Partial Regularity for Nonlinear Subelliptic Systems with Dini Continuous Coefficients in Heisenberg Groups

Abstract

This paper is concerned with partial regularity to nonlinear subelliptic systems with Dini continuous coefficients under quadratic controllable growth conditions in the Heisenberg group ℍ^𝑛. 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 ℍ^𝑛 in divergence form, and more precisely, we consider the following systems: −^2𝑛_𝑖=1𝑋_𝑖𝐴^𝛼_𝑖(𝜉,𝑢(𝜉),𝑋𝑢(𝜉))=𝐵^𝛼(𝜉,𝑢(𝜉),𝑋𝑢(𝜉))inΩ, (1) where Ω is a bounded domain in ℍ^𝑛, 𝑋={𝑋₁,…,𝑋_2𝑛}, the definition of 𝑋_𝑖(𝑖=1,…,2𝑛) is to be seen in the next section (11), 𝑢=(𝑢¹,…,𝑢^𝑁)∶Ω→ℝ^𝑁, 𝐴^𝛼_𝑖(𝜉,𝑢,𝑝)∶ℝ^2𝑛+1×ℝ^𝑁×ℝ^2𝑛𝑁→ℝ^2𝑛𝑁, and 𝐵^𝛼(𝜉,𝑢,𝑝)∶ℝ^2𝑛+1×ℝ^𝑁×ℝ^2𝑛𝑁→ℝ^𝑁.

Under the coefficients 𝐴^𝛼_𝑖 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 ℍ^𝑛. 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 𝐴^𝛼_𝑖 with respect to the variables (𝜉,𝑢) that (1+𝑝)⁻¹𝐴^𝛼_𝑖𝜉,𝑢,𝑝−𝐴^𝛼_𝑖̃𝑢𝜇𝑑𝜉̃+|𝑢|𝜉,𝑢,𝑝≤𝜅(||)𝜉,𝑢− (2) for all 𝜉,𝜉̃∈Ω, 𝑢,𝑢∈ℝ^𝑁, and 𝑝∈ℝ^2𝑛𝑁, 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 𝑀(𝑟)=^𝑟₀𝜇𝜌𝜌𝑑𝜌<∞forsome𝑟>0. (3)

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|)⁻¹𝐴^𝛼_𝑖(𝜉,𝑢,𝑝) satisfies (2) and (3). Let 𝑢∈𝐻𝑊^1,2(Ω,ℝ^𝑁) 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)=𝑟^𝛾, 0<𝛾<1, we have 𝑀(𝑟)=𝛾⁻¹𝑟^𝛾 Hölder continuity Γ^1,𝛾 to be optimal in that case.

As is well known, even under reasonable assumptions on 𝐴^𝛼_𝑖 and 𝐵^𝛼 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 𝐴^𝛼_𝑖 or 𝐵^𝛼 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 ℍ^𝑛, some new difficulties will arise. Already in the first step, trying to apply the standard difference quotient method, the main difference between Euclidean ℝ^𝑛 and Heisenberg groups ℍ^𝑛 becomes clear. Any time we use horizontal difference quotients (i.e., in the directions 𝑋_𝑖), 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, 𝑊^2,2 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∈𝐿^𝑝_loc and 𝑋²𝑢∈𝐿²_loc 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 −∑^𝑘_𝑖=1𝑋_𝑖(𝐴^𝛼_𝑖(𝜉,𝑢)𝑋_𝑗𝑢)=𝐵^𝛼(𝜉,𝑢,𝑋𝑢) with 𝑋_𝑖, 𝑋_𝑗(𝑖,𝑗=1,…,𝑘) being horizontal vector fields in Carnot groups of step two, where 𝐴^𝛼_𝑖 and 𝐵^𝛼 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 −∑^𝑘_𝑖=1𝑋_𝑖(𝐴^𝛼_𝑖(𝜉,𝑢)𝑋_𝑗𝑢)=0 in the Carnot group with general step, where 𝐴^𝛼_𝑖 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 −∑^2𝑛_𝑖=1𝑋_𝑖𝐴^𝛼_𝑖(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 𝐴^𝛼_𝑖. When the assumption of Hölder continuity on 𝐴^𝛼_𝑖 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 𝐴^𝛼_𝑖 and 𝐵^𝛼.(H1)𝐴^𝛼_𝑖(𝜉,𝑢,𝑝) is differentiable in 𝑝, and there exist some constants 𝐿 such that 𝐴^𝛼_{𝑖,𝑝𝑗𝛽}𝜉,𝑢,𝑝≤𝐿,𝜉,𝑢,𝑝∈Ω×ℝ^𝑁×ℝ^2𝑛𝑁. (4) Here, we write down 𝐴^𝛼_{𝑖,𝑝𝑗𝛽}(𝜉,𝑢,𝑝)=(𝜕𝐴^𝛼_𝑖(𝜉,𝑢,𝑝)/𝜕𝑝^𝑗_𝛽). (H2)𝐴^𝛼_𝑖(𝜉,𝑢,𝑝) is uniformly elliptic; that is, for some 𝜆>0, we have 𝐴^𝛼_{𝑖,𝑝𝑗𝛽}𝜉,𝑢,𝑝𝜂^𝛼_𝑖𝜂^𝛽_𝑗≥𝜆𝜂²,∀𝜂∈ℝ^2𝑛𝑁. (5)(H3) There exist a modulus of continuity 𝜇∶(0,+∞)→[0,+∞) and a nondecreasing function 𝜅∶[0,+∞)→[1,+∞) such that (1+𝑝)⁻¹𝐴^𝛼_𝑖𝜉,𝑢,𝑝−𝐴^𝛼_𝑖̃𝑢𝜇𝑑𝜉̃+|𝑢|.𝜉,𝑢,𝑝≤𝜅(||)𝜉,𝑢− (6)(H4)𝐵^𝛼 satisfies quadratic controllable growth condition 𝐵^𝛼𝜉,𝑢,𝑝≤𝐶1+|𝑢|^𝑟−1+𝑝^{2(1−1/𝑟)}, (7) where 𝑟=2𝑄/(𝑄−2) because 𝑄>2; see (16).

Without loss of generality, we can assume that 𝜅≥1 and the following.(𝜇1)𝜇 is nondecreasing with 𝜇(0+)=0.(𝜇2)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.(𝜇3)𝑀(𝑟)=∫^𝑟₀(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 𝑋_𝑖 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 𝐴^𝛼_𝑖 and 𝐵^𝛼 satisfy (H1)–(H4), (μ1)–(μ3) and that 𝑢∈𝐻𝑊^1,2(Ω,ℝ^𝑁) is a weak solution to the system (1); that is, _Ω𝐴^𝛼_𝑖(𝜉,𝑢,𝑋𝑢)𝑋_𝑖𝜙^𝛼𝑑𝜉=_Ω𝐵^𝛼(𝜉,𝑢,𝑋𝑢)𝜙^𝛼𝑑𝜉∀𝜙∈𝐶^∞₀Ω,ℝ^𝑁. (8) Then, there exists a relatively closed set Sing𝑢⊂Ω such that 𝑢∈𝐶¹(Ω⧵Sing𝑢,ℝ^𝑁). Furthermore, Sing𝑢⊂Σ₁∪Σ₂ and Haar meas (Ω⧵Sing𝑢)=0, where Σ₁=𝜉₀∈Ω∶sup_𝑟>0𝑢_𝜉0,𝑟+(𝑋𝑢)_𝜉0,𝑟,Σ=∞₂=𝜉₀∈Ω∶lim_𝑟→0+inf𝐵_𝑟𝜉_0ℍ⁻¹_𝑛×_{𝐵𝑟(𝜉0)}𝑋𝑢−(𝑋𝑢)_𝜉0,𝑟²𝑑𝜉>0. (9) In addition, for 𝜏∈[𝛾,1) and 𝜉₀∈Ω⧵Sing𝑢, the derivative 𝑋𝑢 has the modulus of continuity 𝑟→𝑟^𝜏+𝑀(𝑟) in a neighborhood of 𝜉₀.

2. Preliminaries

The Heisenberg group ℍ^𝑛 is defined as ℝ^2𝑛+1 endowed with the following group multiplication: ⋅∶ℍ^𝑛×ℍ^𝑛⟶ℍ^𝑛,𝜉¹,𝑡,𝜉̃¹,𝑡̃↦𝜉¹+𝜉̃¹̃1,𝑡+𝑡+2^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖−𝑥_𝑖𝑦_𝑖, (10) for all  𝜉=(𝜉¹,𝑡)=(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦₁,𝑦₂,…,𝑦_𝑛,𝑡), 𝜉̃=(𝜉̃¹,𝑡)̃=(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦₁,𝑦₂,…,𝑦_𝑛,𝑡)̃. This multiplication corresponds to addition in Euclidean ℝ^2𝑛+1. Its neutral element is (0,0), and its inverse to (𝜉¹,𝑡) is given by (−𝜉¹,−𝑡). Particularly, the mapping (𝜉,𝜉̃)↦𝜉⋅𝜉̃⁻¹ is smooth, so (ℍ^𝑛,⋅) is a Lie group.

The basic vector corresponding to its Lie algebra can be explicitly calculated by the exponential map and is given by 𝑋_𝑖=𝜕𝜕𝑥_𝑖−𝑦_𝑖2𝜕𝜕𝑡,𝑋_𝑖+𝑛=𝜕𝜕𝑦_𝑖+𝑥_𝑖2𝜕𝜕𝜕𝑡,𝑇=𝜕𝑡 (11) for 𝑖=1,2,…,𝑛, and note that the special structure of the commutators: 𝑋_𝑖,𝑋_𝑖+𝑛=−𝑋_𝑖+𝑛,𝑋_𝑖=𝑇,else𝑋_𝑖,𝑋_𝑗[]==0,𝑇,𝑇𝑇,𝑋_𝑖=0, (12) that is, (ℍ^𝑛,⋅), is a nilpotent Lie group of step 2. 𝑋=(𝑋₁,…,𝑋_2𝑛) is called the horizontal gradient and 𝑇 the vertical derivative.

The pseudonorm is defined by 𝜉¹,𝑡=𝜉¹⁴+𝑡^21/4, (13) and the metric induced by this pseudonorm is given by 𝑑𝜉̃,𝜉=𝜉⁻¹⋅𝜉̃. (14) The measure used on ℍ^𝑛 is Haar measure, and the volume of the pseudoball 𝐵_𝑅(𝜉₀)={𝜉∈ℍ^𝑛∶𝑑(𝜉₀,𝜉)<𝑅} is given by 𝐵_𝑅𝜉_0ℍ𝑛=𝑅^2𝑛+2𝐵₁𝜉_0ℍ𝑛≜𝜔_𝑛𝑅^2𝑛+2. (15) The number 𝑄=2𝑛+2 (16) is called the homogeneous dimension of ℍ^𝑛.

The horizontal Sobolev spaces 𝐻𝑊^1,𝑝(Ω)(1≤𝑝<∞) are defined as 𝐻𝑊^1,𝑝(Ω)=𝑢∈𝐿^𝑝(Ω)∶𝑋_𝑖𝑢∈𝐿^𝑝(Ω),𝑖=1,2,…,2𝑛. (17) Then, 𝐻𝑊^1,𝑝(Ω) is a Banach space with the norm ‖𝑢‖_𝐻𝑊^1,𝑝_(Ω)=‖𝑢‖_𝐿^𝑝_(Ω)+^2𝑛_𝑖=1𝑋_𝑖𝑢_𝐿𝑝(Ω). (18)𝐻𝑊₀^1,𝑝(Ω) is the completion of 𝐶^∞₀(Ω) under norm (18).

Lu [28] showed the following Poincaré type inequality related to Hörmander's vector fields for 𝑢∈𝐻𝑊^1,𝑞(𝐵_𝑅(𝜉₀)), 1<𝑞<𝑄, 1≤𝑝≤𝑞𝑄/(𝑄−𝑞): _{𝐵𝑅(𝜉0)}𝑢−𝑢_𝜉0,𝑅^𝑝𝑑𝜉^1/𝑝≤𝐶_𝑝𝑅_{𝐵𝑅(𝜉0)}|𝑋𝑢|^𝑞𝑑𝜉^1/𝑞, (19) where we write down ∮_{𝐵𝑟(𝜉0)}𝑢𝑑𝜉=|𝐵_𝑟(𝜉₀)|_ℍ⁻¹_𝑛∫_{𝐵𝑟(𝜉0)}𝑢𝑑𝜉 here and there. Note the fact that the horizontal vectors 𝑋_𝑖 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)=[∫^𝑟₀(√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: 𝑠𝜂(𝑡)≤𝑠𝜂(𝑠)+𝑡,𝑠∈[0,1],𝑡>0. (20) The condition (H3) becomes 𝐴^𝛼_𝑖𝜉,𝑢,𝑝−𝐴^𝛼_𝑖̃𝑢√𝜉,𝑢,𝑝≤𝜅(||)𝜂𝑑²𝜉,𝜉̃+|𝑢−𝑢|²1+𝑝. (21) 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, 𝐴^𝛼_{𝑖,𝑝𝑗𝛽}𝜉,𝑢,𝑝−𝐴^𝛼_{𝑖,𝑝𝑗𝛽}𝜉̃,𝑢,𝑝≤𝜔𝑀,𝑑²𝜉,𝜉̃+|𝑢−𝑢|²+𝑝−𝑝². (22) Using (H1) and (H2), we see that 𝐴^𝛼_𝑖𝜉,𝑢,𝑝−𝐴^𝛼_𝑖𝑝𝑝,𝐴𝜉,𝑢,≤𝐿𝑝−_𝛼𝑖𝜉,𝑢,𝑝−𝐴^𝛼_𝑖𝜉,𝑢,𝑝𝑝−𝑝≥𝜆𝑝−𝑝². (23)(24)

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:(I)for any 𝒜∈Bil(ℝ^2𝑛𝑁) satisfying 𝒜(𝜈,𝜈)≥𝜆|𝜈|²,𝒜(𝜈,𝜈)≤𝐿|𝜈||𝜈|,𝜈,𝜈∈ℝ^2𝑛𝑁, (25)(II) for any 𝑤∈𝐻𝑊^1,2(𝐵_𝜌(𝜉₀),ℝ^𝑁) satisfying _{𝐵𝜌(𝜉0)}|𝑋𝑤|²𝑑𝜉≤1,_{𝐵𝜌(𝜉0)}𝒜𝑋𝑤,𝑋𝜑𝑑𝜉≤𝛿sup_{𝐵𝜌𝜉0}𝑋𝜑,∀𝜑∈𝐶¹₀𝐵_𝜌𝜉₀,ℝ^𝑁, (26)then, there exists an 𝒜-harmonic function ℎ such that _{𝐵𝜌(𝜉0)}|𝑋ℎ|²𝑑𝜉≤1,𝜌⁻²_{𝐵𝜌(𝜉0)}|ℎ−𝑤|²𝑑𝜉≤𝜀. (27)

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: sup_{𝐵𝜌/2𝜉0}|𝑢|²+𝜌²|𝑋𝑢|²+𝜌⁴𝑋²𝑢²≤𝐶_{0𝐵𝜌(𝜉0)}|𝑋𝑢|²𝑑𝜉. (28) 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∈𝐻𝑊^1,2(Ω,ℝ^𝑁) be a weak solution to the system (1) under the conditions (H1)–(H4), (μ1)–(μ3). Then, for every 𝜉₀=(𝑥⁰₁,𝑥⁰₂,…,𝑥⁰_𝑛,𝑦⁰₁,𝑦⁰₂,…,𝑦⁰_𝑛,𝑡)∈Ω, 𝑢₀∈ℝ^𝑁, 𝑝₀∈ℝ^2𝑛𝑁, and 0<𝜌<𝑅<𝜌₁(|𝑢₀|,|𝑝₀|)≤1 such that 𝐵_𝑅(𝜉₀)⊂⊂Ω, the inequality _{𝐵𝜌(𝜉0)}𝑋𝑢−𝑝₀²𝑑𝜉≤𝐶_𝑐1(𝑅−𝜌)^2𝐵_𝑅^(𝜉₀⁾𝑢−𝑢₀−𝜉¹−𝜉¹₀𝑝₀²𝑑𝜉+𝐹 (29) holds, where 𝜉¹=(𝑥₁,𝑥₂,…,𝑥_𝑛,𝑦₁,𝑦₂,…,𝑦_𝑛) is the horizontal component of 𝜉=(𝜉¹,𝑡)∈Ω and 𝐹=𝜔_𝑛𝑅^𝑄𝐾₁𝑢₀,𝑝₀𝜂𝑅²+_{𝐵𝑅(𝜉0)}1+𝑢^𝑟+|𝑋𝑢|²𝑑𝜉^{2(1−1/𝑟)}. (30)

Proof. Let 𝑣=𝑢−𝑢₀−(𝜉¹−𝜉¹₀)𝑝₀. Take a test function 𝜑=𝜙²𝑣 in (8) with 𝜙∈𝐶^∞₀(𝐵_𝑅(𝜉₀),ℝ^𝑁) satisfying 0≤𝜙≤1, |∇𝜙|≤𝐶/(𝑅−𝜌), and 𝜙≡1 on 𝐵_𝜌(𝜉₀). Then, we have 𝑋𝑣=𝑋𝑢−𝑝₀, |𝑋𝜑|≤𝜙|𝑋𝑢−𝑝₀|+𝐶/(𝑅−𝜌)|𝑣|, and _{𝐵𝑅(𝜉0)}𝐴^𝛼_𝑖(𝜉,𝑢,𝑋𝑢)𝜙²𝑋𝑢−𝑝₀𝑑𝜉=−2_{𝐵𝑅(𝜉0)}𝜙𝑋𝜙𝐴^𝛼_𝑖()+𝜉,𝑢,𝑋𝑢𝑣𝑑𝜉_{𝐵𝑅(𝜉0)}𝐵^𝛼(𝜉,𝑢,𝑋𝑢)𝜑^𝛼𝑑𝜉. (31) Adding this to the equations −_{𝐵𝑅(𝜉0)}𝐴^𝛼_𝑖𝜉,𝑢,𝑝₀𝜙²𝑋𝑢−𝑝₀𝑑𝜉=2_{𝐵𝑅(𝜉0)}𝜙𝑋𝜙𝐴^𝛼_𝑖𝜉,𝑢,𝑝₀−𝑣𝑑𝜉_{𝐵𝑅(𝜉0)}𝐴^𝛼_𝑖𝜉,𝑢,𝑝₀𝑋𝜑^𝛼𝑑𝜉,0=_{𝐵𝑅(𝜉0)}𝐴^𝛼_𝑖𝜉₀,𝑢₀,𝑝₀𝑋𝜑^𝛼. (32) It follows that by using the hypotheses (H1), (H3) (i.e., (23), (21), resp.), and (H4), _{𝐵𝑅𝜉0}𝐴^𝛼_𝑖(𝜉,𝑢,𝑋𝑢)−𝐴^𝛼_𝑖𝜉,𝑢,𝑝₀𝜙²𝑋𝑢−𝑝₀𝑑𝜉=2_{𝐵𝑅𝜉0}𝐴^𝛼_𝑖𝜉,𝑢,𝑝₀−𝐴^𝛼_𝑖(𝜉,𝑢,𝑋𝑢)𝜙𝑣𝑋𝜙𝑑𝜉+_{𝐵𝑅(𝜉0)}𝐴^𝛼_𝑖𝜉,𝑢₀+𝜉¹−𝜉¹₀𝑝₀,𝑝₀−𝐴^𝛼_𝑖𝜉,𝑢,𝑝₀𝑋𝜑^𝛼+𝑑𝜉_{𝐵𝑅(𝜉0)}𝐴^𝛼_𝑖𝜉₀,𝑢₀,𝑝₀−𝐴^𝛼_𝑖𝜉,𝑢₀+𝜉¹−𝜉¹₀𝑝₀,𝑝₀𝑋𝜑^𝛼𝑑𝜉+_{𝐵𝑅(𝜉0)}𝐵^𝛼(𝜉,𝑢,𝑋𝑢)𝜑^𝛼𝑑𝜉≤𝐼+𝐼𝐼+𝐼𝐼𝐼+𝐼𝑉+𝑉, (33) where 𝐼=2𝐿_{𝐵𝑅𝜉0}𝑋𝑢−𝑝₀𝜙𝑣𝑝||𝑋𝜙𝑑𝜉,𝐼𝐼=1+₀𝜅𝑢₀+𝑅𝑝₀×_{𝐵𝑅𝜉0}√𝜂|𝑣|²𝑋𝑢−𝑝₀𝜙²𝑝𝑑𝜉,𝐼𝐼𝐼=21+₀𝜅𝑢₀+𝑅𝑝₀×_{𝐵𝑅𝜉0}√𝜂|𝑣|²|𝑣𝜙𝑝|𝑋𝜙𝑑𝜉,𝐼𝑉=1+₀𝜅𝑢₀+𝑅𝑝₀×_{𝐵𝑅(𝜉0)}√𝜂𝑅²1+𝑝₀²𝑋𝑢−𝑝₀𝜙²+2𝜙|𝑣|𝑋𝜙𝑑𝜉,𝑉=𝐶_{𝐵𝑅(𝜉0)}1+|𝑢|^𝑟−1+|𝑋𝑢|^{2(1−1/𝑟)}𝜑𝑑𝜉. (34) Applying (H2), the left hand side of (33) can be estimated as 𝜆_{𝐵𝑅(𝜉0)}𝑋𝑢−𝑝₀²𝜙²≤𝑑𝜉_{𝐵𝑅(𝜉0)}𝐴^𝛼_𝑖(𝜉,𝑢,𝑋𝑢)−𝐴^𝛼_𝑖𝜉,𝑢,𝑝₀𝜙²𝑋𝑢−𝑝₀𝑑𝜉. (35) For 𝜀>0 to be fixed later, we have, using Young's inequality, 𝐼≤𝜀_{𝐵𝑅(𝜉0)}𝑋𝑢−𝑝₀²𝜙²𝑑𝜉+𝐶𝐿²𝑅−𝜌^2𝐵_𝑅^𝜉₀|𝑣|²𝑑𝜉. (36) Using Jensen's inequality, (20), and the fact that 𝜂(𝑡𝑠)≤𝑡𝜂(𝑠) for 𝑡≥1, we arrive at 1+𝑝₀²𝜅²(⋅)_{𝐵𝑅(𝜉0)}𝜂|𝑣|²𝑑𝜉≤𝜔_𝑛𝑅^𝑄−21+𝑝₀²𝜅²(⋅)𝑅²𝜂_{𝐵𝑅(𝜉0)}|𝑣|²𝑑𝜉≤𝜔_𝑛𝑅^𝑄−2_{𝐵𝑅(𝜉0)}|𝑣|²+𝑝𝑑𝜉1+₀²𝜅²(⋅)𝑅²𝜂×1+𝑝₀²𝜅²(⋅)𝑅²≤𝑅⁻²_{𝐵𝑅(𝜉0)}|𝑣|²𝑑𝜉+𝜔_𝑛𝑅^𝑄1+𝑝₀⁴𝜅⁴(⋅)𝜂𝑅², (37) 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 𝐼𝐼≤𝜀_{𝐵𝑅(𝜉0)}𝑋𝑢−𝑝₀²𝜙²𝑑𝜉+𝜀⁻¹1+𝑝₀²𝜅²(⋅)_{𝐵𝑅(𝜉0)}𝜂|𝑣|²𝑑𝜉≤𝜀_{𝐵𝑅(𝜉0)}𝑋𝑢−𝑝₀²𝜙²+1𝑑𝜉𝜀𝑅−𝜌^2𝐵_𝑅^(𝜉₀⁾|𝑣|²𝑑𝜉+𝜀⁻¹𝜔_𝑛𝑅^𝑄1+𝑝₀⁴𝜅⁴(⋅)𝜂𝑅². (38) And similarly, we see 𝐼𝐼𝐼≤4𝐶𝑅−𝜌^2𝐵_𝑅^(𝜉₀⁾|𝑣|²+𝑝𝑑𝜉1+₀²𝜅²(⋅)_{𝐵𝑅(𝜉0)}𝜂|𝑣|²≤𝐶𝑑𝜉𝑅−𝜌^2𝐵_𝑅^(𝜉₀⁾|𝑣|²𝑑𝜉+𝜔_𝑛𝑅^𝑄1+𝑝₀⁴𝜅⁴(⋅)𝜂𝑅²,𝐼𝑉≤𝜀_{𝐵𝑅(𝜉0)}𝑋𝑢−𝑝₀²𝜙²+𝑑𝜉4𝐶𝜀𝑅−𝜌^2𝐵_𝑅^(𝜉₀⁾|𝑣|²𝑑𝜉+𝜀⁻¹𝜔_𝑛𝑅^𝑄1+𝑝₀²𝜅²(⋅)𝜂×_{𝐵𝑅(𝜉0)}𝑅²1+𝑝₀²𝑑𝜉≤𝜀_{𝐵𝑅𝜉0}𝑋𝑢−𝑝₀²𝜙²+𝑑𝜉𝐶𝜀𝑅−𝜌^2𝐵_𝑅^𝜉₀|𝑣|²𝑑𝜉+𝜀⁻¹𝜔_𝑛𝑅^𝑄1+𝑝₀⁴𝜅⁴(⋅)𝜂𝑅². (39) Here we have used 𝜅≥1 in the last inequality.
By Hölder's inequality, (19), and Young's inequality, one gets𝑉≤𝐶_{𝐵𝑅𝜉0}𝜑^𝑟𝑑𝜉^1/𝑟×_{𝐵𝑅𝜉0}1+|𝑢|^𝑟+|𝑋𝑢|²𝑑𝜉^{(𝑟−1)/𝑟}≤𝐶_{𝐵𝑅𝜉0}𝑋𝜑²𝑑𝜉^1/2×_{𝐵𝑅𝜉0}1+|𝑢|^𝑟+|𝑋𝑢|²𝑑𝜉^{(𝑟−1)/𝑟}≤𝜀_{𝐵𝑅(𝜉0)}𝑋𝜑²(𝜀)×𝑑𝜉+𝐶_{𝐵𝑅𝜉0}1+|𝑢|^𝑟+|𝑋𝑢|²𝑑𝜉^{2(𝑟−1)/𝑟}≤𝜀_{𝐵𝑅𝜉0}𝑋𝑢−𝑝₀²𝜙²+𝑑𝜉𝐶𝜀𝑅−𝜌^2𝐵_𝑅^𝜉₀|𝑣|²(𝜀)𝑑𝜉+𝐶_{𝐵𝑅𝜉0}1+|𝑢|^𝑟+|𝑋𝑢|²𝑑𝜉^{2(𝑟−1)/𝑟}, (40) where we have used the fact that |𝑋𝜑|≤𝜙|𝑋𝑢−𝑝₀|+𝐶/(𝑅−𝜌)|𝑣|.
Applying these estimates to (37), we obtain (𝜆−4𝜀)_{𝐵𝑅𝜉0}𝑋𝑢−𝑝₀²𝜙²≤𝐶()𝑑𝜉𝐿,𝜀𝑅−𝜌^2𝐵_𝑅^𝜉₀|𝑣|²+𝜀𝑑𝜉₋₁+2𝜔_𝑛𝑅^𝑄1+𝑝₀⁴𝜅⁴(⋅)𝜂𝑅²+𝐶(𝜀)_{𝐵𝑅𝜉0}1+|𝑢|^𝑟+|𝑋𝑢|²𝑑𝜉^{2(𝑟−1)/𝑟}. (41) 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∈𝐻𝑊^1,2(Ω,ℝ^𝑁) is a weak solution to (1) with the assumptions of (H1)–(H4) and (𝜇1)–(𝜇3).

Lemma 4. Let 𝐵_𝜌(𝜉₀)⊂⊂Ω with 𝜌≤𝜌₁(|𝑢₀|,|𝑝₀|) and 𝜑∈𝐶^∞₀(𝐵_𝜌(𝜉₀),ℝ^𝑁) satisfying |𝜑|≤𝜌² and sup_{𝐵𝜌(𝜉0)}|𝑋𝜑|≤1. Then, there exists a constant 𝐶₁≥1 such that _{𝐵𝜌𝜉0}𝐴^𝛼_{𝑖,𝑝𝑗𝛽}𝜉₀,𝑢₀,𝑝₀𝑋𝑢−𝑝₀𝑋𝜑^𝛼𝑑𝜉≤𝐶₁Φ𝜉₀,𝜌,𝑝₀+𝜔𝑢₀+𝑝₀,Φ𝜉₀,𝜌,𝑝₀Φ^1/2𝜉₀,𝜌,𝑝₀+𝐾₁𝑢₀,𝑝₀√𝜂𝜌²sup_{𝐵𝜌𝜉0}𝑋𝜑. (42)

Proof. Using the fact that ∫_{𝐵𝜌(𝜉0)}𝐴^𝛼_𝑖(𝜉₀,𝑢₀,𝑝₀)𝑋𝜑^𝛼𝑑𝜉=0 and the weak form (8), we deduce _{𝐵𝜌(𝜉0)}¹₀𝐴^𝛼_{𝑖,𝑝𝑗𝛽}𝜉₀,𝑢₀,𝜃𝑋𝑢+(1−𝜃)𝑝₀×𝑋𝑢−𝑝₀𝑑𝜃𝑋𝜑^𝛼=𝑑𝜉_{𝐵𝜌𝜉0}𝐴^𝛼_𝑖𝜉₀,𝑢₀,𝑋𝑢−𝐴^𝛼_𝑖𝜉₀,𝑢₀,𝑝₀𝑋𝜑^𝛼=𝑑𝜉_{𝐵𝜌𝜉0}𝐴^𝛼_𝑖𝜉₀,𝑢₀,𝑋𝑢−𝐴^𝛼_𝑖(𝜉,𝑢,𝑋𝑢)𝑋𝜑^𝛼+𝑑𝜉_{𝐵𝜌𝜉0}𝐵^𝛼(𝜉,𝑢,𝑋𝑢)𝜑^𝛼𝑑𝜉. (43) It yields _{𝐵𝜌𝜉0}𝐴^𝛼_{𝑖,𝑝𝑗𝛽}𝜉₀,𝑢₀,𝑝₀𝑋𝑢−𝑝₀𝑋𝜑^𝛼=𝑑𝜉_{𝐵𝜌(𝜉0)10}𝐴^𝛼_{𝑖,𝑝𝑗𝛽}𝜉₀,𝑢₀,𝑝₀−𝐴^𝛼_{𝑖,𝑝𝑗𝛽}𝜉₀,𝑢₀,𝜃𝑋𝑢+(1−𝜃)𝑝₀×𝑋𝑢−𝑝₀𝑑𝜃𝑑𝜉sup_{𝐵𝜌𝜉0}+𝑋𝜑_{𝐵𝜌𝜉0}𝐴^𝛼_𝑖𝜉₀,𝑢₀,𝑋𝑢−𝐴^𝛼_𝑖𝜉,𝑢₀+𝑝₀𝜉−𝜉₀,𝑋𝑢sup_{𝐵𝜌𝜉0}+𝑋𝜑_{𝐵𝜌𝜉0}𝐴^𝛼_𝑖𝜉,𝑢₀+𝑝₀𝜉−𝜉₀,𝑋𝑢−𝐴^𝛼_𝑖(𝜉,𝑢,𝑋𝑢)sup_{𝐵𝜌𝜉0}+𝑋𝜑_{𝐵𝜌(𝜉0)}𝐵^𝛼(𝜉,𝑢,𝑋𝑢)𝜑^𝛼𝑑𝜉∶=𝐼^′+𝐼𝐼^′+𝐼𝐼𝐼^′+𝐼𝑉^′. (44) Using (22), Hölder's inequality, the fact that 𝑡→𝜔²(𝑠,𝑡) is concave, and Jensen's inequality, we have 𝐼^′≤sup_{𝐵𝜌𝜉0}𝑋𝜑_{𝐵𝜌𝜉0}𝜔𝑢₀+𝑝₀,𝑋𝑢−𝑝₀²𝑋𝑢−𝑝₀𝑑𝜉≤sup_{𝐵𝜌𝜉0}𝑋𝜑_{𝐵𝜌𝜉0}𝜔²𝑢₀+𝑝₀,𝑋𝑢−𝑝₀²𝑑𝜉^1/2×_{𝐵𝜌𝜉0}𝑋𝑢−𝑝₀²𝑑𝜉^1/2≤𝜔𝑢₀+𝑝₀,Φ𝜉₀,𝜌,𝑝₀Φ^1/2𝜉₀,𝜌,𝑝₀sup_{𝐵𝜌𝜉0}𝑋𝜑. (45) Similarly, using (21) and the fact that 𝜂(𝑡𝑠)≤𝑡𝜂(𝑠) for 𝑡≥1, we obtain 𝐼𝐼^′≤sup_{𝐵𝜌𝜉0}𝑋𝜑𝜅(⋅)√𝜂𝜌²1+𝑝₀²×_{𝐵𝜌𝜉0}(1+|𝑋𝑢|)𝑑𝜉≤sup_{𝐵𝜌𝜉0}𝑋𝜑𝜅(⋅)√𝜂𝜌²1+𝑝₀²×_{𝐵𝜌𝜉0}1+𝑝₀+𝑋𝑢−𝑝₀𝑑𝜉≤sup_{𝐵𝜌𝜉0}𝑋𝜑_{𝐵𝜌𝜉0}𝑋𝑢−𝑝₀²𝑑𝜉+𝜅²(⋅)1+𝑝₀²𝜂𝜌²+𝜅(⋅)1+𝑝₀³√𝜂𝜌²≤Φ𝜉₀,𝜌,𝑝₀+2𝜅²(⋅)1+𝑝₀³√𝜂𝜌²×sup_{𝐵𝜌𝜉0}𝑋𝜑, (46) 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), 𝐼𝐼𝐼^′≤sup_{𝐵𝜌𝜉0}𝑋𝜑𝜅(⋅)_{𝐵𝜌(𝜉0)}√𝜂|𝑣|²(1+|𝑋𝑢|)𝑑𝜉≤sup_{𝐵𝜌𝜉0}𝑋𝜑_{𝐵𝜌𝜉0}𝑋𝑢−𝑝₀²𝑑𝜉+𝜅²(⋅)_{𝐵𝜌𝜉0}𝜂|𝑣|²(⋅)𝑝𝑑𝜉+𝜅1+_{0𝐵𝜌𝜉0}√𝜂|𝑣|²𝑑𝜉≤sup_{𝐵𝜌𝜉0}𝑋𝜑Φ𝜉₀,𝜌,𝑝₀+2𝜌⁻²_{𝐵𝜌𝜉0}|𝑣|²𝑑𝜉+𝜅⁴(⋅)𝜂𝜌²+𝜅²(⋅)1+𝑝₀²√𝜂𝜌²≤1+2𝐶_𝑝Φ𝜉₀,𝜌,𝑝₀+2𝜅⁴(⋅)1+𝑝₀²√𝜂𝜌². (47) Using Hölder's inequality, (19), and Young's inequality, we have 𝐼𝑉^′≤𝐶_{𝐵𝜌𝜉0}1+|𝑢|^𝑟−1+|𝑋𝑢|^{2(1−1/𝑟)}𝜑𝑑𝜉≤𝐶_{𝐵𝜌(𝜉0)}|𝑋𝑢|^{2(1−1/𝑟)}𝜑𝑑𝜉+𝐶_{𝐵𝜌𝜉0}𝑢−𝑢₀−𝑝₀𝜉¹−𝜉¹₀^𝑟−1𝜑𝑑𝜉+𝐶𝜌²1+𝑢₀+𝑝₀^𝑟−1≤𝐶_{𝐵𝜌𝜉0}|𝑋𝑢|²𝑑𝜉^(1−1/𝑟)_{𝐵𝜌(𝜉0)}𝜑^𝑟𝑑𝜉^1/𝑟+𝐶_{𝐵𝜌𝜉0}𝑢−𝑢₀−𝑝₀𝜉¹−𝜉¹₀^𝑟𝑑𝜉^(1−1/𝑟)×_{𝐵𝜌(𝜉0)}𝜑^𝑟𝑑𝜉^1/𝑟+𝐶𝜌²1+(𝑢₀+𝑝₀)^𝑟−1≤𝐶_{𝐵𝜌𝜉0}|𝑋𝑢|²𝑑𝜉^(1−1/𝑟)_{𝐵𝜌𝜉0}𝜑^𝑟𝑑𝜉^1/𝑟+_{𝐵𝜌𝜉0}𝑋𝑢−𝑝₀²𝑑𝜉^{(𝑟/2)(1−1/𝑟)}×_{𝐵𝜌𝜉0}𝜑^𝑟𝑑𝜉^1/𝑟+𝐶𝜌²1+(𝑢₀+𝑝₀)^𝑟−1≤𝐶‖𝑢‖_𝐻𝑊^1,2_{𝐵𝜌𝜉0}𝑋𝑢−𝑝₀²𝑑𝜉^(1−1/𝑟)×_{𝐵𝜌𝜉0}𝜑^𝑟𝑑𝜉^1/𝑟+𝐶𝜌1+(𝑢₀+𝑝₀)^𝑟−1≤𝐶‖𝑢‖_𝐻𝑊^1,2_{(𝐵𝜌(𝜉0))𝐵𝜌𝜉0}𝑋𝑢−𝑝₀²𝑑𝜉+𝐶𝜌^2𝑟+𝐶𝜌²1+𝑢₀+𝑝₀^𝑟−1≤𝐶₂Φ𝜉₀,𝜌,𝑝₀+𝐶𝜌²1+𝑢₀+𝑝₀^𝑟−1≤𝐶₂Φ𝜉₀,𝜌,𝑝₀+𝐶𝜅(⋅)1+𝑢₀+𝑝₀²√𝜂𝜌², (48) where we have used the assumption (𝜂4) and the fact that 𝑟=2𝑄/(𝑄−2)=(2𝑛+4)/2𝑛≤3 and 𝐶₂=𝐶‖𝑢‖_𝐻𝑊^1,2_{(𝐵𝜌(𝜉0))}≥1. Combining these estimates, we obtain the conclusion with 𝐶₁=(1+𝐶₂+2𝐶_𝑝)≥1.

Lemma 5. Assume that the conditions of Lemma 2 and the following smallness conditions hold: 𝜔𝑢_𝜉0,𝜌+(𝑋𝑢)_𝜉0,𝜌,Φ𝜉₀,𝜌,(𝑋𝑢)_𝜉0,𝜌+Φ^1/2𝜉₀,𝜌,(𝑋𝑢)_𝜉0,𝜌≤𝛿2,𝐶₃𝐾²₁𝑢_𝜉0,𝜌,(𝑋𝑢)_𝜉0,𝜌𝜂𝜌²≤𝛿² (49)(50) with 𝐶₃=8𝐶²₁𝐶₅, together with 𝜌≤𝜌₁1+𝑢_𝜉0,𝜌,1+(𝑋𝑢)_𝜉0,𝜌. (51) Then, the following growth condition holds for 𝜏∈[𝛾,1)Φ𝜉₀,𝜃𝜌≤𝜃^2𝜏Φ𝜉₀,𝜌+𝐾^∗𝑢_{𝜉0,2𝜃𝜌},(𝑋𝑢)_{𝜉0,2𝜃𝜌}𝜂𝜌², (52) 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+2^𝑄𝐶_𝑐)𝐾²₁(1+𝑠,1+𝑡).

Proof. We define 𝑤=[𝑢−𝑢_𝜉0,𝜌−𝑝₀(𝜉¹−𝜉¹₀)]𝜎⁻¹, where 𝜎=𝐶₁Φ𝜉₀,𝜌,𝑝₀+4𝛿⁻²𝐾²₁𝑢₀,𝑝₀𝜂𝜌². (53) Then, we have 𝑋𝑤=𝜎⁻¹(𝑋𝑢−𝑝₀). Now, we consider 𝐵_𝜌(𝜉₀)⊂⊂Ω such that 𝜌≤𝜌₁(|𝑢₀|,|𝑝₀|). Applying Lemma 4 on 𝐵_𝜌(𝜉₀) to 𝑢, we have for any 𝜑∈𝐶^∞₀(𝐵_𝜌(𝜉₀),ℝ^𝑁), _{𝐵𝜌𝜉0}|𝑋𝑤|²𝑑𝜉=𝜎⁻²Φ𝜉₀,𝜌,𝑝₀≤1𝐶²₁≤1,_{𝐵𝜌𝜉0}𝐴^𝛼_{𝑖,𝑝𝑗𝛽}𝜉₀,𝑢₀,𝑝₀≤Φ𝑋𝑤𝑋𝜑𝑑𝜉_1/2𝜉₀,𝜌,𝑝₀+𝜔𝑢₀+𝑝₀,Φ𝜉₀,𝜌,𝑝₀+𝛿2sup_{𝐵𝜌𝜉0}𝑋𝜑. (54)(55) 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 𝐴^𝛼_{𝑖,𝑝𝑗𝛽}(𝜉₀,𝑢₀,𝑝₀)-harmonic function ℎ∈𝐻𝑊^1,2(𝐵_𝜌(𝜉₀),ℝ^𝑁) such that _{𝐵𝜌(𝜉0)}|𝑋ℎ|²𝑑𝜉≤1,𝜌⁻²_{𝐵𝜌(𝜉0)}|𝑤−ℎ|²𝑑𝜉≤𝜀. (56) Taking 𝑢₀=𝑢_{𝜉0,2𝜃𝜌},𝜃∈(0,1/4] and replacing 𝑝₀ by 𝑝₀+𝜎(𝑋ℎ)_{𝜉0,2𝜃𝜌}, we use Lemma 3 to obtain _{𝐵𝜃𝜌(𝜉0)}𝑋𝑢−𝑝₀−𝜎(𝑋ℎ)_{𝜉0,2𝜃𝜌}²𝑑𝜉≤𝐶_𝑐1(𝜃𝜌)^2𝐵_2𝜃𝜌^(𝜉₀⁾𝑢−𝑢_{𝜉0,2𝜃𝜌}−𝑝₀+𝜎(𝑋ℎ)_{𝜉0,2𝜃𝜌}×𝜉¹−𝜉¹₀²𝑑𝜉+𝐹, (57) where 𝐹=𝜔_𝑛2𝜃𝜌^𝑄𝐾₁𝑢_{𝜉0,2𝜃𝜌},𝑝₀+𝜎(𝑋ℎ)_{𝜉0,2𝜃𝜌}𝜂2𝜃𝜌²+_{𝐵2𝜃𝜌(𝜉0)}1+𝑢^𝑟+|𝑋𝑢|²𝑑𝜉^{2(1−1/𝑟)}. (58) Using the fact that 𝑢−(𝑝₀+𝜎(𝑋ℎ)_{𝜉0,2𝜃𝜌})(𝜉¹−𝜉¹₀) has mean value 𝑢_{𝜉0,2𝜃𝜌} on the ball 𝐵_2𝜃𝜌(𝜉₀), the definition of 𝑤, and (19), we have 1𝜃𝜌^2𝐵_2𝜃𝜌^(𝜉₀⁾𝑢−𝑢_{𝜉0,2𝜃𝜌}−𝑝₀+𝜎(𝑋ℎ)_{𝜉0,2𝜃𝜌}𝜉¹−𝜉¹₀²≤𝑑𝜉4𝜎²2𝜃𝜌^2𝐵_2𝜃𝜌^(𝜉₀⁾𝑤−ℎ_{𝜉0,2𝜃𝜌}−(𝑋ℎ)_{𝜉0,2𝜃𝜌}𝜉¹−𝜉¹₀²≤𝑑𝜉4𝜎²2𝜃𝜌^2𝐵_2𝜃𝜌^(𝜉₀⁾|𝑤−ℎ|²𝑑𝜉+_{𝐵2𝜃𝜌(𝜉0)}ℎ−ℎ_{𝜉0,2𝜃𝜌}−(𝑋ℎ)_{𝜉0,2𝜃𝜌}𝜉¹−𝜉¹₀²𝑑𝜉≤4𝜎²(2𝜃)^−𝑄−2𝜀+𝐶_{𝑝𝐵2𝜃𝜌(𝜉0)}𝑋ℎ−(𝑋ℎ)_{𝜉0,2𝜃𝜌}²𝑑𝜉≤4𝜎²(2𝜃)^−𝑄−2𝜀+𝐶²_𝑝(2𝜃𝜌)²_{𝐵2𝜃𝜌(𝜉0)}𝑋²ℎ²𝑑𝜉≤4𝜎²(2𝜃)^−𝑄−2𝜀+𝐶²_𝑝(2𝜃)²𝐶₀≤𝐶₄𝜃^−𝑄−2𝜀+𝜃²Φ𝜉₀,𝜌,𝑝₀+4𝛿⁻²𝐾²₁𝑢_{𝜉0,2𝜃𝜌},𝑝₀𝜂𝜌², (59)
where 𝐶₄∶=𝐶₄(𝑄,𝜆,𝐿)≥1. Note that in the second last inequality we have used the fact that _{𝐵2𝜃𝜌(𝜉0)}𝑋²ℎ𝑑𝜉≤sup_{𝐵𝜌𝜉0}𝑋²ℎ≤𝐶₀𝜌⁻²_{𝐵𝜌(𝜉0)}|𝑋ℎ|²𝑑𝜉≤𝐶₀𝜌⁻². (60) 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 _{𝐵2𝜃𝜌(𝜉0)}1+𝑢^𝑟+|𝑋𝑢|²𝑑𝜉^{2(1−1/𝑟)}≤𝐶_{𝐵2𝜃𝜌𝜉0}𝑋𝑢−𝑝₀²𝑑𝜉^{2(1−1/𝑟)}+𝐶_{𝐵2𝜃𝜌𝜉0}|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}. (61)
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(|𝑢_{𝜉0,2𝜃𝜌}|,|𝑃|); see (64) and (65)), we deduce that Φ𝜉₀,𝜃𝜌≤𝐵_𝜃𝜌𝜉_0ℍ⁻¹_{𝑛𝐵2𝜃𝜌𝜉0}|𝑋𝑢−𝑃|²𝑑𝜉≤𝐶_𝑐2^𝑄(𝜃𝜌)^2𝐵_2𝜃𝜌^𝜉₀𝑢−𝑢_{𝜉0,2𝜃𝜌}−𝑝₀+𝜎(𝑋ℎ)_{𝜉0,2𝜃𝜌}×𝜉¹−𝜉¹₀²𝑑𝜉+2^𝑄𝐶_𝑐𝐾₁𝑢_{𝜉0,2𝜃𝜌},𝑝₀+𝜎(𝑋ℎ)_{𝜉0,2𝜃𝜌}𝜂𝜌²+𝐶_𝑐(2𝜃𝜌)^{2𝑄(1−1/𝑟)}(𝜃𝜌)^𝑄×_{𝐵2𝜃𝜌𝜉0}1+𝑢^𝑟+|𝑋𝑢|²𝑑𝜉^{2(1−1/𝑟)}≤2^𝑄𝐶₄𝐶_𝑐𝜃^−𝑄−2𝜀+𝜃²×Φ𝜉₀,𝜌+4𝛿⁻²𝐾²₁×𝑢_{𝜉0,2𝜃𝜌},(𝑋𝑢)_{𝜉0,2𝜃𝜌}𝜂𝜌²+2^𝑄𝐶_𝑐𝐾₁𝑢_{𝜉0,2𝜃𝜌},(𝑋𝑢)_{𝜉0,2𝜃𝜌}+𝜎(𝑋ℎ)_{𝜉0,2𝜃𝜌}𝜂𝜌²+2^𝑄𝐶_𝑐(2𝜃)^{2−𝑄(𝑟−1)}Φ(𝜉₀,𝜌)^{2(1−1/𝑟)}+1+(𝑋𝑢)_{𝜉0,2𝜃𝜌}^{2(1−1/𝑟)2}𝜌². (62) We now specify 𝜀=𝜃^𝑄+4, 𝜃∈(0,1/4] such that 2^𝑄+1𝐶₄𝐶_𝑐𝜃²≤𝜃^2𝜏. Note that the small condition (50) implies 𝜎²𝐶₅≤1 with 𝐶₅=max{𝐶₀,𝐶_𝑐2^𝑄(2𝜃)^{−(𝑄2+4)/(𝑄−2)}}, and it yields 2^𝑄𝐶_𝑐(2𝜃)^{2−𝑄(𝑟−1)}Φ(𝜉₀,𝜌)^{2(1−1/𝑟)}𝜎()≤1,𝑋ℎ_{𝜉0,2𝜃𝜌}≤𝜎sup_{𝐵2𝜃𝜌𝜉0}|𝑋ℎ|≤𝜎𝐶_{0𝐵𝜌𝜉0}|𝑋ℎ|²𝑑𝜉≤𝜎𝐶₀≤1, (63)(64) 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 𝑢_{𝜉0,2𝜃𝜌}≤𝑢_𝜉0,𝜌+𝑢_{𝜉0,2𝜃𝜌}−𝑢_𝜉0,𝜌≤𝑢_𝜉0,𝜌+(2𝜃)^−𝑄/2×_{𝐵𝜌(𝜉0)}𝑢−(𝑋𝑢)_𝜉0,𝜌𝜉¹−𝜉¹₀−𝑢_𝜉0,𝜌²𝑑𝜉^1/2≤𝑢_𝜉0,𝜌+(2𝜃)^−𝑄/2𝜌𝐶_𝑝Φ^1/2𝜉₀≤𝑢,𝜌_𝜉0,𝜌+𝜎𝐶_𝑝𝐶₁(2𝜃)^𝑄/2≤𝑢_𝜉0,𝜌+𝜎𝐶₅≤𝑢_𝜉0,𝜌+1. (65) Combining these estimates with (62), we have Φ𝜉₀,𝜃𝜌≤𝜃^2𝜏Φ𝜉₀+,𝜌4𝛿⁻²𝐾²₁𝑢_{𝜉0,2𝜃𝜌},(𝑋𝑢)_{𝜉0,2𝜃𝜌}+2^𝑄𝐶_𝑐𝐾₁𝑢_{𝜉0,2𝜃𝜌},(𝑋𝑢)_{𝜉0,2𝜃𝜌}+𝜎(𝑋ℎ)_{𝜉0,2𝜃𝜌}𝜂𝜌²+1+1+(𝑋𝑢)_{𝜉0,2𝜃𝜌}^{2(1−1/𝑟)2}𝜂𝜌²≤𝜃^2𝜏Φ𝜉₀,𝜌+𝐾𝑢_{𝜉0,2𝜃𝜌},(𝑋𝑢)_{𝜉0,2𝜃𝜌}𝜂𝜌²+2+𝑢_{𝜉0,2𝜃𝜌}+(𝑋𝑢)_{𝜉0,2𝜃𝜌}^2(𝑟−1)𝜂𝜌²≤𝜃^2𝜏Φ𝜉₀,𝜌+𝐾^∗𝑢_{𝜉0,2𝜃𝜌},(𝑋𝑢)_{𝜉0,2𝜃𝜌}𝜂𝜌². (66) 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 𝜔²2𝑇,2Φ₀(𝑇)+2Φ₀(𝑇)≤12𝛿²,𝐶₁Φ₀(𝑇)≤𝜃^𝑄1−𝜃^𝜏2𝑇². (67) With Φ₀(𝑇) from (67), we choose 𝜌₀(𝑇)∈(0,1] (depending on 𝑄, 𝑁, 𝜆, 𝐿, 𝜏, 𝜔, 𝜂, and 𝜅) such that 𝜌₀(𝑇)≤𝜌₁(),𝐶1+2𝑇,1+2𝑇₃𝐾²₁(2𝑇,2𝑇)𝜂𝜌₀(𝑇)²≤𝛿²,𝐾₀(𝑇)𝜂𝜌₀(𝑇)²≤𝜃^2𝛾−𝜃^2𝜏Φ₀(𝑇),21+𝐶_𝑝𝐾₀(𝑇)𝐻𝜌₀(𝑇)²≤𝜃^𝑄1−𝜃^𝛾2𝜃^2𝛾−𝜃^2𝜏𝑇², (68) 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 𝐵_𝜌(𝜉₀)⊂⊂Ω one has(1) |𝑢_𝜉0,𝜌|+|(𝑋𝑢)_𝜉0,𝜌|≤𝑇₀, (2)𝜌≤𝜌₀(𝑇₀), (3)Φ(𝜉₀,𝜌)≤Φ₀(𝑇₀). Then, the small conditions (49)–(51) are satisfied on the balls 𝐵_𝜃^𝑗_𝜌(𝜉₀) for 𝑗∈𝑁∪{0}. Moreover, the limit Λ_𝜉0=lim_𝑗→∞(𝑋𝑢)_𝜉0,𝜃^𝑗_𝜌 exists, and the inequality _{𝐵𝜌𝜉0}𝑋𝑢−Λ_𝜉0²𝑑𝜉≤𝐶₆𝑟𝜌^2𝜏Φ𝜉₀,𝜌+𝐻𝑟² (69) 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.