This paper is concerned with partial regularity to nonlinear subelliptic systems with Dini continuous coefficients under quadratic controllable growth conditions in the Heisenberg group ℍn. Based on a generalization of the technique of 𝒜-harmonic approximation introduced by Duzaar and Steffen, partial regularity to the sub-elliptic system is established in the Heisenberg group. Our result is optimal in the sense that in the case of Hölder continuous coefficients we establish the optimal Hölder exponent for the horizontal gradients of the weak solution on its regular set.

1. Introduction and Statements of Main Results

In this paper, we are concerned with partial regularity of weak solutions to nonlinear sub-elliptic systems of equations of second order in the Heisenberg group ℍn in divergence form, and more precisely, we consider the following systems:
(1)-∑i=12nXiAiα(ξ,u(ξ),Xu(ξ))=Bα(ξ,u(ξ),Xu(ξ))inΩ,
where Ω is a bounded domain in ℍn, X={X1,…,X2n}, the definition of Xi(i=1,…,2n) is to be seen in the next section (11), u=(u1,…,uN):Ω→ℝN, Aiα(ξ,u,p):ℝ2n+1×ℝN×ℝ2nN→ℝ2nN, and Bα(ξ,u,p):ℝ2n+1×ℝN×ℝ2nN→ℝN.

Under the coefficients Aiα assumed to be Dini continuous, the aim of this paper is to establish optimal partial regularity to the sub-elliptic system (1) in the Heisenberg group ℍn. Comparing Hölder continuous coefficients (see [1, 2] for the case of sub-elliptic systems), such assumption is weaker. More precisely, we assume for the continuity of Aiα with respect to the variables (ξ,u) that
(2)(1+|p|)-1|Aiα(ξ,u,p)-Aiα(ξ~,u~,p)|≤κ(|u|)μ(d(ξ,ξ~)+|u-u~|)
for all ξ,ξ~∈Ω, u,u~∈ℝN, and p∈ℝ2nN, where κ:(0,+∞)→[1,+∞) is monotone nondecreasing and μ:(0,+∞)→[0,+∞) is monotone nondecreasing and concave with μ(0+)=0. We also required that r→r-γμ(r) be nonincreasing for some γ∈(0,1) and that
(3)M(r)=∫0rμ(ρ)ρdρ<∞forsomer>0.

We adopt the method of 𝒜-harmonic approximation to the case of sub-elliptic systems in the Heisenberg groups and establish the optimal partial regularity result. Roughly speaking, assume additionally to the standard hypotheses (see precisely (H1), (H2), and (H4)) that (1+|p|)-1Aiα(ξ,u,p) satisfies (2) and (3). Let u∈HW1,2(Ω,ℝN) be a weak solution of the sub-elliptic system (1). Then, u is of class C1 outside a closed singular set Singu⊂Ω of the Haar measure 0. Furthermore, for ξ0∈Ω∖Singu, the derivative Xu of u has the modulus of continuity r→M(r) in a neighborhood of ξ0. Our result is optimal in the sense that in the case μ(r)=rγ, 0<γ<1, we have M(r)=γ-1rγ Hölder continuity Γ1,γ to be optimal in that case.

As is well known, even under reasonable assumptions on Aiα and Bα of the systems of equations, one cannot, in general, expect that weak solutions of (1) will be classical, that is, C2-solutions. This was first shown by de Giorgi [3]; we also refer the reader to Giaquinta [4] and Chen and Wu [5] for further discussion and additional examples. Then, the goal is to establish partial regularity theory. Moreover, a new method called 𝒜-harmonic approximation technique is introduced by Duzaar and Steffen in [6] and simplified by Duzaar and Grotowski in [7] to study elliptic systems with quadratic growth case. Then, similar results have been proved for more general Aiα or Bα in the Euclidean setting; see [8–11] for Hölder continuous coefficients and [12–15] for Dini continuous coefficients.

However, turning to sub-elliptic equations and systems in the Heisenberg groups ℍn, some new difficulties will arise. Already in the first step, trying to apply the standard difference quotient method, the main difference between Euclidean ℝn and Heisenberg groups ℍn becomes clear. Any time we use horizontal difference quotients (i.e., in the directions Xi), extra terms with derivatives in the T direction will arise due to noncommutativity (see (12)), but these cannot be controlled by using the initial assumptions on the weak solution. Several results were focused on those equations which have a bearing on basic vector fields on the Heisenberg group or, more generally, the Carnot group. Capogna [16, 17] studied the regularities for weak solutions to quasi-linear equations. Concretely, by a technique combining fractional difference quotients and fractional derivatives defined by Fourier transform, differentiability in the nonhorizontal direction, W2,2 estimate, and C∞ continuity of weak solutions are obtained; see [16] for the case of Heisenberg groups and [17] for Carnot groups. To sub-elliptic p-Laplace equations in Heisenberg groups, Marchi in [18–20] showed that Tu∈Llocp and X2u∈Lloc2 for 1+(1/5)<p<1+5 by using theories of Besov space and Bessel potential space. Domokos in [21, 22] improved these results for 1<p<4 employing the A. Zygmund theory related to vector fields. Recently, by meticulous arguments, Manfredi and Mingione in [23] and Mingione et al. in [24] proved Hölder regularity with regard to full Euclidean gradient for weak solutions and further C∞ continuity under the coefficients assumed to be smooth.

While regularities for weak solutions to sub-elliptic systems concerning vector fields are more complicated, Capogna and Garofalo in [25] showed the partial Hölder regularity for the horizontal gradient of weak solutions to quasilinear sub-elliptic systems -∑i=1kXi(Aiα(ξ,u)Xju)=Bα(ξ,u,Xu) with Xi, Xj(i,j=1,…,k) being horizontal vector fields in Carnot groups of step two, where Aiα and Bα satisfy the quadratic structure conditions. Their way relies mainly on generalization of classical direct method in the Euclidean setting. Shores in [26] considered a homogeneous quasi-linear system -∑i=1kXi(Aiα(ξ,u)Xju)=0 in the Carnot group with general step, where Aiα also satisfies the quadratic growth condition. She established higher differentiability and smoothness for weak solutions of the system with constant coefficients and deduced partial regularity for weak solutions of the original system. With respect to the case of nonquadratic growth, Föglein in [27] treated the homogeneous nonlinear system -∑i=12nXiAiα(ξ,Xu)=0 in the Heisenberg group under superquadratic structure conditions. She got a priori estimates for weak solutions of the system with constant coefficients and partial regularity for the horizontal gradient of weak solutions to the initial system. Later, Wang and Niu [1] and Wang and Liao [2] treated more general nonlinear sub-elliptic system in the Carnot groups under superquadratic growth conditions and subquadratic growth conditions, respectively.

The regularity results for sub-elliptic systems mentioned above require Hölder continuity with respect to the coefficients Aiα. When the assumption of Hölder continuity on Aiα is weakened to Dini continuity, how to establish partial regularity of weak solutions to nonlinear sub-elliptic systems in the Heisenberg group. This paper is devoted to this topic. To define weak solution to (1), we assume the following structure conditions on Aiα and Bα.

Aiα(ξ,u,p) is differentiable in p, and there exist some constants L such that
(4)|Ai,pβjα(ξ,u,p)|≤L,(ξ,u,p)∈Ω×ℝN×ℝ2nN.

Here, we write down Ai,pβjα(ξ,u,p)=(∂Aiα(ξ,u,p)/∂pβj).

Aiα(ξ,u,p) is uniformly elliptic; that is, for some λ>0, we have
(5)Ai,pβjα(ξ,u,p)ηiαηjβ≥λ|η|2,∀η∈ℝ2nN.

There exist a modulus of continuity μ:(0,+∞)→[0,+∞) and a nondecreasing function κ:[0,+∞)→[1,+∞) such that
(6)(1+|p|)-1|Aiα(ξ,u,p)-Aiα(ξ~,u~,p)|≤κ(|u|)μ(d(ξ,ξ~)+|u-u~|).

Without loss of generality, we can assume that κ≥1 and the following.

μ is nondecreasing with μ(0+)=0.

μ is concave; in the proof of the regularity theorem, we have to require that r→r-γμ(r) is nonincreasing for some exponent γ∈(0,1). We also require Dini's condition (2) which was already mentioned in the introduction.

M(r)=∫0r(μ(ρ)/ρ)dρ<∞ for some r>0.

In the present paper, we will apply the method of 𝒜-harmonic approximation adapting to the setting of Heisenberg groups to study partial regularity for the system (1). Since basic vector fields Xi of Lie algebras corresponding to the Heisenberg group are more complicated than gradient vector fields in the Euclidean setting, we have to find a different auxiliary function in proving Caccioppoli type inequality. Besides, the nonhorizontal derivatives of weak solutions will happen in the Taylor type formula in the Heisenberg group and cannot be effectively controlled in the present hypotheses. So, the method employing Taylor's formula in [12] is not appropriate in our setting. In order to obtain the desired decay estimate, we use the Poincaré type inequality in [28] as a replacement. And we obtain the following main result.

Theorem 1.

Assume that coefficients Aiα and Bα satisfy (H1)–(H4), (μ1)–(μ3) and that u∈HW1,2(Ω,ℝN) is a weak solution to the system (1); that is,
(8)∫ΩAiα(ξ,u,Xu)Xiϕαdξ=∫ΩBα(ξ,u,Xu)ϕαdξhhhhhhhhhhhhhhhhhhhhhhh∀ϕ∈C0∞(Ω,ℝN).
Then, there exists a relatively closed set
Sing
u⊂Ω such that u∈C1(Ω∖
Sing
u,ℝN). Furthermore,
Sing
u⊂Σ1∪Σ2 and Haar meas (Ω∖
Sing
u)=0, where
(9)Σ1={ξ0∈Ω:supr>0(|uξ0,r|+|(Xu)ξ0,r|)=∞},Σ2={×∫Br(ξ0)|Xu-(Xu)ξ0,r|2dξ>0ξ0∈Ω:limr→0+inf|Br(ξ0)|ℍn-1×∫Br(ξ0)|Xu-(Xu)ξ0,r|2dξ>0}.
In addition, for τ∈[γ,1) and ξ0∈Ω∖
Sing
u, the derivative Xu has the modulus of continuity r→rτ+M(r) in a neighborhood of ξ0.

2. Preliminaries

The Heisenberg group ℍn is defined as ℝ2n+1 endowed with the following group multiplication:
(10)·:ℍn×ℍn⟶ℍn,((ξ1,t),(ξ~1,t~))↦(ξ1+ξ~1,t+t~+12∑i=1n(xiy~i-x~iyi)),
for all ξ=(ξ1,t)=(x1,x2,…,xn,y1,y2,…,yn,t), ξ~=(ξ~1,t~)=(x~1,x~2,…,x~n,y~1,y~2,…,y~n,t~). This multiplication corresponds to addition in Euclidean ℝ2n+1. Its neutral element is (0,0), and its inverse to (ξ1,t) is given by (-ξ1,-t). Particularly, the mapping (ξ,ξ~)↦ξ·ξ~-1 is smooth, so (ℍn,·) is a Lie group.

The basic vector corresponding to its Lie algebra can be explicitly calculated by the exponential map and is given by
(11)Xi=∂∂xi-yi2∂∂t,Xi+n=∂∂yi+xi2∂∂t,T=∂∂t
for i=1,2,…,n, and note that the special structure of the commutators:
(12)[Xi,Xi+n]=-[Xi+n,Xi]=T,else[Xi,Xj]=0,[T,T]=[T,Xi]=0,
that is, (ℍn,·), is a nilpotent Lie group of step 2. X=(X1,…,X2n) is called the horizontal gradient and T the vertical derivative.

The pseudonorm is defined by
(13)∥(ξ1,t)∥=(|ξ1|4+t2)1/4,
and the metric induced by this pseudonorm is given by
(14)d(ξ~,ξ)=∥ξ-1·ξ~∥.
The measure used on ℍn is Haar measure, and the volume of the pseudoball BR(ξ0)={ξ∈ℍn:d(ξ0,ξ)<R} is given by
(15)|BR(ξ0)|ℍn=R2n+2|B1(ξ0)|ℍn≜ωnR2n+2.
The number
(16)Q=2n+2
is called the homogeneous dimension of ℍn.

The horizontal Sobolev spaces HW1,p(Ω)(1≤p<∞) are defined as
(17)HW1,p(Ω)={u∈Lp(Ω):Xiu∈Lp(Ω),i=1,2,…,2nu∈Lp(Ω):Xiu∈Lp(Ω)(Ω)}.
Then, HW1,p(Ω) is a Banach space with the norm
(18)∥u∥HW1,p(Ω)=∥u∥Lp(Ω)+∑i=12n∥Xiu∥Lp(Ω).HW01,p(Ω) is the completion of C0∞(Ω) under norm (18).

Lu [28] showed the following Poincaré type inequality related to Hörmander's vector fields for u∈HW1,q(BR(ξ0)), 1<q<Q, 1≤p≤qQ/(Q-q):
(19)(∮BR(ξ0)|u-uξ0,R|pdξ)1/p≤CpR(∮BR(ξ0)|Xu|qdξ)1/q,
where we write down ∮Br(ξ0)udξ=|Br(ξ0)|ℍn-1∫Br(ξ0)udξ here and there. Note the fact that the horizontal vectors Xi defined in (11) fit Hörmander's vector fields and that (19) is valid for p=q=2.

Following [12], for technical convenience, letting η(t)=μ2(2t), we have the corresponding properties for η: (η1) η is continuous, nondecreasing and η(0)=0; (η2) η is concave, and r→r-γη(r) is nonincreasing for some exponent γ∈(0,1); (η3) H(r):=4M2(2r)=[∫0r(η(ρ)/ρ)dρ]2<∞ for some r>0. Changing κ by a constant, but keeping κ≥1, we may assume the following: (η4) η(1)=1, implying η(t)≥t for t∈[0,1]. Also note that it implies that from (η2) and (η4), η(t)≤(γ2/4)H(t) for all t≥0.

Furthermore, the following inequality holds:
(20)sη(t)≤sη(s)+t,s∈[0,1],t>0.
The condition (H3) becomes
(21)|Aiα(ξ,u,p)-Aiα(ξ~,u~,p)|≤κ(|u|)η(d2(ξ,ξ~)+|u-u~|2)(1+|p|).
Moreover, we deduce the existence of a nonnegative modulus of continuity with ω(t,0)=0 for all t such that ω(s,t) is nondecreasing with respect to t for fixed s and ω2(s,t) is concave and nondecreasing with respect to s for fixed t. Also, we have for |u|+|Xu|≤M,
(22)|Ai,pβjα(ξ,u,p)-Ai,pβjα(ξ~,u~,p~)|≤ω(M,d2(ξ,ξ~)+|u-u~|2+|p-p~|2).
Using (H1) and (H2), we see that
(23)|Aiα(ξ,u,p)-Aiα(ξ,u,p~)|≤L|p-p~|,(24)(Aiα(ξ,u,p)-Aiα(ξ,u,p~))(p-p~)≥λ|p-p~|2.

In the sequel, the constant C may vary from line to line.

3. Caccioppoli Type Inequality

In this section, we present the following 𝒜-harmonic approximation lemma in the Heisenberg group introduced by Föglein [27] with p=2 as a special case and prove a Caccioppoli type inequality in our setting.

Lemma 2.

Let λ and L be fixed positive numbers and n,N∈ℕ with n≥2. If for any given ε>0, there exists δ=δ(n,N,λ,ε)∈(0,1] with the following properties:

for any 𝒜∈Bil(ℝ2nN) satisfying
(25)𝒜(ν,ν)≥λ|ν|2,𝒜(ν,ν-)≤L|ν||ν-|,ν,ν-∈ℝ2nN,

for any w∈HW1,2(Bρ(ξ0),ℝN) satisfying
(26)∮Bρ(ξ0)|Xw|2dξ≤1,|∮Bρ(ξ0)𝒜(Xw,Xφ)dξ|≤δsupBρ(ξ0)|Xφ|,hhhhhhhhhhhh∀φ∈C01(Bρ(ξ0),ℝN),

then, there exists an 𝒜-harmonic function h such that
(27)∮Bρ(ξ0)|Xh|2dξ≤1,ρ-2∮Bρ(ξ0)|h-w|2dξ≤ε.

Föglein [27] established a priori estimate for the weak solution u to homogeneous sub-elliptic systems with constant coefficients in the Heisenberg group (also see [25] for Carnot groups of step 2). We list it as follows:
(28)supBρ/2(ξ0)(|u|2+ρ2|Xu|2+ρ4|X2u|2)≤C0∮Bρ(ξ0)|Xu|2dξ.
In what follows, we let ρ1(s,t)=(1+s+t)-1κ(s+t)-1 and K1(s,t)=(1+t)4κ(s+t)4 for s,t≥0. Note that ρ1≤1 and that s→ρ1(s,t), t→ρ1(s,t) are nonincreasing functions.

Lemma 3.

Let u∈HW1,2(Ω,ℝN) be a weak solution to the system (1) under the conditions (H1)–(H4), (μ1)–(μ3). Then, for every ξ0=(x10,x20,…,xn0,y10,y20,…,yn0,t)∈Ω, u0∈ℝN, p0∈ℝ2nN, and 0<ρ<R<ρ1(|u0|,|p0|)≤1 such that BR(ξ0)⊂⊂Ω, the inequality
(29)∫Bρ(ξ0)|Xu-p0|2dξ≤Cc[1(R-ρ)2∫BR(ξ0)|u-u0-(ξ1-ξ01)p0|2dξ+F]
holds, where ξ1=(x1,x2,…,xn,y1,y2,…,yn) is the horizontal component of ξ=(ξ1,t)∈Ω and
(30)F=ωnRQK1(|u0|,|p0|)η(R2)+[∫BR(ξ0)(1+ur+|Xu|2)dξ]2(1-1/r).

Proof.

Let v=u-u0-(ξ1-ξ01)p0. Take a test function φ=ϕ2v in (8) with ϕ∈C0∞(BR(ξ0),ℝN) satisfying 0≤ϕ≤1, |∇ϕ|≤C/(R-ρ), and ϕ≡1 on Bρ(ξ0). Then, we have Xv=Xu-p0, |Xφ|≤ϕ|Xu-p0|+C/(R-ρ)|v|, and
(31)∫BR(ξ0)Aiα(ξ,u,Xu)ϕ2(Xu-p0)dξ=-2∫BR(ξ0)ϕXϕAiα(ξ,u,Xu)vdξ+∫BR(ξ0)Bα(ξ,u,Xu)φαdξ.
Adding this to the equations
(32)-∫BR(ξ0)Aiα(ξ,u,p0)ϕ2(Xu-p0)dξ=2∫BR(ξ0)ϕXϕAiα(ξ,u,p0)vdξ-∫BR(ξ0)Aiα(ξ,u,p0)Xφαdξ,0=∫BR(ξ0)Aiα(ξ0,u0,p0)Xφα.
It follows that by using the hypotheses (H1), (H3) (i.e., (23), (21), resp.), and (H4),
(33)∫BR(ξ0)[Aiα(ξ,u,Xu)-Aiα(ξ,u,p0)]ϕ2(Xu-p0)dξ=2∫BR(ξ0)[Aiα(ξ,u,p0)-Aiα(ξ,u,Xu)]ϕvXϕdξ+∫BR(ξ0)[Aiα(ξ,u0+(ξ1-ξ01)p0,p0)-Aiα(ξ,u,p0)(ξ,u0+(ξ1-ξ01)p0,p0)]Xφαdξ+∫BR(ξ0)[(ξ1-ξ01)Aiα(ξ0,u0,p0)-Aiα(ξ,u0+(ξ1-ξ01)p0,p0)]Xφαdξ+∫BR(ξ0)Bα(ξ,u,Xu)φαdξ≤I+II+III+IV+V,
where
(34)I=2L∫BR(ξ0)|Xu-p0||ϕ||v||Xϕ|dξ,II=(1+|p0|)κ(|u0|+R|p0|)×∫BR(ξ0)η(|v|2)|Xu-p0|ϕ2dξ,III=2(1+|p0|)κ(|u0|+R|p0|)×∫BR(ξ0)η(|v|2)|v||Xϕ||ϕ|dξ,IV=(1+|p0|)κ(|u0|+R|p0|)×∫BR(ξ0)η(R2(1+|p0|2))[|Xu-p0|ϕ2+2|ϕ||v||Xϕ||Xu-p0|ϕ2]dξ,V=C∫BR(ξ0)(1+|u|r-1+|Xu|2(1-1/r))φdξ.
Applying (H2), the left hand side of (33) can be estimated as
(35)λ∫BR(ξ0)|Xu-p0|2ϕ2dξ≤∫BR(ξ0)[Aiα(ξ,u,Xu)-Aiα(ξ,u,p0)]ϕ2(Xu-p0)dξ.
For ε>0 to be fixed later, we have, using Young's inequality,
(36)I≤ε∫BR(ξ0)|Xu-p0|2|ϕ|2dξ+CL2(R-ρ)2∫BR(ξ0)|v|2dξ.
Using Jensen's inequality, (20), and the fact that η(ts)≤tη(s) for t≥1, we arrive at
(37)(1+|p0|)2κ2(·)∫BR(ξ0)η(|v|2)dξ≤ωnRQ-2(1+|p0|)2κ2(·)R2η(∮BR(ξ0)|v|2dξ)≤ωnRQ-2[∮BR(ξ0)|v|2dξ+(1+|p0|)2κ2(·)R2η×((1+|p0|)2κ2(·)R2)∮BR(ξ0)|v|2dξ]≤R-2∫BR(ξ0)|v|2dξ+ωnRQ(1+|p0|)4κ4(·)η(R2),
where κ(·) is an abbreviation of the function κ(|u0|+|p0|). Also, note that the application of (20) in the second last inequality is possible by our choice R≤ρ1(|u0|+|p0|).

Using Young's inequality and (37) in II, we obtain
(38)II≤ε∫BR(ξ0)|Xu-p0|2|ϕ|2dξ+ε-1(1+|p0|)2κ2(·)∫BR(ξ0)η(|v|2)dξ≤ε∫BR(ξ0)|Xu-p0|2|ϕ|2dξ+1ε(R-ρ)2∫BR(ξ0)|v|2dξ+ε-1ωnRQ(1+|p0|)4κ4(·)η(R2).
And similarly, we see
(39)III≤4C(R-ρ)2∫BR(ξ0)|v|2dξ+(1+|p0|)2κ2(·)∫BR(ξ0)η(|v|2)dξ≤C(R-ρ)2∫BR(ξ0)|v|2dξ+ωnRQ(1+|p0|)4κ4(·)η(R2),IV≤ε∫BR(ξ0)|Xu-p0|2|ϕ|2dξ+4Cε(R-ρ)2∫BR(ξ0)|v|2dξ+ε-1ωnRQ(1+|p0|)2κ2(·)η×(∮BR(ξ0)R2(1+|p0|2)dξ)≤ε∫BR(ξ0)|Xu-p0|2|ϕ|2dξ+Cε(R-ρ)2∫BR(ξ0)|v|2dξ+ε-1ωnRQ(1+|p0|)4κ4(·)η(R2).
Here we have used κ≥1 in the last inequality.

By Hölder's inequality, (19), and Young's inequality, one gets(40)V≤C(∫BR(ξ0)|φ|rdξ)1/r×(∫BR(ξ0)(1+|u|r+|Xu|2)dξ)(r-1)/r≤C(∫BR(ξ0)|Xφ|2dξ)1/2×(∫BR(ξ0)(1+|u|r+|Xu|2)dξ)(r-1)/r≤ε∫BR(ξ0)|Xφ|2dξ+C(ε)×(∫BR(ξ0)(1+|u|r+|Xu|2)dξ)2(r-1)/r≤ε∫BR(ξ0)|Xu-p0|2|ϕ|2dξ+Cε(R-ρ)2∫BR(ξ0)|v|2dξ+C(ε)(∫BR(ξ0)(1+|u|r+|Xu|2)dξ)2(r-1)/r,
where we have used the fact that |Xφ|≤ϕ|Xu-p0|+C/(R-ρ)|v|.

Applying these estimates to (37), we obtain
(41)(λ-4ε)∫BR(ξ0)|Xu-p0|2ϕ2dξ≤C(L,ε)(R-ρ)2∫BR(ξ0)|v|2dξ+(ε-1+2)ωnRQ(1+|p0|)4κ4(·)η(R2)+C(ε)(∫BR(ξ0)(1+|u|r+|Xu|2)dξ)2(r-1)/r.
Choosing ε=λ/8, we obtain the desired inequality (29).

4. Proof of the Main Theorem

In this section, we will complete the proof of the partial regularity results via the following lemmas. In the sequel, we always suppose that u∈HW1,2(Ω,ℝN) is a weak solution to (1) with the assumptions of (H1)–(H4) and (μ1)–(μ3).

Lemma 4.

Let Bρ(ξ0)⊂⊂Ω with ρ≤ρ1(|u0|,|p0|) and φ∈C0∞(Bρ(ξ0),ℝN) satisfying |φ|≤ρ2 and supBρ(ξ0)|Xφ|≤1. Then, there exists a constant C1≥1 such that
(42)∮Bρ(ξ0)Ai,pβjα(ξ0,u0,p0)(Xu-p0)Xφαdξ≤C1[η(ρ2)Φ(ξ0,ρ,p0)+ω(|u0|+|p0|,Φ(ξ0,ρ,p0))Φ1/2(ξ0,ρ,p0)+K1(|u0|,|p0|)η(ρ2)]supBρ(ξ0)|Xφ|.

Proof.

Using the fact that ∫Bρ(ξ0)Aiα(ξ0,u0,p0)Xφαdξ=0 and the weak form (8), we deduce
(43)∮Bρ(ξ0)[∫01Ai,pβjα(ξ0,u0,θXu+(1-θ)p0)×(Xu-p0)dθ∫01Ai,pβjα(ξ0,u0,θXu+(1-θ)p0)]Xφαdξ=∮Bρ(ξ0)[Aiα(ξ0,u0,Xu)-Aiα(ξ0,u0,p0)]Xφαdξ=∮Bρ(ξ0)[Aiα(ξ0,u0,Xu)-Aiα(ξ,u,Xu)]Xφαdξ+∮Bρ(ξ0)Bα(ξ,u,Xu)φαdξ.
It yields
(44)∮Bρ(ξ0)Ai,pβjα(ξ0,u0,p0)(Xu-p0)Xφαdξ=∮Bρ(ξ0)[∫01(Ai,pβjα(ξ0,u0,p0)-Ai,pβjα(ξ0,u0,θXu+(1-θ)p0))×(Xu-p0)dθ∫01(Ai,pβjα(ξ0,u0,p0)]dξsupBρ(ξ0)|Xφ|+∮Bρ(ξ0)[Aiα(ξ0,u0,Xu)-Aiα(ξ,u0+p0(ξ-ξ0),Xu)]supBρ(ξ0)|Xφ|+∮Bρ(ξ0)[Aiα(ξ,u0+p0(ξ-ξ0),Xu)-Aiα(ξ,u,Xu)]supBρ(ξ0)|Xφ|+∮Bρ(ξ0)Bα(ξ,u,Xu)φαdξ∶=I′+II′+III′+IV′.
Using (22), Hölder's inequality, the fact that t→ω2(s,t) is concave, and Jensen's inequality, we have
(45)I′≤supBρ(ξ0)|Xφ|∮Bρ(ξ0)ω(|u0|+|p0|,|Xu-p0|2)|Xu-p0|dξ≤supBρ(ξ0)|Xφ|[∮Bρ(ξ0)ω2(|u0|+|p0|,|Xu-p0|2)dξ]1/2×[∮Bρ(ξ0)|Xu-p0|2dξ]1/2≤ω(|u0|+|p0|,Φ(ξ0,ρ,p0))Φ1/2(ξ0,ρ,p0)supBρ(ξ0)|Xφ|.
Similarly, using (21) and the fact that η(ts)≤tη(s) for t≥1, we obtain
(46)II′≤supBρ(ξ0)|Xφ|κ(·)η(ρ2(1+|p0|2))×∮Bρ(ξ0)(1+|Xu|)dξ≤supBρ(ξ0)|Xφ|κ(·)η(ρ2(1+|p0|)2)×∮Bρ(ξ0)(1+|p0|+|Xu-p0|)dξ≤supBρ(ξ0)|Xφ|[(∮Bρ(ξ0)|Xu-p0|2dξ)+κ2(·)(1+|p0|)2η(ρ2)+κ(·)(1+|p0|)3η(ρ2)(∮Bρ(ξ0)|Xu-p0|2dξ)]≤[Φ(ξ0,ρ,p0)+2κ2(·)(1+|p0|)3η(ρ2)]×supBρ(ξ0)|Xφ|,
where we have used the fact that η(ρ2)≤η(ρ2) which follows from the nondecreasing property of the function η(t), (η4), and our assumption ρ≤ρ1≤1.

In the same way, it follows that by using (21), (37), and (19),
(47)III′≤supBρ(ξ0)|Xφ|κ(·)∮Bρ(ξ0)η(|v|2)(1+|Xu|)dξ≤supBρ(ξ0)|Xφ|[∮Bρ(ξ0)|Xu-p0|2dξ+κ2(·)∮Bρ(ξ0)η(|v|2)dξ+κ(·)(1+|p0|)∮Bρ(ξ0)η(|v|2)dξ]≤supBρ(ξ0)|Xφ|[+2ρ-2∮Bρ(ξ0)|v|2dξ+κ4(·)η(ρ2)Φ(ξ0,ρ,p0)+2ρ-2∮Bρ(ξ0)|v|2dξ+κ4(·)η(ρ2)+κ2(·)(1+|p0|)2η(ρ2)+2ρ-2∮Bρ(ξ0)|v|2dξ+κ4(·)η(ρ2)]≤[(1+2Cp)Φ(ξ0,ρ,p0)+2κ4(·)(1+|p0|)2η(ρ2)].
Using Hölder's inequality, (19), and Young's inequality, we have
(48)IV′≤C∮Bρ(ξ0)(1+|u|r-1+|Xu|2(1-1/r))|φ|dξ≤C∮Bρ(ξ0)|Xu|2(1-1/r)|φ|dξ+C∮Bρ(ξ0)|u-u0-p0(ξ1-ξ01)|r-1|φ|dξ+Cρ2[1+(|u0|+|p0|)r-1]≤C(∮Bρ(ξ0)|Xu|2dξ)(1-1/r)(∮Bρ(ξ0)|φ|rdξ)1/r+C(∮Bρ(ξ0)|u-u0-p0(ξ1-ξ01)|rdξ)(1-1/r)×(∮Bρ(ξ0)|φ|rdξ)1/r+Cρ2[1+(|u0|+|p0|)r-1]≤C(∮Bρ(ξ0)|Xu|2dξ)(1-1/r)(∮Bρ(ξ0)|φ|rdξ)1/r+(∮Bρ(ξ0)|Xu-p0|2dξ)(r/2)(1-1/r)×(∮Bρ(ξ0)|φ|rdξ)1/r+Cρ2[1+(|u0|+|p0|)r-1]≤C∥u∥HW1,2(∮Bρ(ξ0)|Xu-p0|2dξ)(1-1/r)×(∮Bρ(ξ0)|φ|rdξ)1/r+Cρ[1+(|u0|+|p0|)r-1]≤C∥u∥HW1,2(Bρ(ξ0))(∮Bρ(ξ0)|Xu-p0|2dξ)+Cρ2r+Cρ2[1+(|u0|+|p0|)r-1]≤C2Φ(ξ0,ρ,p0)+Cρ2(1+|u0|+|p0|)r-1≤C2Φ(ξ0,ρ,p0)+Cκ(·)(1+|u0|+|p0|)2η(ρ2),
where we have used the assumption (η4) and the fact that r=2Q/(Q-2)=(2n+4)/2n≤3 and C2=C∥u∥HW1,2(Bρ(ξ0))≥1. Combining these estimates, we obtain the conclusion with C1=(1+C2+2Cp)≥1.

Lemma 5.

Assume that the conditions of Lemma 2 and the following smallness conditions hold:
(49)ω(|uξ0,ρ|+|(Xu)ξ0,ρ|,Φ(ξ0,ρ,(Xu)ξ0,ρ))+Φ1/2(ξ0,ρ,(Xu)ξ0,ρ)≤δ2,(50)C3K12(|uξ0,ρ|,|(Xu)ξ0,ρ|)η(ρ2)≤δ2
with C3=8C12C5, together with
(51)ρ≤ρ1(1+|uξ0,ρ|,1+|(Xu)ξ0,ρ|).
Then, the following growth condition holds for τ∈[γ,1)(52)Φ(ξ0,θρ)≤θ2τΦ(ξ0,ρ)+K*(|uξ0,2θρ|,|(Xu)ξ0,2θρ|)η(ρ2),
where one abbreviates Φ(ξ0,r)=Φ(ξ0,r,(Xu)ξ0,r) and K*(s,t)=K(s,t)+(2+s+t)2(r-1) with K(s,t)=(4δ-2+2QCc)K12(1+s,1+t).

Proof.

We define w=[u-uξ0,ρ-p0(ξ1-ξ01)]σ-1, where
(53)σ=C1Φ(ξ0,ρ,p0)+4δ-2K12(|u0|,|p0|)η(ρ2).
Then, we have Xw=σ-1(Xu-p0). Now, we consider Bρ(ξ0)⊂⊂Ω such that ρ≤ρ1(|u0|,|p0|). Applying Lemma 4 on Bρ(ξ0) to u, we have for any φ∈C0∞(Bρ(ξ0),ℝN),
(54)∮Bρ(ξ0)|Xw|2dξ=σ-2Φ(ξ0,ρ,p0)≤1C12≤1,(55)∮Bρ(ξ0)Ai,pβjα(ξ0,u0,p0)XwXφdξ≤[δ2Φ1/2(ξ0,ρ,p0)+ω(|u0|+|p0|,Φ(ξ0,ρ,p0))+δ2]supBρ(ξ0)|Xφ|.
In consideration of the small condition (49), we see that (54) and (55) imply conditions (26) in Lemma 2. Also note that (H1) and (H3) imply condition (25). So, there exists an Ai,pβjα(ξ0,u0,p0)-harmonic function h∈HW1,2(Bρ(ξ0),ℝN) such that
(56)∮Bρ(ξ0)|Xh|2dξ≤1,ρ-2∮Bρ(ξ0)|w-h|2dξ≤ε.
Taking u0=uξ0,2θρ,θ∈(0,1/4] and replacing p0 by p0+σ(Xh)ξ0,2θρ, we use Lemma 3 to obtain
(57)∫Bθρ(ξ0)|Xu-p0-σ(Xh)ξ0,2θρ|2dξ≤Cc[1(θρ)2∫B2θρ(ξ0)|u-uξ0,2θρ-(p0+σ(Xh)ξ0,2θρ)×(ξ1-ξ01)|2dξ+F∫B2θρ(ξ0)1(θρ)2],
where
(58)F=ωn(2θρ)QK1(|uξ0,2θρ|,|p0+σ(Xh)ξ0,2θρ|)η((2θρ)2)+[∫B2θρ(ξ0)(1+ur+|Xu|2)dξ]2(1-1/r).
Using the fact that u-(p0+σ(Xh)ξ0,2θρ)(ξ1-ξ01) has mean value uξ0,2θρ on the ball B2θρ(ξ0), the definition of w, and (19), we have
(59)1(θρ)2∮B2θρ(ξ0)|u-uξ0,2θρ-(p0+σ(Xh)ξ0,2θρ)(ξ1-ξ01)|2dξ≤4σ2(2θρ)2∮B2θρ(ξ0)|w-hξ0,2θρ-(Xh)ξ0,2θρ(ξ1-ξ01)|2dξ≤4σ2(2θρ)2[∮B2θρ(ξ0)|w-h|2dξ+∮B2θρ(ξ0)|h-hξ0,2θρ-(Xh)ξ0,2θρ(ξ1-ξ01)|2dξ∮B2θρ(ξ0)]≤4σ2[(2θ)-Q-2ε+Cp∮B2θρ(ξ0)|Xh-(Xh)ξ0,2θρ|2dξ]≤4σ2[(2θ)-Q-2ε+Cp2(2θρ)2∮B2θρ(ξ0)|X2h|2dξ]≤4σ2[(2θ)-Q-2ε+Cp2(2θ)2C0]≤C4(θ-Q-2ε+θ2)[(|uξ0,2θρ|,|p0|)Φ(ξ0,ρ,p0)+4δ-2K12(|uξ0,2θρ|,|p0|)η(ρ2)],

where C4:=C4(Q,λ,L)≥1. Note that in the second last inequality we have used the fact that
(60)∮B2θρ(ξ0)|X2h|dξ≤supBρ(ξ0)|X2h|≤C0ρ-2∮Bρ(ξ0)|Xh|2dξ≤C0ρ-2.
In consideration of the fact that r=2Q/(Q-2)>2, Q≥4 and the assumptions θ∈(0,1/4] and Φ≤1, it follows that
(61)[∮B2θρ(ξ0)(1+ur+|Xu|2)dξ]2(1-1/r)≤C[∮B2θρ(ξ0)|Xu-p0|2dξ]2(1-1/r)+C(∮B2θρ(ξ0)|Xu|2dξ)r-1+(1+|p0|4(1-1/r))≤C[(2θ)-2Q(1-1/r)Φ(ξ0,ρ,p0)2(1-1/r)+(2θ)-Q(r-1)Φ(ξ0,ρ,p0)r-1]+(1+|p0|2(1-1/r)+|p0|4(1-1/r))≤C(2θ)-Q(r-1)Φ(ξ0,ρ,p0)2(1-1/r)+(1+|p0|2(1-1/r))2.

Let P=p0+σ(Xh)ξ0,2θρ with p0=(Xu)ξ0,2θρ. Combining these estimates (57)–(61) and considering the small condition (51) (it implies ρ≤ρ1(|uξ0,2θρ|,|P|); see (64) and (65)), we deduce that
(62)Φ(ξ0,θρ)≤|Bθρ(ξ0)|ℍn-1∫B2θρ(ξ0)|Xu-P|2dξ≤Cc2Q(θρ)2∮B2θρ(ξ0)|u-uξ0,2θρ-(p0+σ(Xh)ξ0,2θρ)×(ξ1-ξ01)|2dξ+2QCcK1(|uξ0,2θρ|,|p0+σ(Xh)ξ0,2θρ|)η(ρ2)+Cc(2θρ)2Q(1-1/r)(θρ)Q×[∮B2θρ(ξ0)(1+ur+|Xu|2)dξ]2(1-1/r)≤2QC4Cc(θ-Q-2ε+θ2)×[(|uξ0,2θρ|,|(Xu)ξ0,2θρ|)η(ρ2))Φ(ξ0,ρ)+4δ-2K12×(|uξ0,2θρ|,|(Xu)ξ0,2θρ|)η(ρ2)Φ(ξ0,ρ)+4δ-2K12]+2QCcK1(|uξ0,2θρ|,|(Xu)ξ0,2θρ+σ(Xh)ξ0,2θρ|)η(ρ2)+[(1+|(Xu)ξ0,2θρ|2(1-1/r))(1+|(Xu)ξ0,2θρ|2(1-1/r))22QCc(2θ)2-Q(r-1)Φ(ξ0,ρ)2(1-1/r)+(1+|(Xu)ξ0,2θρ|2(1-1/r))2]ρ2.
We now specify ε=θQ+4, θ∈(0,1/4] such that 2Q+1C4Ccθ2≤θ2τ. Note that the small condition (50) implies σ2C5≤1 with C5=max{C0,Cc2Q(2θ)-(Q2+4)/(Q-2)}, and it yields
(63)2QCc(2θ)2-Q(r-1)Φ(ξ0,ρ)2(1-1/r)≤1,(64)|σ(Xh)ξ0,2θρ|≤σsupB2θρ(ξ0)|Xh|≤σC0(∮Bρ(ξ0)|Xh|2dξ)≤σC0≤1,
where we have used the a priori estimate (28) for the 𝒜-harmonic function h. Furthermore, using (19) and recalling the definition of σ and C1, we have
(65)|uξ0,2θρ|≤|uξ0,ρ|+|uξ0,2θρ-uξ0,ρ|≤|uξ0,ρ|+(2θ)-Q/2×(∮Bρ(ξ0)|u-(Xu)ξ0,ρ(ξ1-ξ01)-uξ0,ρ|2dξ)1/2≤|uξ0,ρ|+(2θ)-Q/2ρCpΦ1/2(ξ0,ρ)≤|uξ0,ρ|+σCpC1(2θ)Q/2≤|uξ0,ρ|+σC5≤|uξ0,ρ|+1.
Combining these estimates with (62), we have
(66)Φ(ξ0,θρ)≤θ2τΦ(ξ0,ρ)+[4δ-2K12(|uξ0,2θρ|,|(Xu)ξ0,2θρ|)+2QCcK1(|uξ0,2θρ|,|(Xu)ξ0,2θρ+σ(Xh)ξ0,2θρ|)]η(ρ2)+[1+(1+|(Xu)ξ0,2θρ|2(1-1/r))2]η(ρ2)≤θ2τΦ(ξ0,ρ)+K(|uξ0,2θρ|,|(Xu)ξ0,2θρ|)η(ρ2)+(2+|uξ0,2θρ|+|(Xu)ξ0,2θρ|)2(r-1)η(ρ2)≤θ2τΦ(ξ0,ρ)+K*(|uξ0,2θρ|,|(Xu)ξ0,2θρ|)η(ρ2).
Then, the proof of Lemma 5 is complete.

For T>0, we find Φ0(T)>0 (depending on Q, N, λ, L, τ, and ω) such that
(67)ω2(2T,2Φ0(T))+2Φ0(T)≤12δ2,C1Φ0(T)≤θQ(1-θτ)2T2.
With Φ0(T) from (67), we choose ρ0(T)∈(0,1] (depending on Q, N, λ, L, τ, ω, η, and κ) such that
(68)ρ0(T)≤ρ1(1+2T,1+2T),C3K12(2T,2T)η(ρ0(T)2)≤δ2,K0(T)η(ρ0(T)2)≤(θ2γ-θ2τ)Φ0(T),2(1+Cp)K0(T)H(ρ0(T)2)≤θQ(1-θγ)2(θ2γ-θ2τ)T2,
where K0(T):=K*(2T,2T).

By the proof method of of Lemma 5.1 in [12] and conditions (67)-(68), Lemma 6 can be proved. As is well known, it is sufficient to complete the proof of Theorem 1 once we obtain Lemma 6.

Lemma 6.

Assume that for some T0>0 and Bρ(ξ0)⊂⊂Ω one has

|uξ0,ρ|+|(Xu)ξ0,ρ|≤T0,

ρ≤ρ0(T0),

Φ(ξ0,ρ)≤Φ0(T0).

Then, the small conditions (49)–(51) are satisfied on the balls Bθjρ(ξ0) for j∈N∪{0}. Moreover, the limit Λξ0=limj→∞(Xu)ξ0,θjρ exists, and the inequality
(69)∮Bρ(ξ0)|Xu-Λξ0|2dξ≤C6((rρ)2τΦ(ξ0,ρ)+H(r2))
is valid for 0<r≤ρ with a constant C6=C6(Q,N,λ,L,τ,andT0).
Proof.

The proof is very similar to the proof of Lemma 5.1 in [12]. We omit it here.

Acknowledgments

The project was supported by the National Natural Science Foundation of China (no. 11201081 and no. 11126294) and by the Science and Technology Planning Project of Jiangxi Province, China, no. GJJ13657.

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