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Partial interior regularity for sub-elliptic systems with Dini continuous coefficients in Carnot groups: the sub-quadratic controllable case
- Dongni Liao1,
- Jialin Wang1Email author,
- Qiang Yang1 and
- Shimin Wu2
- Received: 12 August 2017
- Accepted: 5 October 2017
- Published: 16 October 2017
Abstract
We consider nonlinear sub-elliptic systems with Dini continuous coefficients for the case \(1< m<2\) in Carnot groups and prove a \(C^{1}\)-partial regularity result for weak solutions under the controllable growth conditions. Our method of proof for sub-elliptic systems is based on a generalization of the technique of \(\mathcal {A}\)-harmonic approximation. It is interesting to point out that our result is optimal in the sense that in the case of Hölder continuous coefficients we get directly the optimal Hölder exponent on its regular set.
Keywords
- nonlinear sub-elliptic system
- Dini continuous coefficients
- sub-quadratic controllable growth
- optimal partial regularity
- Carnot groups
MSC
- 35H20
- 35B65
1 Introduction and the main result
In this paper, we are focused on the nonlinear sub-elliptic systems involving sub-quadratic (\(1< m<2\)) controllable growth terms in divergence form in Carnot groups G, where Ω is a bounded domain in G, \(X = \{X_{1} , \ldots,X_{k} \}\) with \(X_{i}\) (\(i = 1, \ldots,k\)) to see (2.1) below, \(u: \Omega\to{ \mathbb{R}}^{N}\), \(A_{i}^{\alpha}:\Omega\times{ \mathbb{R}}^{N}\times{\mathbb{R}}^{k\times N} \to{\mathbb{ R}}^{k\times N}\), and \(B^{\alpha}:\Omega\times{ \mathbb{R}}^{N}\times{\mathbb{R}}^{k\times N} \to{ \mathbb{R}}^{N}\).
$$ - \sum_{i = 1}^{k} {X_{i} A_{i}^{\alpha}(\xi,u,Xu)} = B^{\alpha}(\xi,u,Xu), \quad \xi\in\Omega, \alpha= 1, \ldots,N, $$
(1.1)
The aim is to weaken assumptions on coefficients \(A_{i}^{\alpha}\) with Hölder continuity in the variables \((\xi,u)\) to the assumptions of Dini continuity, and to show a partial regularity result with optimal estimates for the modulus of continuity for the horizontal derivative Xu; see [1, 2] for the case of sub-elliptic systems with Hölder continuous coefficients.
As is well known, even under reasonable assumptions on the coefficients \(A_{i}^{\alpha}\), \(B^{\alpha}\) in the systems, people cannot in general expect that weak solutions of nonlinear elliptic systems of equations will be classical (i.e., \(C^{2}\)-solutions) like elliptic scalar equations; see [3] by De Giorgi. Then the goal is to establish partial regularity of weak solutions for systems; see monographs [4–6] for more details. Moreover, Duzaar and Steffen in [7] introduced a new method called \(\mathcal {A}\)-harmonic approximation technique, which was then simplified by Duzaar and Grotowski in [8], to investigate elliptic systems with quadratic growth case. This technique avoids proving a reverse Hölder inequality. Later, many results have been obtained for more general coefficients \(A_{i}^{\alpha}\) or \(B^{\alpha}\) in the Euclidean setting; see [9–12] for Hölder continuous coefficients, and [13–15] for Dini continuous coefficients.
With respect to sub-elliptic systems in Carnot groups, there are some new difficulties and challenges which remain due to the lack of commutation and homogeneity of the horizontal vector fields. We list several results for sub-elliptic systems with Hölder continuous coefficients. Capogna and Garofalo [16], and Shores [17] considered the quadratic growth case, whereas Föglein [18], Wang and Niu [1], and Wang and Liao [2] treated non-quadratic cases. It is remarkable that Zheng and Feng [19] showed everywhere regularity for weak solutions of sub-elliptic p-harmonic systems while p is closed to 2 in Carnot groups. We also refer readers to [20–25] for regularity of weak solutions and to [26–30] for other interesting topics such as variational principle, uniqueness, existence and nonexistence of solutions to sub-elliptic equations on Carnot groups.
Recently, Wang and Liao [31] considered sub-elliptic systems with Dini continuous coefficients under super-quadratic conditions (\(m\geq2\)). In this paper, we treat the sub-quadratic case (\(1< m< 2\)). It is worth mentioning that the key point is to establish a certain excess-decay estimate for the excess functional Φ. In the case \(m\geq2\), this functional is given by where we have used the notation \(\fint_{B_{r}(\xi_{0})}u(\xi) \,d\xi= \frac {1}{|B_{r}(\xi_{0})|_{G}}\int_{B_{r}(\xi_{0})}u(\xi) \,d\xi\). However, in the sub-quadratic case \(1< m<2\), one should establish the excess-decay estimate for the following functional: where \(V(A)=(1+|A|^{2})^{\frac{4}{m-2}}\) for \(A\in\mathbb{R}^{k\times N}\), which itself leads to the continuity of the horizontal gradient Xu of the weak solution via the integral characterization of continuity by Campanato. For \(1< m<2\), we first have to generalize the approximation lemma directly (see Lemma 4 below), and the proof requires a Sobolev-Poincaré type inequality related to the functions V (see Lemma 5 below).
$$ \Phi(\xi_{0} ,\rho,p_{0} ) = \fint_{B_{\rho}(\xi_{0} )} \bigl[ \vert {Xu - p_{0} } \vert ^{2} + \vert {Xu - p_{0} } \vert ^{m} \bigr] \, d\xi, $$
$$ \Phi(\xi_{0} ,\rho,p_{0} ) = \fint_{B_{\rho}(\xi_{0} )} { \bigl\vert {V(Xu) - V(p_{0}) } \bigr\vert ^{2} } \,d\xi, $$
(1.2)
In the sequel, let \(\Omega\subset{G}\) be a bounded domain in Carnot groups G with general step and consider weak solutions of the sub-elliptic systems (1.1), i.e., with sub-quadratic controllable structure conditions (H1)-(H4), where:
$$ \int_{\Omega}{A_{i}^{\alpha}(\xi,u,Xu)} X_{i} \varphi^{\alpha}\,d\xi= \int_{\Omega}{B^{\alpha}(\xi,u,Xu)} \varphi^{\alpha}\,d\xi,\quad \forall \varphi\in C_{0}^{\infty}\bigl(\Omega,{ \mathbb{R}}^{N} \bigr) $$
(1.3)
(H1): \(A_{i}^{\alpha}(\xi ,u,p)\) is differentiable with respect to p, with bounded and continuous derivatives, that is, there exists a constant C such that where we denote by \(A_{i,p_{\beta}^{j} }^{\alpha}( \cdot) = \frac{\partial A_{i}^{\alpha}( \cdot)}{\partial p_{\beta}^{j} }\). Furthermore, (H1) implies that there exists a continuously nonnegative and bounded function \(\omega(s,t): [0,\infty)\times[0,\infty) \to[0,\infty)\), where \(\omega(s,0) = 0\) for all s, and \(\omega(s,t)\) is monotonously nondecreasing in s for fixed t; \(\omega(s,t)\) is concave and monotonously nondecreasing in t for fixed s such that
$$ \bigl\vert {A_{i,p_{\beta}^{j} }^{\alpha}(\xi,u,p)} \bigr\vert \le C \bigl(1 + \vert p \vert ^{2} \bigr)^{\frac{m - 2}{2}}, \quad 1< m< 2, $$
(1.4)
$$ \bigl\vert {A_{i,p_{\beta}^{j} }^{\alpha}(\xi,u,p) - A_{i,p_{\beta}^{j} }^{\alpha}(\xi,u,\tilde{p})} \bigr\vert \le C \bigl(1 + \vert p \vert ^{2} + \vert \tilde{p} \vert ^{2} \bigr)^{\frac{m - 2}{2}}\omega \bigl( \vert p \vert , \vert {p - \tilde{p}} \vert \bigr). $$
(1.5)
(H2): \(A_{i}^{\alpha}(\xi,u,p)\) satisfies the following ellipticity condition: where λ is a positive constant.
$$ A_{i,p_{\beta}^{j} }^{\alpha}(\xi,u,p)\eta_{i}^{\alpha}\eta_{j}^{\beta}\ge \lambda \bigl(1 + \vert p \vert ^{2} \bigr)^{\frac{m - 2}{2}} \vert \eta \vert ^{2}, \quad \forall\eta\in{ \mathbb{R}}^{k\times N}, $$
(1.6)
(H3): There exists a modulus of continuity \(\mu:(0, \infty )\rightarrow[0,\infty)\) such that where \(K(\cdot): [0,\infty) \to[0,\infty)\) is monotonously nondecreasing. Without loss of generality, we assume \(K(\cdot) \ge1\) and that (μ1) μ is nondecreasing with \(\mu{(0 + ) = 0} \mu(1)=1\); (μ2) μ is concave, and \(r \to r^{ - \gamma}\mu(r)\) is nonincreasing for some exponent \(\gamma\in(0,1)\); (μ3) Dini condition \(H(r) = \int_{0}^{r} {\frac{\sqrt{\mu(\rho )}}{\rho}}\, d\rho< \infty\) for some \(r > 0\).
$$ \bigl\vert {A_{i}^{\alpha}(\xi,u,p) - A_{i}^{\alpha}(\tilde{\xi},\tilde {u},p)} \bigr\vert \le K \bigl( \vert u \vert \bigr)\mu \bigl( \bigl(d^{m}( {\xi},\tilde{\xi} ) + \vert {u - \tilde{u}} \vert ^{m} \bigr)^{\frac{1}{m}} \bigr) \bigl(1 + \vert p \vert \bigr)^{\frac{m}{2}}, $$
(1.7)
(H4) (Controllable growth condition): The term \(B^{\alpha}\) satisfies sub-quadratic controllable growth condition where C is a positive constant, and \(r=\frac{mQ}{Q-m}\) if \(m< Q\); or \(Q\leq r < +\infty\) if \(m=Q\). We note that \(Q \geq3\) is the homogeneous dimension in non-Abelian Carnot groups (see (2.3) below), and the exponent m satisfies \(1< m<2\). So those infer that \(m< Q\), and then \(r=\frac{mQ}{Q-m}\) in our setting.
$$ \bigl\vert {B^{\alpha}(\xi,u,p)} \bigr\vert \le C \bigl(1+|u|^{r-1}+|p|^{m(r-1)/r} \bigr),\quad 1< m< 2, $$
(1.8)
We follow the strategy in the Euclidean case used by Duzaar et al. in [14] and Duzaar and Grotowski in [8] with the necessary modification to handle the sub-elliptic case in Carnot groups. First, inspired by [18], we choose horizontal variables to construct an auxiliary function to establish Caccioppoli type inequality; see Lemma 7 below. The method of using Taylor’s expansion in [14] cannot be easily adapted to our situation. Instead, we choose different auxiliary functions and apply the Sobolev-Poincaré type inequality (3.6) to obtain the desired excess improvement estimates.
The main result in this paper is as follows.
Theorem 1
Assume that coefficients
\(A_{i}^{\alpha}\)
and
\(B^{\alpha}\)
satisfy conditions (H1)-(H4) with (μ1)-(μ3), and
\(u \in HW^{1,m}(\Omega,{ \mathbb{R}}^{N})\)
is a weak solution to (1.3) with bounded domain
\(\Omega\subset G\). Then there exists an open subset
\(\Omega_{0} \subset\Omega\)
such that
\(u \in C^{1 }(\Omega_{0} ,{ \mathbb{R}}^{N})\). Furthermore, the closed singular set
\(\Omega \setminus \Omega_{0} \subset\Sigma_{1} \cup\Sigma_{2} \)
of Lebesgue measure zero, where
In addition, for
\(\tau\in[\gamma, 1)\)
and
\(\xi_{0} \in\Omega \setminus \Omega_{0}\), the derivative
Xu
has the modulus of continuity
\(r \rightarrow r^{\tau}+M(r)\)
in a neighborhood of
\(\xi_{0}\).
$$\begin{aligned} &\Sigma_{1} = \Bigl\{ {\xi_{0} \in\Omega: \lim _{r \to0^{+} } \sup \bigl( \bigl\vert {(Xu)_{\xi _{0} ,r} } \bigr\vert \bigr) = \infty} \Bigr\} , \\ &\Sigma_{2} = \biggl\{ {\xi_{0} \in\Omega: \lim _{r \to0^{+} } \inf \fint_{B_{r} (\xi _{0} )} { \bigl\vert {Xu - (Xu)_{\xi_{0} ,r} } \bigr\vert } ^{m}\, d\xi> 0} \biggr\} . \end{aligned}$$
It is worth pointing out that the Haar measure in Carnot groups \(\mathbb {G}\) with the underlying manifold \(\mathbb{R}^{n}\) is just the Lebesgue measure in \(\mathbb{R}^{n}\). Our result is optimal in the sense that when \(\mu(\rho) = \rho^{\gamma}\), \(0 < \gamma< 1\), we have \(M(r) = \int_{0}^{r} {\frac{{\mu(\rho)}}{\rho}} \, d\rho= \gamma^{ - 1}r^{\gamma}\), and \(C ^{1,\gamma} \) regularity is known to be optimal in that case; see the reference [2] by Wang and Liao.
2 Preliminaries
A Carnot group G of step r is a simply connected, nilpotent Lie group whose Lie algebra \(\mathfrak{g}\) admits a stratification, i.e., \(\mathfrak{g}=\bigoplus_{j=1}^{r} V^{j} \) such that \([V^{1},V^{j}]=V^{j+1}\), \(j=1,\ldots,r-1 \) and \([V^{1},V^{r}]={\{0\}}\). Let \(X_{i}^{l}\) be a left-invariant basis vector field of \(V^{l}\) with \(1\leq l\leq r\) and \(1\leq i\leq m_{l}\), where \(m_{l}\) is the dimension of \(V^{l}\). For simplicity, we write \(X_{i}=X_{i}^{1}\), \(k=m_{1}\), and denote by \(X=(X_{1},\ldots,X_{k})\) the horizontal gradient. We say that \({X_{i}}\) (\(i=1,\ldots,k\)) are the horizontal vector fields with the form where \(a_{ij}(\xi)\) is a polynomial.
$$ X_{i}=\partial_{i}+\sum_{j=i+1}^{n}a_{ij}( \xi)\partial _{j},\qquad X_{i}(0)=\partial_{i}, $$
(2.1)
We write and the distance from origin is defined by For any \(\xi, \eta\in G\), we set \(d(\xi,\eta)=d(\eta^{-1}\circ \xi)\), where \(\eta^{-1}=-\eta=(-\eta^{1},\ldots,-\eta^{r})\) is the reverse of η, and ∘ is the multiplication rule in G defined by \(\xi\circ\widetilde{\xi}=\xi+\widetilde{\xi}+P(\xi,\widetilde{\xi} ), \xi, \widetilde{\xi}\in G\), where \(P:G\times G\mapsto G\) has polynomial components. Following [32], we denote by \(\omega_{G}= \vert {B_{1}(0)} \vert _{G} \) the volume of unit ball. Then \(\vert {B_{r}(\xi)} \vert _{G} = \omega_{G} r^{Q} \), where is the homogeneous dimension of G.
$$ \xi= \bigl(\xi^{1},\xi^{2},\ldots,\xi^{r} \bigr)= \bigl( {x_{1}^{1},x_{2}^{1}, \ldots ,x_{m_{1}}^{1};x_{1}^{2}, \ldots,x_{m_{2}}^{2};\ldots;x_{1}^{r}, \ldots ,x_{m_{l}}^{r}} \bigr) \in G, $$
$$ d(\xi)= \Biggl[ {\sum_{l=1}^{r}{ \Biggl( {\sum_{i=1}^{m_{l}}{ \bigl\vert {x_{i}^{l}} \bigr\vert ^{2}}} \Biggr) }^{\frac {{r!}}{l}}} \Biggr] ^{\frac{1}{{2r!}}}. $$
(2.2)
$$ Q = \sum_{l = 1}^{r} {l m_{l} } $$
(2.3)
The non-Abelian Heisenberg group \(\mathbb{H}^{n}\) is the simplest and the most important prototype of the Carnot group with step \(r=2\), which is defined as \(\mathbb{R}^{2n + 1}\) endowed with the following group multiplication: The basic vector fields corresponding to its Lie algebra \(\mathfrak{g}\) can be explicitly calculated by the exponential map and are given by for \(i = 1,2, \ldots,n\). The homogeneous dimension Q in the Heisenberg group \(H^{n}\) is \(2n+2\geq4\). Given the property of the Heisenberg algebra \(\mathfrak{g}\), one has \(\mathfrak{g} =\mathbb{R}^{2n+1}=V_{1}\oplus V_{2}\) with \(V_{1}=\mathbb {R}^{2n}\times\{0\}\) and \(V_{2}=\{0\}_{ \mathbb{R}^{2n}} \times\mathbb {R}\) such that \([X_{i}, X_{n+j}] =T\delta_{ij}\) and \([V_{1}, V_{1}] =V_{2}\).
$$ { \bigl(\xi^{1},t \bigr)\cdot \bigl(\tilde{\xi}^{1}, \tilde{t} \bigr)} := \Biggl( {\xi^{1} + \tilde{\xi}^{1},t + \tilde{t} + \frac{1}{2}\sum_{i = 1}^{n} { ( {x_{i} \tilde{y}_{i} - \tilde {x}_{i} y_{i} } )} } \Biggr). $$
(2.4)
$$ X_{i} = \frac{\partial}{\partial x_{i} } - \frac{y_{i} }{2}\frac{\partial }{\partial t}, \qquad X_{i + n} = \frac{\partial}{\partial y_{i} } + \frac{x_{i} }{2} \frac{\partial}{\partial t},\qquad T = \frac{\partial }{\partial t} $$
(2.5)
Let \(1 \leq m < \infty\) and \(\Omega\subset{ G}\) be an open set. If \(u \in L^{m}(\Omega)\) satisfies we say that u belongs to the horizontal Sobolev space. The space \(HW_{0}^{1,m}(\Omega) \) is the completion of \(C_{0}^{\infty}(\Omega)\) under the norm (2.6).
$$ \Vert u \Vert _{HW^{1,m}(\Omega)} = \Vert u \Vert _{L^{m}(\Omega )} + \sum_{i = 1}^{k} { \Vert {X_{i} u} \Vert _{L^{m}(\Omega)}< \infty}, $$
(2.6)
Throughout the paper, we shall use the functions \(V, W:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) defined by for each \(\varsigma\in\mathbb{R}^{n}\) and \(m>1\). By the elementary inequality \(\Vert x \Vert _{\frac{2}{{2-m}}}\leq \Vert x \Vert _{1}\leq2^{\frac{m}{2}} \Vert x \Vert _{\frac {2}{{2-m}}}\) applied to the vector \(x=( {1, \vert {\varsigma} \vert ^{2-m}}) \in\mathbb{R}^{2}\), we conclude that which immediately yields The purpose of introducing W is the fact that in contrast to \(\vert V \vert ^{\frac{2}{m}}\), the function \(\vert W \vert ^{\frac{2}{m}}\) is convex (see [9]).
$$ V(\varsigma)={\varsigma}/{{ \bigl( {1+ \vert \varsigma \vert ^{2}} \bigr) ^{\frac{{2-m}}{4}}}},\qquad W(\varsigma)={ \varsigma}/{{ \bigl( {1+ \vert \varsigma \vert ^{2-m}} \bigr) ^{\frac{1}{2}}}} $$
(2.7)
$$ \bigl( {1+ \vert {\varsigma} \vert ^{2}} \bigr) ^{\frac {{2-m}}{2}} \leq1+ \vert {\varsigma} \vert ^{2-m}\leq2^{\frac{m}{2}} \bigl( {1+ \vert {\varsigma} \vert ^{2}} \bigr) ^{\frac {{2-m}}{2}}, $$
(2.8)
$$ \bigl\vert {W(\varsigma)} \bigr\vert \leq \bigl\vert {V( \varsigma)} \bigr\vert \leq2^{\frac{m}{4}} \bigl\vert {W( \varsigma)} \bigr\vert . $$
(2.9)
The following lemma includes some useful properties of the function V, which can be found in Lemma 2.1 of [33].
Lemma 2
Let
\(m\in(1,2)\)
and
\(V, W : \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\)
be the functions defined in (2.7). Then there holds for any
\(\varsigma_{1},\varsigma_{2}\in\mathbb{R}^{n}\)
and
\(t>0\):
- (1)
\(\frac{1}{\sqrt{2} }\min ( { \vert {\varsigma_{1}} \vert , \vert {\varsigma_{1}} \vert ^{\frac{m}{2}} } ) \leq \vert {V ( {\varsigma_{1}} )} \vert \leq\min ( { \vert {\varsigma_{1}} \vert , \vert {\varsigma_{1}} \vert ^{\frac{m}{2}} } )\);
- (2)
\(\vert {V(t\varsigma_{1})} \vert \leq\max ( {t,t^{\frac {m}{2}} } ) \vert {V(\varsigma_{1})} \vert \);
- (3)
\(\vert {V(\varsigma_{1} + \varsigma_{2})} \vert \leq C(m)( \vert {V(\varsigma_{1})} \vert + \vert {V(\varsigma_{2})} \vert )\);
- (4)
\(\frac{m}{2} \vert \varsigma_{1} - \varsigma_{2} \vert \leq{{ \vert {V(\varsigma_{1}) - V(\varsigma_{2})} \vert }}/{{( {1 + \vert {\varsigma_{1}} \vert ^{2} + \vert {\varsigma_{2}} \vert ^{2} })^{\frac{{m - 2}}{4}} }} \leq C(m,n) \vert \varsigma_{1} - \varsigma_{2} \vert \);
- (5)
\(\vert {V(\varsigma_{1} ) - V(\varsigma_{2})} \vert \leq C(m,n) \vert {V( \varsigma_{1} - \varsigma_{2})} \vert \);
- (6)
\(\vert {V(\varsigma_{1} - \varsigma_{2})} \vert \leq C(m,M) \vert { V(\varsigma_{1} ) - V(\varsigma_{2})} \vert \) for all \(\varsigma_{1}\) with \(|\varsigma_{2}|\leq M\).
Let \(\varsigma_{1},\varsigma_{2}\in\mathbb{R}^{n}\) with \(|\varsigma _{2}|\leq M\), and we mention that the following two estimates can be deduced from Lemma 2(1) and (6): for \(\vert {\varsigma_{1}-\varsigma_{2}} \vert \leq1\), while for \(\vert {\varsigma_{1}-\varsigma_{2}} \vert >1\),
$$ \vert \varsigma_{1}-\varsigma_{2} \vert ^{2}\leq C(m,M) \bigl\vert {V(\varsigma_{1})-V( \varsigma_{2})} \bigr\vert ^{2} $$
(2.10)
$$ \vert \varsigma_{1}-\varsigma_{2} \vert ^{m}\leq C(m,M) \bigl\vert {V(\varsigma_{1})-V( \varsigma_{2})} \bigr\vert ^{2}. $$
(2.11)
The next lemma is an algebraic fact from [34].
Lemma 3
For every
\(t \in ( { - \frac{1}{2},0} )\)
and
\(\eta\geq0\), it holds
for any
\(p,\tilde{p}\in{\mathbb{R}}^{k\times N}\), not both zero if
\(\eta=0\).
$$ 1 \leqslant\frac{{\int_{0}^{1} { ( {\eta^{2} + \vert {p + s(\tilde{p} - p)} \vert ^{2} } )^{t} \,ds } }}{{ ( {\eta^{2} + \vert p \vert ^{2} + \vert {\tilde{p}} \vert ^{2} } )^{t} }} \leqslant\frac{8}{{2t + 1}}, $$
(2.12)
The nondecreasing property of μ yields \(s\mu(t)\le s\mu(t)\) for all \(0\le t \le s\). By the nonincreasing property of \(r\mapsto\frac{\mu(r)}{r}\) and \(\mu(1)\le1\), we conclude that By (μ2), we further obtain for \(\theta\in(0,1)\), \(t>0\), \(j\in\mathbb {N}\cup\{0\}\), which implies It yields particularly that \(\mu(t)\le\frac{\gamma^{2}}{4}H(t)\) for all \(t\le0\), and \(t\mapsto t^{-\gamma}H(t)\) is also nonincreasing.
$$ s\mu(t)\leq s\mu(s)+t,\quad s\in[0,1], t>0. $$
(2.13)
$$\frac{2}{\gamma} \bigl(1-\theta^{\gamma}\bigr)\mu^{1/2} \bigl( \theta^{2j}t \bigr)={ \int_{\theta ^{2(j+1)}t}^{\theta^{2j}t}\tau^{\frac{\gamma}{2}-1}\frac{\mu^{1/2}(\theta ^{2j}t)}{(\theta^{2j}t)^{\gamma/2}}} \,d\tau\le \int_{\theta ^{2(j+1)}t}^{\theta^{2j}t}\frac{\mu^{1/2}(\tau)}{\tau} \,d\tau, $$
$$\sum_{j = 0}^{\infty}{\mu^{1/2} \bigl( \theta^{2j} t \bigr)} \le\frac {\gamma}{{2(1 - \theta^{\gamma})}}H^{1/2} (t). $$
In what follows, we denote \(\rho_{1} (s,t) = (1 + s+t)^{ - 1}K(s + t)^{ - 1}\), and \(K_{1}(s,t)=(1+t)^{2m}K^{4}(s+t)\) for \(s, t\ge0\), and note that \(\rho_{1} \le 1\) and that \(s \to\rho_{1} (s,t)\), \(t \to\rho_{1} (s,t)\) are nonincreasing functions.
3 Caccioppoli type inequality
We first generalize an \(\mathcal{A} \)-harmonic approximation lemma in our setting. Then a Sobolev-Poincaré type inequality (Lemma 5) and a prior estimate (Lemma 6) are introduced for the sub-quadratic case in Carnot groups, and the detailed proofs can be found in [2] by Wang, Liao and Yu. The last and key point is to prove a Caccioppoli type inequality.
By \(\operatorname{Bil}( {\mathbb{R}}^{k\times N})\) we denote the collection of bi-linear forms defined in \({\mathbb{R}}^{k\times N}\). Let \(\mathcal{A}\in\operatorname{Bil}({\mathbb{R}}^{k\times N})\), we say that a function \(h\in HW^{1,m}(\Omega ,{\mathbb{R}}^{N})\) is \(\mathcal{A}\)-harmonic if h satisfies
$$ \int_{\Omega}\mathcal{A}(Xh,X\varphi) \,d\xi=0,\quad \forall \varphi\in C_{0}^{1} \bigl(\Omega,{\mathbb{R}}^{N} \bigr). $$
(3.1)
Similarly to [9], one can establish the following \(\mathcal{A} \)-harmonic approximation lemma for the case \(1< m<2\) in Carnot groups.
Lemma 4
Let
λ
and
L
be fixed positive numbers, \(1< m<2\), and
\(k, N\in\mathbb{N}\)
with
\(k\geq2\). Suppose that for any given
\(\varepsilon>0\), there exists
\(\delta=\delta(k,N,\lambda ,L,\varepsilon)\in(0,1]\)
with the following properties:
such that
- (I)for any bi-linear form \(\mathcal{A}\in\operatorname {Bil}({\mathbb{R}}^{k\times N})\) satisfying$$ \mathcal{A}(\nu,\nu)\geq\lambda \vert \nu \vert ^{2}\quad \textit{and} \quad \mathcal{A}(\nu,\bar{\nu})\leq L \vert \nu \vert \vert \bar{\nu} \vert ,\quad \nu,\bar{\nu}\in{ \mathbb{R}}^{k\times N}, $$(3.2)
- (II)for any function \(g\in HW^{1,m}(B_{\rho}(\xi _{0}),{\mathbb{R}}^{N})\) satisfying$$\begin{aligned}& \fint_{B_{\rho}(\xi _{0})}{ \bigl\vert W({Xg}) \bigr\vert }^{2} \,d\xi\leq\Upsilon^{2}\leq 1 \quad \textit{and} \end{aligned}$$(3.3)$$\begin{aligned}& \biggl\vert { \fint_{B_{\rho }(\xi_{0})}{\mathcal{A}(Xg,X\varphi) \,d\xi}} \biggr\vert \leq\Upsilon \delta \sup_{B_{\rho}(\xi_{0})} \vert {X\varphi} \vert , \quad \forall\varphi\in C_{0}^{1} \bigl(B_{\rho}( \xi_{0}),{ \mathbb{R}}^{N} \bigr). \end{aligned}$$(3.4)
$$h\in H= \biggl\{ {h\in HW^{1,m}\bigl(B_{\rho}( \xi_{0}),{ \mathbb{R}}^{N}\bigr)\Big|{\fint_{B_{\rho}(\xi _{0})}{ \bigl\vert {W(Xh)} \bigr\vert }^{2} \,d\xi\leq1}} \biggr\} $$
$$ \fint_{B_{\rho}(\xi _{0})}{ \biggl\vert W \biggl( \frac{g-\Upsilon h}{\rho} \biggr) \biggr\vert }^{2} \,d\xi\leq\Upsilon^{2} \varepsilon. $$
(3.5)
Lemma 5
(Sobolev-Poincaré type inequality)
Let
\(m\in(1,2)\)
and
\(u\in HW^{1,m}(B_{\rho }(\xi_{0}),\mathbb{R}^{N})\)
with
\(B_{\rho}(\xi_{0})\subset\Omega\), then
Furthermore, the analogous inequality is valid with
W
replaced by
V
defined in (2.7), and in particular, the inequality also holds if we substitute 2 for
\(\frac{2Q}{Q-m}\).
$$ \biggl( \fint_{B_{\rho }(\xi_{0})} \biggl\vert W \biggl( \frac{u-u_{\xi_{0},\rho}}{\rho} \biggr) \biggr\vert ^{\frac{2Q}{Q-m}}\,d\xi \biggr) ^{\frac{Q-m}{2Q}} \leq C_{P} \biggl( \fint_{B_{\rho}(\xi_{0})} \bigl\vert W ( Xu ) \bigr\vert ^{2} \,d\xi \biggr) ^{\frac{1}{2}}. $$
(3.6)
Lemma 6
Let
\(u \in HW^{1,1} (\Omega ,\mathbb{R}^{N} )\)
be a weak solution of the systems with constant coefficients
Then
u
is smooth and there exists
\(C_{0}\geq1\)
such that for any
\(B_{\rho}(\xi_{0} ) \subset\Omega\),
$$ \int_{\Omega}{A_{ij}^{\alpha\beta} X_{j} u^{\beta}X_{i} \phi^{\alpha}\,d\xi= 0}, \quad \phi \in C_{0}^{\infty}\bigl(\Omega,\mathbb{R}^{N} \bigr). $$
(3.7)
$$ \sup_{B_{\rho/ 2}(\xi_{0} )} \bigl( { \vert u -u_{\xi_{0}, \rho} \vert ^{2}}+\rho^{2} \vert {Xu} \vert ^{2} + \rho^{4} \bigl\vert {X^{2}u} \bigr\vert ^{2} \bigr) \le C_{0} \rho^{2} \fint_{B_{\rho}(\xi_{0} )} { \vert {Xu} \vert } ^{2} \,d\xi. $$
(3.8)
To establish the main result, a crucial ingredient in the local regularity of weak solutions to systems (1.3) is a Caccioppoli type inequality. We prove a version which is adapted to our situation.
Lemma 7
(Caccioppoli type inequality)
Let
\(u\in HW^{1,m}(\Omega,{ \mathbb{R}}^{N})\)
be a weak solution to system (1.3) under conditions (H1)-(H4) with (μ1)-(μ3). Then, for any
\(\xi _{0}=(\xi_{0}^{1},\ldots,\xi_{0}^{r})\in\Omega\), and
\(0< t< s \le \rho_{1}^{\frac{m}{(2-m)(m-1)}}(|u_{0}|,|p_{0}|)\), it holds
with
where we denote
\(K(\cdot)=K ( { \vert {u_{0} } \vert + \vert {p_{0} } \vert } ) \), \(C_{c}=C_{c}(Q,N,m,L,\lambda,M)\), and
\(\xi^{1}=(\xi_{1}^{1},\xi _{2}^{1},\ldots,\xi_{k}^{1})\)
is the horizontal component of
\(\xi\in G\).
$$ \fint_{B_{t}(\xi_{0})} \bigl\vert {V(Xu-p_{0})} \bigr\vert ^{2} \,d\xi\le C_{c} \biggl( \fint _{B_{s}(\xi_{0})} \biggl\vert {V \biggl(\frac{u-u_{0}-p_{0}(\xi^{1} -\xi _{0}^{1})}{s-t} \biggr)} \biggr\vert ^{2} \,d\xi+U \biggr) $$
(3.9)
$$\begin{aligned} U =& \bigl[ {K(\cdot) \bigl( {1 + \vert {p_{0} } \vert } \bigr)} \bigr]^{2m/(m-1)} \mu^{2} \bigl(s^{(2-m)(m-1)/m } \bigr) + \bigl( {1+2M + \vert p_{0} \vert s} \bigr)^{m/(m-1)^{2} } s^{m - 1} \\ &{}+ \mu^{2} \bigl(s^{(2-m)(m-1)/m } \bigr) \biggl[ { \fint_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{ ( {1 - \frac{1}{r}} )\frac{m}{{m - 1}}}, \end{aligned}$$
(3.10)
Proof
Let \(\eta\in C_{0}^{\infty}(B_{s}(\xi_{0}))\) be a standard cut-off function satisfying \(0\leq\eta \leq1, \vert {X \eta} \vert <\frac{C}{s-t}\) and \(\eta \equiv1\) on \(B_{t}(\xi_{0})\). Inspired by the way of [18], we let \(v=u(\xi)-u_{0}-p_{0}(\xi ^{1}-\xi_{0}^{1})\), and then define two functions one has and Using hypothesis (H2), Lemma 3 and the elementary inequality we have
$$ \varphi=\eta v, \qquad \phi =(1-\eta)v, $$
(3.11)
$$ X\varphi+X\phi=Xu-p_{0} $$
(3.12)
$$ \bigl\vert {V(X\varphi) } \bigr\vert , \bigl\vert {V(X\phi) } \bigr\vert \leqslant C(m) \biggl( { \bigl\vert {V(Xv)} \bigr\vert + \biggl\vert V \biggl({\frac{v}{{s-t}}} \biggr) \biggr\vert } \biggr). $$
(3.13)
$$ 1+ \vert a \vert ^{2}+ \vert {b-a} \vert ^{2}\leq3 \bigl( {1+ \vert a \vert ^{2}+ \vert b \vert ^{2}} \bigr) , $$
(3.14)
$$\begin{aligned} & \int_{B_{s}(\xi_{0})}{ \bigl[ {A_{i}^{\alpha}( \xi,u,p_{0}+X\varphi )-A_{i}^{\alpha}( \xi,u,p_{0})} \bigr] }X_{i}\varphi^{\alpha}\,d\xi \\ &\quad = \int_{B_{s}(\xi_{0})}{ \int_{0}^{1}{\frac{{\partial A_{i}^{\alpha }(\xi ,u,p_{0}+\theta X\varphi)}}{{\partial p_{j}^{\beta}}}}}\, d\theta X_{j}\varphi^{\beta}X_{i}\varphi^{\alpha}\,d \xi \\ &\quad \geqslant \lambda \int_{B_{s}(\xi_{0})} \biggl[{ \int_{0}^{1}{ \bigl( {1+ \vert {p_{0}+\theta X\varphi} \vert ^{2}} \bigr) }}^{(m-2)/2}\,d\theta \biggr] \vert {X\varphi} \vert ^{2}\,d\xi \\ &\quad \geqslant 3^{(m-2)/2}\lambda \int_{B_{s}(\xi_{0})}{ \bigl( {1+ \vert {p_{0}} \vert ^{2}+ \vert {X\varphi} \vert ^{2}} \bigr) ^{(m-2)/2}} \vert {X\varphi} \vert ^{2}\,d\xi. \end{aligned}$$
(3.15)
From (3.15), it follows that where we have used (3.12), (1.3) and the fact that
$$\begin{aligned} & 3^{(m-2)/2}\lambda \int_{B_{s}(\xi_{0})}{ \bigl( {1+ \vert {p_{0}} \vert ^{2}+ \vert {X\varphi} \vert ^{2}} \bigr) ^{(m-2)/2}} \vert {X\varphi} \vert ^{2}\,d\xi \\ &\quad \leq \int_{B_{s}(\xi_{0})}{ \bigl[ {A_{i}^{\alpha}(\xi ,u,p_{0}+X\varphi)-A_{i}^{\alpha}( \xi,u,p_{0})} \bigr] }X_{i}\varphi ^{\alpha}\,d\xi \\ &\quad\leq - \int_{B_{s}(\xi_{0})}{ \bigl[ {A_{i}^{\alpha}(\xi ,u,p_{0})-A_{i}^{\alpha} \bigl( { \xi,u_{0}+p_{0} \bigl(\xi^{1}-\xi _{0}^{1} \bigr),p_{0}} \bigr) } \bigr] }X_{i}\varphi^{\alpha}\,d\xi \\ &\qquad{} - \int_{B_{s}(\xi_{0})}{ \bigl[ {A_{i}^{\alpha} \bigl( { \xi ,u_{0}+p_{0} \bigl(\xi^{1}- \xi_{0}^{1} \bigr),p_{0}} \bigr) -A_{i}^{\alpha}(\xi _{0},u_{0},p_{0})} \bigr] }X_{i}\varphi^{\alpha}\,d\xi \\ &\qquad {} - \int_{B_{s}(\xi_{0})}{ \int_{0}^{1}{\frac{{\partial A_{i}^{\alpha} ( {\xi,u,Xu-\theta(Xu-X\varphi-p_{0})} ) }}{{\partial p_{j}^{\beta}}}}}\,d\theta X_{j}\phi^{\beta}X_{i}\varphi ^{\alpha }\,d \xi \\ &\qquad {} + \int_{B_{s}(\xi_{0})}{B_{i}(\xi,u,Xu)\varphi^{\alpha}\,d \xi} \\ &\quad := {I}+\mathit{II}+\mathit{III}+\mathit{IV}, \end{aligned}$$
(3.16)
$$ \int_{B_{s}(\xi_{0})}{A_{i}^{\alpha}(\xi _{0},u_{0},p_{0})}X\varphi^{\alpha} \,d \xi=0. $$
Noting that \(\varphi=v\) on \(B_{t}(\xi_{0})\), \(m-2<0\) and (3.14), the left-hand side in (3.16) can be estimated by where we have applied Lemma 2(4) in the second inequality and Lemma 2(6) for the final inequality.
$$\begin{aligned} & 3^{(m-2)/2}\lambda \int_{B_{t}(\xi_{0})}{ \bigl( {1+ \vert {p_{0}} \vert ^{2}+ \vert {X\varphi} \vert ^{2}} \bigr) ^{(m-2)/2} \vert {X\varphi} \vert ^{2}}\,d\xi \\ &\quad = 3^{(m-2)/2}\lambda \int_{B_{t}(\xi_{0})}{ \bigl( {1+ \vert {p_{0}} \vert ^{2}+ \vert {Xv} \vert ^{2}} \bigr) ^{(m-2)/2} \vert {Xv} \vert ^{2}}\,d\xi \\ &\quad \geq C(m)\lambda \int_{B_{t}(\xi_{0})}{ \bigl[ { \bigl( {1+ \vert {p_{0}} \vert ^{2}+ \vert {Xu} \vert ^{2}} \bigr) ^{(m-2)/4} \vert {Xu-p_{0}} \vert } \bigr] }^{2}\,d\xi \\ &\quad \geq C(k,N,m,\lambda) \int_{B_{t}(\xi_{0})}{ \bigl\vert {V(Xu)-V(p_{0})} \bigr\vert ^{2}}\,d\xi \\ &\quad \geq C(k,N,m,\lambda,M) \int_{B_{t}(\xi_{0})} \bigl\vert {V(Xv)} \bigr\vert ^{2}\,d\xi, \end{aligned}$$
(3.17)
The structure condition (H3) yields where we have used the inequality \(s\mu(t)\le s\mu(s)+t\) for \(s\in [0,1]\) and \(t>0\).
$$\begin{aligned} I \le& \int_{B_{s} (\xi_{0} )} {K( \cdot)} \bigl(1 + \vert p_{0} \vert \bigr)^{m/2} \mu \bigl( {|v|} \bigr) \vert {X\varphi} \vert \,d\xi \\ \le& \frac{1}{{s^{(2 - m)(m - 1)/m} }} \int_{B_{s} (\xi_{0} )} { \bigl[ {K( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m/2} s^{(2 - m)(m - 1)/m} \mu \bigl( {|v|} \bigr)} \bigr]} \vert {X\varphi} \vert \,d\xi \\ \le& \frac{1}{{s^{(2 - m)(m - 1)/m} }} \int_{B_{s} (\xi_{0} )} \bigl[ K( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m/2} s^{(2 - m)(m - 1)/m} \\ &{}\times \mu \bigl( {K( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m/2} s^{(2 - m)(m - 1)/m} } \bigr) + |v| \bigr] \vert {X\varphi} \vert \,d\xi \\ \le& \frac{1}{{s^{(2 - m)(m - 1)/m} }} \int_{B_{s} (\xi_{0} )} { \bigl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} s^{(2 - m)(m - 1)/m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr) + |v|} \bigr]} \vert {X\varphi} \vert \,d\xi \\ \le& \int_{B_{s} (\xi_{0} )} { \biggl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr) + \frac{{|v|}}{{s^{(2 - m)(m - 1)/m} }}} \biggr]} \vert {X\varphi} \vert \,d\xi, \end{aligned}$$
(3.18)
To obtain a suitable estimate for I, we need to take the domain \(B_{s} (\xi_{0} )\) into four parts: \(B_{s} (\xi_{0} ) \cap \{ { \vert {v/s} \vert > 1} \} \cap \{ { \vert {X\varphi} \vert \leq1} \}\), \(B_{s}(\xi_{0})\cap \{ { \vert {v/s} \vert >1} \} \cap \{ { \vert {X\varphi} \vert >1} \} \), \(B_{s}(\xi_{0})\cap \{ { \vert {v/s} \vert \leq1} \} \cap \{ { \vert {X\varphi} \vert >1} \} \), \(B_{s}(\xi_{0})\cap \{ { \vert {v/s} \vert \leq1} \} \cap \{ { \vert {X\varphi} \vert \leq1} \} \), and we will use Young’s inequality, (2.10) and (2.11), repeatedly.
Case 1: On the part \(B_{s} (\xi_{0} ) \cap \{ { \vert {v/s} \vert > 1} \} \cap \{ { \vert {X\varphi} \vert > 1} \}\), we see
$$\begin{aligned} & \biggl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr) + \frac{{|v|}}{{s^{(2 - m)(m - 1)/m} }}} \biggr] \vert {X\varphi} \vert \\ &\quad \le 2\varepsilon \vert {X\varphi} \vert ^{m} + \varepsilon^{ - 1} \bigl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr)} \bigr]^{m/(m - 1)} + \varepsilon^{ - 1} \biggl\vert { \frac{v}{s}} \biggr\vert \vert v \vert ^{1/(m - 1)} s^{m - 1} \\ &\quad \le 2\varepsilon \vert {X\varphi} \vert ^{m} + \varepsilon^{ - 1} \bigl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr)} \bigr]^{m/(m - 1)} \\ &\qquad {}+ \varepsilon^{ - 1} \biggl\vert { \frac{v}{s}} \biggr\vert ^{m} + \varepsilon^{ - 1} \vert v \vert ^{m/(m - 1)^{2} } s^{m} \\ &\quad \le 2\varepsilon C(m,M) \bigl\vert {V(X\varphi)} \bigr\vert ^{2} + \varepsilon ^{ - 1} C(m,M) \biggl\vert {V \biggl( { \frac{v}{s}} \biggr)} \biggr\vert ^{2} \\ &\qquad {}+ \varepsilon^{ - 1} \bigl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr)} \bigr]^{m/(m - 1)} + \varepsilon^{ - 1} \vert {2M + p_{0} s} \vert ^{m/(m - 1)^{2} } s^{m}. \end{aligned}$$
(3.19)
Case 2: On the set \(B_{s}(\xi_{0})\cap \{ { \vert {v/s} \vert >1} \} \cap \{ { \vert {X\varphi} \vert \le1} \} \), it holds where we have used the inequality \(2(m-1)/m<1\).
$$\begin{aligned} & \biggl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr) + \frac{{|v|}}{{s^{(2 - m)(m - 1)/m} }}} \biggr] \vert {X\varphi} \vert \\ &\quad \le 2\varepsilon \vert {X\varphi} \vert ^{2} + \varepsilon^{ - 1} \bigl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr)} \bigr]^{2} + \varepsilon^{ - 1} \biggl\vert { \frac{v}{s}} \biggr\vert ^{m} + \varepsilon^{ - 1} \vert v \vert ^{m/(m - 1)} s^{m} \\ &\quad \le 2\varepsilon C(m,M) \bigl\vert {V(X\varphi)} \bigr\vert ^{2} + \varepsilon ^{ - 1} C(m,M) \biggl\vert {V \biggl( { \frac{v}{s}} \biggr)} \biggr\vert ^{2} \\ &\qquad {}+ \varepsilon^{ - 1} \bigl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr)} \bigr]^{2} + \varepsilon^{ - 1} \vert {2M + p_{0} s} \vert ^{m/(m - 1)} s^{m}, \end{aligned}$$
(3.20)
Case 3: On the set \(B_{s}(\xi_{0})\cap \{ { \vert {v/s} \vert \leq1} \} \cap \{ { \vert {X\varphi} \vert >1} \} \), observing \(m/(m-1)>2\), one has
$$\begin{aligned} & \biggl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr) + \frac{{|v|}}{{s^{(2 - m)(m - 1)/m} }}} \biggr] \vert {X\varphi} \vert \\ &\quad \le 2\varepsilon \vert {X\varphi} \vert ^{m} + \varepsilon^{ - 1} \bigl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr)} \bigr]^{m/(m - 1)} + \varepsilon^{ - 1} \biggl\vert { \frac{v}{s}} \biggr\vert \vert v \vert ^{1/(m - 1)} s^{m - 1} \\ &\quad \le 2\varepsilon C(m,M) \bigl\vert {V(X\varphi)} \bigr\vert ^{2} + \varepsilon ^{ - 1} \bigl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr)} \bigr]^{m/(m - 1)} \\ &\qquad {}+ \varepsilon^{ - 1} \vert {2M + p_{0} s} \vert ^{1/(m - 1)} s^{m - 1}. \end{aligned}$$
(3.21)
Case 4: For \(B_{s}(\xi_{0})\cap \{ { \vert {v/s} \vert \leq1} \} \cap \{ { \vert {X\varphi} \vert \leq1} \} \), there is
$$\begin{aligned} & \biggl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr) + \frac{{|v|}}{{s^{(2 - m)(m - 1)/m} }}} \biggr] \vert {X\varphi} \vert \\ &\quad \le 2\varepsilon C(m,M) \bigl\vert {V(X\varphi)} \bigr\vert ^{2} + \varepsilon ^{ - 1} \bigl[ {K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m} \mu \bigl( {s^{(2 - m)(m - 1)/m} } \bigr)} \bigr]^{2} + \varepsilon^{ - 1} s^{m}. \end{aligned}$$
(3.22)
Combining these estimations with (3.18), we get
$$\begin{aligned} I \le& \varepsilon C(m,M) \int_{B_{s}(\xi_{0})} \bigl\vert {V(Xv)} \bigr\vert ^{2} \,d \xi + C(\varepsilon,m,M) \int_{B_{s}(\xi_{0})} \biggl\vert {V \biggl( {\frac{v}{{s - t}}} \biggr)} \biggr\vert ^{2} \,d\xi \\ &{}+ \varepsilon^{ - 1} K^{2m/(m - 1)} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m^{2} /(m - 1)} \mu^{2} \bigl( {s^{(2 - m)(m - 1)/m} } \bigr) \bigl\vert {B_{s} (\xi_{0} )} \bigr\vert _{G} \\ &{}+ \varepsilon^{ - 1} \bigl( {1+2M + \vert p_{0} \vert s} \bigr)^{m/(m-1)^{2} } \bigl\vert {B_{s} (\xi_{0} )} \bigr\vert _{G} s^{m - 1}. \end{aligned}$$
(3.23)
Similarly to I, it follows that
$$\begin{aligned} \mathit{II} \le& \int_{B_{s} (\xi_{0} )} {K( \cdot)} \bigl(1 + \vert p_{0} \vert \bigr)^{(m + 2)/2} \mu ( s ) \vert {X\varphi} \vert \,d\xi \\ \le& \int_{B_{1} (\xi_{0} )} { \bigl[ {\varepsilon \vert {X\varphi} \vert ^{m} + \varepsilon^{ - 1} K^{m/(m - 1)} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr) ^{m(m + 2)/2(m - 1)} \mu^{m/(m - 1)} ( s )} \bigr]} \,d\xi \\ &{}+ \int_{B_{2} (\xi_{0} )} { \bigl[ {\varepsilon \vert {X\varphi} \vert ^{2} + \varepsilon^{ - 1} K^{2} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{(m + 2)} \mu^{2} ( s )} \bigr]} \,d\xi \\ \le& 2\varepsilon C(m,M) \int_{B_{s} (\xi_{0} )} { \bigl\vert {V(Xv)} \bigr\vert ^{2} \,d\xi} + 2\varepsilon C(m,M) \int_{B_{s} (\xi_{0} )} { \biggl\vert {V \biggl( {\frac {v}{{s - t}}} \biggr)} \biggr\vert ^{2} \,d\xi} \\ &{} + \varepsilon^{ - 1} K^{m/(m - 1)} ( \cdot) \bigl(1 + \vert p_{0} \vert \bigr)^{m(m + 2)/2(m - 1)} \mu^{2} ( s ) \bigl\vert {B_{s} (\xi_{0} )} \bigr\vert _{G}. \end{aligned}$$
(3.24)
By (H1), Lemma 3 and (3.14), it holds
$$\begin{aligned} \mathit{III} \leqslant& L \int_{B_{s}(\xi_{0})}{ \biggl[ { \int_{0}^{1}{ \bigl( {1+ \bigl\vert {Xu+\theta \bigl[ { ( {Xu-X\phi} ) -Xu} \bigr] } \bigr\vert ^{2}} \bigr) }^{(m-2)/2}\,d\theta} \biggr] \vert {X\phi} \vert \vert {X\varphi} \vert \,d\xi} \\ \leqslant& \frac{8L}{m-1} \int_{B_{s}(\xi_{0})}{ \bigl( {1+ \vert {Xu} \vert ^{2}+ \vert {Xu-X\phi} \vert ^{2}} \bigr) ^{(m-2)/2}} \vert {X\phi} \vert \vert {X\varphi} \vert \,d\xi \\ \leqslant& \frac{8L}{m-1} \int_{B_{s}(\xi_{0})}{ \bigl( {1+ \vert {X\phi} \vert ^{2}} \bigr) ^{(m-2)/2}} \vert {X\phi} \vert \vert {X\varphi} \vert \,d\xi. \end{aligned}$$
(3.25)
Noting that \(X\phi\subset B_{s}\setminus B_{t}\) and \(-1/2<(m-2)/2<0\), we split the domain \(B_{s}(\xi_{0})\) into four parts: \(B_{s}(\xi_{0})\cap \{ { \vert {X\phi} \vert >1} \} \cap \{ { \vert {X\varphi} \vert >1} \} \), \(B_{s}(\xi_{0})\cap \{ { \vert {X\phi} \vert \leq1} \} \cap \{ { \vert {X\varphi} \vert \leq1} \} \), \(B_{s}(\xi_{0})\cap \{ { \vert {X\phi} \vert >1} \} \cap \{ { \vert {X\varphi} \vert \leq1} \} \) and \(B_{s}(\xi_{0})\cap \{ { \vert {X\phi} \vert \leq1} \} \cap \{ { \vert {X\varphi} \vert >1} \} \). Similarly to I, it follows that
$$ \mathit{III}\leqslant\frac{C(L,m,M)}{m-1} \biggl( { \int_{B_{s}(\xi _{0})\setminus B_{t}(\xi_{0})}{ \bigl\vert {V(Xv)} \bigr\vert ^{2}\,d\xi+ \int_{B_{s}(\xi_{0})}{ \biggl\vert {V \biggl( { \frac{v}{s-t}} \biggr) } \biggr\vert ^{2}\,d\xi }}} \biggr) . $$
(3.26)
Using Hölder’s inequality, one has
$$\begin{aligned} \begin{aligned}[b] \mathit{IV} &\le C \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)^{1 - \frac{1}{r}} \vert \varphi \vert \,d\xi } \\ &\le C \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{1 - \frac{1}{r}} \biggl( { \int_{B_{s} (\xi_{0} )} { \vert \varphi \vert ^{r} \,d\xi} } \biggr)^{\frac{1}{r}}. \end{aligned} \end{aligned}$$
(3.27)
Analogously to I, we take the domain \(B_{s}(\xi_{0})\) into two parts.
Case 1: For \(B_{s} (\xi_{0} ) \cap \{ { \vert {Xv } \vert > 1} \} \), by Sobolev type inequality, Young’s inequality and (2.11), it follows that where we have used \(|\varphi|^{r}<|v|^{r}\).
$$\begin{aligned}& \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{1 - \frac{1}{r}} \biggl( { \int _{B_{s} (\xi_{0} )} { \vert \varphi \vert ^{r} \,d\xi} } \biggr)^{\frac {1}{r}} \\& \quad \le C \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{1 - \frac{1}{r}} \biggl( { \int_{B_{s} (\xi_{0} )} { \vert {Xv } \vert ^{m} \,d\xi} } \biggr)^{\frac {1}{m}} \\& \quad \le C(\varepsilon) \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{ ( {1 - \frac{1}{r}} )\frac{{m }}{m-1}} + C \varepsilon \biggl( { \int _{B_{s} (\xi_{0} )} { \vert {Xv } \vert ^{m} \,d\xi} } \biggr) \\& \quad \le C(\varepsilon) \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{ ( {1 - \frac{1}{r}} )\frac{{m }}{m-1}} \\& \qquad {}+ C(m,M)\varepsilon \biggl( { \int _{B_{s} (\xi_{0} )} { \bigl\vert {V ( {Xv } )} \bigr\vert ^{2} \,d\xi} } \biggr), \end{aligned}$$
(3.28)
Case 2: On the set \(B_{s} (\xi_{0} ) \cap \{ { \vert {Xv } \vert \leq1} \} \), where we have used (2.10).
$$\begin{aligned}& \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{1 - \frac{1}{r}} \biggl( { \int _{B_{s} (\xi_{0} )} { \vert \varphi \vert ^{r} \,d\xi} } \biggr)^{\frac {1}{r}} \\& \quad \le C(\varepsilon) \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{2 ( {1 - \frac{1}{r}} )} + C(m,M)\varepsilon \biggl( { \int_{B_{s} (\xi_{0} )} { \bigl\vert {V ( {Xv } )} \bigr\vert ^{2} \,d\xi} } \biggr), \end{aligned}$$
(3.29)
Combining these estimates in IV, we have
$$\begin{aligned} \mathit{IV} \le& C(m,M)\varepsilon \biggl( { \int_{B_{s} (\xi_{0} )} { \bigl\vert {V ( {Xv } )} \bigr\vert ^{2} \,d\xi} } \biggr) + C(\varepsilon) \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{ ( {1 - \frac{1}{r}} )\frac {m}{{m - 1}}} \\ &{}+C(\varepsilon) \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{2 ( {1 - \frac{1}{r}} )} . \end{aligned}$$
(3.30)
Substituting (3.17), (3.23), (3.24), (3.26) and (3.30) into (3.16), we finally arrive at We let \(\varepsilon=[3^{m-2}\lambda C(m,M)]/5\). ‘Filling the hole’ with \(\theta=\frac{\frac {C(L,m,M)}{m-1}+4\varepsilon C(m,M)}{\frac {C(L,m,M)}{m-1}+3^{m-2}\lambda C^{2}(m,M)}<1\) yields where \(C=C(N,L,\lambda,m,M)\), and The proof is completed by taking mean values of integral and noting that \([m(r-1)/r(m-1)-1]Q=m/(m-1)\) and \(s^{m/(m-1)}\le s^{2(2-m)(m-1)/m}\le\mu^{2} (s^{(2-m)(m-1)/m } )\). □
$$\begin{aligned}& \biggl[3^{(m-2)}\lambda C^{2}(m,M)+\frac{C(L,m,M)}{m-1} \biggr] \int_{B_{t} (\xi_{0} )} { \bigl\vert {V(Xv)} \bigr\vert ^{2} \,d\xi} \\& \quad \leqslant \biggl[\frac{C(L,m,M)}{m-1}+4\varepsilon C(m,M) \biggr] \int _{B_{s} (\xi_{0} )} {{ \bigl\vert {V(Xv)} \bigr\vert ^{2} }\,d\xi} \\& \qquad {}+ C(\varepsilon ,L,m,M) \int_{B_{s} (\xi_{0} )} { \biggl\vert {V \biggl( { \frac{v}{s-t}} \biggr)} \biggr\vert ^{2} \,d\xi} \\& \qquad {} +C(\varepsilon) \bigl[ {K(\cdot) \bigl( {1 + \vert {p_{0} } \vert } \bigr)} \bigr]^{2m/(m-1)} \mu^{2} \bigl(s^{(2-m)(m-1)/m } \bigr) \bigl\vert B_{s} (\xi_{0} ) \bigr\vert _{G} \\& \qquad {}+C(\varepsilon ) \bigl( {1+2M + |p_{0}| s} \bigr)^{m/(m-1)^{2} } \bigl\vert B_{s} (\xi_{0} ) \bigr\vert _{G} s^{m - 1} \\& \qquad {} + C(\varepsilon) \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{ ( {1 - \frac{1}{r}} )\frac{m}{{m - 1}}} \\& \qquad {}+ C( \varepsilon) \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{2 ( {1 - \frac{1}{r}} )}. \end{aligned}$$
$$\begin{aligned} \int_{B_{t}(\xi_{0})} \bigl\vert {V(Xu-p_{0})} \bigr\vert ^{2} \,d\xi \le& C \biggl( \int _{B_{s}(\xi_{0})} \biggl\vert {V \biggl(\frac{v}{s-t} \biggr)} \biggr\vert ^{2} \,d\xi+\hat {U} \biggr) \\ &{}+\theta \int_{B_{s}(\xi_{0})} \bigl\vert {V(Xu-p_{0})} \bigr\vert ^{2} \,d\xi, \end{aligned}$$
$$\begin{aligned} \hat{U} =& \bigl[ {K(\cdot) \bigl( {1 + \vert {p_{0} } \vert } \bigr)} \bigr]^{2m/(m-1)} \mu^{2} \bigl(s^{(2-m)(m-1)/m } \bigr) \bigl\vert B_{s} (\xi_{0} ) \bigr\vert _{G} \\ &{}+ \bigl( {1+2M + |p_{0}| s} \bigr)^{m/(m-1)^{2} } \bigl\vert B_{s} (\xi_{0} ) \bigr\vert _{G} s^{m - 1} \\ &{}+ \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{ ( {1 - \frac{1}{r}} )\frac{m}{{m - 1}}} \\ &{}+ \biggl[ { \int_{B_{s} (\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{2 ( {1 - \frac{1}{r}} )}. \end{aligned}$$
(3.31)
4 Proof of the main result
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
where we denote
\(F(s,t )=K^{4/(2-m)}(s+t)(2+t)^{2}+(1+s+t)^{r-1}\).
$$\begin{aligned}& \fint_{B_{\rho}(\xi _{0})}{A_{i,p_{\beta}^{j}}^{\alpha}(\xi _{0},u_{0},p_{0}) (Xu-p_{0})X \varphi^{\alpha}} \,d\xi \\& \quad \leq C_{1} \sup_{B_{\rho}(\xi_{0})} \vert {X\varphi } \vert \bigl[ {\omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert ,\Phi^{\frac{1}{2}}(\xi_{0},\rho,p_{0})} \bigr) \Phi ^{\frac{1}{2}}(\xi_{0},\rho,p_{0})} \\& \qquad {}+ { \Phi( \xi_{0},\rho,p_{0})+\mu (\sqrt{\rho}) F \bigl( \vert {u_{0}} \vert , \vert {p_{0}} \vert \bigr)} \bigr] , \end{aligned}$$
(4.1)
Proof
Noting the fact we have
$$\begin{aligned}& \int_{B_{\rho}(\xi_{0} )} \biggl[ \int_{0}^{1} A_{i,p_{\beta}^{j} }^{\alpha}\bigl(\xi_{0} ,u_{0} ,\theta Xu + (1 - \theta)p_{0} \bigr) (Xu - p_{0} ) \,d\theta \biggr]X \varphi^{\alpha}\,d\xi \\& \quad = \int_{B_{\rho}(\xi_{0} )} \bigl[A_{i}^{\alpha}( \xi_{0} ,u_{0} ,Xu) - A_{i}^{\alpha}( \xi _{0} ,u_{0} ,p_{0} ) \bigr]X \varphi^{\alpha}\,d\xi \\& \quad = \int_{B_{\rho}(\xi_{0} )} \bigl[A_{i}^{\alpha}( \xi_{0} ,u_{0} ,Xu) - A_{i}^{\alpha}( \xi ,u,Xu) \bigr]X\varphi^{\alpha}\,d\xi \\& \qquad {}+ \int_{B_{\rho}(\xi_{0} )} B^{\alpha} (\xi,u,Xu)\varphi^{\alpha}\,d\xi, \end{aligned}$$
(4.2)
$$\begin{aligned}& \int_{B_{\rho}(\xi_{0})}{A_{i,p_{\beta}^{j}}^{\alpha}(\xi _{0},u_{0},p_{0}) (Xu-p_{0})X \varphi^{\alpha}} \,d\xi \\& \quad = \int_{B_{\rho}(\xi_{0} )} { \biggl[ { \int_{0}^{1} {A_{i,p_{\beta}^{j} }^{\alpha}( \xi_{0} ,u_{0} ,p_{0} )} \,d\theta(Xu - p_{0} )} \biggr]} X\varphi ^{\alpha}\,d\xi \\& \quad \leq \int_{B_{\rho}(\xi_{0})} \biggl[{ \int_{0}^{1}} \bigl\vert {{{{ A_{i,p_{\beta}^{j}}^{\alpha}(\xi_{0},u_{0},p_{0})-A_{i,p_{\beta }^{j}}^{\alpha} \bigl(\xi_{0},u_{0},\theta Xu+(1-\theta)p_{0} \bigr)}}}} \bigr\vert \vert {{{Xu-p_{0}}}} \vert { \,d\theta} \biggr] \\& \qquad {}\times\sup_{B_{\rho}(\xi_{0})} \vert {X\varphi} \vert \,d\xi \\& \qquad {} + \int_{B_{\rho}(\xi_{0})}{ \bigl\vert {A_{i}^{\alpha}(\xi _{0},u_{0},Xu)-A_{i}^{\alpha} \bigl( \xi,u_{0}+p_{0} \bigl(\xi^{1}-\xi _{0}^{1} \bigr),Xu \bigr)} \bigr\vert } \sup _{B_{\rho}(\xi_{0})} \vert {X\varphi} \vert \,d\xi \\& \qquad {} + \int_{B_{\rho}(\xi_{0})}{ \bigl\vert {A_{i}^{\alpha} \bigl(\xi ,u_{0}+p_{0} \bigl(\xi^{1}- \xi_{0}^{1} \bigr),Xu \bigr)-A_{i}^{\alpha}( \xi,u,Xu)} \bigr\vert } \sup_{B_{\rho}(\xi_{0})} \vert {X\varphi} \vert \,d\xi \\& \qquad {} +C \int_{B_{\rho}(\xi_{0})}{ \bigl( \vert p \vert ^{m(1 - \frac{1}{r})} + \vert u \vert ^{r - 1} + 1 \bigr) \vert \varphi \vert } \,d\xi \\& \quad := I^{\prime}+\mathit{II}^{\prime}+\mathit{III}^{\prime}+ \mathit{IV}^{\prime}. \end{aligned}$$
(4.3)
Using (H1) and estimate (1.5) yields (note that \(m-2<0\)) Let Then it follows that, by first Hölder’s inequality and then Jensen’s inequality, we have where we have used estimates (2.10) and (2.11) in the last inequality.
$$\begin{aligned} I^{\prime} \leqslant&C(L) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \int _{B_{\rho}(\xi_{0})} \biggl\{ { \int_{0}^{1}{ \bigl[ { \bigl( {1+ \vert {p_{0}} \vert ^{2}} \bigr) ^{\frac{{m-2}}{2}}+ \bigl( {1+ \bigl\vert {p_{0}+\theta(Xu-p_{0})} \bigr\vert ^{2}} \bigr) ^{\frac {{m-2}}{2}}} \bigr] ^{\frac{1}{2}}}} \\ &{} \times \bigl[ { \bigl( {1+ \vert {p_{0}} \vert ^{2}+ \bigl\vert {p_{0}+\theta(Xu-p_{0})} \bigr\vert ^{2}} \bigr) ^{\frac{{m-2}}{2}}\omega \bigl( { \vert {p_{0}} \vert , \bigl\vert {\theta (Xu-p_{0})} \bigr\vert } \bigr) } \bigr] ^{\frac{1}{2}}\,d\theta \biggr\} \vert {Xu-p_{0}} \vert \,d\xi \\ \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \int _{B_{\rho}(\xi_{0})}{ \bigl( {1+ \vert {Xu-p_{0}} \vert ^{2}} \bigr) }^{\frac{{m-2}}{4}}\omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert , \vert {Xu-p_{0}} \vert } \bigr) \vert {Xu-p_{0}} \vert \,d\xi \\ \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \int _{B_{\rho}(\xi_{0})}{ \bigl( {1+ \vert {Xu-p_{0}} \vert ^{\frac{{m-2}}{2}}} \bigr) } \vert {Xu-p_{0}} \vert \omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert , \vert { Xu-p_{0}} \vert } \bigr) \,d\xi. \end{aligned}$$
(4.4)
$$ B_{1}=:B_{\rho}(\xi_{0})\cap \bigl\{ { \vert {Xu-p_{0}} \vert \leq1} \bigr\} ,\qquad B_{2}=:B_{\rho}( \xi_{0})\cap \bigl\{ { \vert {Xu-p_{0}} \vert >1} \bigr\} . $$
(4.5)
$$\begin{aligned} I^{\prime} \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \biggl[ \int_{B_{1}}{ \vert {Xu-p_{0}} \vert }\omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert , \vert { Xu-p_{0}} \vert } \bigr) \,d\xi \\ &{}+ \int_{B_{2}}{ \vert {Xu-p_{0}} \vert ^{\frac{m}{2}}}\omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert , \vert {Xu-p_{0}} \vert } \bigr) \,d\xi \biggr] \\ \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi | \biggl( { \int_{B_{1}}{ \vert {Xu-p_{0}} \vert ^{2}}\,d\xi } \biggr) ^{\frac{1}{2}} \biggl( { \int_{B_{\rho}(\xi_{0})}{\omega \bigl( { \vert {p_{0}} \vert , \vert {Xu-p_{0}} \vert } \bigr) \,d \xi}} \biggr) ^{\frac{1}{2}} \\ &{} +C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \biggl( { \int _{B_{2}}{ \vert {Xu-p_{0}} \vert ^{m}}\,d\xi} \biggr) ^{\frac{1}{2}} \biggl( { \int_{B_{\rho}(\xi_{0})}{\omega \bigl( { \vert {p_{0}} \vert , \vert {Xu-p_{0}} \vert } \bigr) \,d\xi }} \biggr) ^{\frac{1}{2}} \\ \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi | \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G} \biggl( {\fint_{B_{\rho}(\xi _{0})}{ \bigl\vert {V(Xu)-V(p_{0})} \bigr\vert ^{2}}\,d\xi} \biggr) ^{\frac {1}{2}} \\ &{}\times\biggl( { \fint_{B_{\rho}(\xi_{0})}{\omega \bigl( { \vert {p_{0}} \vert , \vert {Xu-p_{0}} \vert } \bigr) \,d\xi}} \biggr) ^{ \frac{1}{2}} \\ \leqslant& C(L,m) \sup_{B_{\rho}(\xi_{0} )}|X\varphi | \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G} \Phi^{\frac{1}{2}}(\xi_{0},\rho,p_{0}) \\ &{}\times\omega^{\frac{1}{2}}\biggl( { \vert {p_{0}} \vert , \biggl( { \fint_{B_{1}}{ \vert {Xu-p_{0}} \vert ^{2}\,d\xi}} \biggr) ^{\frac{1}{2}}} { + \biggl( { \int_{B_{2}}{ \vert {Xu-p_{0}} \vert ^{m}\,d\xi}} \biggr) ^{\frac{1}{m}}} \biggr) \\ \le& C(L,m,M) \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G} \Phi^{\frac{1}{2}}(\xi _{0},\rho,p_{0}) \omega^{\frac{1}{2}} \bigl( { \vert {p_{0}} \vert , \Phi^{\frac{1}{2}}(\xi_{0},\rho,p_{0})} \bigr), \end{aligned}$$
(4.6)
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 where we have used \(4/(4-m)<2\), \(1+m/2<2\) and \(\mu(\rho)\le1\) for \(\rho \in[0,1]\).
$$\begin{aligned} \mathit{II}^{\prime} \leqslant& \int_{B_{\rho}(\xi_{0})}{K(\cdot) \mu(\rho ) \bigl(1+ \vert {p_{0}} \vert \bigr) \bigl(1+ \vert {Xu} \vert \bigr)^{\frac {m}{2}}\,d\xi} \\ \leqslant& K(\cdot) \mu(\rho) \bigl(1+ \vert {p_{0}} \vert \bigr)^{1 +\frac{m}{2}} \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G}+ \int_{B_{1 }+B_{2}}{K(\cdot) \mu(\rho) \bigl(1+ \vert {p_{0}} \vert \bigr) \vert {Xu-p_{0}} \vert ^{\frac{m}{2}}\,d\xi} \\ \leqslant& K(\cdot) \mu(\rho) \bigl(1+ \vert {p_{0}} \vert \bigr)^{1 +\frac{m}{2}} \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G}+ \bigl[ {K(\cdot) \mu(\rho) \bigl(1+ \vert {p_{0}} \vert \bigr)} \bigr] ^{2} \vert {B_{2}} \vert _{G} \\ &{} + \bigl[ {K(\cdot) \mu(\rho) \bigl(1+ \vert {p_{0}} \vert \bigr)} \bigr] ^{\frac{4}{{4-m}}} \vert {B_{1}} \vert _{G} + \int_{B_{2}}{ \vert {Xu-p_{0}} \vert ^{m}\,d\xi}+ \int_{B_{1}}{ \vert {Xu-p_{0}} \vert ^{2}\,d\xi} \\ \leqslant& \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G}\Phi(\xi _{0},\rho,p_{0})+3 \bigl[ {K( \cdot) \bigl(1+ \vert {p_{0}} \vert \bigr)} \bigr] ^{2} \bigl\vert {B_{\rho}(\xi_{0})} \bigr\vert _{G}\mu(\rho), \end{aligned}$$
(4.7)
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, 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
$$\begin{aligned} \mathit{III}' \le& \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \int _{B_{\rho}(\xi_{0})} K( \cdot) \bigl( {1 + |Xu|} \bigr)^{m/2} \mu \bigl( {|v|} \bigr) \,d\xi \\ \le& \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \int_{B_{\rho}(\xi _{0})} \biggl[\frac{{|v|}}{\rho}\sqrt{\rho}+ K^{2} ( \cdot) ( {1 + p_{0} } )^{m} \mu ( {\sqrt{\rho}} ) + K^{2} ( \cdot)|Xu - p_{0} |^{m} \mu ( {\sqrt{\rho}} ) \biggr]\,d\xi \\ \le& \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \biggl[ C(m,M) \bigl(C_{P} + K^{2} ( \cdot) \bigr) \int_{B_{\rho}(\xi_{0} )} { \bigl\vert V ( {Xu - p_{0} } ) \bigr\vert ^{2} } \,d\xi \\ &{}+ K^{4/(2 - m)} ( \cdot) ( {2 + p_{0} } )^{m} \mu ( {\sqrt{\rho}} ) \bigl\vert B_{\rho}(\xi_{0} ) \bigr\vert _{G} \biggr] \\ \le& \sup_{B_{\rho}(\xi_{0} )}|X\varphi| \bigl[C(m,M) \bigl(C_{P} + K^{2} ( \cdot) \bigr)\Phi(\xi_{0} ,p_{0} , \rho) \bigl\vert B_{\rho}(\xi_{0} ) \bigr\vert _{G} \\ &{}+ K^{4/(2 - m)} ( \cdot) ( {2 + p_{0} } )^{m} \mu ( {\sqrt{\rho}} ) \bigl\vert B_{\rho}( \xi_{0} ) \bigr\vert _{G} \bigr]. \end{aligned}$$
(4.8)
$$\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 and then
$$ \vert {Xu-p_{0}} \vert ^{m}\leq \vert {Xu-p_{0}} \vert ^{2}+1\leq \bigl\vert {V ( {Xu} ) -V ( {p_{0}} ) } \bigr\vert ^{2}+1, $$
$$\begin{aligned} \mbox{(4.9)} \leq& C\mu (\sqrt{\rho}) \bigl\vert {B_{\rho}(\xi_{0} )} \bigr\vert \bigl[ { \bigl( {1 + \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{r - 1} + \bigl( {1 + \vert {u_{0} } \vert + \vert {p_{0} } \vert } \bigr)^{m} } \bigr] \\ &{}+C \bigl\vert {B_{\rho}(\xi_{0} )} \bigr\vert _{G} \Phi(\xi_{0} ,\rho,p_{0} ). \end{aligned}$$
(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:
with
\(C_{2}=8C_{1}^{2}C_{4}\), together with the condition
Then the following growth inequality is valid for
\(\tau\in[\gamma, 1)\):
where
\(\sigma=\min\{(2-m)(m-1)/m, (m-1)/2\}\), and
\(K^{\ast }(s,t)=C_{8}F^{\frac{2m}{m-1}}(1+s,M+t) \).
$$\begin{aligned}& \omega^{\frac{1}{2}} \bigl( { \vert {u_{\xi_{0} ,\rho} } \vert + \bigl\vert {(Xu)_{\xi_{0} ,\rho} } \bigr\vert ,\Phi^{\frac{1}{2}} \bigl(\xi_{0} ,\rho ,(Xu)_{\xi_{0} ,\rho} \bigr)} \bigr)+ \Phi^{1 / 2} \bigl(\xi_{0} ,\rho,(Xu)_{\xi_{0} ,\rho} \bigr) \le\frac{\delta}{2}, \end{aligned}$$
(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)
$$ \rho\le\rho^{\frac{m}{(2-m)(m-1)}} _{1} \bigl(1+\vert u_{\xi_{0} ,\rho} \vert, 1+ \bigl\vert (Xu)_{\xi _{0} ,\rho} \bigr\vert \bigr). $$
(4.15)
$$ \Phi \bigl(\xi_{0},\theta\rho,(Xu)_{\xi_{0},\theta\rho} \bigr)\leq\theta ^{2\tau} {\Phi (\rho)+K^{\ast} \bigl(|u_{\xi_{0},\rho}|, \bigl\vert (Xu)_{\xi_{0},\rho} \bigr\vert \bigr) } \mu^{2} \bigl( {\rho}^{\sigma} \bigr), $$
(4.16)
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 and
$$ w=u-(u_{\xi_{0},\rho}-\Upsilon h_{\xi_{0},2\theta\rho})- ( {Xu} ) _{\xi_{0,}\rho} \bigl(\xi^{1}-\xi_{0}^{1} \bigr) $$
$$ \Upsilon={C(m,M)}\Gamma(\rho),\quad \Gamma(\rho )=C_{1}\sqrt{\Phi(\rho)+4 \delta^{-2}\mu^{2} (\sqrt{\rho}) F^{2} \bigl( \vert {u_{\xi_{0},\rho}} \vert , \bigl\vert {(Xu)_{\xi_{0},\rho}} \bigr\vert \bigr)}. $$
Noting (4.13) and (4.14) yields and by (4.1), we derive (note the definition of ϒ and \(\Gamma(\rho)\))
$$ \omega^{1/2} \bigl( { \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }} \bigr\vert ,\Phi^{1/2} ( \rho ) } \bigr) +\Phi ^{1/2} ( \rho ) \leq \frac{\delta}{2}, $$
(4.17)
$$\begin{aligned}& \biggl\vert {\fint_{B_{\rho}(\xi_{0})}{ \bigl[ {{{ A_{i,p_{\beta}^{j}}^{\alpha }}} \bigl( {\xi_{0},u_{\xi_{0},\rho}, ( {Xu} ) _{\xi_{0},\rho}} \bigr) Xw} \bigr] X_{i}\varphi^{\alpha }\,d \xi}} \biggr\vert \\& \quad \leq \Upsilon\frac{{\omega^{1/2} ( { \vert { ( {Xu} ) _{\xi_{0},\rho}} \vert ,\Phi^{1/2}(\rho )} ) \Phi^{1/2}(\rho)+\Phi(\rho)+\mu (\sqrt{\rho}) F ( \vert {u_{\xi_{0},\rho}} \vert , \vert {(Xu)_{\xi_{0},\rho}} \vert )}}{{C(m,M)\Gamma(\rho)}} \sup _{B_{\rho}(\xi_{0})} \vert {X \varphi} \vert \\& \quad \leq \Upsilon \biggl[ {\omega^{1/2} \bigl( { \bigl\vert { ( {Xu} ) _{\xi_{0},\rho}} \bigr\vert ,\Phi^{1/2}(\rho )} \bigr) + \Phi^{1/2}(\rho)+\frac{\delta}{2}} \biggr] \sup_{B_{\rho}(\xi _{0})} \vert {X\varphi} \vert \\& \quad \leq \Upsilon\delta\sup_{B_{\rho}(\xi _{0})} \vert {X\varphi} \vert . \end{aligned}$$
(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 where the constant C depends only on m, k and N.
$$\begin{aligned} \Phi(\theta\rho) =& \fint_{B_{\theta\rho}(\xi_{0})}{ \bigl\vert {V(Xu)-V \bigl( { ( {Xu} ) _{\xi_{0,\theta\rho}}} \bigr) } \bigr\vert ^{2}\,d\xi} \\ \leq& C \fint_{B_{\theta\rho}(\xi_{0})}{ \bigl\vert {V \bigl( {Xu- ( {Xu} ) _{\xi_{0,\theta\rho}}} \bigr) } \bigr\vert ^{2}\,d\xi } \\ \leq& C \fint_{B_{\theta\rho}(\xi_{0})}{ \bigl\vert {V \bigl( {Xu- ( {Xu} ) _{\xi_{0, \rho}}-\Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr) } \bigr\vert ^{2}\,d\xi} \\ &{} +C \bigl\vert {V \bigl( { ( {Xu} ) _{\xi_{0,\theta\rho }}-(Xu)_{\xi_{0,}\rho}- \Upsilon(Xh)_{(\xi _{0},2\theta\rho)}} \bigr) } \bigr\vert ^{2}, \end{aligned}$$
(4.21)
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 where we have denoted Using Lemma 2(1), we deduce that \(V^{2} (\Xi ^{1/2}+\Xi^{1/m} )\le C(\Xi+\Xi^{2/m}) \). So we get 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 where we have used Lemma 2(1).
$$\begin{aligned}& \bigl\vert { ( {Xu} ) _{\xi_{0,\theta}\rho}-(Xu)_{\xi _{0,}\rho}- \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert \\& \quad \leq \fint_{B_{\theta\rho}(\xi_{0})}{ \bigl\vert {Xu-(Xu)_{\xi _{0,}\rho}- \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert \,d\xi } \\& \quad \leq \sqrt{2} \fint_{B_{1}}{ \bigl\vert V \bigl({Xu-(Xu)_{\xi _{0,}\rho}- \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr) \bigr\vert \,d\xi} \\& \qquad {}+ \sqrt[m]{2} \fint_{B_{2}}{ \bigl\vert V \bigl({Xu-(Xu)_{\xi _{0,}\rho}- \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr) \bigr\vert ^{2/m} \,d\xi} \\& \quad \leq \sqrt[m]{2} \bigl({\Xi} ^{1/2}+{\Xi}^{1/m} \bigr), \end{aligned}$$
(4.22)
$$ {\Xi}=: \fint_{B_{\theta\rho}(\xi_{0})}{ \bigl\vert {V \bigl( {Xu-(Xu)_{\xi_{0,}\rho}- \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr) } \bigr\vert ^{2}\,d\xi}. $$
$$ \Phi(\theta\rho)\leq C \bigl( {\Xi+V^{2} \bigl(\Xi ^{1/2}+\Xi^{1/m} \bigr)} \bigr) \leq C_{3} \bigl( { \Xi+\Xi ^{2/m}} \bigr) , $$
(4.23)
$$ \fint_{B_{\rho}(\xi_{0})}{ \vert {Xh} \vert }\,d\xi\leq2 \sqrt{2} \fint_{B_{\rho}(\xi _{0})}{ \bigl\vert W({Xh}) \bigr\vert ^{2} }\,d\xi\le2\sqrt{2}, $$
(4.24)
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) 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 where
$$ \Upsilon \bigl\vert {(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert \leq \Upsilon\sup_{B_{\rho/2} (\xi_{0} )} \vert {Xh} \vert \le\Upsilon C_{0} \fint_{B_{\rho}(\xi_{0})}{ \vert {Xh} \vert }\,d\xi\le2\sqrt{2}\Upsilon C_{0} \leq1. $$
(4.25)
$$\begin{aligned} \Xi \leq& C_{c} \biggl[ \bigl\vert B_{2\theta\rho}( \xi _{0}) \bigr\vert _{G}^{-1} \int_{B_{2\theta\rho}(\xi_{0})} \biggl\vert {V \biggl( {\frac {{u-u_{\xi_{0},\rho}- ( { ( {Xu} ) _{\xi_{0},\rho }+\Upsilon (Xh)_{(\xi_{0},2\theta\rho)}} ) (\xi^{1}-\xi _{0}^{1})}}{{2\theta\rho}}} \biggr) } \biggr\vert ^{2}\,d\xi \\ &{}+U \biggr], \end{aligned}$$
(4.26)
$$\begin{aligned} U =& \bigl[ {K \bigl( { \vert {u_{\xi_{0},\rho}} \vert + \bigl\vert { ( {Xu} ) _{\xi_{0},\rho}+\Upsilon(Xh)_{(\xi_{0},2\theta \rho)}} \bigr\vert } \bigr) \bigl( {1+ \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }+ \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert } \bigr) } \bigr] ^{2m/(m-1) } \\ &{}\times \mu^{2} \bigl( {(2\theta \rho) }^{(2-m)(m-1)/m} \bigr) \\ &{}+ \bigl( {1+2M + \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }+ \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert } \bigr)^{m/(m-1)^{2} } (2\theta \rho)^{m - 1} \\ &{}+\mu^{2} \bigl( {(2\theta \rho) }^{(2-m)(m-1)/m} \bigr) \biggl[ { \fint_{B_{\rho}(\xi_{0} )} { \bigl( { \vert {Xu} \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)\,d\xi} } \biggr]^{\frac{m}{{m - 1}} ( {1 - \frac{1}{r}} )}. \end{aligned}$$
By Lemma 2(3), one gets 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
$$\begin{aligned}& \int_{B_{2\theta\rho} (\xi_{0} )} \biggl\vert {V \biggl( {\frac{{u - u_{\xi_{0} ,\rho} - ( { ( {Xu} )_{\xi_{0} ,\rho} + \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} )(\xi^{1} - \xi_{0}^{1} )}}{ {2\theta\rho}}} \biggr)} \biggr\vert ^{2} \,d\xi \\& \quad \leq \int_{B_{2\theta\rho} (\xi_{0} )} \biggl\vert V \biggl( \frac{{u - ( {u_{\xi_{0} ,\rho} - \Upsilon h_{\xi_{0} ,2\theta \rho} } ) - ( {Xu} )_{\xi_{0} ,\rho} (\xi^{1} - \xi _{0}^{1} ) - \Upsilon h(\xi)}}{{2\theta\rho}} \\& \qquad {}+ \frac{{\Upsilon h(\xi) - \Upsilon h_{\xi_{0} ,2\theta\rho} - \Upsilon (Xh)_{(\xi_{0},2\theta\rho)}(\xi^{1} - \xi_{0}^{1} )}}{{2\theta\rho}} \biggr) \biggr\vert ^{2} \,d\xi \\& \quad \leq C \biggl[ \int_{B_{2\theta\rho} (\xi_{0} )} \biggl( \biggl\vert {V \biggl( { \frac{{w - \Upsilon h(\xi)}}{{2\theta\rho}}} \biggr)} \biggr\vert ^{2} \\& \qquad {}+ \biggl\vert {V \biggl( {\Upsilon\frac{{h(\xi) - h_{\xi_{0} ,2\theta\rho } - (Xh)_{(\xi_{0},2\theta\rho)}(\xi^{1} - \xi_{0}^{1} )}}{{2\theta\rho }}} \biggr)} \biggr\vert ^{2} \biggr)\,d\xi \biggr]. \end{aligned}$$
(4.27)
$$\begin{aligned} \fint_{B_{2\theta\rho}(\xi_{0})}{ \biggl\vert {V \biggl( {\frac{{w-\Upsilon h(\xi)}}{{2\theta\rho}}} \biggr) } \biggr\vert ^{2}\,d\xi } \le& C(2\theta)^{-Q-2} \fint_{B_{\rho}(\xi_{0})}{ \biggl\vert {W \biggl( {\frac{{w-\Upsilon h(\xi)}}{\rho}} \biggr) } \biggr\vert ^{2}\,d\xi} \\ \le& C(2\theta)^{-Q-2} \Upsilon^{2}\varepsilon. \end{aligned}$$
(4.28)
Using Lemma 2, Sobolev-Poincare type inequality (3.6), Lemma 6, (2.9) and (4.20) leads to where we denote \(C_{5}=4CC_{0}C_{P}^{4}\).
$$\begin{aligned}& \fint_{B_{2\theta\rho} (\xi_{0} )} { \biggl\vert {V \biggl( {\Upsilon \frac{{h(\xi) - h_{\xi_{0} ,2\theta\rho} - (Xh)_{(\xi_{0},2\theta\rho )}(\xi^{1} - \xi_{0}^{1} )}}{{2\theta\rho}}} \biggr)} \biggr\vert ^{2} \,d\xi} \\& \quad \leq C_{P}^{2}\Upsilon^{2} \fint_{B_{2\theta\rho} (\xi_{0} )} { \bigl\vert {V \bigl( {{{Xh(\xi) - (Xh)_{(\xi_{0},2\theta\rho)}}} } \bigr)} \bigr\vert ^{2} \,d\xi } \\& \quad \leq C_{P}^{4}(2\theta\rho)^{2} \Upsilon^{2} \fint_{B_{2\theta\rho} (\xi _{0} )} { \bigl\vert {V \bigl( {X^{2}h} \bigr)} \bigr\vert ^{2} \,d\xi} \\& \quad \leq C_{P}^{4}(2\theta\rho)^{2} \Upsilon^{2} \sup_{B_{\rho /2} (\xi_{0} )} \bigl\vert {X^{2} h} \bigr\vert ^{2} \\& \quad \leq C_{0}C_{P}^{4}(2 \theta)^{2}\Upsilon^{2} \fint_{B_{\rho} (\xi_{0} )} { \bigl\vert { ( {Xh} )} \bigr\vert ^{2} \,d\xi} \\& \quad \leq CC_{0}C_{P}^{4}(2 \theta)^{2} \Upsilon^{2} \fint_{B_{\rho} (\xi_{0} )} { \bigl\vert {W ( {Xh} )} \bigr\vert ^{2} \,d\xi} \\& \quad \leq C_{5}\theta^{2} \Upsilon^{2}, \end{aligned}$$
(4.29)
Using (4.25), we get and Applying Sobolev type inequality, we have The smallness conditions imply And then it follows where we have used the definition of F in the first step.
$$\begin{aligned} \begin{aligned} &\bigl[ {K \bigl( { \vert {u_{\xi_{0},\rho}} \vert + \bigl\vert { ( {Xu} ) _{\xi_{0},\rho}+\Upsilon(Xh)_{(\xi_{0},2\theta \rho)}} \bigr\vert } \bigr) \bigl( {1+ \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }+ \Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert } \bigr) } \bigr] ^{2m/(m-1) } \\ &\qquad {}\times\mu^{2} \bigl( {(2\theta \rho) }^{(2-m)(m-1)/m} \bigr) \\ &\quad \leq \bigl[ {K \bigl( {|u_{\xi_{0},\rho}|+ \bigl\vert { ( {Xu} ) _{\xi _{0},\rho}} \bigr\vert +1} \bigr) \bigl( {2+ \bigl\vert { ( {Xu} ) _{\xi_{0},\rho}} \bigr\vert } \bigr) } \bigr] ^{{2m/(m-1)} } \mu^{2} \bigl( { \rho}^{(2-m)(m-1)/m} \bigr) \\ &\quad \leq F^{2m/(m-1)} \bigl( 1+|u_{\xi_{0},\rho}|,{ \bigl\vert { ( {Xu} ) _{\xi_{0},\rho}} \bigr\vert } \bigr) \mu^{2} \bigl( { \rho}^{(2-m)(m-1)/m} \bigr) \end{aligned} \end{aligned}$$
$$\begin{aligned}& \bigl( {1+2M + \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }+\Upsilon(Xh)_{(\xi_{0},2\theta\rho)}} \bigr\vert } \bigr)^{m/(m-1)^{2} } (2\theta\rho)^{m - 1} \\& \quad \le \bigl( {2+2M + \bigl\vert { ( {Xu} ) _{\xi_{0},\rho }} \bigr\vert } \bigr)^{m/(m-1)^{2} } \mu^{2} \bigl(\rho^{(m - 1)/2} \bigr). \end{aligned}$$
$$\begin{aligned}& \biggl[ { \fint_{B_{2\theta\rho} (\xi_{0} )} { \bigl( { \vert Xu \vert ^{m} + \vert u \vert ^{r} + 1} \bigr)} \,d\xi} \biggr]^{m(r - 1)/r(m - 1)} \\& \quad \le \biggl[ {2^{m - 1} \fint_{B_{2\theta\rho} (\xi_{0} )} { \bigl( { \bigl\vert Xu - (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{m} } \bigr)} \,d\xi} \biggr]^{m(r - 1)/r(m - 1)} \\& \qquad {}+ \bigl( {2^{m - 1} \bigl\vert (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{m} } \bigr)^{m(r - 1)/r(m - 1)} \\& \qquad {} + \biggl[ { \fint_{B_{2\theta\rho} (\xi_{0} )} {2^{r - 1} \bigl( { \bigl\vert u - u_{\xi_{0} ,\rho} - (Xu)_{\xi_{0} ,\rho} \bigl(\xi^{1} - \xi_{0}^{1} \bigr) \bigr\vert ^{r} } \bigr)} \,d \xi} \biggr]^{m(r - 1)/r(m - 1)} \\& \qquad {}+ \bigl( {2^{r - 1} \bigl\vert u_{\xi _{0} ,\rho} + (Xu)_{\xi_{0} ,\rho} \bigl(\xi^{1} - \xi_{0}^{1} \bigr) \bigr\vert ^{r} + 1} \bigr)^{m(r - 1)/r(m - 1)} \\& \quad \le \bigl( {2^{m - 1} } \bigr)^{m(r - 1)/r(m - 1)} \biggl[(2\theta )^{ - Q} { \fint_{B_{\rho}(\xi_{0} )} { { \bigl\vert Xu - (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{m} } } \,d\xi} \biggr]^{m(r - 1)/r(m - 1)} \\& \qquad {}+ \bigl( {2^{m - 1} \bigl\vert (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{m} } \bigr)^{m(r - 1)/r(m - 1)} \\& \qquad {} + \bigl( {2^{r - 1} } \bigr)^{m(r - 1)/r(m - 1)} \biggl[(2 \theta)^{ - Q} { \fint_{B_{\rho}(\xi_{0} )} { \bigl\vert Xu - (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{m} } \,d\xi } \biggr]^{(r - 1)/(m - 1)} \\& \qquad {}+ \bigl( {2^{r - 1} \bigl\vert u_{\xi_{0} ,\rho} + (Xu)_{\xi_{0} ,\rho} \bigr\vert ^{r} + 1} \bigr)^{m(r - 1)/r(m - 1)} \\& \quad \le C(m,M,Q) \bigl[ {(2\theta)^{ - Q} \Phi \bigl( \xi_{0} ,\rho,(Xu)_{\xi _{0},\rho} \bigr)} \bigr]^{m(r - 1)/r(m - 1)} \\& \qquad {}+ C(m,Q) \bigl( {1 + \vert u_{\xi_{0} ,\rho} \vert + \bigl\vert (Xu)_{\xi_{0} ,\rho} \bigr\vert } \bigr)^{m(r - 1)/(m - 1)}. \end{aligned}$$
(4.30)
$$ (2\theta) ^{ - Q} \Phi \bigl(\xi_{0} ,\rho,(Xu)_{\xi_{0},\rho} \bigr) \leq1. $$
$$\begin{aligned}& \biggl[ { \fint_{B_{2\theta\rho} (\xi_{0} )} { \bigl( {|Xu|^{m} + |u|^{r} + 1} \bigr)} \,d\xi} \biggr]^{m(r - 1)/r(m - 1)} \mu^{2} \bigl( {(2\theta \rho) }^{(2-m)(m-1)/m} \bigr) \\& \quad \leq CF^{\frac{m}{m-1}} \bigl(|u_{\xi_{0},\rho }|, \bigl\vert (Xu)_{\xi_{0},\rho} \bigr\vert \bigr)\mu^{2} \bigl( { \rho}^{(2-m)(m-1)/m} \bigr), \end{aligned}$$
Combining all the above estimates with (4.26) and letting \(\varepsilon=\theta^{Q+4}\), we arrive at 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) 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 \(C_{6}\). □
$$ \Xi\leq C_{6} \bigl[ {\theta^{2} \Upsilon^{2}+F^{\frac{2m}{m-1}} \bigl(1+|u_{\xi_{0},\rho}|,M+ \bigl\vert (Xu)_{\xi_{0},\rho} \bigr\vert \bigr)\mu^{2} \bigl( {\rho }^{\sigma} \bigr)} \bigr] , $$
(4.31)
$$\begin{aligned} \Phi(\theta\rho) \leq& \theta^{2\tau} \bigl[ {\Phi ( \rho)+C_{7}F^{\frac{2m}{m-1}} \bigl(1+ \vert u_{\xi_{0},\rho} \vert ,M+ \bigl\vert (Xu)_{\xi_{0},\rho } \bigr\vert \bigr) } \mu^{2} \bigl( {\rho}^{\sigma} \bigr) \bigr] \\ :=& \theta^{2\tau} {\Phi (\rho)+K^{\ast} \bigl( \vert u_{\xi_{0},\rho} \vert , \bigl\vert (Xu)_{\xi_{0},\rho} \bigr\vert \bigr) }\mu^{2} \bigl( {\rho}^{\sigma} \bigr), \end{aligned}$$
(4.32)
For \(T > 0\), we find \(\Phi_{0} (T) > 0\) (depending on Q, N, λ, L, τ and ω) such that
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
where \(K_{0} (T): = K^{*}(2T,2T)\).
$$\begin{aligned}& \omega^{\frac{1}{2}} \bigl(2T,2\Phi_{0}^{\frac{1}{2}} (T) \bigr) + 2\Phi_{0}^{\frac{1}{2}} (T) \le \frac{1}{2} \delta, \quad \mbox{and} \end{aligned}$$
(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)
$$\begin{aligned}& \rho_{0} (T) \le\rho^{(2-m)(m-1)/m} _{1} (1 + 2T,1 + 2T), \end{aligned}$$
(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)
The rest of the process to obtain Theorem 1 is very similar to [13]. We omit it here.
5 Conclusion
For ELLIPTIC systems in Euclidean spaces, there are many literature sources to study partial regularity of weak solutions by the so-called \(\mathcal{A}\)-harmonic approximation method. With respect to SUB-ELLIPTIC equations and systems in Heisenberg groups, or general Carnot groups, there are some new difficulties and challenges which remain due to the lack of commutation and homogeneity of the horizontal vector fields. This work is concerned with nonlinear sub-elliptic systems (1.1) with Dini continuous coefficients which are weaker than Hölder continuous coefficients. Under sub-quadratic controllable growth conditions, a \(C^{1}\) partial regularity result is obtained by adapting the technique of \(\mathcal{A}\)-harmonic approximation to our setting. The new result generalizes the corresponding result for elliptic systems in Euclidean spaces and enriches regularity theory for nonlinear sub-elliptic systems in Carnot groups. The authors also believe that this work helps further study nonlinear sub-elliptic systems structured on Hörmander vector fields.
Declarations
Acknowledgements
We would like to express our gratitude to anonymous reviewers for their valuable comments.
Availability of data and materials
Not applicable.
Funding
This paper was partially supported by the National Natural Science Foundation of China (No. 11661006), supported by the Natural Science Foundation of Jiangxi Province (No. 20161BAB201032), and supported by the Science and Technology Planning Project of Jiangxi Province, China (No. GJJ160954).
Authors’ contributions
The work presented here was carried out in collaboration among all authors. JL found the motivation of this paper and suggested the outline of the proofs. DN provided good ideas for completing this paper. YQ and SM gave some valuable suggestions for completing the paper. All authors have contributed to, read and approved the final version.
Competing interests
The authors declare that there are no competing interests regarding the publication of this article.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
School of Mathematics and Computer Science, Gannan Normal University, Ganzhou, Jiangxi, 341000, P.R. China
(2)
School of Physics and Electronic Information Technology, Gannan Normal University, Ganzhou, Jiangxi, 341000, P.R. China
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