In this section, we are devoted to enhancing the integrable index of X-gradient of their weak solutions based on \(L^{p}\)-estimates of linear subelliptic equations, perturbation argument and the bootstrap argument. First of all, we prove the boundedness of the weak solutions to equation (1.1) by way of the idea from De Giorgi’s iteration; also see [25]. To this end, let us consider in \(HW^{1,2}(\Omega)\) the following linear subelliptic equations in divergence form:
$$ \sum_{i,j}^{m} X^{*}_{i} \bigl(a_{ij}(x)X_{j}u \bigr)=\sum_{i}^{m}X^{*}_{i}f_{i}, \quad \mbox{a.e. }x\in\Omega, $$
(3.1)
where \(a_{ij}\in \operatorname{VMO}(\Omega)\) and satisfy uniformly ellipticity H1, and \({\mathbf{f}(x)}=(f_{1}(x),f_{2}(x),\ldots,f_{m}(x))\in [L^{p}(\Omega ) ]^{m}\) with \(p>2\). We recall an interior \(L^{p}\)-estimate of X-gradient to equation (3.1), which can be referred to Theorem 2.8 in [37].
Lemma 3.1
Let
\(u\in HW^{1,2}_{\mathrm{loc}}(\Omega)\)
be any weak solution to linear subelliptic equations (3.1). Suppose that the leading coefficients
\(a_{ij}\in \operatorname{VMO}(\Omega)\)
and satisfy uniformly ellipticity H1, and
\({\mathbf{f}(x)}\in[L^{p}(\Omega)]^{m}\)
with
\(2< p<+\infty\). Then
\(u\in HW^{1,p}(\Omega')\)
for any
\(\Omega '\subset\subset\Omega\). Moreover, there exists a positive constant
\(C=C(\mu,L, Q,p,R)\)
such that for any
\(B_{R}\subset\Omega\)
we have
$$ \Vert Xu\Vert _{L^{p}(B_{R/2})}\leq C \bigl( \bigl\Vert { \mathbf{f}(x)} \bigr\Vert _{L^{p}(B_{R})}+\Vert u\Vert _{L^{2}(B_{R})} \bigr). $$
(3.2)
For the convenience of studying quasilinear subelliptic equations (1.1), we here give the following conclusion, which can be found a similar conclusion from [43] in the case of Euclidean metric and usual gradient.
Lemma 3.2
Let Ω be a bounded Lipschitz open set in
\(\mathbb{R}^{n}\). For any
\(g(x)\in L^{q}(\Omega)\)
with
\(q>1\), there exists a vector-valued function
\(G(x):\Omega\rightarrow\mathbb{R}^{m}\)
with
\(G(x)=\{G^{1}(x),G^{2}(x),\ldots, G^{m}(x)\}\in [L^{q^{*}}(\Omega ) ]^{m}\)
such that
\(g(x)=\sum_{i}^{m}X^{*}_{i}(G^{i}(x))\); and we have
$$ \|G\|_{(L^{q^{*}}(\Omega))^{m}}\leq C(Q,q,\partial\Omega)\|g\| _{L^{q}(\Omega)}, $$
(3.3)
where
$$q^{*}=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{Qq}{Q-q}, & 1\le q< Q, \\ \textit{any }q^{*}\ge q,& q\ge Q, \end{array}\displaystyle \right . $$
is the Sobolev conjugate index of
q.
Proof
Given any fixed point \(y\in\mathbb{R}^{n}\), let \(\Gamma(x,y)\) be the fundamental solution to sub-Laplacian equations \(\sum_{i=1}^{k} X^{*}_{i}X_{i}u=0\) in \(\mathbb{R}^{n}\). By Theorem 2.2 in [4], its fundamental solution \(\Gamma(x,y)\) deserves the following local properties:
$$ \Gamma(x,y)\simeq\frac{\rho(x,y)^{2}}{|B(x,\rho(x,y))|}\simeq\rho (x,y)^{2-Q}, $$
(3.4)
and there exists a positive constant \(C=C(Q)\) such that
$$ \bigl\vert X^{s}\Gamma(x,y) \bigr\vert \le C \frac{\rho(x,y)^{2-s}}{\vert B(x,\rho(x,y))\vert }\le C\rho(x,y)^{2-s-Q},\quad s=1,2,\ldots. $$
(3.5)
Note that ∂Ω is Lipschitz continuous, there exists an extending function \(\tilde{g}(x)\) defined in \(\mathbb{R}^{n}\) such that \(\tilde{g}(x)=g(x)\) on Ω and \(\|\tilde{g}\| _{L^{q}(\mathbb{R}^{n})}\le C(\Omega) \|g\|_{L^{q}(\Omega)}\); moreover, \(\tilde{g}(x)\) has compact support in \(\mathbb{R}^{n}\), namely, there exists \(\Omega\subset\subset V\subset\mathbb{R}^{n}\) such that \(\operatorname{supp}(\tilde{g})\subset V\). Therefore, it is easy to see that \(N\tilde{g}(x)=\int_{\mathbb {R}^{n}}\Gamma(x,y)\tilde{g}(y)\,dy\) satisfies \(\sum_{i=1}^{k} X^{*}_{i}X_{i}(N\tilde{g}(x))=g(x)\), a.e. \(x\in\Omega\).
Thanks to the Calderon-Zygmund theory of a singular integral operator, it follows that
$$\bigl\Vert X^{2} \bigl(N\tilde{g}(x) \bigr) \bigr\Vert _{L^{q}(V)}\leq C(Q,q,\Omega) \Vert g\Vert _{L^{q}(\Omega)}. $$
Now let us restrict \(G(x):=\{X_{1}(N\tilde{g}(x)),\ldots ,X_{m}(N\tilde{g}(x))\}\) to Ω and employ the Sobolev embedding inequality of X-gradient of Theorem 2.1, then it yields
$$\bigl\Vert G(x) \bigr\Vert _{L^{q^{*}}(\Omega)}= \bigl\Vert X \bigl(N \tilde{g}(x) \bigr) \bigr\Vert _{L^{q^{*}}(\Omega )}\leq C \bigl\Vert X^{2} \bigl(N\tilde{g}(x) \bigr) \bigr\Vert _{L^{q}(V)}\leq C \Vert g \Vert _{L^{q}(\Omega)}. $$
This lemma is proved. □
Further, we consider the following linear subelliptic equations in divergence form:
$$ -X^{*}_{i} \bigl(a_{ij}(x)X_{j}u \bigr)=-X^{*}_{i}f_{i}(x)+g(x) , \quad \mbox{a.e. }x\in \Omega, $$
(3.6)
where \(a_{ij}\in \operatorname{VMO}(\Omega)\) satisfies the uniform ellipticity H1, and \({\mathbf{f}(x)}\in[L^{p}(\Omega)]^{m}\), \(g(x)\in L^{q}(\Omega)\) with p, q satisfying \(p>2\) and \(q>\frac{pQ}{Q+p}\). By a simply computation we have the following.
Lemma 3.3
Let
\(u\in HW^{1,2}_{\mathrm{loc}}(\Omega)\)
be any weak solution to equation (3.6). Suppose that the leading coefficients
\(a_{ij}\in \operatorname{VMO}(\Omega)\)
satisfy the uniform ellipticity H1 and H2, and
\({\mathbf{f}(x)}\in[L^{p}(\Omega)]^{m}\), \(g(x)\in L^{q}(\Omega)\)
with
\(p, q\)
satisfying
\(p>2\)
and
\(q>\frac{2Q}{Q+2}\). Then, for any
\(\Omega'\subset\subset\Omega\), we have
\(u\in HW^{1,r}(\Omega')\)
with
\(r=\min\{p,q^{*}\}\). Moreover, there exists a positive constant
\(C=C(\mu,L, Q, p, q, R)\)
such that for any
\(B_{R}\subset\Omega\), we have
$$ \Vert Xu\Vert _{L^{r}(B_{R/2})}\leq C \bigl( \bigl\Vert {\mathbf{f}(x)} \bigr\Vert _{L^{r}(B_{R})}+\Vert g\Vert _{L^{q}(B_{R})}+\Vert u\Vert _{L^{2}(B_{R})} \bigr) $$
(3.7)
with
\(r=\min\{p,q^{*}\}\).
Proof
On the basis of Lemma 3.2, we know that for \(g(x)\in L^{q}(\Omega)\) there exists a vectorial-valued function \(G(x)=(G_{1}(x),\ldots,G_{m}(x))\in[L^{q^{*}}(\Omega)]^{m}\) such that \(g(x)=-\sum_{i=1}^{m} X^{*}_{i}G_{i}(x)\) for a.e. \(x\in\Omega\), and
$$\|G\|_{L^{q^{*}}(B_{R})}\leq C \|g\|_{L^{q}(B_{R})}. $$
In this way, equation (3.6) can be rewritten as
$$ -X^{*}_{i} \bigl(a_{ij}(x)X_{j}u \bigr)=-X^{*}_{i}({\mathbf{f}}+G),\quad \mbox{a.e. }x\in B_{R}, $$
(3.8)
where \({\mathbf{f}}+G\in[L^{r}(B_{R})]^{m}\) and \(r=\min\{p,q^{*}\}\). Using Lemma 3.1 it yields
$$\begin{aligned} \Vert Xu\Vert _{L^{r}(B_{R/2})} \leq& C \bigl(\Vert { \mathbf{f}}+G \Vert _{L^{r}(B_{R})}+\Vert u\Vert _{L^{2}(B_{R})} \bigr) \\ \leq& C \bigl(\Vert {\mathbf{f}}\Vert _{L^{r}(B_{R})}+\Vert G\Vert _{L^{q^{*}}(B_{R})}+\Vert u\Vert _{L^{2}(B_{R})} \bigr) \\ \leq& C \bigl(\Vert {\mathbf{f}}\Vert _{L^{r}(B_{R})}+\Vert g\Vert _{L^{q}(B_{R})}+\Vert u\Vert _{L^{2}(B_{R})} \bigr), \end{aligned}$$
where \(C=C(\mu,Q,p,q,R)\). This completes the proof of Lemma 3.3. □
In order to get the boundedness of weak solutions to equation (1.1) under the controllable growth, we will use so-called De Giorgi’s iteration argument (cf. Lemma 5.1 in Chapter 5 of [44]). We denote by \(A_{k}=\{x\in\Omega: u(x)>k\} \) the distributional function of u on Ω, and by \(|A_{k}|\) denote the measure of \(A_{k}\) with the C.-C. metric.
Lemma 3.4
Let
\(u(x)\)
be a measurable function defined on Ω. If, for any
\(k\geq k_{0}>0\)
there exist constants
γ, α, and
ε
satisfying
\(\gamma, \varepsilon>0\), \(0\leq\alpha\leq 1+\varepsilon\)
such that
$$ \int_{A_{k}}(u-k)\,dx \leq N_{0} k^{\alpha} |A_{k}|^{1+\varepsilon}. $$
(3.9)
Then
\(u(x)\)
is essentially bounded on Ω; namely, there exists a positive constant
\(N=N(\gamma,\alpha,\varepsilon,k_{0},\|u\| _{L^{1}_{(A_{k_{0}})}})\)
such that
$$\operatorname{ess}\sup_{x\in\Omega} u(x)\leq N. $$
Based on Lemma 3.4 above and Lemma 2.1 (Sobolev inequality of X-gradient), we obtain the following useful conclusion (cf. Lemma 5.2 in Chapter 2 of [44]).
Lemma 3.5
Let
\(u(x)\in HW^{1, 2}(\Omega)\)
and
\(Q\ge2\). If, for any
\(k\geq k_{0}>0\), there exist constants
γ, ε, α
satisfying
\(\gamma, \varepsilon>0\)
and
\(0\leq\alpha\leq 2+\varepsilon\)
such that
$$ \int_{A_{k}}|X u|^{2}\,dx\leq N_{0} k^{\alpha}|A_{k}|^{1-\frac {2}{Q}+\varepsilon}. $$
(3.10)
Then
\(u(x)\)
is essentially bounded on Ω, and there exists a positive constant
\(N=N(\gamma,\alpha,\varepsilon,k_{0}, \|u\| _{L^{1}_{(A_{k_{0}})}})\)
such that
$$\operatorname{ess}\sup_{x\in\Omega} \bigl\vert u(x) \bigr\vert \leq N. $$
Hence, in order to establish the boundedness of weak solutions to equation (1.1) we only need to prove that the weak solutions of equation (1.1) satisfy inequality (3.10).
Lemma 3.6
(Boundedness of weak solutions)
Let
\(u(x)\in HW^{1,2}(\Omega)\)
be any weak solution to quasilinear subelliptic equations (1.1). Suppose that the leading coefficients and lower terms satisfy the structural assumptions H1 and H3. Then
\(u(x)\)
is essentially bounded, and
$$\|u\|_{L^{\infty}(\Omega)}=\operatorname{ess}\sup_{x \in\Omega} \bigl\vert u(x) \bigr\vert \leq M, $$
where
\(M=M(Q,\mu,\mu_{1},\mu_{2},\|{\mathbf{f}}\|_{L^{p}},\|g\|_{L^{q}})>0\).
Proof
Notice that the assumptions of H1 and H3 on \(a_{i}(x,u)\), by Young’s inequality, yield
$$\begin{aligned}& A_{ij}(x,u)X_{i}u X_{j}u \ge \mu \vert Xu \vert ^{2}, \\& \Biggl\vert \sum_{i=1}^{m} a_{i}(x,u)X_{i}u \Biggr\vert \le \mu_{1} \bigl(\vert u\vert ^{\frac{\gamma}{2}}\vert Xu\vert + \bigl\vert { \mathbf{f}(x)} \bigr\vert \vert Xu\vert \bigr)\le\frac{\mu }{4}\vert Xu \vert ^{2}+\frac{8\mu_{1}}{\mu} \vert u\vert ^{\gamma}+ \mu_{1} \bigl\vert {\mathbf{f}(x)} \bigr\vert \vert Xu\vert , \end{aligned}$$
which implies
$$ \sum_{i,j=1}^{m} \bigl( A_{ij}(x,u)X_{j}u+a_{i}(x,u) \bigr)X_{i}u\geq \frac{3}{4}\mu \vert Xu\vert ^{2}-C(\mu, \mu_{1})\vert u\vert ^{\gamma}- \mu_{1} \bigl\vert {\mathbf{f}(x)} \bigr\vert \vert Xu\vert . $$
(3.11)
Using the controllable growth of \(b(x,u,Xu)\) and Young’s inequality, it follows that
$$\begin{aligned} \bigl\vert b(x,u,Xu)u \bigr\vert \leq&\mu_{2} \vert u\vert \bigl(\vert Xu\vert ^{2(1-\frac{1}{\gamma })}+\vert u\vert ^{\gamma-1}+ \bigl\vert g(x) \bigr\vert \bigr) \\ \le& \frac{\mu}{4} \vert Xu\vert ^{2}+C(\mu, \mu_{2})\vert u\vert ^{\gamma}+ \mu_{2}\vert u \vert \bigl\vert g(x) \bigr\vert . \end{aligned}$$
(3.12)
Let us combine (3.11) and (3.12) and take \(\varphi=(u-k)_{+}\) with \(k>0\) determined later as the test function. By integrating on the distributional function \(A_{k}\) we have
$$\begin{aligned} \int_{A_{k}} \vert Xu\vert ^{2}\,dx \le& C(\mu, \mu_{1},\mu_{2}) \int _{A_{k}}\vert u\vert ^{\gamma}\,dx \\ &{}+ \mu_{1} \int_{A_{k}} \bigl\vert {\mathbf{f}(x)} \bigr\vert \vert Xu \vert \,dx+\mu _{2} \int_{A_{k}}\vert u\vert \bigl\vert g(x) \bigr\vert \,dx \\ :=& C(\mu,\mu_{1},\mu_{2})K_{1}+ \mu_{1}K_{2}+ \mu_{2}K_{3}. \end{aligned}$$
(3.13)
Now let us estimate \(K_{1}\), \(K_{2}\), \(K_{3}\), respectively, as follows.
To estimate \(K_{2}\), we have
$$\begin{aligned} K_{2} \le& \vert A_{k}\vert ^{1-\frac{1}{p}-\frac{1}{2}} \biggl( \int _{A_{k}} \bigl\vert {\mathbf{f}(x)} \bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}} \biggl( \int _{A_{k}}\vert Xu\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \\ \le& \frac{1}{4\mu_{1}} \int_{A_{k}}\vert Xu\vert ^{2}\,dx+C\vert A_{k}\vert ^{\frac{p-2}{p}} \biggl( \int_{A_{k}} \bigl\vert {\mathbf{f}(x)} \bigr\vert ^{p}\,dx \biggr)^{\frac{2}{p}}. \end{aligned}$$
(3.14)
To estimate \(K_{3}\), we get
$$\begin{aligned} K_{3} \le& \vert A_{k}\vert ^{1-\frac{1}{2^{*}}-\frac{1}{q}} \biggl( \int _{A_{k}}\vert u\vert ^{2^{*}}\,dx \biggr)^{\frac{1}{2^{*}}} \biggl( \int _{A_{k}} \bigl\vert g(x) \bigr\vert ^{q}\,dx \biggr)^{\frac{1}{q}} \\ \le& C\vert A_{k}\vert ^{\frac{Q+2}{2Q}-\frac{1}{q}} \biggl( \int_{A_{k}}\vert Xu\vert ^{2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int_{A_{k}} \bigl\vert g(x) \bigr\vert ^{q}\,dx \biggr)^{\frac {1}{q}} \\ \le& \frac{1}{4\mu_{2}} \int_{A_{k}}\vert Xu\vert ^{2}\,dx+C\vert A_{k}\vert ^{\frac{Q+2}{Q}-\frac {2}{q}} \biggl( \int_{A_{k}} \bigl\vert g(x) \bigr\vert ^{q}\,dx \biggr)^{\frac{2}{q}}. \end{aligned}$$
(3.15)
To estimate \(K_{1}\), by the Sobolev inequality in Lemma 2.1 we deduce
$$ K_{1}\le C |A_{k}|^{\gamma\kappa_{1}} \biggl( \int_{A_{k}}|Xu|^{2}\,dx \biggr)^{\frac{\gamma}{2}-1} \int_{A_{k}}|Xu|^{2}\,dx $$
(3.16)
with \(\kappa_{1}=(\frac{1}{\gamma}-\frac{1}{2})+\frac{1}{Q}\ge0\) and \(\frac{\gamma}{2}-1>0\). Now we put (3.14), (3.15), and (3.16) into (3.13), then by using Lebesgue’s absolute continuity on the integral domain and choosing a suitable large \(k>0\) we derive
$$C |A_{k}|^{\gamma\kappa} \biggl( \int_{B_{R}}|Xu|^{2}\,dx \biggr)^{\frac {\gamma}{2}-1}\le \frac{1}{4}, $$
which implies
$$\int_{A_{k}} |Xu|^{2}\,dx \le C(\mu, \mu_{1}, \mu_{2})M|A_{k}|^{\varrho}, $$
where \(\varrho=\min\{\frac{p-2}{p},\frac{Q+2}{Q}-\frac{2}{q}\} \) and \(M=(\|{\mathbf{f}(x)}\|_{L^{p}}+\|g(x)\|_{L^{q}})^{2}\).
Considering \(p>Q\) and \(q>\frac{pQ}{Q+p}\), it yields \(\frac {Q+2}{Q}-\frac{2}{q}>\frac{Q+2}{Q}-\frac{2}{\frac{Qp}{Q+p}}=\frac {p-2}{p}\). Then we have
$$\varrho=\min \biggl\{ \frac{p-2}{p},\frac{Q+2}{Q}-\frac{2}{q} \biggr\} =\frac{p-2}{p}=1-\frac{2}{Q}+\varepsilon_{1}, $$
where \(\varepsilon_{1}=\frac{2}{Q}-\frac{2}{p}>0\). Hence, the boundedness of u on Ω is obtained due to Lemma 3.5. This lemma is proved. □
Proof of Theorem 1.4
Let us prove it in two steps by semilinear setting and quasilinear setting.
Step 1. First let us consider the following semilinear subelliptic equations:
$$ -\sum_{i,j}^{m} X^{*}_{i} \bigl(A_{ij}(x)X_{j}u+a_{i}(x,u) \bigr)=b(x,u,Xu),\quad \mbox{a.e. }x\in\Omega, $$
(3.17)
where \(b(x,u,Xu)\) is under the controllable growth. Our idea is to use the bootstrap argument to improve the integrable index of X-gradient of weak solutions. It is easily seen that \(\sup_{x\in\Omega}|u|\le M\) due to Lemma 3.6. Considering \(b(x,u,Xu)\) satisfies the controllable growth H3, we derive
$$ \left \{ \textstyle\begin{array}{l} |a_{i}(x,u) |\leq\mu_{1} (M^{\frac{\gamma }{2}}+f_{i}(x) ) \in L^{p}(\Omega), \\ |b(x,u,Xu) | \leq\mu_{2} (|Xu|^{2(1-\frac{1}{\gamma })}+M^{\gamma-1}+g(x) ). \end{array}\displaystyle \right . $$
(3.18)
Setting \(\chi=2(1-\frac{1}{\gamma})\), by Lemma 2.3 it follows that there exists an integrable index \(p_{0}>2\) such that \(Xu\in L^{p_{0}}_{\mathrm{loc}}(\Omega)\), which implies
$$b(x,u,Xu)\in L^{q_{1}}_{\mathrm{loc}}(\Omega),\qquad q_{1}= \min \biggl\{ \frac {p_{0}}{\chi},q \biggr\} . $$
Thanks to Lemma 3.3, it yields
$$ Xu\in L^{r_{1}} \bigl(\Omega' \bigr), \qquad r_{1}= \min \bigl\{ p,q^{*}_{1} \bigr\} ,\quad \mbox{for any }\Omega' \subset\subset\Omega. $$
(3.19)
(i) If \(q\leq\frac{p_{0}}{\chi}\), then \(q_{1}=q\) and \(r_{1}=r=\min \{p,q^{*}\}\). Thus, Theorem 1.4 is proved.
(ii) If \(q>\frac{p_{0}}{\chi}\), then \(q_{1}=\frac{p_{0}}{\chi}\), and
$$ q^{*}_{1}= \textstyle\begin{cases} \frac{Qp_{0}}{Q\chi-p_{0}}, & \frac{p_{0}}{\chi} < Q, \\ q^{*}_{1}>q_{1}, & \frac{p_{0}}{\chi} \geq Q. \end{cases} $$
(3.20)
If now \(\frac{p_{0}}{\chi}\geq Q\), then \(r_{1}=\min\{p,q^{*}\}\), Thus, Theorem 1.4 holds again.
If instead \(\frac{p_{0}}{\chi}< Q\), then \(q_{1}^{*}=\frac {Qp_{0}}{Q\chi-p_{0}}< q^{*}\), so \(r_{1}=\min\{p,\frac{Qp_{0}}{Q\chi -p_{0}}\}\). For the case of \(p\le\frac{Qp_{0}}{Q\chi-p_{0}}\), we can also obtain Theorem 1.4. For the other case with \(p> \frac{Qp_{0}}{Q\chi-p_{0}}\), we have \(Xu\in L^{\frac{Qp_{0}}{Q\chi -p_{0}}}_{\mathrm{loc}}(\Omega')\), namely, \(|Xu|^{\chi}\in L_{\mathrm{loc}}^{\frac {Qp_{0}}{(Q\chi-p_{0})\chi}}(\Omega')\). Again using the controllable growth (3.18), we have
$$b(x,u,Xu)\in L_{\mathrm{loc}}^{q_{2}}(\Omega),\qquad q_{2}= \min \biggl\{ \frac {Qp_{0}}{(Q\chi-p_{0})\chi},q \biggr\} \ge q_{1}. $$
Thus, by Lemma 3.3 it follows that
$$Xu\in L^{r_{2}}_{\mathrm{loc}} \bigl(\Omega' \bigr), \qquad r_{2}= \min \bigl\{ p,q^{*}_{2} \bigr\} \ge r_{1}. $$
Iterating the above procedure we can arrive at \(Xu\in L^{r}(\Omega') \) with \(r=\min\{p,q^{*}\}\) after finite steps. This is because the integral index of Xu is improved by a fixed step length χ. This completes Step 1.
Step 2. We now consider quasilinear subelliptic equation (1.1) under the controllable growth H3. For any \(x_{0}\in\Omega\), let \(B_{R}=B_{R}(x_{0})\subset\Omega\) be a ball centered at \(x_{0}\) with radii R in Ω. We set \(\bar{u}_{R}=\fint_{B_{R}}u\,dx\), then equation (1.1) can be rewritten as
$$ -X^{*}_{i} \bigl(A_{ij}(x,\bar{u}_{R})X_{j}u \bigr)=X^{*}_{i} \bigl(a_{i}(x,u)- \bigl(A_{ij}(x, \bar{u}_{R})+A_{ij}(x,u) \bigr)X_{j}u \bigr)+b(x,u,Xu), \quad x\in B_{R}. $$
By Lemma 3.6 it implies that \(\sup_{x\in\Omega}|u|\le M\). Combining the controllable growth (3.18) and the \(L^{p}\)-estimates of semilinear subelliptic equations in the step 1 above, we conclude that for \(r=\min \{p,q^{*}\}\) we have
$$\begin{aligned} \Vert Xu\Vert _{L^{r}(B_{\frac{R}{2}})} \le& C \bigl( \bigl\Vert a_{i}(x,u)- \bigl(A_{ij}(x,\bar{u}_{R})-A_{ij}(x,u) \bigr)Xu \bigr\Vert _{L^{r}(B_{R})} \\ &{}+ \bigl\Vert b(x,u,Xu) \bigr\Vert _{L^{q}(B_{R})}+\Vert u\Vert _{L^{2}(B_{R})} \bigr) \\ \le& C \bigl( \bigl\Vert {\mathbf{f}(x)} \bigr\Vert _{L^{p}(B_{R})}+ \bigl\Vert g(x) \bigr\Vert _{L^{q}(B_{R})}+\Vert u\Vert _{L^{2}(B_{R})} \bigr) \\ &{}+C\sup_{x\in B_{R}} \bigl\vert A_{ij}(x, \bar{u}_{R})-A_{ij}(x,u) \bigr\vert \Vert Xu\Vert _{L^{r}(B_{R})} \\ \le& C \bigl( \bigl\Vert {\mathbf{f}(x)} \bigr\Vert _{L^{p}(B_{R})}+ \bigl\Vert g(x) \bigr\Vert _{L^{q}(B_{R})}+\Vert u\Vert _{L^{2}(B_{R})} \bigr)+C\omega \bigl(\vert u-\bar{u}_{R}\vert \bigr)\Vert Xu \Vert _{L^{r}(B_{R})} \\ \le& C \bigl( \bigl\Vert {\mathbf{f}(x)} \bigr\Vert _{L^{p}(\Omega)}+ \bigl\Vert g(x) \bigr\Vert _{L^{q}(\Omega )}+\Vert u \Vert _{L^{2}(\Omega)} \bigr)+\vartheta \Vert Xu\Vert _{L^{r}(B_{R})}, \end{aligned}$$
where we used the uniform continuity H2 on \(A_{ij}(x,u)\) with respect to u in the second last step. If we choose a suitable small \(R>0\) such that the continuity modulus \(\omega(\cdot)\) satisfying \(C\omega (|u-u_{R}|)\le\vartheta<1\), then by employing a standard iteration argument we get
$$ \Vert Xu\Vert _{L^{r}(B_{\frac{R}{2}})}\le C \bigl( \bigl\Vert { \mathbf{f}(x)} \bigr\Vert _{L^{p}(\Omega)}+ \bigl\Vert g(x) \bigr\Vert _{L^{q}(\Omega)}+\Vert u\Vert _{L^{2}(\Omega)} \bigr). $$
(3.21)
In fact, let us denote
$$ r_{0}=\frac{1}{2}R,\qquad r_{k}= \Biggl( \frac{1}{2}+\sum^{k}_{l=1} \frac {1}{2^{l+1}} \Biggr)R,\qquad B^{(k)}=B_{r_{k}},\quad k=1,2,\ldots, $$
and set
$$ A_{k}=\Vert Xu\Vert _{L^{r}(B^{(k)})},\qquad B= \bigl( \bigl\Vert {\mathbf{f}(x)} \bigr\Vert _{L^{p}(\Omega)}+ \bigl\Vert g(x) \bigr\Vert _{L^{q}(\Omega)}+\Vert u\Vert _{L^{2}(\Omega)} \bigr). $$
Then
$$ A_{k}\leq\theta A_{k+1}+CB. $$
Now, multiplying both sides by \(\theta^{k}\) with \(\theta\in(0,1)\), and summing up with respect to k, it follows that
$$ \sum^{\infty}_{k=0}\theta^{k}A_{k}= \sum^{\infty}_{k=1}\theta ^{k}A_{k}+ \sum^{\infty}_{k=0}\theta^{k}CB. $$
Since \(\theta<1\), the summation \(\sum^{\infty}_{k=0}\theta^{k}\) is finite. Therefore,
which implies (3.21). Theorem 1.4 is completely proved. □