This section will proceed to the proof of Theorem 1 by Lemmas 8-9. We first give a linearization strategy for nonlinear systems.
Lemma 8
We claim that if
\(\rho\leq\rho_{1}^{\frac {m}{(2-m)(m-1)}}(|u_{0}|,|p_{0}|)\)
and
\(\varphi\in C_{0}^{\infty}(B_{\rho}(\xi_{0}),{ \mathbb{R}}^{N})\)
with
\(\sup_{B_{\rho}(\xi_{0})} \vert {X\varphi } \vert \leq1\), then there exist some constants
\(C_{1}=C_{1}(L,m,M,C_{P},K)>1\)
such that
$$\begin{aligned}& \fint_{B_{\rho}(\xi _{0})}{A_{i,p_{\beta}^{j}}^{\alpha}(\xi _{0},u_{0},p_{0}) (Xu-p_{0})X \varphi^{\alpha}} \,d\xi \\& \quad \leq C_{1} \sup_{B_{\rho}(\xi_{0})} \vert {X\varphi } \vert \bigl[ {\omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert ,\Phi^{\frac{1}{2}}(\xi_{0},\rho,p_{0})} \bigr) \Phi ^{\frac{1}{2}}(\xi_{0},\rho,p_{0})} \\& \qquad {}+ { \Phi( \xi_{0},\rho,p_{0})+\mu (\sqrt{\rho}) F \bigl( \vert {u_{0}} \vert , \vert {p_{0}} \vert \bigr)} \bigr] , \end{aligned}$$
(4.1)
where we denote
\(F(s,t )=K^{4/(2-m)}(s+t)(2+t)^{2}+(1+s+t)^{r-1}\).
Proof
Noting the fact
$$\begin{aligned}& \int_{B_{\rho}(\xi_{0} )} \biggl[ \int_{0}^{1} A_{i,p_{\beta}^{j} }^{\alpha}\bigl(\xi_{0} ,u_{0} ,\theta Xu + (1 - \theta)p_{0} \bigr) (Xu - p_{0} ) \,d\theta \biggr]X \varphi^{\alpha}\,d\xi \\& \quad = \int_{B_{\rho}(\xi_{0} )} \bigl[A_{i}^{\alpha}( \xi_{0} ,u_{0} ,Xu) - A_{i}^{\alpha}( \xi _{0} ,u_{0} ,p_{0} ) \bigr]X \varphi^{\alpha}\,d\xi \\& \quad = \int_{B_{\rho}(\xi_{0} )} \bigl[A_{i}^{\alpha}( \xi_{0} ,u_{0} ,Xu) - A_{i}^{\alpha}( \xi ,u,Xu) \bigr]X\varphi^{\alpha}\,d\xi \\& \qquad {}+ \int_{B_{\rho}(\xi_{0} )} B^{\alpha} (\xi,u,Xu)\varphi^{\alpha}\,d\xi, \end{aligned}$$
(4.2)
we have
$$\begin{aligned}& \int_{B_{\rho}(\xi_{0})}{A_{i,p_{\beta}^{j}}^{\alpha}(\xi _{0},u_{0},p_{0}) (Xu-p_{0})X \varphi^{\alpha}} \,d\xi \\& \quad = \int_{B_{\rho}(\xi_{0} )} { \biggl[ { \int_{0}^{1} {A_{i,p_{\beta}^{j} }^{\alpha}( \xi_{0} ,u_{0} ,p_{0} )} \,d\theta(Xu - p_{0} )} \biggr]} X\varphi ^{\alpha}\,d\xi \\& \quad \leq \int_{B_{\rho}(\xi_{0})} \biggl[{ \int_{0}^{1}} \bigl\vert {{{{ A_{i,p_{\beta}^{j}}^{\alpha}(\xi_{0},u_{0},p_{0})-A_{i,p_{\beta }^{j}}^{\alpha} \bigl(\xi_{0},u_{0},\theta Xu+(1-\theta)p_{0} \bigr)}}}} \bigr\vert \vert {{{Xu-p_{0}}}} \vert { \,d\theta} \biggr] \\& \qquad {}\times\sup_{B_{\rho}(\xi_{0})} \vert {X\varphi} \vert \,d\xi \\& \qquad {} + \int_{B_{\rho}(\xi_{0})}{ \bigl\vert {A_{i}^{\alpha}(\xi _{0},u_{0},Xu)-A_{i}^{\alpha} \bigl( \xi,u_{0}+p_{0} \bigl(\xi^{1}-\xi _{0}^{1} \bigr),Xu \bigr)} \bigr\vert } \sup _{B_{\rho}(\xi_{0})} \vert {X\varphi} \vert \,d\xi \\& \qquad {} + \int_{B_{\rho}(\xi_{0})}{ \bigl\vert {A_{i}^{\alpha} \bigl(\xi ,u_{0}+p_{0} \bigl(\xi^{1}- \xi_{0}^{1} \bigr),Xu \bigr)-A_{i}^{\alpha}( \xi,u,Xu)} \bigr\vert } \sup_{B_{\rho}(\xi_{0})} \vert {X\varphi} \vert \,d\xi \\& \qquad {} +C \int_{B_{\rho}(\xi_{0})}{ \bigl( \vert p \vert ^{m(1 - \frac{1}{r})} + \vert u \vert ^{r - 1} + 1 \bigr) \vert \varphi \vert } \,d\xi \\& \quad := I^{\prime}+\mathit{II}^{\prime}+\mathit{III}^{\prime}+ \mathit{IV}^{\prime}. \end{aligned}$$
(4.3)
Using (H1) and estimate (1.5) yields (note that \(m-2<0\))
$$\begin{aligned} I^{\prime} \leqslant&C(L) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \int _{B_{\rho}(\xi_{0})} \biggl\{ { \int_{0}^{1}{ \bigl[ { \bigl( {1+ \vert {p_{0}} \vert ^{2}} \bigr) ^{\frac{{m-2}}{2}}+ \bigl( {1+ \bigl\vert {p_{0}+\theta(Xu-p_{0})} \bigr\vert ^{2}} \bigr) ^{\frac {{m-2}}{2}}} \bigr] ^{\frac{1}{2}}}} \\ &{} \times \bigl[ { \bigl( {1+ \vert {p_{0}} \vert ^{2}+ \bigl\vert {p_{0}+\theta(Xu-p_{0})} \bigr\vert ^{2}} \bigr) ^{\frac{{m-2}}{2}}\omega \bigl( { \vert {p_{0}} \vert , \bigl\vert {\theta (Xu-p_{0})} \bigr\vert } \bigr) } \bigr] ^{\frac{1}{2}}\,d\theta \biggr\} \vert {Xu-p_{0}} \vert \,d\xi \\ \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \int _{B_{\rho}(\xi_{0})}{ \bigl( {1+ \vert {Xu-p_{0}} \vert ^{2}} \bigr) }^{\frac{{m-2}}{4}}\omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert , \vert {Xu-p_{0}} \vert } \bigr) \vert {Xu-p_{0}} \vert \,d\xi \\ \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \int _{B_{\rho}(\xi_{0})}{ \bigl( {1+ \vert {Xu-p_{0}} \vert ^{\frac{{m-2}}{2}}} \bigr) } \vert {Xu-p_{0}} \vert \omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert , \vert { Xu-p_{0}} \vert } \bigr) \,d\xi. \end{aligned}$$
(4.4)
Let
$$ B_{1}=:B_{\rho}(\xi_{0})\cap \bigl\{ { \vert {Xu-p_{0}} \vert \leq1} \bigr\} ,\qquad B_{2}=:B_{\rho}( \xi_{0})\cap \bigl\{ { \vert {Xu-p_{0}} \vert >1} \bigr\} . $$
(4.5)
Then it follows that, by first Hölder’s inequality and then Jensen’s inequality, we have
$$\begin{aligned} I^{\prime} \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \biggl[ \int_{B_{1}}{ \vert {Xu-p_{0}} \vert }\omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert , \vert { Xu-p_{0}} \vert } \bigr) \,d\xi \\ &{}+ \int_{B_{2}}{ \vert {Xu-p_{0}} \vert ^{\frac{m}{2}}}\omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert , \vert {Xu-p_{0}} \vert } \bigr) \,d\xi \biggr] \\ \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi | \biggl( { \int_{B_{1}}{ \vert {Xu-p_{0}} \vert ^{2}}\,d\xi } \biggr) ^{\frac{1}{2}} \biggl( { \int_{B_{\rho}(\xi_{0})}{\omega \bigl( { \vert {p_{0}} \vert , \vert {Xu-p_{0}} \vert } \bigr) \,d \xi}} \biggr) ^{\frac{1}{2}} \\ &{} +C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \biggl( { \int _{B_{2}}{ \vert {Xu-p_{0}} \vert ^{m}}\,d\xi} \biggr) ^{\frac{1}{2}} \biggl( { \int_{B_{\rho}(\xi_{0})}{\omega \bigl( { \vert {p_{0}} \vert , \vert {Xu-p_{0}} \vert } \bigr) \,d\xi }} \biggr) ^{\frac{1}{2}} \\ \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi | \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G} \biggl( {\fint_{B_{\rho}(\xi _{0})}{ \bigl\vert {V(Xu)-V(p_{0})} \bigr\vert ^{2}}\,d\xi} \biggr) ^{\frac {1}{2}} \\ &{}\times\biggl( { \fint_{B_{\rho}(\xi_{0})}{\omega \bigl( { \vert {p_{0}} \vert , \vert {Xu-p_{0}} \vert } \bigr) \,d\xi}} \biggr) ^{ \frac{1}{2}} \\ \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi | \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G} \Phi^{\frac{1}{2}}(\xi_{0},\rho,p_{0}) \\ &{}\times\omega^{\frac{1}{2}}\biggl( { \vert {p_{0}} \vert , \biggl( { \fint_{B_{1}}{ \vert {Xu-p_{0}} \vert ^{2}\,d\xi}} \biggr) ^{\frac{1}{2}}} { + \biggl( { \int_{B_{2}}{ \vert {Xu-p_{0}} \vert ^{m}\,d\xi}} \biggr) ^{\frac{1}{m}}} \biggr) \\ \le& C(L,m,M) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G} \Phi^{\frac{1}{2}}(\xi _{0},\rho,p_{0}) \omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert , \Phi^{\frac{1}{2}}(\xi_{0},\rho,p_{0})} \bigr), \end{aligned}$$
(4.6)
where we have used estimates (2.10) and (2.11) in the last inequality.
By employing (H3), estimates (2.10) and (2.11), Young’s inequality, and noting the fact that \(K(\cdot)\) is monotone nondecreasing and \(K(\cdot)>1\) and that \(\rho\leq1\), we deduce
$$\begin{aligned} \mathit{II}^{\prime} \leqslant& \int_{B_{\rho}(\xi_{0})}{K(\cdot) \mu(\rho ) \bigl(1+ \vert {p_{0}} \vert \bigr) \bigl(1+ \vert {Xu} \vert \bigr)^{\frac {m}{2}}\,d\xi} \\ \leqslant& K(\cdot) \mu(\rho) \bigl(1+ \vert {p_{0}} \vert \bigr)^{1 +\frac{m}{2}} \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G}+ \int_{B_{1 }+B_{2}}{K(\cdot) \mu(\rho) \bigl(1+ \vert {p_{0}} \vert \bigr) \vert {Xu-p_{0}} \vert ^{\frac{m}{2}}\,d\xi} \\ \leqslant& K(\cdot) \mu(\rho) \bigl(1+ \vert {p_{0}} \vert \bigr)^{1 +\frac{m}{2}} \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G}+ \bigl[ {K(\cdot) \mu(\rho) \bigl(1+ \vert {p_{0}} \vert \bigr)} \bigr] ^{2} \vert {B_{2}} \vert _{G} \\ &{} + \bigl[ {K(\cdot) \mu(\rho) \bigl(1+ \vert {p_{0}} \vert \bigr)} \bigr] ^{\frac{4}{{4-m}}} \vert {B_{1}} \vert _{G} + \int_{B_{2}}{ \vert {Xu-p_{0}} \vert ^{m}\,d\xi}+ \int_{B_{1}}{ \vert {Xu-p_{0}} \vert ^{2}\,d\xi} \\ \leqslant& \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G}\Phi(\xi _{0},\rho,p_{0})+3 \bigl[ {K( \cdot) \bigl(1+ \vert {p_{0}} \vert \bigr)} \bigr] ^{2} \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G}\mu(\rho), \end{aligned}$$
(4.7)
where we have used \(4/(4-m)<2\), \(1+m/2<2\) and \(\mu(\rho)\le1\) for \(\rho \in[0,1]\).
Similarly to (3.23) to estimate \(\mathit{III}^{\prime}\), the domain \(B_{\rho}(\xi_{0})\) is divided into four parts as previously mentioned. Then we obtain that, by Lemma 2(6) and Lemma 5,
$$\begin{aligned} \mathit{III}' \le& \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \int _{B_{\rho}(\xi_{0})} K( \cdot) \bigl( {1 + |Xu|} \bigr)^{m/2} \mu \bigl( {|v|} \bigr) \,d\xi \\ \le& \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \int_{B_{\rho}(\xi _{0})} \biggl[\frac{{|v|}}{\rho}\sqrt{\rho}+ K^{2} ( \cdot) ( {1 + p_{0} } )^{m} \mu ( {\sqrt{\rho}} ) + K^{2} ( \cdot)|Xu - p_{0} |^{m} \mu ( {\sqrt{\rho}} ) \biggr]\,d\xi \\ \le& \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \biggl[ C(m,M) \bigl(C_{P} + K^{2} ( \cdot) \bigr) \int_{B_{\rho}(\xi_{0} )} { \bigl\vert V ( {Xu - p_{0} } ) \bigr\vert ^{2} } \,d\xi \\ &{}+ K^{4/(2 - m)} ( \cdot) ( {2 + p_{0} } )^{m} \mu ( {\sqrt{\rho}} ) \bigl\vert B_{\rho}(\xi_{0} ) \bigr\vert _{G} \biggr] \\ \le& \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \bigl[C(m,M) \bigl(C_{P} + K^{2} ( \cdot) \bigr)\Phi(\xi_{0} ,p_{0} , \rho) \bigl\vert B_{\rho}(\xi_{0} ) \bigr\vert _{G} \\ &{}+ K^{4/(2 - m)} ( \cdot) ( {2 + p_{0} } )^{m} \mu ( {\sqrt{\rho}} ) \bigl\vert B_{\rho}( \xi_{0} ) \bigr\vert _{G} \bigr]. \end{aligned}$$
(4.8)
With the help of the assumptions that \(\sup_{B_{\rho}(\xi_{0} )} \vert \varphi \vert \le\rho\le1\), Hölder’s inequality, Sobolev type inequality and Young’s inequality, we get
$$\begin{aligned} \mathit{IV}' \leq& C \int_{B_{\rho}(\xi_{0} )} { \bigl( { \vert {{Xu} } \vert ^{m(1 - \frac{1}{r})} + \vert u \vert ^{r - 1} + 1} \bigr)} \vert \varphi \vert \,d\xi \\ \leq& C \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert {{Xu} } \vert ^{m} } \,d\xi} \biggr)^{(1 - \frac{1}{r})} \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert \varphi \vert ^{r} } \,d\xi} \biggr)^{\frac{1}{r}} + C\rho \bigl\vert {B_{\rho}( \xi_{0} )} \bigr\vert _{G} \bigl[ {1 + \bigl( { \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} } \bigr] \\ &{}+ C \biggl( { \int_{B_{\rho}(\xi_{0} )} { \bigl\vert {u - u_{0} - p_{0} \bigl(\xi^{1} - \xi_{0}^{1} \bigr)} \bigr\vert ^{r} } \,d\xi} \biggr)^{(1 - \frac{1}{r})} \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert \varphi \vert ^{r} } \,d\xi} \biggr)^{\frac{1}{r}} \\ \leq& C \biggl( { \int_{B_{\rho}(\xi_{0} )} {{ \bigl( \vert {{Xu} } -p_{0} \vert ^{m} + \vert p_{0} \vert ^{m} \bigr)}} \,d \xi} \biggr)^{(1 - \frac{1}{r})} \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert \varphi \vert ^{r} } \,d\xi} \biggr)^{\frac{1}{r}} \\ &{}+ C \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert {Xu - p_{0} } \vert ^{m} } \,d\xi} \biggr)^{\frac{r-1}{m} } \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert \varphi \vert ^{r} } \,d\xi} \biggr)^{\frac{1}{r}} \\ &{}+ C\rho \bigl\vert {B_{\rho}( \xi_{0} )} \bigr\vert _{G} \bigl[ {1 + \bigl( { \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} } \bigr] \\ \le& C \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert {Xu - p_{0} } \vert ^{m} } \,d\xi} \biggr)^{\frac{{r - 1}}{r}} \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert \varphi \vert ^{r} \,d\xi} } \biggr)^{\frac{1}{r}} \\ &{}+ C \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert {p_{0} } \vert ^{m} \,d\xi } } \biggr)^{\frac{{r - 1}}{r}} \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert \varphi \vert ^{r} \,d\xi} } \biggr)^{\frac{1}{r}} \\ &{}+ C \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert {Xu - p_{0} } \vert ^{m} } \,d\xi} \biggr)^{\frac{r-1}{m} } \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert \varphi \vert ^{r} } \,d\xi} \biggr)^{\frac{1}{r}} \\ &{}+ C\rho \bigl\vert {B_{\rho}( \xi_{0} )} \bigr\vert _{G} \bigl[ {1 + \bigl( { \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} } \bigr] \\ \leq& C \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert {{Xu-p_{0}} } \vert ^{m} } \,d\xi} \biggr)^{ \frac{r-1}{r}} \biggl( { \int_{B_{\rho}(\xi_{0} )} { \vert \varphi \vert ^{r} } \,d\xi} \biggr)^{\frac{1}{r}} \\ &{}+ C\rho \bigl\vert {B_{\rho}( \xi_{0} )} \bigr\vert _{G} \bigl[ {1 + \bigl( { \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} + \vert {p_{0} } \vert ^{m(1 - \frac{1}{r})} } \bigr] \\ \leq& C \int_{B_{\rho}(\xi_{0} )} { \vert {Xu - p_{0} } \vert ^{m} } \,d\xi+ C\rho^{r} \bigl\vert {B_{\rho}( \xi_{0} )} \bigr\vert _{G} \\ &{}+ C\rho \bigl\vert {B_{\rho}(\xi_{0} )} \bigr\vert _{G} \bigl[ {1 + \bigl( { \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} + \vert {p_{0} } \vert ^{m(1 - \frac{1}{r})} } \bigr]. \end{aligned}$$
(4.9)
On the case \(B_{1}=:B_{\rho}(\xi_{0})\cap \{ { \vert {Xu-p_{0}} \vert \leq1} \} \), by (2.10) and Young’s inequality, one gets
$$ \vert {Xu-p_{0}} \vert ^{m}\leq \vert {Xu-p_{0}} \vert ^{2}+1\leq \bigl\vert {V ( {Xu} ) -V ( {p_{0}} ) } \bigr\vert ^{2}+1, $$
and then
$$\begin{aligned} \mbox{(4.9)} \leq& C\mu (\sqrt{\rho}) \bigl\vert {B_{\rho}(\xi_{0} )} \bigr\vert \bigl[ { \bigl( {1 + \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} + \bigl( {1 + \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{m} } \bigr] \\ &{}+C \bigl\vert {B_{\rho}(\xi_{0} )} \bigr\vert _{G} \Phi(\xi_{0} ,\rho,p_{0} ). \end{aligned}$$
(4.10)
On the other hand, on \(B_{2}=:B_{\rho}(\xi_{0})\cap \{ { \vert {Xu-p_{0}} \vert >1} \} \), by (2.11) and Young’s inequality, one has
$$\begin{aligned} \mbox{(4.9)} \leq& C\mu (\sqrt{\rho}) \bigl\vert {B_{\rho}(\xi_{0} )} \bigr\vert \bigl[ { \bigl( {1 + \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} + \bigl( {1 + \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{m} } \bigr] \\ &{}+C \bigl\vert {B_{\rho}(\xi_{0} )} \bigr\vert _{G} \Phi(\xi_{0} ,\rho,p_{0} ). \end{aligned}$$
(4.11)
Thus we infer that, by combining these estimates and noting the definition of \(F(s,t)\), we have
$$ \mathit{IV}^{\prime}\leq C\Phi(\xi_{0}, \rho,p_{0}) \bigl\vert {B_{\rho}(\xi_{0} )} \bigr\vert _{G} + CF \bigl( \vert {u_{0} } \vert , \vert {p_{0} } \vert \bigr) \bigl\vert {B_{\rho}( \xi_{0} )} \bigr\vert _{G} \mu (\sqrt{\rho}). $$
(4.12)
Combining the estimates \(I^{\prime}\), \(\mathit{II}^{\prime}\), \(\mathit{III}^{\prime}\) and \(\mathit{IV}^{\prime}\) with (4.3), we immediately conclude (4.1). □
We next establish an initial excess improvement estimate assuming that the excess Φ is initially small enough. Precisely,
Lemma 9
(Excess improvement)
Let
\(u\in HW^{1,m}(\Omega,\mathbb{R}^{N})\)
satisfy the conditions of Theorem
1. Assume that Lemma
4
and the following smallness conditions hold:
$$\begin{aligned}& \omega^{\frac{1}{2}} \bigl( { \vert {u_{\xi_{0} ,\rho} } \vert + \bigl\vert {(Xu)_{\xi_{0} ,\rho} } \bigr\vert ,\Phi^{\frac{1}{2}} \bigl(\xi_{0} ,\rho ,(Xu)_{\xi_{0} ,\rho} \bigr)} \bigr)+ \Phi^{1 / 2} \bigl(\xi_{0} ,\rho,(Xu)_{\xi_{0} ,\rho} \bigr) \le\frac{\delta}{2}, \end{aligned}$$
(4.13)
$$\begin{aligned}& C_{2}F^{2} \bigl(\vert u_{\xi_{0} , \rho} \vert,\bigl\vert (Xu)_{\xi_{0} , \rho}\bigr\vert \bigr)\mu (\sqrt{\rho} )\leq \delta^{2}, \end{aligned}$$
(4.14)
with
\(C_{2}=8C_{1}^{2}C_{4}\), together with the condition
$$ \rho\le\rho^{\frac{m}{(2-m)(m-1)}} _{1} \bigl(1+\vert u_{\xi_{0} ,\rho} \vert, 1+ \bigl\vert (Xu)_{\xi _{0} ,\rho} \bigr\vert \bigr). $$
(4.15)
Then the following growth inequality is valid for
\(\tau\in[\gamma, 1)\):
$$ \Phi \bigl(\xi_{0},\theta\rho,(Xu)_{\xi_{0},\theta\rho} \bigr)\leq\theta ^{2\tau} {\Phi (\rho)+K^{\ast} \bigl(|u_{\xi_{0},\rho}|, \bigl\vert (Xu)_{\xi_{0},\rho} \bigr\vert \bigr) } \mu^{2} \bigl( {\rho}^{\sigma} \bigr), $$
(4.16)
where
\(\sigma=\min\{(2-m)(m-1)/m, (m-1)/2\}\), and
\(K^{\ast }(s,t)=C_{8}F^{\frac{2m}{m-1}}(1+s,M+t) \).
Proof
For simplicity, we will use the abbreviation \(\Phi(\rho)=\Phi(\xi_{0} ,\rho,(Xu)_{\xi_{0} ,\rho} )\) in the sequel. For \(\varepsilon>0\) to be determined later, we take \(\delta\in(0,1)\) and \(\Upsilon\in[0,1]\) to be the corresponding constant from the \(\mathcal{A}\)-harmonic approximation lemma (Lemma 4), and set
$$ w=u-(u_{\xi_{0},\rho}-\Upsilon h_{\xi_{0},2\theta\rho})- ( {Xu} ) _{\xi_{0,}\rho} \bigl(\xi^{1}-\xi_{0}^{1} \bigr) $$
and
$$ \Upsilon={C(m,M)}\Gamma(\rho),\quad \Gamma(\rho )=C_{1}\sqrt{\Phi(\rho)+4 \delta^{-2}\mu^{2} (\sqrt{\rho}) F^{2} \bigl( \vert {u_{\xi_{0},\rho}} \vert , \bigl\vert {(Xu)_{\xi_{0},\rho}} \bigr\vert \bigr)}. $$
Noting (4.13) and (4.14) yields
$$ \omega^{1/2} \bigl( { \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }} \bigr\vert ,\Phi^{1/2} ( \rho ) } \bigr) +\Phi ^{1/2} ( \rho ) \leq \frac{\delta}{2}, $$
(4.17)
and by (4.1), we derive (note the definition of ϒ and \(\Gamma(\rho)\))
$$\begin{aligned}& \biggl\vert {\fint_{B_{\rho}(\xi_{0})}{ \bigl[ {{{ A_{i,p_{\beta}^{j}}^{\alpha }}} \bigl( {\xi_{0},u_{\xi_{0},\rho}, ( {Xu} ) _{\xi_{0},\rho}} \bigr) Xw} \bigr] X_{i}\varphi^{\alpha }\,d \xi}} \biggr\vert \\& \quad \leq \Upsilon\frac{{\omega^{1/2} ( { \vert { ( {Xu} ) _{\xi_{0},\rho}} \vert ,\Phi^{1/2}(\rho )} ) \Phi^{1/2}(\rho)+\Phi(\rho)+\mu (\sqrt{\rho}) F ( \vert {u_{\xi_{0},\rho}} \vert , \vert {(Xu)_{\xi_{0},\rho}} \vert )}}{{C(m,M)\Gamma(\rho)}} \sup _{B_{\rho}(\xi_{0})} \vert {X \varphi} \vert \\& \quad \leq \Upsilon \biggl[ {\omega^{1/2} \bigl( { \bigl\vert { ( {Xu} ) _{\xi_{0},\rho}} \bigr\vert ,\Phi^{1/2}(\rho )} \bigr) + \Phi^{1/2}(\rho)+\frac{\delta}{2}} \biggr] \sup_{B_{\rho}(\xi _{0})} \vert {X\varphi} \vert \\& \quad \leq \Upsilon\delta\sup_{B_{\rho}(\xi _{0})} \vert {X\varphi} \vert . \end{aligned}$$
(4.18)
Then from the definition of ϒ, Lemma 2(6) and (2.9), we have
$$ \fint_{B_{\rho}(\xi_{0})}{ \bigl\vert {W(Xw)} \bigr\vert ^{2}\,d\xi}\leq \fint_{B_{\rho}(\xi _{0})}{ \bigl\vert {V(Xw)} \bigr\vert ^{2}\,d\xi}\leq C(m,M)\Phi(\rho)\leq \Upsilon^{2}. $$
(4.19)
Inequalities (4.18) and (4.19) fulfill the conditions of \(\mathcal{A}\)-harmonic approximation lemma, which ensures than we find an \(\mathcal{A}=A_{i,p_{\beta}^{j} }^{\alpha}(\xi _{0} ,u_{\xi_{0}, \rho} , (Xu)_{\xi_{0}, \rho} )\)-harmonic function \(h\in W^{1,m}(B_{\rho}(\xi_{0}),\mathbb{R}^{N})\) such that
$$ \fint_{B_{\rho}(\xi_{0})}{ \bigl\vert {W(Xh)} \bigr\vert ^{2}\,d\xi}\leq1,\qquad \fint_{B_{\rho}(\xi_{0})}{ \biggl\vert {W \biggl( { \frac{{w-\Upsilon h}}{\rho}} \biggr) } \biggr\vert ^{2}\,d\xi}\leq \Upsilon^{2}\varepsilon. $$
(4.20)
Using Lemma 2(3) and (6), we have
$$\begin{aligned} \Phi(\theta\rho) =& \fint_{B_{\theta\rho}(\xi_{0})}{ \bigl\vert {V(Xu)-V \bigl( { ( {Xu} ) _{\xi_{0,\theta\rho}}} \bigr) } \bigr\vert ^{2}\,d\xi} \\ \leq& C \fint_{B_{\theta\rho}(\xi_{0})}{ \bigl\vert {V \bigl( {Xu- ( {Xu} ) _{\xi_{0,\theta\rho}}} \bigr) } \bigr\vert ^{2}\,d\xi } \\ \leq& C \fint_{B_{\theta\rho}(\xi_{0})}{ \bigl\vert {V \bigl( {Xu- ( {Xu} ) _{\xi_{0, \rho}}-\Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr) } \bigr\vert ^{2}\,d\xi} \\ &{} +C \bigl\vert {V \bigl( { ( {Xu} ) _{\xi_{0,\theta\rho }}-(Xu)_{\xi_{0,}\rho}- \Upsilon(Xh)_{(\xi _{0},2\theta\rho)}} \bigr) } \bigr\vert ^{2}, \end{aligned}$$
(4.21)
where the constant C depends only on m, k and N.
Next, we proceed to estimate the right-hand side of (4.21). Decomposing \(B_{\theta\rho}(\xi_{0})\) into two parts: \(B_{1}=B_{\theta\rho}(\xi_{0})\cap \{ \vert { Xu-(Xu)_{\xi_{0,}\rho}-\Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \vert \leq 1 \}\) and \(B_{2}=B_{\theta\rho}(\xi_{0})\cap \{ \vert {Xu-(Xu)_{\xi_{0,}\rho}-\Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \vert >1 \}\). Then by Lemma 2(1) and Hölder inequality, we obtain
$$\begin{aligned}& \bigl\vert { ( {Xu} ) _{\xi_{0,\theta}\rho}-(Xu)_{\xi _{0,}\rho}- \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert \\& \quad \leq \fint_{B_{\theta\rho}(\xi_{0})}{ \bigl\vert {Xu-(Xu)_{\xi _{0,}\rho}- \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert \,d\xi } \\& \quad \leq \sqrt{2} \fint_{B_{1}}{ \bigl\vert V \bigl({Xu-(Xu)_{\xi _{0,}\rho}- \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr) \bigr\vert \,d\xi} \\& \qquad {}+ \sqrt[m]{2} \fint_{B_{2}}{ \bigl\vert V \bigl({Xu-(Xu)_{\xi _{0,}\rho}- \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr) \bigr\vert ^{2/m} \,d\xi} \\& \quad \leq \sqrt[m]{2} \bigl({\Xi} ^{1/2}+{\Xi}^{1/m} \bigr), \end{aligned}$$
(4.22)
where we have denoted
$$ {\Xi}=: \fint_{B_{\theta\rho}(\xi_{0})}{ \bigl\vert {V \bigl( {Xu-(Xu)_{\xi_{0,}\rho}- \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr) } \bigr\vert ^{2}\,d\xi}. $$
Using Lemma 2(1), we deduce that \(V^{2} (\Xi ^{1/2}+\Xi^{1/m} )\le C(\Xi+\Xi^{2/m}) \). So we get
$$ \Phi(\theta\rho)\leq C \bigl( {\Xi+V^{2} \bigl(\Xi ^{1/2}+\Xi^{1/m} \bigr)} \bigr) \leq C_{3} \bigl( { \Xi+\Xi ^{2/m}} \bigr) , $$
(4.23)
where the constant \(C_{3}\) depends only on m, Q and N. Then it remains for us to estimate Ξ. By considering the cases \(\vert {Xh} \vert \leq1\) and \(\vert {Xh} \vert >1\), separately and keeping in mind (4.20), we obtain
$$ \fint_{B_{\rho}(\xi_{0})}{ \vert {Xh} \vert }\,d\xi\leq2 \sqrt{2} \fint_{B_{\rho}(\xi _{0})}{ \bigl\vert W({Xh}) \bigr\vert ^{2} }\,d\xi\le2\sqrt{2}, $$
(4.24)
where we have used Lemma 2(1).
Note that the smallness conditions (4.13) and (4.14) imply \(C_{4}\Upsilon^{2}\le1\) with \(C_{4}=\max\{ 8C_{0}^{2},(2\theta)^{-Q}\}\), where we have assumed \(\frac {1}{2}C_{1}^{2}C_{4}\delta^{2}\le1\), which is no restriction. Then it follows by applying the priori estimate for constant coefficients sub-elliptic systems (see Lemma 6)
$$ \Upsilon \bigl\vert {(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert \leq \Upsilon\sup_{B_{\rho/2} (\xi_{0} )} \vert {Xh} \vert \le\Upsilon C_{0} \fint_{B_{\rho}(\xi_{0})}{ \vert {Xh} \vert }\,d\xi\le2\sqrt{2}\Upsilon C_{0} \leq1. $$
(4.25)
Caccioppoli type inequality applied on \(B_{2\theta\rho}(\xi_{0})\) with \(u_{0}=u_{\xi_{0},\rho}\), and \(p_{0}=(Xu)_{\xi_{0},\rho}+\Upsilon(Xh)_{\xi _{0},2\theta\rho}\), \(\theta\in(0,1/4]\) yields
$$\begin{aligned} \Xi \leq& C_{c} \biggl[ \bigl\vert B_{2\theta\rho}( \xi _{0}) \bigr\vert _{G}^{-1} \int_{B_{2\theta\rho}(\xi_{0})} \biggl\vert {V \biggl( {\frac {{u-u_{\xi_{0},\rho}- ( { ( {Xu} ) _{\xi_{0},\rho }+\Upsilon (Xh)_{(\xi_{0},2\theta\rho)}} ) (\xi^{1}-\xi _{0}^{1})}}{{2\theta\rho}}} \biggr) } \biggr\vert ^{2}\,d\xi \\ &{}+U \biggr], \end{aligned}$$
(4.26)
where
$$\begin{aligned} U =& \bigl[ {K \bigl( { \vert {u_{\xi_{0},\rho}} \vert + \bigl\vert { ( {Xu} ) _{\xi_{0},\rho}+\Upsilon(Xh)_{(\xi_{0},2\theta \rho)}} \bigr\vert } \bigr) \bigl( {1+ \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }+ \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert } \bigr) } \bigr] ^{2m/(m-1) } \\ &{}\times \mu^{2} \bigl( {(2\theta \rho) }^{(2-m)(m-1)/m} \bigr) \\ &{}+ \bigl( {1+2M + \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }+ \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert } \bigr)^{m/(m-1)^{2} } (2\theta \rho)^{m - 1} \\ &{}+\mu^{2} \bigl( {(2\theta \rho) }^{(2-m)(m-1)/m} \bigr) \biggl[ { \fint_{B_{\rho}(\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{\frac{m}{{m - 1}} ( {1 - \frac{1}{r}} )}. \end{aligned}$$
By Lemma 2(3), one gets
$$\begin{aligned}& \int_{B_{2\theta\rho} (\xi_{0} )} \biggl\vert {V \biggl( {\frac{{u - u_{\xi_{0} ,\rho} - ( { ( {Xu} )_{\xi_{0} ,\rho} + \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} )(\xi^{1} - \xi_{0}^{1} )}}{ {2\theta\rho}}} \biggr)} \biggr\vert ^{2} \,d\xi \\& \quad \leq \int_{B_{2\theta\rho} (\xi_{0} )} \biggl\vert V \biggl( \frac{{u - ( {u_{\xi_{0} ,\rho} - \Upsilon h_{\xi_{0} ,2\theta \rho} } ) - ( {Xu} )_{\xi_{0} ,\rho} (\xi^{1} - \xi _{0}^{1} ) - \Upsilon h(\xi)}}{{2\theta\rho}} \\& \qquad {}+ \frac{{\Upsilon h(\xi) - \Upsilon h_{\xi_{0} ,2\theta\rho} - \Upsilon (Xh)_{(\xi_{0},2\theta\rho)}(\xi^{1} - \xi_{0}^{1} )}}{{2\theta\rho}} \biggr) \biggr\vert ^{2} \,d\xi \\& \quad \leq C \biggl[ \int_{B_{2\theta\rho} (\xi_{0} )} \biggl( \biggl\vert {V \biggl( { \frac{{w - \Upsilon h(\xi)}}{{2\theta\rho}}} \biggr)} \biggr\vert ^{2} \\& \qquad {}+ \biggl\vert {V \biggl( {\Upsilon\frac{{h(\xi) - h_{\xi_{0} ,2\theta\rho } - (Xh)_{(\xi_{0},2\theta\rho)}(\xi^{1} - \xi_{0}^{1} )}}{{2\theta\rho }}} \biggr)} \biggr\vert ^{2} \biggr)\,d\xi \biggr]. \end{aligned}$$
(4.27)
To estimate the right-hand side, we employ (2.9), Lemma 2(2) (note that \(\frac{1}{2\theta}\geq1\)) and (4.20) to infer
$$\begin{aligned} \fint_{B_{2\theta\rho}(\xi_{0})}{ \biggl\vert {V \biggl( {\frac{{w-\Upsilon h(\xi)}}{{2\theta\rho}}} \biggr) } \biggr\vert ^{2}\,d\xi } \le& C(2\theta)^{-Q-2} \fint_{B_{\rho}(\xi_{0})}{ \biggl\vert {W \biggl( {\frac{{w-\Upsilon h(\xi)}}{\rho}} \biggr) } \biggr\vert ^{2}\,d\xi} \\ \le& C(2\theta)^{-Q-2} \Upsilon^{2}\varepsilon. \end{aligned}$$
(4.28)
Using Lemma 2, Sobolev-Poincare type inequality (3.6), Lemma 6, (2.9) and (4.20) leads to
$$\begin{aligned}& \fint_{B_{2\theta\rho} (\xi_{0} )} { \biggl\vert {V \biggl( {\Upsilon \frac{{h(\xi) - h_{\xi_{0} ,2\theta\rho} - (Xh)_{(\xi_{0},2\theta\rho )}(\xi^{1} - \xi_{0}^{1} )}}{{2\theta\rho}}} \biggr)} \biggr\vert ^{2} \,d\xi} \\& \quad \leq C_{P}^{2}\Upsilon^{2} \fint_{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {V \bigl( {{{Xh(\xi) - (Xh)_{(\xi_{0},2\theta\rho)}}} } \bigr)} \bigr\vert ^{2} \,d\xi } \\& \quad \leq C_{P}^{4}(2\theta\rho)^{2} \Upsilon^{2} \fint_{B_{2\theta\rho} (\xi _{0} )} { \bigl\vert {V \bigl( {X^{2}h} \bigr)} \bigr\vert ^{2} \,d\xi} \\& \quad \leq C_{P}^{4}(2\theta\rho)^{2} \Upsilon^{2} \sup_{B_{\rho /2} (\xi_{0} )} \bigl\vert {X^{2} h} \bigr\vert ^{2} \\& \quad \leq C_{0}C_{P}^{4}(2 \theta)^{2}\Upsilon^{2} \fint_{B_{\rho} (\xi_{0} )} { \bigl\vert { ( {Xh} )} \bigr\vert ^{2} \,d\xi} \\& \quad \leq CC_{0}C_{P}^{4}(2 \theta)^{2} \Upsilon^{2} \fint_{B_{\rho} (\xi_{0} )} { \bigl\vert {W ( {Xh} )} \bigr\vert ^{2} \,d\xi} \\& \quad \leq C_{5}\theta^{2} \Upsilon^{2}, \end{aligned}$$
(4.29)
where we denote \(C_{5}=4CC_{0}C_{P}^{4}\).
Using (4.25), we get
$$\begin{aligned} \begin{aligned} &\bigl[ {K \bigl( { \vert {u_{\xi_{0},\rho}} \vert + \bigl\vert { ( {Xu} ) _{\xi_{0},\rho}+\Upsilon(Xh)_{(\xi_{0},2\theta \rho)}} \bigr\vert } \bigr) \bigl( {1+ \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }+ \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert } \bigr) } \bigr] ^{2m/(m-1) } \\ &\qquad {}\times\mu^{2} \bigl( {(2\theta \rho) }^{(2-m)(m-1)/m} \bigr) \\ &\quad \leq \bigl[ {K \bigl( {|u_{\xi_{0},\rho}|+ \bigl\vert { ( {Xu} ) _{\xi _{0},\rho}} \bigr\vert +1} \bigr) \bigl( {2+ \bigl\vert { ( {Xu} ) _{\xi_{0},\rho}} \bigr\vert } \bigr) } \bigr] ^{{2m/(m-1)} } \mu^{2} \bigl( { \rho}^{(2-m)(m-1)/m} \bigr) \\ &\quad \leq F^{2m/(m-1)} \bigl( 1+|u_{\xi_{0},\rho}|,{ \bigl\vert { ( {Xu} ) _{\xi_{0},\rho}} \bigr\vert } \bigr) \mu^{2} \bigl( { \rho}^{(2-m)(m-1)/m} \bigr) \end{aligned} \end{aligned}$$
and
$$\begin{aligned}& \bigl( {1+2M + \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }+\Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert } \bigr)^{m/(m-1)^{2} } (2\theta\rho)^{m - 1} \\& \quad \le \bigl( {2+2M + \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }} \bigr\vert } \bigr)^{m/(m-1)^{2} } \mu^{2} \bigl(\rho^{(m - 1)/2} \bigr). \end{aligned}$$
Applying Sobolev type inequality, we have
$$\begin{aligned}& \biggl[ { \fint_{B_{2\theta\rho} (\xi_{0} )} { \bigl( { \vert Xu \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)} \,d\xi} \biggr]^{m(r - 1)/r(m - 1)} \\& \quad \le \biggl[ {2^{m - 1} \fint_{B_{2\theta\rho} (\xi_{0} )} { \bigl( { \bigl\vert Xu - (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{m} } \bigr)} \,d\xi} \biggr]^{m(r - 1)/r(m - 1)} \\& \qquad {}+ \bigl( {2^{m - 1} \bigl\vert (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{m} } \bigr)^{m(r - 1)/r(m - 1)} \\& \qquad {} + \biggl[ { \fint_{B_{2\theta\rho} (\xi_{0} )} {2^{r - 1} \bigl( { \bigl\vert u - u_{\xi_{0} ,\rho} - (Xu)_{\xi_{0} ,\rho} \bigl(\xi^{1} - \xi_{0}^{1} \bigr) \bigr\vert ^{r} } \bigr)} \,d \xi} \biggr]^{m(r - 1)/r(m - 1)} \\& \qquad {}+ \bigl( {2^{r - 1} \bigl\vert u_{\xi _{0} ,\rho} + (Xu)_{\xi_{0} ,\rho} \bigl(\xi^{1} - \xi_{0}^{1} \bigr) \bigr\vert ^{r} + 1} \bigr)^{m(r - 1)/r(m - 1)} \\& \quad \le \bigl( {2^{m - 1} } \bigr)^{m(r - 1)/r(m - 1)} \biggl[(2\theta )^{ - Q} { \fint_{B_{\rho}(\xi_{0} )} { { \bigl\vert Xu - (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{m} } } \,d\xi} \biggr]^{m(r - 1)/r(m - 1)} \\& \qquad {}+ \bigl( {2^{m - 1} \bigl\vert (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{m} } \bigr)^{m(r - 1)/r(m - 1)} \\& \qquad {} + \bigl( {2^{r - 1} } \bigr)^{m(r - 1)/r(m - 1)} \biggl[(2 \theta)^{ - Q} { \fint_{B_{\rho}(\xi_{0} )} { \bigl\vert Xu - (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{m} } \,d\xi } \biggr]^{(r - 1)/(m - 1)} \\& \qquad {}+ \bigl( {2^{r - 1} \bigl\vert u_{\xi_{0} ,\rho} + (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{r} + 1} \bigr)^{m(r - 1)/r(m - 1)} \\& \quad \le C(m,M,Q) \bigl[ {(2\theta)^{ - Q} \Phi \bigl( \xi_{0} ,\rho,(Xu)_{\xi _{0},\rho} \bigr)} \bigr]^{m(r - 1)/r(m - 1)} \\& \qquad {}+ C(m,Q) \bigl( {1 + \vert u_{\xi_{0} ,\rho} \vert + \bigl\vert (Xu)_{\xi_{0} ,\rho} \bigr\vert } \bigr)^{m(r - 1)/(m - 1)}. \end{aligned}$$
(4.30)
The smallness conditions imply
$$ (2\theta) ^{ - Q} \Phi \bigl(\xi_{0} ,\rho,(Xu)_{\xi_{0},\rho} \bigr) \leq1. $$
And then it follows
$$\begin{aligned}& \biggl[ { \fint_{B_{2\theta\rho} (\xi_{0} )} { \bigl( {|Xu|^{m} + |u|^{r} + 1} \bigr)} \,d\xi} \biggr]^{m(r - 1)/r(m - 1)} \mu^{2} \bigl( {(2\theta \rho) }^{(2-m)(m-1)/m} \bigr) \\& \quad \leq CF^{\frac{m}{m-1}} \bigl(|u_{\xi_{0},\rho }|, \bigl\vert (Xu)_{\xi_{0},\rho} \bigr\vert \bigr)\mu^{2} \bigl( { \rho}^{(2-m)(m-1)/m} \bigr), \end{aligned}$$
where we have used the definition of F in the first step.
Combining all the above estimates with (4.26) and letting \(\varepsilon=\theta^{Q+4}\), we arrive at
$$ \Xi\leq C_{6} \bigl[ {\theta^{2} \Upsilon^{2}+F^{\frac{2m}{m-1}} \bigl(1+|u_{\xi_{0},\rho}|,M+ \bigl\vert (Xu)_{\xi_{0},\rho} \bigr\vert \bigr)\mu^{2} \bigl( {\rho }^{\sigma} \bigr)} \bigr] , $$
(4.31)
where \(\sigma=\min\{(2-m)(m-1)/m, (m-1)/2\}\), \(C_{6}\) depends only on Q, N, m, M, L, λ and \(C_{P}\). For given \(\tau\in[\gamma,1)\), choosing \(\theta\in (0,\frac{1}{4}) \) suitable such that \(C_{3}C_{6}\theta^{2}\leq \theta^{2\tau}\), we easily find (note the definition of ϒ)
$$\begin{aligned} \Phi(\theta\rho) \leq& \theta^{2\tau} \bigl[ {\Phi ( \rho)+C_{7}F^{\frac{2m}{m-1}} \bigl(1+ \vert u_{\xi_{0},\rho} \vert ,M+ \bigl\vert (Xu)_{\xi_{0},\rho } \bigr\vert \bigr) } \mu^{2} \bigl( {\rho}^{\sigma} \bigr) \bigr] \\ :=& \theta^{2\tau} {\Phi (\rho)+K^{\ast} \bigl( \vert u_{\xi_{0},\rho} \vert , \bigl\vert (Xu)_{\xi_{0},\rho} \bigr\vert \bigr) }\mu^{2} \bigl( {\rho}^{\sigma} \bigr), \end{aligned}$$
(4.32)
where we have used that \((2-m)(m-1)/m \le3-2\sqrt{2}<1/2\), \(K^{\ast }(s,t)=C_{7}F^{\frac{2m}{m-1}}(1+s,M+t) \), and the constant \(C_{7}\) has the same dependencies as \(C_{6}\). □
For \(T > 0\), we find \(\Phi_{0} (T) > 0\) (depending on Q, N, λ, L, τ and ω) such that
$$\begin{aligned}& \omega^{\frac{1}{2}} \bigl(2T,2\Phi_{0}^{\frac{1}{2}} (T) \bigr) + 2\Phi_{0}^{\frac{1}{2}} (T) \le \frac{1}{2} \delta, \quad \mbox{and} \end{aligned}$$
(4.33)
$$\begin{aligned}& 2(1+\sqrt{C_{p}}) \sqrt{\Phi_{0} (T)} \le \theta^{Q/2} \bigl(1 - \theta^{\tau}\bigr)T. \end{aligned}$$
(4.34)
With \(\Phi_{0} (T)\) from (4.33) and (4.34), we choose \(\rho_{0} (T) \in(0,1]\) (depending on Q, N, λ, L, τ, ω, η and κ) such that
$$\begin{aligned}& \rho_{0} (T) \le\rho^{(2-m)(m-1)/m} _{1} (1 + 2T,1 + 2T), \end{aligned}$$
(4.35)
$$\begin{aligned}& C_{2} F^{2} (2T,2T)\mu^{2} \bigl( \rho_{0} (T) \bigr) \le\delta^{2}, \end{aligned}$$
(4.36)
$$\begin{aligned}& K_{0} (T)\mu^{2} \bigl(\rho_{0} (T) ^{\sigma} \bigr) \le \bigl(\theta^{2\gamma} - \theta ^{2\tau } \bigr)\Phi_{0} (T), \quad \mbox{and} \end{aligned}$$
(4.37)
$$\begin{aligned}& 2(1 + C_{p} )K_{0} (T)H \bigl( \rho_{0} (T)^{2} \bigr) \le\theta^{Q} \bigl(1 - \theta^{\gamma}\bigr)^{2} \bigl(\theta^{2\gamma} - \theta^{2\tau} \bigr)T^{2}, \end{aligned}$$
(4.38)
where \(K_{0} (T): = K^{*}(2T,2T)\).
The rest of the process to obtain Theorem 1 is very similar to [13]. We omit it here.