In this section, we deal with the \(\mathfrak{K}_{\psi }^{\theta }\)-obstacle problem related to the homogeneous nondiagonal quasilinear degenerate elliptic system
$$ X_{\alpha }^{\ast } \bigl({A_{ij}^{\alpha \beta } (x,u)X_{\beta }u ^{j}} \bigr)=0, $$
(3.1)
where \(i,j=1,2,\ldots ,N\), \(\alpha ,\beta =1,2,\ldots ,m\), coefficients \(A_{ij}^{\alpha \beta } (x,u)\) satisfy (H1). The main results are the higher integrability and a Campanato type estimate for the gradients of weak solutions. Let us recall that a function \(u\in S_{X,\mathrm{loc}} ^{1}(\varOmega ,\mathbb{R}^{N})\) is called a weak solution to (3.1) if
$$ \int _{\varOmega }A_{ij}^{\alpha \beta }(x,u)X_{\beta }u^{j}X_{\alpha } \phi ^{i}\,dx=0 $$
for all \(\phi \in C_{0}^{\infty }(\varOmega ,\mathbb{R}^{N})\); a function \(u\in \mathfrak{K}_{\psi }^{\theta } ( {\varOmega ,\mathbb{R}^{N}} )\) is called a weak solution to the \(\mathfrak{K}_{\psi }^{\theta }\)-obstacle problem for (3.1) if
$$ \int _{\varOmega }A_{ij}^{\alpha \beta }(x,u)X_{\beta }u^{j}X_{\alpha } \phi ^{i}\,dx\ge 0 $$
(3.2)
for all \(\phi \in C_{0}^{\infty }(\varOmega ,\mathbb{R}^{N})\) with \(\phi +u\geq \psi \) a.e. Ω.
For the diagonal homogeneous linear degenerate elliptic system with constant coefficients, we have the following estimates (see [35, Theorem 3.2]).
Lemma 3.1
Let
\(u\in S_{X,\mathrm{loc}}^{1}(\varOmega ,\mathbb{R}^{N})\)
be a weak solution to the linear system
$$X_{\alpha }^{\ast } \bigl(a^{\alpha \beta }X_{\beta }u^{i} \bigr)=0, \quad i=1,2,\ldots ,N, $$
with constant coefficients
\(a^{\alpha \beta }\in \mathbb{R}\)
for which (1.2) holds. Then, for any
\(x_{0}\in \varOmega \), there exist
\(c>0\)
and
\(0< R_{0}<\min \{d_{0},\operatorname{dist}(x_{0},\partial \varOmega ) \}/2\)
such that, for any
ρ, R
with
\(0<\rho \leq R\leq R_{0}\), it follows
$$ \int _{B_{\rho }(x_{0})} \vert Xu \vert ^{2}\,dx\leq c \biggl( \frac{\rho }{R} \biggr) ^{Q} \int _{B_{R}(x_{0})} \vert Xu \vert ^{2}\,dx. $$
(3.3)
In order to prove the higher integrability for gradients of weak solutions to the \(\mathfrak{K}_{\psi }^{\theta }\)-obstacle problem for (3.1), we need the Gehring lemma on the metric measure space \((Y,d,\mu )\), where d is a metric and μ is a doubling measure (see [36]).
Lemma 3.2
Let
\(q\in [\bar{q},2Q]\), where
\(\bar{q}>1\)
is fixed. Assume that functions
F, G
are nonnegative and
\(G\in L_{\mathrm{loc}}^{q} (Y, \mu )\), \(F\in L_{\mathrm{loc}}^{r_{0}}(Y,\mu )\)
for some
\(r_{0}>q\). If there exists a constant
\(b>1\)
such that, for every ball
\(B\subset \sigma B\subset Y\), the following inequality holds:
$$ \fint _{B}G^{q}d\mu \leq b \biggl[ \biggl(\fint _{\sigma B}Gd \mu \biggr)^{q}+\fint _{\sigma B}F^{q}\,d \mu \biggr], $$
then there exists a nonnegative constant
\(\varepsilon _{0} =\varepsilon _{0}(b,\bar{q},Q,C_{d},\sigma )\)
such that
\(G\in L_{\mathrm{loc}}^{p} (Y,\mu )\)
for
\(p\in [q,q+\varepsilon _{0})\). Moreover,
$$ \biggl(\fint _{B}G^{p}d\mu \biggr)^{\frac{1}{p}} \leq C \biggl[ \biggl(\fint _{\sigma B}G^{q}\,d\mu \biggr)^{\frac{1}{q}}+ \biggl(\fint _{\sigma B}F^{p}\,d\mu \biggr)^{\frac{1}{p}} \biggr] $$
for some positive constant
\(C=C(b,\bar{q},Q,C_{d},\sigma )\).
Theorem 3.3
(Higher integrability)
Let
\(u\in S_{X,\mathrm{loc}}^{1}(\varOmega ,\mathbb{R}^{N})\)
be a weak solution to the
\(\mathfrak{K}_{\psi }^{\theta }\)-obstacle problem for (3.1) and
\(X\psi \in L_{X}^{\sigma }(\varOmega , \mathbb{R}^{mN})(\sigma >2)\). Then there exists
\(p>2\)
such that
\(u\in S_{X,\mathrm{loc}}^{1,p}(\varOmega ,\mathbb{R}^{N})\). Furthermore, for any
\(B_{R}\subset \subset \varOmega \), we have
$$ \biggl(\fint _{B_{R/2}} \vert Xu \vert ^{p}\,dx \biggr)^{\frac{1}{p}}\leq c \biggl[ \biggl(\fint _{B_{R}} \vert Xu \vert ^{2}\,dx \biggr)^{\frac{1}{2}} + \biggl(\fint _{B_{R}} \vert X \psi \vert ^{p}\,dx \biggr)^{\frac{1}{p}} \biggr], $$
(3.4)
where the constant
\(c>0\)
does not depend on
R.
Proof
For the weak solution \(u\in S_{X,\mathrm{loc}}^{1}(\varOmega ,\mathbb{R} ^{N})\) and \(B_{R}\subset \subset \varOmega \), consider the function
$$\phi =-\eta ^{2} \bigl(u-\psi -(u-\psi )_{R} \bigr), $$
where η is a cutoff function on \(B_{R}\), i.e., \(\eta \in C_{0} ^{\infty }(B_{R})\) such that \(0\leq \eta \leq 1\), \(\eta =1\) in \(B_{R/2} \), and \(|X\eta |\leq c/R\). Since \(u_{R} \geq \psi _{R}\), we have
$$\phi =\eta ^{2}(\psi -u)+\eta ^{2}(u_{R} -\psi _{R})\geq \eta ^{2}(\psi -u)+ \eta ^{2}(\psi _{R} -\psi _{R} )\geq \psi -u $$
a.e. in Ω and it is an admissible function for (3.2). Taking \(\phi =-\eta ^{2}(u-\psi -(u- \psi )_{R})\) in (3.2), we immediately get
$$\begin{aligned} & \int _{B_{R}} a^{\alpha \beta }(x)\delta _{ij} \eta ^{2}X_{\beta }u^{j}X_{\alpha }u^{i}\,dx \\ &\quad\leq - \int _{B_{R}} B_{ij}^{\alpha \beta } (x,u)\eta ^{2}X_{\beta }u ^{j}X_{\alpha }u^{i}\,dx\\ &\qquad {}+ \int _{B_{R}} A_{ij}^{\alpha \beta }(x,u) \eta ^{2}X_{\beta }u^{j}X_{\alpha }\psi ^{i}\,dx \\ &\qquad {}+2 \int _{B_{R}}A_{ij}^{\alpha \beta }(x,u)\eta (\psi -\psi _{R})^{i}X _{\beta }u^{j}X_{\alpha } \eta \,dx\\ &\qquad {} -2 \int _{B_{R}}A_{ij}^{\alpha \beta }(x,u) \eta (u-u_{R})^{i}X_{\beta }u^{j}X_{\alpha } \eta \,dx. \end{aligned}$$
By means of assumption (H1), Young and Sobolev inequalities, we see that
$$\begin{aligned} \nu \int _{B_{R}}\eta ^{2} \vert Xu \vert ^{2}\,dx\leq{}& (\varepsilon +\delta \nu ) \int _{B_{R}}\eta ^{2} \vert Xu \vert ^{2}\,dx+\frac{c_{\varepsilon } \vert B_{R} \vert }{R^{2}}\fint _{B_{R}} \vert u-u_{R} \vert ^{2}\,dx \\ &{} +\frac{c_{\varepsilon }}{R^{2}} \int _{B_{R}} \vert \psi -\psi _{R} \vert ^{2}\,dx+c _{\varepsilon } \int _{B_{R}} \vert X\psi \vert ^{2}\,dx \\ \leq {}&(\varepsilon +\delta \nu ) \int _{B_{R}}\eta ^{2} \vert Xu \vert ^{2}\,dx+c_{ \varepsilon } \vert B_{R} \vert \biggl( \fint _{B_{R}} \vert Xu \vert ^{2q_{0}}\,dx \biggr) ^{\frac{1}{q_{0}}}\\ &{}+c_{\varepsilon } \int _{B_{R}} \vert X\psi \vert ^{2}\,dx. \end{aligned}$$
Choosing \(\varepsilon =\nu /3\) and noting \(\eta =1\) on \(B_{R/2}\), it follows
$$ \int _{B_{R/2}} \vert Xu \vert ^{2}\,dx\leq c \vert B_{R} \vert \biggl(\fint _{B_{R}} \vert Xu \vert ^{2q _{0}}\,dx \biggr)^{\frac{1}{q_{0}}}+c \int _{B_{R}} \vert X\psi \vert ^{2}\,dx. $$
Dividing by \(|B_{R/2}|\) on both sides, we get
$$ \fint _{B_{R/2}} \vert Xu \vert ^{2}\,dx \leq c \biggl( \fint _{B_{R}} \vert Xu \vert ^{2q _{0}}\,dx \biggr)^{\frac{1}{q_{0}}} + c\fint _{B_{R}} \vert X\psi \vert ^{2}\,dx. $$
Now in Lemma 3.2 we set \(G=|Xu|^{2q_{0}}\), \(F=|X\psi |^{2q _{0}}\) and \(q=1/{q_{0}}\). Then
$$\vert Xu \vert ^{2q_{0} }\in L_{\mathrm{loc}}^{r} ( \varOmega ),\quad \forall r \in [1/{q_{0}},1/{q_{0}}+ \varepsilon _{0} ), $$
and
$$ \biggl(\fint _{B_{R/2}} \vert Xu \vert ^{2q_{0}r}\,dx \biggr)^{\frac{1}{r}} \leq c \biggl(\fint _{B_{R}} \vert Xu \vert ^{2}\,dx \biggr)^{q_{0}} + c \biggl(\fint _{B_{R}} \vert X\psi \vert ^{2q_{0}r}\,dx \biggr)^{\frac{1}{r}}. $$
If we set \(p=2q_{0} r\), then \(p\in [2,2+2q_{0}\varepsilon _{0})\), and
$$ \biggl(\fint _{B_{R/2}} \vert Xu \vert ^{p}\,dx \biggr)^{\frac{1}{p}}\leq c \biggl[ \biggl(\fint _{B_{R}} \vert Xu \vert ^{2}\,dx \biggr)^{\frac{1}{2}} + \biggl(\fint _{B_{R}} \vert X \psi \vert ^{p}\,dx \biggr)^{\frac{1}{p}} \biggr], $$
where c does not depend on R. The proof is complete. □
By virtue of the above result, we can establish a Campanato type estimate for the gradients of weak solutions to the \(\mathfrak{K}_{ \psi }^{\theta }\)-obstacle problem for (3.1).
Theorem 3.4
Let
\(u\in S_{X,\mathrm{{loc}}}^{1}(\varOmega ,\mathbb{R}^{N})\)
be a weak solution to the
\(\mathfrak{K}_{\psi }^{\theta }\)-obstacle problem for (3.1) and
\(X\psi \in L_{X}^{\sigma ,\lambda } ( \varOmega ,\mathbb{R}^{mN})\), \(\sigma >2\). Then, for any
\(x_{0}\in \varOmega \), there exist
\(c>0\)
and
\(0< R_{0}<\min \{d_{0},\operatorname{dist}(x _{0},\partial \varOmega )\}/2\)
such that, for any
ρ, R
with
\(0<\rho \leq R\leq R_{0}\), it follows
$$ \int _{B_{\rho }} \vert Xu \vert ^{2}\,dx\leq c \biggl[ \biggl(\frac{\rho }{R} \biggr) ^{Q} + \bigl(\eta _{R} \bigl(a^{\alpha \beta }\bigr) \bigr)^{\frac{p-2}{p}} + \delta \biggr] \int _{B_{R}} \vert Xu \vert ^{2}\,dx +c \frac{ \vert B_{R} \vert }{R^{\lambda }} \Vert X\psi \Vert _{L^{p,\lambda }}^{2}, $$
(3.5)
where
\(2< p<\sigma \).
Proof
Let \(B_{R} =B(x_{0},R)\subset \subset \varOmega \). In \(B_{R/2} \) we split u as \(u=U+w\), where \(U\in S_{X}^{1}(B_{R/2},\mathbb{R}^{N})\) is the weak solution to the following boundary value problem for homogeneous system with constant coefficients:
$$ \textstyle\begin{cases} X_{\alpha }^{\ast } ((a^{\alpha \beta }(x))_{R/2}\delta _{ij} X _{\beta }U^{j} )=0, \quad \mbox{in } B_{R/2}, \\ U-u\in S_{X,0}^{1}(B_{R/2}, \mathbb{R}^{N}). \end{cases} $$
(3.6)
Denote \((a^{\alpha \beta })_{R/2}:=(a^{\alpha \beta }(x))_{R/2}\). Therefore, by Lemma 3.1, there exist \(c>0\) and \(0< R_{0}< \min \{d_{0},\operatorname{dist}(x_{0},\partial \varOmega )\}/2\) such that, for all \(0<\rho <R/2\),
$$ \int _{B_{\rho }} \vert XU \vert ^{2}\,dx\leq c \biggl( \frac{\rho }{R} \biggr)^{Q} \int _{B_{R/2}} \vert XU \vert ^{2}\,dx. $$
(3.7)
Using (3.6), we know from (3.2) that, for all \(\phi \in S_{X,0}^{1}(B_{R/2},\mathbb{R}^{N})\) with \(\phi +u \geq \psi \) a.e. \(B_{R/2}\),
$$\begin{aligned} \int _{B_{R/2}} \bigl(a^{\alpha \beta }\bigr)_{R/2}\delta _{ij} X_{\beta }w^{j}X _{\alpha }\phi ^{i}\,dx &\geq \int _{B_{R/2}} \bigl(\bigl(a^{\alpha \beta }\bigr)_{R/2}-a ^{\alpha \beta }(x) \bigr)\delta _{ij} X_{\beta }u^{j}X_{\alpha } \phi ^{i}\,dx \\ & \quad - \int _{B_{R/2}}B_{ij}^{\alpha \beta }(x,u)X_{\beta }u^{j}X_{\alpha } \phi ^{i}\,dx. \end{aligned}$$
(3.8)
Since \(U-u\in S_{X,0}^{1}(B_{R/2},\mathbb{R}^{N})\), we can choose \(\phi =U\vee \psi -u\in S_{X,0}^{1} (B_{R/2},\mathbb{R}^{N})\) as a test function in (3.8), where \(U\vee \psi \) is a vector-valued function with components \(U^{i}\vee \psi ^{i}=\max \{U^{i},\psi ^{i}\}\). Then
$$\begin{aligned} & \int _{B_{R/2}} \bigl(a^{\alpha \beta }\bigr)_{R/2} \delta _{ij}X_{\beta }w ^{j} X_{\alpha }(u- U\vee \psi )^{i}\,dx \\ &\quad\leq \int _{B_{R/2}} \bigl(\bigl(a^{\alpha \beta }\bigr)_{R/2}-a^{\alpha \beta }(x) \bigr)\delta _{ij}X_{\beta }u^{j}X_{\alpha }(u-U \vee \psi )^{i}\,dx \\ & \qquad {}- \int _{B_{R/2}} B_{ij}^{\alpha \beta } (x,u)X_{\beta }u^{j}X_{\alpha }({u-U \vee \psi })^{i}\,dx. \end{aligned}$$
(3.9)
Noting \((u-U\vee \psi )^{i}=w^{i}+(U-U\vee \psi )^{i}\), it follows by using the Hölder inequality that
$$\begin{aligned} & \int _{B_{R/2}} \bigl(a^{\alpha \beta }\bigr)_{R/2}\delta _{ij} X_{\beta }w ^{j}X_{\alpha }w^{i}\,dx \\ &\quad\leq \int _{B_{R/2}} \bigl(a^{\alpha \beta }\bigr)_{R/2}\delta _{ij} X_{\beta }w ^{j}X_{\alpha }(U\vee \psi -U)^{i}\,dx \\ &\qquad {}+ \int _{B_{R/2}} \bigl(\bigl(a^{\alpha \beta }\bigr)_{R/2}-a^{\alpha \beta }(x) \bigr)\delta _{ij}X_{\beta }u^{j}X_{\alpha }w^{i}\,dx \\ &\qquad {}+ \int _{B_{R/2}} \bigl(\bigl(a^{\alpha \beta }\bigr)_{R/2}-a^{\alpha \beta }(x) \bigr)\delta _{ij} X_{\beta }u^{j}X_{\alpha }(U-U \vee \psi )^{i}\,dx \\ & \qquad{} - \int _{B_{R/2}}B_{ij}^{\alpha \beta } X_{\beta }u^{j}X_{ \alpha }w^{i}\,dx - \int _{B_{R/2}}B_{ij}^{\alpha \beta } X_{\beta }u^{j}X _{\alpha }(U-U\vee \psi )^{i}\,dx \\ &\quad \leq \biggl(\varepsilon +\frac{\delta \nu }{2}\biggr) \int _{B_{R/2}} \vert Xw \vert ^{2}\,dx+\biggl(c _{\varepsilon }+\frac{\delta \nu }{2}\biggr) \int _{B_{R/2}} \bigl\vert X(U-U\vee \psi ) \bigr\vert ^{2}\,dx \\ &\qquad {}+c_{\varepsilon } \int _{B_{R/2}} \bigl\vert \bigl(a^{\alpha \beta } \bigr)_{R/2}-a ^{\alpha \beta }(x) \bigr\vert ^{2} \vert Xu \vert ^{2}\,dx+\delta \nu \int _{B_{R/2}} \vert Xu \vert ^{2}\,dx. \end{aligned}$$
(3.10)
Recalling (1.2) and taking \(\varepsilon =\frac{\nu }{3}\), we have
$$\begin{aligned} \int _{B_{R/2}} \vert Xw \vert ^{2}\,dx\leq{}& c \int _{B_{R/2}} \bigl\vert X(U-U\vee \psi ) \bigr\vert ^{2}\,dx \\ &{} +c \int _{B_{R/2}} \bigl\vert \bigl(a^{\alpha \beta } \bigr)_{R/2}-a^{\alpha \beta }(x) \bigr\vert ^{2} \vert Xu \vert ^{2}\,dx+\delta \int _{B_{R/2}} \vert Xu \vert ^{2}\,dx. \end{aligned}$$
(3.11)
Since \(a^{\alpha \beta }(x)\in L^{\infty }\cap \text{VMO}\) and invoking (3.4) in Theorem 3.3, we conclude that there exists \(p>2\) such that
$$\begin{aligned} & \int _{B_{R/2}} \bigl\vert \bigl(a^{\alpha \beta } \bigr)_{R/2}-a^{\alpha \beta }(x) \bigr\vert ^{2} \vert Xu \vert ^{2}\,dx \\ &\quad\leq \vert B_{R/2} \vert \biggl(\fint _{B_{R/2}} \bigl\vert \bigl(a^{\alpha \beta }\bigr)_{R/2}-a ^{\alpha \beta }(x) \bigr\vert ^{\frac{2p}{p-2}}\,dx \biggr)^{ \frac{p-2}{p}} \biggl(\fint _{B_{R/2}} \vert Xu \vert ^{p}\,dx \biggr)^{ \frac{2}{p}} \\ &\quad \leq c \bigl(\eta _{R}\bigl(a^{\alpha \beta }\bigr) \bigr)^{\frac{p-2}{p}} \vert B _{R/2} \vert \biggl(\fint _{B_{R/2}} \vert Xu \vert ^{p}\,dx \biggr)^{\frac{2}{p}} \\ &\quad \leq c \bigl(\eta _{R}\bigl(a^{\alpha \beta }\bigr) \bigr)^{\frac{p-2}{p}} \int _{B_{R}} \vert Xu \vert ^{2}\,dx +c \vert B_{R} \vert \biggl(\fint _{B_{R}} \vert X\psi \vert ^{p}\,dx \biggr) ^{\frac{2}{p}}. \end{aligned}$$
(3.12)
On the other hand, \(U-U\vee \psi \in S_{X,0}^{1}(B_{R/2},\mathbb{R} ^{N})\) satisfies
$$\begin{aligned} &\int _{B_{R/2}}\bigl(a^{\alpha \beta }\bigr)_{R/2}\delta _{ij} X_{\beta }(U-U \vee \psi )^{j}X_{\alpha } \varphi ^{i}\,dx \\ &\quad =- \int _{B_{R/2}}\bigl(a^{\alpha \beta }\bigr)_{R/2}\delta _{ij} X_{\beta }(U\vee \psi )^{j}X_{\alpha } \varphi ^{i}\,dx \end{aligned}$$
(3.13)
for all \(\varphi \in S_{X,0}^{1}(B_{R/2},\mathbb{R}^{N})\). Thus choosing \(\varphi =U-U\vee \psi \) in (3.13) and noting \((U\vee \psi )^{i}= \psi ^{i}\) for \(x\in {\mathrm{supp}}(U-U\vee \psi )^{i}\), we obtain from (1.2) that
$$\begin{aligned} \nu \int _{B_{R/2}} \bigl\vert X(U-U\vee \psi ) \bigr\vert ^{2}\,dx &\leq \int _{B_{R/2}}\bigl(a^{ \alpha \beta }\bigr)_{R/2} X_{\beta }(U-U\vee \psi )^{i}X_{\alpha }(U-U \vee \psi )^{i}\,dx \\ &=- \int _{B_{R/2}}\bigl(a^{\alpha \beta }\bigr)_{R/2} X_{\beta }(U\vee \psi )^{i}X _{\alpha }(U-U\vee \psi )^{i}\,dx \\ &=- \int _{\mathrm{supp}(U-U\vee \psi )^{i}} \bigl(a^{\alpha \beta }\bigr)_{R/2} X_{\beta }\psi ^{i}X_{\alpha }(U-U\vee \psi )^{i}\,dx \\ &\leq \varepsilon \int _{B_{R/2}} \bigl\vert X(U-U\vee \psi ) \bigr\vert ^{2}\,dx+c_{\varepsilon } \int _{B_{R/2}} \vert X\psi \vert ^{2}\,dx. \end{aligned}$$
Letting \(\varepsilon =\frac{\nu }{2}\), we get
$$ \int _{B_{R/2}} \bigl\vert X(U-U\vee \psi ) \bigr\vert ^{2}\,dx\leq c \int _{B_{R/2}} \vert X\psi \vert ^{2}\,dx. $$
(3.14)
Inserting (3.12) and (3.14) into (3.11), we have
$$ \int _{B_{R/2}} \vert Xw \vert ^{2}\,dx\leq c \bigl[ \bigl(\eta _{R}\bigl(a^{\alpha \beta }\bigr) \bigr)^{\frac{p-2}{p}}+ \delta \bigr] \int _{B_{R}} \vert Xu \vert ^{2}\,dx+c \vert B _{R} \vert \biggl(\fint _{B_{R}} \vert X\psi \vert ^{p}\,dx \biggr)^{\frac{2}{p}}. $$
(3.15)
Then it follows by using (3.7) and (3.15) that, for any \(0<\rho <R/2\),
$$\begin{aligned} \int _{B_{\rho }} \vert Xu \vert ^{2}\,dx &\leq 2 \int _{B_{\rho }} \vert XU \vert ^{2}\,dx+2 \int _{B_{\rho }} \vert Xw \vert ^{2}\,dx \\ &\leq c \biggl(\frac{\rho }{R} \biggr)^{Q} \int _{B_{R/2}} \vert XU \vert ^{2}\,dx+2 \int _{B_{R/2}} \vert Xw \vert ^{2}\,dx \\ &\leq c \biggl(\frac{\rho }{R} \biggr)^{Q} \int _{B_{R/2}} \vert Xu \vert ^{2}\,dx+c \int _{B_{R/2}} \vert Xw \vert ^{2}\,dx \\ &\leq c \biggl[ \biggl(\frac{\rho }{R} \biggr)^{Q}+ \bigl(\eta _{R} \bigl(a ^{\alpha \beta }\bigr) \bigr)^{\frac{p-2}{p}}+\delta \biggr] \int _{B_{R}} \vert Xu \vert ^{2}\,dx+c \vert B_{R} \vert \biggl(\fint _{B_{R}} \vert X\psi \vert ^{p}\,dx \biggr)^{ \frac{2}{p}} \\ &\leq c \biggl[ \biggl(\frac{\rho }{R} \biggr)^{Q}+ \bigl(\eta _{R} \bigl(a ^{\alpha \beta }\bigr) \bigr)^{\frac{p-2}{p}}+\delta \biggr] \int _{B_{R}} \vert Xu \vert ^{2}\,dx+c \frac{ \vert B_{R} \vert }{R^{\lambda }} \Vert X\psi \Vert _{L ^{p,\lambda }}^{2} . \end{aligned}$$
(3.16)
It is obvious that the above inequality is valid for \(R/2\leq \rho \leq R\). Thus, for all \(0<\rho \leq R\), we have
$$\int _{B_{\rho }} \vert Xu \vert ^{2}\,dx\leq c \biggl[ \biggl(\frac{\rho }{R} \biggr) ^{Q}+ \bigl(\eta _{R} \bigl(a^{\alpha \beta }\bigr) \bigr)^{\frac{p-2}{p}}+ \delta \biggr] \int _{B_{R}} \vert Xu \vert ^{2}\,dx+c \frac{ \vert B_{R} \vert }{R^{\lambda }} \Vert X\psi \Vert _{L^{p,\lambda }}^{2}. $$
□