^{1-\delta}\,dx. \end{aligned}$$

(3.23)

To estimate the last integral in (3.23), let $0<\tau<\frac {1}{2}$ and $x\in D\cap E_{\mu_{0}}$. If $\vert Xu \vert \geq \tau^{-1}\mu_{0}$, then $M\vert X\hat{u} \vert \leq\mu_{0}\leq \tau \vert Xu \vert$ and

$$ \vert Xu{ \vert ^{p - 1}} {\bigl(M\vert X\hat{u} \vert \bigr)^{1 - \delta}} \le \vert Xu{ \vert ^{p - 1}} {\bigl( \tau \vert Xu \vert \bigr)^{1 - \delta}} = {\tau ^{1 - \delta}} \vert Xu{ \vert ^{p - \delta}}; $$

(3.24)

if $\vert Xu \vert <\tau^{-1}\mu_{0}$, then

$$ \vert Xu{ \vert ^{p - 1}} {\bigl(M\vert X\hat{u} \vert \bigr)^{1 - \delta}} \le{ \bigl( {{\tau^{ - 1}} {\mu _{0}}} \bigr) ^{p - 1}}\mu_{0}^{1 - \delta} \le{ \tau^{1 - p}}\mu_{0}^{p - \delta}. $$

(3.25)

By (3.24) and (3.25), we deduce that, for any $x\in D\cap E_{\mu_{0}}$,

$$ \vert Xu \vert ^{p-1}\bigl(M\vert X\hat{u} \vert \bigr)^{1-\delta}\leq c \bigl( \tau^{1-\delta} \vert Xu \vert ^{p-\delta}+\tau^{1-p}\mu_{0}^{p-\delta} \bigr) . $$

(3.26)

For the second term in (3.26), we first observe from the proof of (3.11) that

$$\frac{1}{R}{({⨍}_{2B}|u-u_0{|}^{s^{\prime }}dx)}^{\frac{1}{s^{\prime }}}\le c{(\frac{1}{|2B|}{\int }_D|Xu-X{u}_0{|}^{s^{\prime }}dx)}^{\frac{1}{s^{\prime }}}.$$

Noticing $\mu_{0}=\frac{c}{\vert 2B \vert }\int_{D}\vert X\hat {u} \vert \,dx$, we have from Hölder’s inequality

$$\begin{array}{rl}{\tau }^{1-p}{\mu }_0^{p-\delta }& \le c{\tau }^{1-p}{\left(\frac{1}{|2B|}{\int }_D\right|X\eta (u-u_0)+\eta X(u-u_0)\left|dx\right)}^{p-\delta }\\ & \le c{\tau }^{1-p}{\left(\frac{1}{R}{({⨍}_{2B}|u-u_0{|}^{s^{\prime }}dx)}^{\frac{1}{s^{\prime }}}\right)}^{p-\delta }\\ & +c{\tau }^{1-p}{\left(\frac{1}{|2B|}{\int }_D\right|X(u-u_0){|}^{s^{\prime }}dx)}^{\frac{p-\delta }{s^{\prime }}}\\ & \le c{\tau }^{1-p}{\left(\frac{1}{|2B|}{\int }_D\right|X(u-u_0){|}^{s^{\prime }}dx)}^{\frac{p-\delta }{s^{\prime }}}\\ & \le c{\tau }^{1-p}(\frac{1}{|2B|}{\int }_D|X{u}_0{|}^{p-\delta }dx)+c{\tau }^{1-p}{(\frac{1}{|2B|}{\int }_D|Xu{|}^{s^{\prime }}dx)}^{\frac{p-\delta }{s^{\prime }}}.\end{array}$$

(3.27)

By (3.26) and (3.27), it follows that

$$\begin{aligned} & \int_{D\cap E_{\mu_{0}}}\vert Xu \vert ^{p-1}\bigl(M\vert X \hat{u} \vert \bigr)^{1-\delta}\,dx \\ &\quad\leq c\tau^{1-\delta} \int_{D}\vert Xu \vert ^{p-\delta}\,dx+c \int_{D}\vert Xu_{0} \vert ^{p-\delta}\,dx +c\tau^{1-p}\vert 2B \vert \biggl( \frac{1}{\vert 2B \vert } \int_{D}\vert Xu \vert ^{s'}\,dx \biggr) ^{\frac{p-\delta}{s'}}. \end{aligned}$$

(3.28)

Taking (3.28) into (3.23), we have

$$\begin{aligned} I_{2}&\leq c \int_{D}\vert u \vert ^{p-\delta}\,dx+ c \int_{D}\vert Xu_{0} \vert ^{p-\delta}\,dx \\ &\quad{} +c \bigl( \varepsilon+\tau^{1-\delta} \bigr) \int_{D}\vert Xu \vert ^{p-\delta}\,dx+c\tau ^{1-p}\vert 2B \vert \biggl( \frac{1}{\vert 2B \vert } \int_{D}\vert Xu \vert ^{s'}\,dx \biggr) ^{\frac{p-\delta}{s'}}. \end{aligned}$$

(3.29)

For the estimation of $I_{3}$: From (2.3), Lemma 2.8 and a similar process to the proof of (3.18), we have

$$\begin{aligned} I_{3}&\leq c \int_{D}\bigl(\vert u \vert ^{p-1}+\vert Xu \vert ^{p-1}\bigr)\vert \hat{u} \vert ^{1-\delta}\,dx \\ &\leq \int_{D} \vert u{ \vert ^{p - \delta}}+ c\varepsilon \int_{D} \vert Xu{ \vert ^{p - \delta}}\,dx + c \int_{D}\vert u - u_{0} \vert ^{p -\delta}\,dx \\ &\leq c \int_{D}\bigl(\vert u \vert ^{p - \delta}+ \vert X{u_{0}} { \vert ^{p - \delta}}\bigr)\,dx \\ &\quad{} + c\varepsilon \int_{D}\vert Xu \vert ^{p -\delta}\,dx + c\vert 2B \vert \biggl( \frac{1}{\vert 2B \vert } \int_{D}\vert Xu \vert ^{t}\,dx \biggr) ^{\frac{{p -\delta} }{{t}}}, \end{aligned}$$

(3.30)

where $t=\max\{s',s''\}< p-\delta$.

Step 3. Taking into account (3.4), (3.8), substituting (3.22), (3.29) and (3.30) into (3.9), and letting $\varepsilon=\tau^{1-\delta}$, it follows that

$$\begin{aligned} &\int_{\frac{B}{2}\cap\Omega} \vert Xu \vert ^{p-\delta}\, dx \\ & \quad \leq c \int_{3B \cap\Omega} \bigl( \vert u \vert ^{p - \delta} +\vert {u_{0}} \vert ^{p - \delta} +\vert X{u_{0}} \vert ^{p - \delta} \bigr)\,dx \\ &\quad \quad{}+c \bigl( \delta+\tau^{1-\delta} \bigr) \int_{3B\cap\Omega} \vert Xu \vert ^{p-\delta}\,dx +c\tau ^{1-p}\vert 2B \vert \biggl( \frac{1}{\vert 2B \vert } \int_{3B\cap\Omega} \vert Xu \vert ^{t}\,dx \biggr) ^{\frac{p-\delta}{t}}. \end{aligned}$$

(3.31)

To sum up the cases $3B\subset\Omega$ and $3B\backslash\Omega\neq \emptyset$, we let

$$ g(x)= \textstyle\begin{cases} \vert Xu \vert ^{t}, &x\in\Omega,\\ 0, & x\in\mathbf{R}^{n}\backslash\Omega, \end{cases} $$

and

$$ f(x)= \textstyle\begin{cases} ( \vert u-u_{0} \vert +\vert u_{0} \vert +\vert Xu_{0} \vert ) ^{t}, & x\in\Omega,\\ 0, & x\in\mathbf{R}^{n}\backslash\Omega. \end{cases} $$

Thus we have from (3.1) and (3.31)

$${⨍}_{\frac{B}{2}}{g}^{q}dx\le b[{\left({⨍}_{3B}gdx\right)}^q+{⨍}_{3B}{f}^{q}dx]+\theta {⨍}_{3B}{g}^{q}dx,$$

where $q=\frac{p-\delta}{t}$, $\theta=c ( \delta+\tau^{1-\delta} )$ and $b = c{\tau^{1 - p}}$. Choosing τ, δ small enough, we see by Lemma 2.3 that there exists $t_{1}=p-\delta+\varepsilon_{0}$, for some $\varepsilon_{0}>0$, such that $\vert Xu \vert \in L^{t_{1}}(\Omega)$.

Furthermore, we will show that there exists ${t_{2}}>r=p-\delta$ such that $u \in L^{t_{2}}(\Omega)$. Since $u-u_{0}\in W_{X,0}^{1,r}(\Omega)$, we obtain from Lemma 2.2 that, for $r< Q$, $r^{*}=Qr/(Q-r)$,

$$ \biggl( \int_{\Omega} \vert u-u_{0} \vert ^{r^{*}}\,dx \biggr) ^{\frac{1}{r^{*}}}\leq C(\Omega) \biggl( \int_{\Omega}\bigl\vert X(u-u_{0}) \bigr\vert ^{r}\,dx \biggr) ^{\frac{1}{r}}< \infty. $$

Taking $t_{2}=\min\{s,r^{*}\}>r$, we have

$$\begin{aligned} \biggl( \int_{\Omega} \vert u \vert ^{t_{2}}\,dx \biggr) ^{\frac{1}{t_{2}}} &\leq\biggl( \int_{\Omega} \vert u-u_{0} \vert ^{t_{2}}\,dx \biggr) ^{\frac{1}{t_{2}}}+ \biggl( \int_{\Omega} \vert u_{0} \vert ^{t_{2}}\,dx \biggr) ^{\frac{1}{t_{2}}} \\ &\leq C \biggl( \int_{\Omega} \vert u-u_{0} \vert ^{r^{*}}\,dx \biggr) ^{\frac{1}{r^{*}}}+ \biggl( \int_{\Omega} \vert u_{0} \vert ^{t_{2}}\,dx \biggr) ^{\frac{1}{t_{2}}} \end{aligned}$$

and then $u\in L^{t_{2}}(\Omega)$ by $u_{0}\in L^{s}(\Omega)$. If $r\geq Q$ then we can apply the above reasoning for any $r^{*}<\infty$ to obtain $u\in L^{t_{2}}(\Omega)$.

We set $\tilde{p}=\min\{t_{1},t_{2}\}>p-\delta$ and $u\in W_{X}^{1,\tilde{p}}(\Omega)$. Repeating the preceding reasoning, we know that there exists $\tilde{\delta}>0$ such that $u\in W_{X}^{1,p+\tilde{\delta}}(\Omega)$ and the proof is complete.

4 Conclusions

In this paper, we obtained the global higher integrability for very weak solutions to the Dirichlet problem for a nonlinear subelliptic equation on Carnot-Carathéodory spaces which implies that such solutions are classical weak solutions. It is a generalization of the corresponding result in the classical Euclidean setting.