On Carlson's Type Removability Test for the Degenerate Quasilinear Elliptic Equations

Abstract

Carlson's type theorem on removable sets for 𝛼-Holder continuous solutions is investigated for the quasilinear elliptic equations div𝐴(𝑥,𝑢,∇𝑢)=0, having degeneration 𝜔 in the Muckenhoupt class. In partial, when 𝛼 is sufficiently small and the operator is weighted 𝑝-Laplacian, we show that the compact set 𝐸 is removable if and only if the Hausdorff measure Λ_𝜔^{−𝑝+(𝑝−1)𝛼}(𝐸)=0.

1. Introduction

In this paper, we will consider questions of a removable singularity for the class of quasilinear elliptic equations of the form div(𝐴(𝑥,𝑢,∇𝑢))=0, (1.1) where 𝐴(𝑥,𝜉,𝜁)={𝐴₁(𝑥,𝜉,𝜁),𝐴₂(𝑥,𝜉,𝜁),…,𝐴_𝑛(𝑥,𝜉,𝜁)}∶𝐷×ℝ×ℝ^𝑛→ℝ^𝑛 are continuous with respect to 𝜉, continuously differentiable with respect to 𝜁 functions. For 𝜉∈ℝ¹, 𝜁∈ℝ^𝑛, it is assumed that 𝐴_𝑗(𝑥,𝜉,𝜁);𝑗=1,2,…𝑛 are measurable functions with respect to a variable 𝑥 in the open domain 𝐷⊂ℝ^𝑛. Let the following growth conditions be satisfied: ^𝑛_𝑖,𝑘=1𝜕𝐴_𝑖𝜕𝜁_𝑘(𝑥,𝜉,𝜁)𝜂_𝑖𝜂_𝑘≥𝜆𝜔(𝑥)𝜁^𝑝−2𝜂²,𝜕𝐴_𝑖𝜕𝜁_𝑘(𝑥,𝜉,𝜁)≤𝜆⁻¹𝜔(𝑥)𝜁^𝑝−2,𝑘,𝑖=1,2,…,𝑛, (1.2) where 1<𝑝<∞,𝜆∈(0,1). Throughout the paper, 𝜔∶ℝ^𝑛→[0,∞] is a measurable function satisfying the doubling condition: for any ball 𝐵=𝐵(𝑥,𝑟) with centre at a point 𝑥 and of radius 𝑟>0, the inequality _2𝐵𝜔𝑑𝑥≤𝐶_𝐵𝜔𝑑𝑥 (1.3) is satisfied, where the constant 𝐶 is positive and does not depend on the ball 𝐵⊂ℝ^𝑛. For the system of functions 𝐴_𝑗(𝑥,𝜉,𝜁);𝑗=1,2,…𝑛, we can write the following expression: 𝐴_𝑖(𝑥,𝑢(𝑥),∇𝑢(𝑥))=¹₀^𝑛_𝑘=1𝜕𝐴_𝑖𝜕𝜁_𝑘(𝑥,𝑢(𝑥),𝑡∇𝑢(𝑥))𝑢_𝑥𝑘=𝑑𝑡_𝑛𝑘=1𝑏_𝑖𝑘(𝑥)𝜔(𝑥)|∇𝑢(𝑥)|^𝑝−2𝑢_𝑥𝑘,𝑖=1,2,…,𝑛, (1.4) where 𝑏_𝑖𝑘(𝑥)=𝜔(𝑥)⁻¹|∇𝑢(𝑥)|^2−𝑝∫¹₀∑^𝑛_𝑘=1𝜕𝐴_𝑖𝜕𝜁_𝑘(𝑥,𝑢(𝑥),𝑡∇𝑢(𝑥))𝑑𝑡;𝑖,𝑘=1,2,…,𝑛. Therefore, (1.1) can be written in the form ^𝑛_𝑖,𝑘=1𝜕𝜕𝑥_𝑖𝜔|∇𝑢|^𝑝−2𝑏_𝑖𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘=0. (1.5) By virtue of (1.2) and (1.4), the system of functions {𝑏_𝑖𝑘(𝑥)}_{𝑖,𝑘=1,2,…,𝑛} is bounded and measurable. Moreover, the condition of uniform ellipticity is satisfied: for a.e. 𝑥∈𝐷,𝜂∈ℝ^𝑛 there exist positive constants 𝐶₁,𝐶₂ such that 𝐶₁𝜂²≤^𝑛_𝑖,𝑘=1𝑏_𝑖𝑘(𝑥)𝜂_𝑖𝜂_𝑘=^𝑛_𝑖,𝑘=1𝜔(𝑥)⁻¹|∇𝑢(𝑥)|^2−𝑝1₀𝜕𝐴_𝑖𝜕𝜁_𝑘(𝑥,𝑢(𝑥),𝑡∇𝑢(𝑥))𝜂_𝑖𝜂_𝑘𝑑𝑡≤𝐶₂𝜂². (1.6)

Denote by 𝐶^𝛼(𝐷),0<𝛼≤1, the class of continuous in 𝐷 functions 𝑓∶𝐷→ℝ satisfying the condition𝑓(𝑥)−𝑓𝑦≤𝐾𝑥−𝑦^𝛼 (1.7) with some 𝐾>0 not depending on the points 𝑥,𝑦∈𝐷. Denote by 𝑊¹_𝑝𝜔(𝐷) the space of measurable functions in 𝐷, which have the finite norm ‖𝑢‖=‖𝑢‖_{𝐿𝑝𝜔(𝐷)}+^𝑛_𝑗=1𝑢_{𝑥𝑗𝐿𝑝𝜔(𝐷)}, (1.8) where 𝑢_𝑥𝑗 are the derivatives of a function 𝑢∈𝐿_𝑝𝜔(𝐷) in the sense of the distribution theory, which belong to the space 𝐿_𝑝𝜔(𝐷). The norm of the space 𝐿_𝑝𝜔(𝐷) is given in the form ‖𝑓‖_{𝐿𝑝𝜔(𝐷)}=(∫_𝐷𝜔|𝑓|^𝑝𝑑𝑥)^1/𝑝 for 𝑝≥1; ‖𝑓‖_{𝐿𝑝𝜔(𝐷)}=esssup_𝑥∈𝐷|𝑓(𝑥)| for 𝑝=∞. Denote by 𝑊̇¹_𝑝𝜔(𝐷) a subspace of the space 𝑊¹_𝑝𝜔(𝐷), where the class of functions 𝐶^∞₀(𝐷) is an everywhere dense set. Denote by 𝑊¹_𝑝𝜔(𝐷) the closure of the set of functions 𝐶^∞(𝐷) with respect to the norm 𝑊¹_𝑝𝜔(𝐷). The spaces 𝑊¹_𝑝𝜔(𝐷) and 𝑊¹_𝑝𝜔(𝐷) coincide and are completely reflexive [1, 2] if the conditions 1<𝑝<∞ and 𝐴_𝑝-Muckenhoupt are fulfilled for 𝜔:_𝐵𝜔(𝑥)𝑑𝑥_𝐵𝜔^{−1/(𝑝−1)}(𝑥)𝑑𝑥^𝑝−1≤𝐶_𝑝|𝐵|^𝑝, (1.9) where |𝐵| denotes the Lebesgue measure of an arbitrary ball 𝐵⊂ℝ^𝑛. In the sequel, we will also use the 𝐴₁-condition:_{𝐵(𝑥,𝜌)}𝜔(𝑥)𝑑𝑥≤𝐶𝜌^𝑛inf_𝑥∈𝐵𝜔. (1.9^′)

Definition 1.1. A function 𝑢∈𝑊¹_𝑝𝜔(𝐷) is called a solution of (1.1) if the integral identity _𝐷𝐴(𝑥,𝑢,∇𝑢)⋅∇𝜑𝑑𝑥=0 (1.10) is fulfilled for any test function 𝜑∈𝑊̇¹_𝑝𝜔(𝐷).

Definition 1.2. Let 𝐸⊂⊂𝐷 be a compact subset of the bounded domain 𝐷⊂ℝ^𝑛. One will say that the set 𝐸 is removable for the class of 𝐶^𝛼(𝐷) of solutions of (1.1) if any solution of (1.1) in 𝐷⧵𝐸 from the space 𝑊¹_𝑝𝜔,loc(𝐷⧵𝐸) belongs to the space 𝑊¹_𝑝𝜔(𝐷) throughout the domain 𝐷 and is extendable inside the compactum 𝐸 as solution.

Definition 1.3. Let 𝐸⊂ℝ^𝑛 be a bounded closed subset, ℎ∶ℝ→(0,∞) a continuous function, and ℎ(0)=0,𝜇 some Radon measure. A finite system of balls {𝐵_𝜈=𝐵(𝑥_𝜈,𝑟_𝜈)}_{𝜈=1,2,…,𝑁}, the radii of which do not exceed 𝛿>0, covers the set 𝐸, that is, 𝐸⊂⋃_𝜈𝐵_𝜈. Assume that Λ_𝜇^ℎ,𝛿(𝐸)=inf{∑_𝜈ℎ(𝑟_𝜈)𝜇(𝐵_𝜈)}, where the lower bound is taken with respect to all the mentioned balls.
Assume that Λ^ℎ_𝜇(𝐸)=lim_𝛿→∞Λ_𝜇^ℎ,𝛿(𝐸). (1.11) In the case 𝜇=𝑑𝑥,ℎ(𝑡)=𝑡^{−𝑝+(𝑝−1)𝛼}, the number Λ^ℎ_𝜇(𝐸) is a Hausdorff measure of order 𝑛−𝑝+(𝑝−1)𝛼 of the set 𝐸. We will sometimes denote it by mes_{𝑛−𝑝+(𝑝−1)𝛼}(𝐸). Denote by Λ_𝜔^{−𝑝+(𝑝−1)𝛼}(𝐸) the term Λ_𝜇^ℎ(𝑟)(𝐸) for ℎ(𝑡)=𝑡^{−𝑝+(𝑝−1)𝛼}, 𝑑𝜇=𝜔𝑑𝑥.

By Carlson’s theorem [3], a necessary and sufficient condition for the compact set 𝐸 to be removable in the class of harmonic outside 𝐸 functions belonging to the class 𝐶^𝛼(𝐷) is expressed in terms of a Hausdorff measure of order 𝑛−2+𝛼 having the formmes_{𝑛−2+𝛼}𝐸=0,0<𝛼<1. (1.12) (For the case 𝛼=1, the same result was proved in [4, 5].) In [6], the corresponding result was proved for a general linear elliptic equation with variable coefficients (see also [7, 8]). In [9], for the case 𝑝≥2, a sufficient condition was proved for a solution of the 𝑝-Laplace equation (𝐴=|∇𝑢|^𝑝−2∇𝑢) to be removable in the class 𝐶^𝛼(𝐷) (0<𝛼≤1) in the form mes_{𝑛−𝑝+(𝑝−1)𝛼}𝐸=0,0<𝛼≤1. (1.13) Furthermore, a complete analogue of Carlson’s result was proved in [10], where the authors not only proved the necessity of condition (1.13), but also gave another proof of sufficiency that includes also range of exponent 1<𝑝<2. Their approach was applied to the case of a metric measure space in [11].

It should be said that in [9] a somewhat general result was in fact obtained for the compact set 𝐸 to be removable for the class 𝐶^𝛼(𝐷) of solutions of the equation ^𝑛_𝑖=1𝜕𝜕𝑥_𝑖𝜔|∇𝑢|^𝑝−2𝜕𝑢𝜕𝑥_𝑖=0 (1.14) in the form of a sufficient condition Λ_𝜔^{−𝑝+(𝑝−1)𝛼}(𝐸)=0. (1.15) Note, in [9], the case 𝑝≥2 was considered and it was required that the function 𝜔 to satisfy the doubling condition.

The present paper continues the development of the approach of [9]. We show that condition (1.15) is also the necessary one for the compact set 𝐸 to be removable. Moreover, imposing some restrictions on the degeneration function, we manage to make the proof embrace a range of the exponent 1<𝑝<2.

We will use the following auxiliary statements.

Lemma 1.4 (see [12]). Assume that a function 𝑢∈𝐿¹(𝐷) satisfies the inequality 𝑢−(_{𝐵(𝑥,𝑟)}𝑢)_𝑥,𝑟𝑑𝑥≤𝑀𝑟^𝑛+𝛼 (1.16) for any ball 𝐵(𝑥,𝑟)⊂𝐷, where 𝛼∈(0,1). Then, 𝑢∈𝐶^𝛼(𝐷) and for any 𝐷^′⊂⊂𝐷 the estimate sup_𝐷′|𝑢|+sup_{𝑥,𝑦∈𝐷′}𝑢(𝑥)−𝑢𝑦𝑥−𝑦^𝛼≤𝐶𝑀+‖𝑢‖_𝐿¹_(𝐷), (1.17) where 𝐶=𝐶(𝑛,𝛼,𝐷^′,𝐷), is satisfied.

We also need the following analogue of the well-known Giaquinta’s lemma [13].

Lemma 1.5. Let 𝜙(𝑡), 𝜔(𝑡) be nonnegative nondecreasing functions on [0,𝑅]. Assume that 𝑠>0 is such that 𝜔(𝜆𝑟)𝜔(𝑟)≥𝜆^𝑠 (1.18) for all 𝑟>0 and 0<𝜆<1. Suppose that 𝜙𝜌𝜔𝜌≤𝐴𝜔(𝑟)𝜌𝑟^𝛼+𝜀𝜙(𝑟)+𝐵𝜔(𝑟)𝑟^𝛽 (1.19) for any 0<𝜌<𝑟<𝑅, with 𝐴,𝐵,𝛼,𝛽 nonnegative constants and 𝛽<𝛼. Then, for any 𝛾∈(𝛽,𝛼), there exists a constant 𝜀₀=𝜀₀(𝐴,𝛼,𝛽,𝛾,𝑠) such that if 𝜀<𝜀₀, then one has, far all 0<𝜌<𝑟<𝑅, 𝜙𝜌𝜔𝜌≤𝑐𝜔(𝑟)𝜌𝑟^𝛾𝜙(𝑟)+𝐵𝜔𝜌𝜌^𝛽, (1.20) where 𝑐=𝑐(𝛽,𝛼,𝐴,𝑠,𝛾)>0.

Proof. For 𝜏∈(0,1) and 𝑟<𝑅, we have 𝜙(𝜏𝑟)≤𝐴𝜏^𝛼𝜔(𝜏𝑟)𝜔(𝑟)1+𝜀𝜏^{−𝛼−𝑠}𝜙(𝑟)+𝐵𝜏^−𝑠𝜔(𝜏𝑟)𝑟^𝛽. (1.21) Choose 𝜏<1 in such a way that 2𝐴𝜏^𝛼=𝜏^𝛾 and assume 𝜀₀𝜏^{−𝛼−𝑠}≤1. Then, we get, for every 𝑟<𝑅, 𝜙(𝜏𝑟)≤𝜏^𝛾𝜔(𝜏𝑟)𝜔(𝑟)𝜙(𝑟)+𝐵𝜏^−𝑠𝜔(𝜏𝑟)𝑟^𝛽 (1.22) and therefore, for all integers 𝑘>0, 𝜙𝜏^𝑘+1𝑟≤𝜏^𝛾𝜔𝜏^𝑘+1𝑟𝜔𝜏^𝑘𝑟𝜙𝜏^𝑘𝑟+𝐵𝜏^−𝑠𝜔𝜏^𝑘+1𝑟𝜏^𝑘𝛽𝑟^𝛽≤𝜏^(𝑘+1)𝛾𝜔𝜏^𝑘+1𝑟𝜔(𝑟)𝜙(𝑟)+𝐵𝜏^−𝑠𝜔𝜏^𝑘+1𝑟𝜏^𝑘𝛽𝑟^𝛽𝑘_𝑗=0𝜏^{𝑗(𝛾−𝛽)}≤𝜏^(𝑘+1)𝛾𝜔𝜏^𝑘+1𝑟𝜔(𝑟)𝜙(𝑟)+𝐵𝜏^−𝑠𝜏^𝑘𝛽1−𝜏^𝛾−𝛽𝑟^𝛽𝜔𝜏^𝑘+1𝑟. (1.23) Choosing 𝑘 such that 𝜏^𝑘+2𝑟<𝜌≤𝜏^𝑘+1𝑟, the last inequality gives 𝜙𝜌≤1𝜏^𝛾𝜌𝑟^𝛾𝜔𝜌𝜔(𝑟)𝜙(𝑟)+𝐵𝜏^−𝑠1−𝜏^𝛾−𝛽𝜏^𝑠+2𝛽𝜌^𝛽𝜔𝜌. (1.24) This proves Lemma 1.5.

We did not find the proof of the next inequality in the literature and therefore give here our proof.

Lemma 1.6. Let 1≤𝑝≤2. Let 𝑥,𝑦∈ℝ^𝑛 be arbitrary points. Then, the estimate |𝑥|^𝑝−2𝑥−𝑦^𝑝−2𝑦≤2^2−𝑝𝑥−𝑦^𝑝−1 (1.25) is valid.

Proof. Let us introduce the vector function 𝜑(𝜃)=𝜃𝑥+(1−𝜃)𝑦^𝑝−2𝜃𝑥+(1−𝜃)𝑦,0≤𝜃≤1 (1.26) acting from [0,1] into ℝ^𝑛. Applying the methods of differential calculus for the vector function, we obtain |𝑥|^𝑝−2𝑥−𝑦^𝑝−2𝑦=𝜑(1)−𝜑(0)=¹₀𝑑𝜑=𝑑𝜃𝑑𝜃𝑝−1¹₀𝜃𝑥+(1−𝜃)𝑦^𝑝−2≤𝑥−𝑦𝑑𝜃𝑝−1𝑥−𝑦¹₀𝜃𝑥+(1−𝜃)𝑦^𝑝−2𝑑𝜃. (1.27)
The set of points {𝑙(𝜃)∈ℝ^𝑛∶𝑙(𝜃)=𝜃𝑥+(1−𝜃)𝑦;0≤𝜃≤1} in ℝ^𝑛 forms a segment of the straight line that connects the point 𝑥 with the point 𝑦. We denote this segment by [𝑥,𝑦]. Let |𝑑𝑙| be a length element of this segment. It is obvious that |𝑑𝑙|=|𝑥−𝑦|𝑑𝜃. Therefore, for the above integral expression, we have the estimate ≤𝑝−1_𝑥,𝑦|𝑙(𝜃)|^𝑝−2|𝑑𝑙(𝜃)|=𝑝−1₀^{|𝑥−𝑦|}|𝑑𝑙(𝜃)|dist(𝑙(𝜃),0)^2−𝑝. (1.28)
To proceed with the estimation of this expression, we introduce into consideration the triangle, the base of which is the segment [𝑥,𝑦] and the vertex lies at the point 0. Now, the integration in the preceding estimate will be carried out with respect to the base of the triangle. It is not difficult to verify that the above integral expression takes a maximal value when the point 0 lies in the middle of the segment [𝑥,𝑦], which means that for it we have the estimate ≤𝑝−1₀^{|𝑥−𝑦|/2}𝑑𝑠𝑠^2−𝑝=2^2−𝑝𝑥−𝑦^𝑝−1. (1.29)
To show that this is true, let us choose a new coordinate system, where the 𝑥_𝑛-axis is directed along the segment [𝑥,𝑦]. Let (𝑢₁,𝑢₂,…,𝑢_𝑛) be the coordinates of the the point 0 in the new coordinate system. Then, the preceding integral expression is equal to 𝑝−1₋^{|𝑥−𝑦|/2}_{𝑥−𝑦/2}𝑢²₁+𝑢²₂+⋯+𝑢²_𝑛−1+𝑢_𝑛−𝑥_𝑛^2𝑝−2𝑑𝑥_𝑛=𝑝−1₋^{|𝑥−𝑦|/2}_{𝑥−𝑦/2}𝑢_𝑛−𝑠^𝑝−2𝑑𝑠≤2𝑝−1₀^{|𝑥−𝑦|/2}𝑠^𝑝−2𝑑𝑠=2^2−𝑝𝑥−𝑦^𝑝−1. (1.30)

The main result of this paper is contained in the following statements.

Theorem 1.7. Let 𝐷⊂ℝ^𝑛 be a bounded domain, 𝐸⊂⊂𝐷 be a compact subset. Let 2<𝑝<∞ and 𝜔 be a positive, locally integrable function satisfying condition (1.3) or 1<𝑝<2 and let any of the following conditions be fulfilled for the function 𝜔: (1)the function 𝜔 is integrable along any finite smooth 𝑛−1-dimensional surface and condition (1.9^′) is fulfilled for it; (2)for any 𝑥∈𝐷 and sufficiently small 𝜌>0, the condition ∫_{𝐵(𝑥,𝜌)}𝜔𝑑𝑦≤𝐶𝜌^𝑠 is fulfilled for some 𝑠>𝑛−𝑝+1, where the constant 𝐶>0 does not depend on 𝑥.
Then, for a compact set 𝐸 to be removable in the class 𝐶^𝛼(𝐷), 0<𝛼≤1 of solutions of (1.1) in 𝐷⧵𝐸, 𝑢∈𝑊¹_𝑝𝜔,loc(𝐷⧵𝐸), it is sufficient that condition (1.15) be fulfilled.

Here, we will use also the fact that a solution of generating equations of the form (1.1) is Hölderian. According to [14], when a weight 𝜔 belongs to the Muckenhoupt 𝐴_𝑝-class, a solution of (1.1) belongs to the class 𝐶^𝜅(𝐷_𝜌) in any subdomain 𝐷_𝜌={𝑥∈𝐷∶dist(𝑥,𝜕𝐷)>𝜌} of the domain 𝐷. For solutions, we have the estimate osc_{𝐵(𝑥,𝜌)}𝜌𝑢≤𝐶𝑟^𝜅osc_{𝐵(𝑥,𝑟)}𝑢,0<𝜌<𝑟, (1.31) where 𝜅=𝜅(𝑛,𝑝,𝐶_𝑝,𝜆)∈(0,1] and 𝐶=𝐶(𝑛,𝑝,𝐶_𝑝,𝜆). Let 𝜅 denote a maximal number 𝜅=𝜅(𝑛,𝑝,𝐶_𝑝,𝜆), for which the estimate (1.31) holds for solutions of (1.1). The following statement is valid.

Theorem 1.8. Let 𝜔∈𝐴_𝑝, 𝐸⊂⊂𝐷 be a compact subset of the domain 𝐷. Let 0<𝛼<𝜅 be some number. In that case, if Λ_𝜔^{−𝑝+(𝑝−1)𝛼}(𝐸)>0, then the set 𝐸 is not removable in the class of 𝑢∈𝑊¹_𝑝𝜔,loc(𝐷⧵𝐸) solutions of (1.1) which belong to 𝐶^𝛼(𝐷).

The foregoing statements give rise to the following corollaries.

Corollary 1.9. Let 0<𝛼<𝜅, 2≤𝑝<∞, 𝜔∈𝐴_𝑝, or 1<𝑝<2 and any of the following conditions be fulfilled:(1)the function 𝜔 satisfies condition (1.9^′) and is integrable along any finite smooth 𝑛−1-dimensional surface; (2)for any 𝑥∈𝐷 and sufficiently small 𝜌>0, the condition ∫_{𝐵(𝑥,𝜌)}𝜔𝑑𝑦≤𝐶𝜌^𝑠 is fulfilled by some 𝑠>𝑛−𝑝+1, where the constant 𝐶>0 does not depend on 𝑥.
Then, for the compact set 𝐸 to be removable for the class of 𝑊¹_𝑝𝜔(𝐷⧵𝐸) solutions of (1.1) belonging to 𝐶^𝛼(𝐷), it is necessary and sufficient that condition (1.15) be fulfilled.

Corollary 1.10. Let 0<𝛼≤1, 1<𝑝<∞. Then, for the compact set 𝐸 to be removable in the class 𝑊¹_𝑝,loc(𝐷⧵𝐸) of solutions of the equation ^𝑛_𝑖=1𝜕𝜕𝑥_𝑖|∇𝑢|^𝑝−2𝜕𝑢𝜕𝑥_𝑖=0 (1.32) belonging to the class 𝐶^𝛼(𝐷) throughout the domain 𝐷, it is necessary and sufficient that condition (1.13) be fulfilled.

2. Proof of the Main Results

In [9], the method of proving Theorem 1.7 was based on the application of an analogue of Landis-Gerver’s mean value theorem [15]. The restrictive condition 𝑝≥2 used in [9] was necessitated by the proof of Lemma 2.1 below (see also [15, Lemma  1]). Below, we prove a such type lemma for the case 1<𝑝<2, ignoring some smoothness of the function 𝑓 and making some additional assumptions for the function 𝜔.

Lemma 2.1. Let 𝐷 be a bounded domain. Let 2≤𝑝<∞ and the function 𝜔∶ℝ^𝑛→[0,∞] satisfy condition (1.3) or 1<𝑝<2, and let any of the following conditions be fulfilled for the function 𝜔: (1)condition (1.9^′) is fulfilled and 𝜔 is integrable along any finite smooth 𝑛−1-dimensional surface; (2)for any 𝑥∈𝐷 and sufficiently small 𝜌>0, the condition ∫_{𝐵(𝑥,𝜌)}𝜔𝑑𝑦≤𝐶𝜌^𝑠, where the constant 𝐶>0 does not depend on 𝑥, is fulfilled for some 𝑠>𝑛−𝑝+1.
Assume that 𝑓∶𝐷→ℝ is a sufficiently smooth function (one can also assume the condition 𝑓(𝑥)∈𝐶^𝛽(𝐷), where 𝛽≥min{𝑝^′,1}). Then, for any 𝜀>0, there exist a finite number of balls {𝐵_𝜈}, 𝜈=1,2,…,𝑁, such that ^𝑁_{𝜈=1𝜕𝐵𝜈}𝜔∇𝑓^𝑝−1𝑑𝑠<𝜀. (2.1)

Proof. We will follow the same reasoning as that used in proving Lemma  1  in [9] (see also [3], Lemma 2.1). The set 𝑂_𝑓={𝑥∈𝐷∶∇𝑓(𝑥)=0} is divided into two parts 𝑂_𝑓=𝑂^′_𝑓∪𝑂_𝑓^′′; here, 𝑂^′_𝑓 is the set of points where ∇²𝑓(𝑥)≠0, and 𝑂_𝑓^′′ is the set of points where ∇²𝑓(𝑥)=0. Let 1<𝑝<2. Then, for the set 𝑂^′_𝑓, our reasoning is as follows. By virtue of the implicit function theorem, the set 𝑂^′_𝑓 lies on a countable quantity of smooth 𝑛−1-dimensional surfaces {𝑆_𝑗};𝑗=1,2,…. Let 𝑥∈𝑆_𝑗 be a fixed point on the 𝑗-th surface. For sufficiently small 𝑟>0, we have _{𝑆(𝑥,𝑟)}𝜔∇𝑓^𝑝−1𝑑𝑠≤2𝐶₁𝑟^𝑝−1𝜔(𝑆(𝑥,𝑟)), (2.2) where 𝐶₁=sup_𝐷|∇²𝑓| and 𝜔(𝑆(𝑥,𝑟)) is integral omega over the 𝑛−1 dimensional surface 𝑆(𝑥,𝑟). By virtue of Fubini’s formula, ^2𝑟𝑥_{𝑟𝑥𝑦−𝑥=𝑡}𝜔𝑑𝑠_𝑦𝑑𝑡≤_{𝑦−𝑥<2𝑟𝑥}𝜔𝑦𝑑𝑦. (2.3) Let 𝐸 be the set of points 𝑡∈(𝑟_𝑥,2𝑟_𝑥), for which the following condition is fulfilled: _{𝑦−𝑥=𝑡}2𝜔𝑑𝑠>𝑟_{𝑥𝑦−𝑥<2𝑟𝑥}𝜔(𝑥)𝑑𝑥. (2.4) Then 2𝑟_{𝑥𝑦−𝑥<2𝑟𝑥}𝜔𝑦𝑑𝑦𝜇₁(𝐸)≤_{𝐸𝑦−𝑥=𝑡}𝜔𝑑𝑠𝑑𝑡≤^2𝑟𝑥_{𝑟𝑥𝑦−𝑥=𝑡}≤𝜔𝑑𝑠𝑑𝑡_{𝑦−𝑥<2𝑟𝑥}𝜔𝑦𝑑𝑦, (2.5) whence for the one-dimensional Lebesgue measure of the set 𝐸 we obtain the estimate 𝜇₁(𝐸)≤𝑟_𝑥/2. Hence, by virtue of the doubling condition, there exists a point 𝑡_𝑥∈(𝑟_𝑥,2𝑟_𝑥), for which _{𝑦−𝑥=𝑡𝑥}2𝜔𝑑𝑠<𝑟_{𝑥𝑦−𝑥<2𝑟𝑥}𝜔𝑦𝑑𝑦≤2𝐶𝑟_{𝑥𝑦−𝑥<𝑟𝑥}𝜔𝑦𝑑𝑦≤4𝐶𝑡_{𝑥𝑦−𝑥<𝑡𝑥}𝜔(𝑥)𝑑𝑥. (2.6) Then, for sufficiently small 𝑡_𝑗>0, for any 𝑥∈𝑆_𝑗, it can be assumed that there exists a number 𝜌_𝑥∈(𝑡_𝑗,2𝑡_𝑗), for which 𝜔𝑆𝑥,𝜌_𝑥≤4𝐶𝜌_𝑥𝜔𝐵𝑥,𝜌_𝑥. (2.7) Therefore, _{𝑆(𝑥,𝜌𝑥)}𝜔∇𝑓^𝑝−1𝑑𝑠≤4𝐶₁𝜌_𝑥^𝑝−1𝜔𝑆𝑥,𝜌_𝑥≤16𝐶𝐶₁𝜌_𝑥^𝑝−2𝜔𝐵𝑥,𝜌_𝑥. (2.8) For the surface 𝑆_𝑗, from the system of balls {𝐵(𝑥,𝜌_𝑥);𝑥∈𝑆_𝑗}, we can extract, by virtue of Besicovtich theorem [16], a subcovering {𝐵(𝑥_𝜈,𝜌_𝜈);𝑥_𝜈∈𝑆_𝑗,𝜈∈ℕ} with finite intersections: _𝜈𝜒_{𝐵(𝑥𝜈,𝜌𝜈)}(𝑥)≤𝐶_𝑛,_𝜈𝐵𝑥_𝜈,𝜌_𝜈⊃𝑆_𝑗. (2.9) Therefore and by construction, for 𝑥_𝜈∈𝑆_𝑗,𝜈=1,2,…, we have 𝜌_𝜈∈(𝑡_𝑗,2𝑡_𝑗). Thus, _{𝜈𝑆(𝑥𝜈,𝜌𝜈)}𝜔∇𝑓^𝑝−1𝑑𝑠≤_𝜈16𝐶𝐶₁𝜌_𝜈^𝑝−2𝜔𝐵𝑥_𝜈,𝜌_𝜈≤_𝜈16𝐶𝐶₁𝐶₂𝜌_𝜈^{𝑛+𝑝−2}inf_{𝑥∈𝐵(𝑥𝜈,𝜌𝜈)}𝜔≤_𝜈32𝐶𝐶₁𝐶₂𝜌_𝜈^𝑝−1𝜔𝐵𝑥_𝜈,𝜌_𝜈∩𝑆_𝑗≤32⋅2^𝑝−1𝐶𝑡_𝑗^𝑝−1𝜔𝑆_𝑗,𝑗=1,2,…. (2.10) Here, we have used the sufficient smallness of 𝑡_𝑗, condition (1.9^′), and the inequality 𝜌_𝜈^𝑛−1inf_{𝑥∈𝐵(𝑥𝜈,𝜌𝜈)}𝜔≤2𝜔(𝐵(𝑥_𝜈,𝜌_𝜈)∩𝑆_𝑗). Choosing now 𝑡_𝑗, 32⋅2^𝑝−1𝐶𝐶₁𝐶₂𝑡_𝑗^𝑝−1𝜔(𝑆_𝑗)≤𝜀/2^𝑗, where 𝜀>0 is an arbitrary number, we obtain ∑_𝜈∫_{𝑆(𝑥𝜈,𝜌𝜈)}𝜔|∇𝑓|^𝑝−1𝑑𝑠<𝜀/2^𝑗, whence, after summing the inequalities over all surfaces 𝑆_𝑗,𝑗∈ℕ, we find _{𝑗𝑥𝜈∈𝑆𝑗𝑆(𝑥𝜈,𝜌𝜈)}𝜔∇𝑓^𝑝−1𝑑𝑠<𝜀. (2.11)
In the case of the second condition 1<𝑝<2, using ∫_{𝐵(𝑥,𝜌)}𝜔𝑑𝑦≤𝐶𝜌^𝑠, we immediately pass from the inequality (2.8) to (2.10) as 𝜔𝑆𝑥,𝜌_𝑥≤4𝐶𝜌_𝑥𝜔𝐵𝑥,𝜌_𝑥≤4𝐶𝐶₄𝜌_𝑥^𝑛−𝑝=8𝐶𝐶₄𝜌_𝑥^2−𝑝mes_𝑛−1𝐵𝑥,𝜌_𝑥∩𝑆_𝑗. (2.12) Due to the latter inequality, an estimate analogous to (2.10) will have the form _{𝜈𝑆(𝑥𝜈,𝜌𝜈)}𝜔∇𝑓^𝑝−1𝑑𝑠≤_𝜈2𝐶₁𝜌_𝜈^𝑝−2𝜔𝐵𝑥_𝜈,𝜌_𝜈≤_𝜈4𝐶₁𝜌_𝜈^{𝑝−2+𝑠},_𝜈8𝐶₁𝜌_𝜈^{𝑠−𝑛+𝑝−1}mes_𝑛−1𝐵𝑥_𝜈,𝜌_𝜈∩𝑆_𝑗≤16⋅2^𝑝−1𝐶𝑡_𝑗^{𝑠−𝑛+𝑝−1}𝜔𝑆_𝑗,𝑗=1,2,…. (2.13) After choosing 𝑡_𝑗 sufficiently small and taking the condition 𝑠>𝑛−𝑝+1 into account, we can make the right-hand part smaller than 𝜀/2^𝑗.
In the case 𝑝≥2, the whole reasoning of [9] is applicable. Note that only instead of the inequality (2.10) we will have _{𝜈𝑆(𝑥𝜈,𝜌𝜈)}𝜔∇𝑓^𝑝−1𝑑𝑠≤𝐶_𝜈𝜌_𝜈^𝑝−1𝜔𝐵𝑥_𝜈,𝜌_𝜈≤𝐶₁𝑡_𝑗^𝑝−2𝜔𝑆^𝑗_𝑡𝑗, (2.14) where 𝑆^𝑗_𝑡𝑗 is the 𝑡_𝑗 neighborhood of the surface 𝑆^𝑗. After choosing a sufficiently small 𝑡_𝑗, we can make the right-hand part of this inequality smaller than 𝜀/2^𝑗. This is possible because the 𝑛-dimensional Lebesgue measure of the surface 𝑆^𝑗 is equal to zero.
Now, it remains to obtain the covering for the set of points 𝑂_𝑓^′′. Let 1<𝑝<2. Let us decompose 𝑂_𝑓^′′=𝑂_𝑓^′′′∪𝑂_𝑓^′′′′, where 𝑂_𝑓^′′′ is the set of points 𝑂_𝑓^′′, for which ∇³𝑓≠0. Here, we repeat the reasoning for 𝑂^′_𝑓. As above, the set 𝑂^′′′′ is divided into two parts. In one part, we have ∇⁴𝑓(𝑥)≠0, to which we apply the same reasoning as for 𝑂^′_𝑓. The second part of 𝑂^′′′′, where ∇⁴𝑓(𝑥)=0, is again divided into two parts. At the 𝑘-th step, when 𝑘(𝑝−1)≥1 and 𝑡>0 is sufficiently small, this process yields the estimate _{𝜈𝑆(𝑥𝜈,𝜌𝜈)}𝜔∇𝑓^𝑝−1𝑑𝑠≤𝜂_𝜈𝐶𝜌_𝜈^{𝑘(𝑝−1)}𝜔𝑆𝑥_𝜈,𝜌_𝜈≤𝜂_𝜈2𝐶𝜌_𝜈^{𝑘(𝑝−1)−1}𝜔𝐵𝑥_𝜈,𝜌_𝜈≤2𝜂𝐶𝐶₃𝜔(𝐷), (2.15) where 𝜂>0 is arbitrary.
Note that, in the case 𝑝≥2, the foregoing estimate gives the desired results immediately at the first step (𝑘=1).

Remark 2.2. It is not difficult to verify that under the assumptions of Lemma 2.1, instead of the condition of sufficient smoothness it sufficed to assume that 𝑓(𝑥)∈𝐶^𝛽(𝐷), where 𝛽≥min{𝑝^′,1}.

Applying approaches similar to those in [15, Theorem  2.2, page 128] and [9], we prove the following analogue of Landis-Gerver’s lemma.

Lemma 2.3. Let 2≤𝑝<∞ and the function 𝜔∶ℝ^𝑛→[0,∞] satisfy condition (1.3) or 1<𝑝<2 and let any of the following conditions be fulfilled for the function 𝜔: (1)condition (1.9^′) is fulfilled and 𝜔 is integrable along any finite smooth 𝑛−1-dimensional surface; (2)for any 𝑥∈𝐷 and sufficiently small 𝜌>0, the condition ∫_{𝐵(𝑥,𝜌)}𝜔𝑑𝑦≤𝐶𝜌^𝑠 is fulfilled by some 𝑠>𝑛−𝑝+1, where the constant 𝐶>0 does not depend on 𝑥.
Let 𝐷 be some domain lying in the spherical layer 𝐵(𝑥₀,2𝑟)⧵𝐵(𝑥₀,𝑟) and having limit points on the surfaces of the spheres 𝑆(𝑥₀,2𝑟) and 𝑆(𝑥₀,𝑟). Let ∑^𝑛_𝑖,𝑘=1𝑎_𝑖𝑘(𝑥)𝜂_𝑖𝜂_𝑘 be the quadratic form, the coefficients of which are well defined and continuously differentiable in the domain 𝐷 and for which the inequalities 𝜆𝜂²≤^𝑛_𝑖,𝑘=1𝑎_𝑖𝑘(𝑥)𝜂_𝑖𝜂_𝑘≤𝜆⁻¹𝜂² (2.16) are fulfilled for any 𝑥∈𝐷,𝜂∈ℝ^𝑛 for some 𝜆∈(0,1). Assume that 𝑓∶𝐷→ℝ is a sufficiently smooth function.
Then, there exists a piecewise-smooth surface Σ, separating, in the domain 𝐷, the surfaces of the spheres 𝑆(𝑥₀,𝑟) and 𝑆(𝑥₀,2𝑟) and being such that _Σ𝜔∇𝑓^𝑝−2𝜕𝑓𝜕𝜈𝑑𝑠≤𝐾osc_𝐷𝑓^𝑝−1𝜔(𝐷)𝑟^𝑝, (2.17) where 𝜕𝑓/𝜕𝜈=∑^𝑛_𝑗=1𝑎_𝑖𝑗(𝑥)𝑓_𝑥𝑖𝑛_𝑗 is conormal derivative on Σ, 𝑛=(𝑛₁,𝑛₂,…,𝑛_𝑛) is unit orthogonal vector to the surface Σ, and the constant 𝐾 depends on 𝑝,𝜆, and the dimension 𝑛.

Proof. It suffices to consider the case 𝑟=1. Indeed, after the change of variables 𝑥=𝑟𝑦, the function 𝑓∶𝐷→𝑅 transforms to the function 𝑓∶𝐷→𝑅, where 𝑓(𝑦)=𝑓(𝑟𝑦). Also, |∇_𝑦𝑓|=|∇_𝑥𝑓|𝑟,𝜕𝑓/𝜕𝑣_𝑦=(𝜕𝑓/𝜕𝑣_𝑥)𝑟,𝜔(𝐷)=𝜔(𝐷)𝑟^−𝑛. 𝐷 lies in the spherical layer 𝐵(0,2)⧵𝐵(0,1). It suffices to show that 𝜕𝑓/𝜕𝑣_𝑦=(𝜕𝑓/𝜕𝑣_𝑥)𝑟. Indeed, let a sufficiently small element of the surface Σ satisfy the equation 𝜑(𝑥)=0 in coordinates 𝑥. Then, after the change of variables, this equation takes the form 𝜑(𝑦)=0, where 𝜑(𝑦)=𝜑(𝑟𝑦). In other words, the normals of the surfaces Σ and Σ are related by 𝑛_𝑥=∇_𝑥𝜑/|∇_𝑥𝜑|=∇_𝑦𝜑/|∇_𝑦𝜑|=𝑛_𝑦. Therefore, 𝜕𝑓𝜕𝑣_𝑦=^𝑛_𝑖,𝑘=1𝑎_𝑖𝑘𝑦𝜕𝑓𝜕𝑦_𝑖𝑛_𝑘=𝑟^𝑛_𝑖,𝑘=1𝑎_𝑖𝑘(𝑥)𝜕𝑓𝜕𝑥_𝑖𝑛_𝑘=𝑟𝜕𝑓𝜕𝑣_𝑥,𝑎_𝑖𝑘𝑦=𝑎_𝑖𝑘𝑟𝑦,𝑖,𝑘=1,2,…,𝑛. (2.18) Applying these equalities, from the estimate _Σ∇_𝑦𝑓^𝑝−2𝜕𝑓𝜕𝑣_𝑦𝑑𝑠_𝑦𝐷≤𝐾osc𝑓^𝑝−1𝜔𝐷, (2.19) we obtain (2.17).
Let us now prove (2.19). Following the notation and reasoning of [15] (see also [9]), we assume that 𝜀=𝜔(𝐷)(osc𝑓)^𝑝−1. For this 𝜀, we find the corresponding balls 𝑄₁,𝑄₂,…,𝑄_𝑁 of Lemma 2.1 and remove them from the domain 𝐷. Assume that 𝐷^∗=𝐷⧵⋃^𝑁_𝑚=1𝑄_𝑚 and intersect 𝐷^∗ with the closed layer (1+1/4)≤|𝑥|≤(1+3/4). Denote this intersection by 𝐷^′. On the closed set 𝐷^′, we have ∇𝑓≠0. Let us choose some 𝛿-neighborhood 𝐷^′_𝛿 with 𝛿<1/4 so small that in 𝐷^′_𝛿 we would have |∇𝑓|>𝛼>0. We consider on 𝐷^′_𝛿 the system of equations 𝑑𝑥_𝑖=𝑑𝑡^𝑛_𝑘=1𝑎_𝑖𝑘(𝑥)𝜕𝑓𝜕𝑥_𝑘,𝑖=1,2,…,𝑛. (2.20) In 𝐷^′_𝛿, there are no stationary points of the system (2.20), and at every point 𝑥∈𝐷^′_𝛿 the direction of the field forms with the direction of the gradient an angle different from the straight angle. Let 𝑙(𝑥) be the vector of the field at the point 𝑥. Then, using cos(𝑙(𝑥),∇𝑓)=(∑^𝑛_𝑘=1𝑎_𝑖𝑘(𝑥)(𝜕𝑓/𝜕𝑥_𝑘),∇𝑓)/|∑^𝑛_𝑘=1𝑎_𝑖𝑘(𝑥)(𝜕𝑓/𝜕𝑥_𝑘)||∇𝑓|>𝜆|∇𝑓|²/𝜆⁻¹|∇𝑓|²=𝜆², we obtain 𝜕𝑓𝜕𝑙>𝜆²∇𝑓>𝛾𝛼>0,𝛾=𝜆². (2.21) From this inequality, it follows that in 𝐷^′_𝛿 there are no closed trajectories and all the trajectories have the uniformly bounded length.
Let some surface 𝑆 be tangential, at each of its points, to the field direction. Then, _𝑆𝜔∇𝑓^𝑝−2𝜕𝑓𝜕𝑣𝑑𝑠=0, (2.22) since the integrand is identically zero. We will use this fact in constructing the needed surface Σ. The base of Σ consists of ruled surfaces, while the generatrices are the trajectories of the system (2.20). Note that they will add nothing to the integral in which we are interested. These surfaces will have the form of fine tubes which will cover the entire 𝐷^′. Let us insert partitions into some of the tubes. The integral over these partitions will not any longer be equal to zero, but we can make it infinitesimal. The construction of tubes practically repeats that given in [15, pages 129–132].
Denote by 𝐸 the intersection of 𝐷^′ with the sphere 𝑆⁰_(1+3/4). Let 𝑁 be the set of points 𝑥∈𝐸, where the direction of the field of the system (2.20) is tangential to the sphere 𝑆⁰_(1+3/4). At the points 𝑥∈𝑁, we have 𝜕𝑓/𝜕𝑣=0, where 𝜕/𝜕𝑣 is the derivative with respect to the conormal to the sphere 𝑆⁰_(1+3/4). Cover 𝑁 by a set 𝐺, open on the sphere 𝑆⁰_(1+3/4) and being such that _𝐺𝜔∇𝑓^𝑝−2𝜕𝑓(𝐷)𝜕𝑣𝑑𝑠≤𝜔osc𝑓^𝑝−1. (2.23) Put 𝐸^′=𝐸⧵𝐺. At the points 𝑥∈𝐸^′, the direction of the field is transversal to the sphere. Cover 𝐸′ on the sphere 𝑆⁰_(1+3/4) by a finite number of uncovered domains with piecewise-smooth boundaries. We will call them cells. We will choose their diameters so small that at the points of the cells the field would be transversal to the sphere and the bundle of trajectories passing through each of the cells would diverge by 𝛿/2𝑛 at most. The surface with trajectories lying inside the ball |𝑥|<(1+3/4) and passing through the cell boundary will be called a tube. Thus, we obtain a finite number of tubes. We will call a tube a through tube if, without intersecting this tube, we can connect by a broken line a point of its corresponding cell with a point of the sphere 𝑆⁰_{(1+1/4)−(𝛿/2)} within the limits of the intersection of 𝐷′ with a spherical layer 1+1/4−𝛿<|𝑥|<1+3/4. Such through tubes are denoted by 𝑇₁,𝑇₂,…,𝑇_𝑠. If every through tube is partitioned, then the spheres 𝑆⁰₁ and 𝑆⁰₂ are separated in 𝐷 by the set-theoretic sum of nonthrough tubes, partitions 𝑇₁,𝑇₂,…,𝑇_𝑠, the spheres 𝑆₁,𝑆₂,…,𝑆_𝑁, and the set 𝐺 on the sphere 𝑆⁰_(1+3/4).
Let us now take care to choose partitions in such a way that the integral ∫𝜔|∇𝑓|^𝑝−2|𝜕𝑓/𝜕𝑣|𝑑𝑠 over them would have the value which we need. Denote by 𝑈_𝑖 the domain bounded by 𝑇_𝑖. Choose any trajectory on this tube. Denote it by 𝐿_𝑖. The length 𝜇₁𝐿_𝑖 of the curve 𝐿_𝑖 satisfies the inequality 𝜇₁𝐿_𝑖>1/2. Introduce, on 𝐿_𝑖, the parameter 𝑙 which is the length of the arc counted from 𝑆⁰_(1+1/4). Denote by 𝜎_𝑖(𝑙) the section 𝑈_𝑖 with a hypersurface which is orthogonal, at the point 𝑙, to the trajectory 𝐿_𝑖. Let the diameter at the beginning of the tube be so small that ∫_𝐿𝑖(∫_{𝜎𝑖(𝑙)}𝜔𝑑𝑠)𝑑𝑙≤2𝜔(𝑈_𝑖). Then, the set 𝐻 of points 𝑙∈𝐿_𝑖, where ∫_{𝜎𝑖(𝑙)}𝜔𝑑𝑠>8𝜔(𝑈_𝑖), satisfies the inequality 𝜇₁𝐿_𝑖<1/4. Thus, for 𝐸=𝐿_𝑖⧵𝐻, the inequality 𝜇₁𝐿_𝑖>1/4 is valid and 𝜔𝑑𝑠<8𝜔𝑈_{𝜎𝑖(𝑙)𝑖}for𝑙∈𝐸. (2.24) At the points of the curve 𝐿_𝑖, the derivative 𝜕𝑓/𝜕𝑙 preserves the sign and therefore _𝐸𝜕𝑓𝜕𝑙𝑑𝑙≤_𝐿𝑖𝜕𝑓𝜕𝑙𝑑𝑙<osc_𝐷′𝛿𝑓. (2.25) Hence, using 𝜇₁𝐿_𝑖>1/4 and the mean value theorem, we see that there exists a point 𝑙₀∈𝐸 such that |𝜕𝑓/𝜕𝑙|_𝑙=𝑙0≤4osc𝑓. On the other hand, since, by virtue of (2.21), |𝜕𝑓/𝜕𝑙|_𝑙=𝑙0≥𝛾|∇𝑓|_𝑙=𝑙0, we have |∇𝑓|^𝑝−1|_𝑙=𝑙0≤(4osc𝑓)^𝑝−1𝛾^1−𝑝. This together with (2.24) gives the estimate ∇𝑓^𝑝−1_{𝑙=𝑙0𝜎𝑖(𝑙0)}𝜔𝑑𝑠≤𝐶𝑝,𝛾𝜔𝑈_𝑖osc𝑓^𝑝−1. (2.26) Let us now choose a cell diameter so small that _{𝜎𝑖(𝑙0)}∇𝑓^𝑝−1𝜔𝑑𝑠≤2𝐶𝑝,𝛾𝜔𝑈_𝑖osc𝑓^𝑝−1. (2.27) This can be done since the derivatives 𝜕𝑓/𝜕𝑥_𝑘,𝑘=1,2,…,𝑛, are uniformly continuous. Therefore, ^𝑠_{𝑖=1𝜎𝑖(𝑙0)}∇𝑓^𝑝−1𝜔𝑑𝑠≤4𝐶𝑝,𝛾𝜔𝑈_𝑖osc𝑓^𝑝−1. (2.28) Denote by Σ the set-theoretic sum of all nonthrough tubes, all 𝜎_𝑖(𝑙₀), all spheres 𝑆_𝑖, and the set 𝐺 on the sphere 𝑆⁰_(1+3/4)𝑟. Then, from Lemma 2.1 and (2.22)–(2.28), we obtain _Σ∇𝑓^𝑝−2𝜕𝑓𝜔𝑈𝜕𝑣𝜔𝑑𝑠≤𝐶𝑝,𝑛,𝛾_𝑖osc𝑓^𝑝−1. (2.29)
Lemma 2.3 is proved.

In this paper, we give the complete proof of Theorem 1.7. Some part of the proof of sufficiency is in fact identical to the proof given in [9]. The method of proving Theorem 1.8 is analogous to the method [10], where the nonweight case was considered.

Proof of Theorem 1.7 (Approximation). Let Λ_𝜔^{−𝑝+(𝑝−1)𝛼}(𝐸)=0 and 𝑢∈𝐶^𝛼(𝐷); let 𝑢∈𝑊¹_𝑝𝜔,loc(𝐷⧵𝐸) be a solution of (1.1). Denote by 𝑢^𝑗 a mean value of the function 𝑢 with smooth kernel 𝜌 with finite support, 𝑢^(𝑗)=𝜌_1/𝑗∗𝑢=𝑗^𝑛∫_ℝ𝑛𝜌((𝑥−𝑦)𝑗)𝑢(𝑦)𝑑𝑦,∫_ℝ𝑛𝜌(𝑥)𝑑𝑥=1,𝑗∈ℕ. Then, it is obvious that 𝑢^(𝑗)∈𝐶^∞(𝐷), 𝑗=1,2,…. Moreover, 𝑢^𝑗→𝑢 uniformly in any subdomain 𝐺⊂𝐷. Also, for any open set 𝐸′⊃𝐸 contained in 𝐺, 𝑢^(𝑗)→𝑢 in the norm of the space 𝑊¹_𝑝𝜔(𝐺⧵𝐸^′) (see [2, 16]). Since, by condition (1.15), we have mes_𝑛𝐸=0, it can be assumed that mes_𝑛𝐸^′<𝜂, where 𝜂>0 is an arbitrary number.
Let 𝜀>0 be an arbitrary number. Cover the set 𝐸 by a finite system of balls {𝐵_𝜈}_{𝜈=1,2,…,𝑁}, ⋃^𝑁_𝜈=1𝐵_𝜈⊃𝐸 such that diam𝐵_𝜈<𝛿, ^𝑁_𝜈=1𝑟_𝜈^{−𝑝+(𝑝−1)𝛼}𝜔𝐵_𝜈<𝜀. (2.30) Assume that the number 𝛿=𝛿(𝜀,𝜂) is so small that the set Γ^′′=⋃^𝑁_𝜈=1(4𝐵_𝜈) lies in 𝐸^′.
For every 𝜈, there exists, by virtue of Lemma 2.3 and inequality (1.31), a piecewise-smooth surface 𝛾_𝜈^(𝑗),𝜈=1,2,…,𝑁,𝑗=1,2,…, separating the surfaces of the spheres 𝜕(2𝐵_𝜈) and 𝜕(4𝐵_𝜈), such that _𝛾𝑗𝜈𝜔∇𝑢^{(𝑗)𝑝−2}𝜕𝑢^(𝑗)𝜕𝑣𝑑𝑠≤𝐾𝑟_𝜈^−𝑝osc_2𝐵𝜈𝑢^{(𝑗)𝑝−1}𝜔4𝐵_𝜈. (2.31) Denote by Γ_𝜈^(𝑗) the interiority of the surface 𝛾_𝜈^(𝑗). Then, Γ^(𝑗)=⋃_𝜈Γ_𝜈^(𝑗)⊃Γ^′=⋃_𝜈(2𝐵_𝜈). Assume that 𝜎_𝜈^(𝑗)=Γ^(𝑗)∩𝛾_𝜈. Let 𝜎_𝜈^(𝑗)≠∅ for some 𝜈. Then, for 𝜈, the inequality (2.31) implies the estimate _{𝜎𝜈(𝑗)}𝜔∇𝑢^{(𝑗)𝑝−2}𝜕𝑢^(𝑗)𝜕𝑣𝑑𝑠≤𝐾𝑟_𝜈^−𝑝osc_2𝐵𝜈𝑢^{(𝑗)𝑝−1}𝜔4𝐵_𝜈. (2.32)
It is obvious that the set 𝐺⧵Γ′ is a strictly interior subdomain of the domain 𝐷⧵𝐸. Thus, we have the identity ^𝑛_{𝑖,𝑘=1𝐺⧵Γ′}𝜔|∇𝑢|^𝑝−2𝑏_𝑖𝑘𝑢_𝑥𝑘𝜓_𝑥𝑖𝑑𝑥=0, (2.33) for any 𝜓∈𝐶¹₀(𝐷⧵Γ′). From this, by virtue of the convergence ‖𝑢^(𝑗)−𝑢‖_{𝑊1𝑝𝜔(𝐺⧵Γ′)}→0 as 𝑗→∞ and the fact that the ‖𝑏^(𝑗)_𝑖𝑘‖ satisfies (2.16) by some 𝜆>0, we find ^𝑛_{𝑖,𝑘=1𝐺⧵Γ′}𝜔∇𝑢^{(𝑗)𝑝−2}𝑏^(𝑗)_𝑖𝑘𝑢_𝑥^(𝑗)_𝑘𝜓_𝑥𝑖𝑑𝑥=𝛿_𝑗. (2.34) This follows from the fact that the integrand is a system of equi-integrable functions: for any subset 𝑔⊂𝐷⧵Γ^′, we have ^𝑛_{𝑖,𝑘=1𝑔}𝜔𝑏^(𝑗)_𝑖𝑘∇𝑢^{(𝑗)𝑝−2}𝑢_𝑥^(𝑗)_𝑖𝜓_𝑥𝑘𝑑𝑥≤𝐶_𝑔𝜔∇𝑢^{(𝑗)𝑝−1}∇𝜓𝑑𝑥≤2𝐶_𝑔𝜔|∇𝑢|^𝑝𝑑𝑥^1/𝑝′_𝑔𝜔∇𝜓^𝑝𝑑𝑥^1/𝑝⟶0 (2.35) as mes_𝑛𝑔→0. Here and in the sequel, speaking in general, we denote by 𝛿_𝑗 different sequences tending to zero as 𝑗→∞.
Green’s Formulae for Approximations
Let now 𝜑∈𝐶¹₀(𝐷) be an arbitrary function. Assume that 𝜓=𝜑𝜉(𝑑(𝑥)/𝜏), where 0≤𝜉(𝑠)≤1 is an infinitely differentiable function equal to zero for 𝑠≤0 and to one for 𝑠≥1 and 𝜏>0 is a parameter, forall𝜑∈𝐶¹₀(𝐷), 𝑑(𝑥)=dist(𝑥,Γ^𝑗). It is obvious that 𝜓∈𝐶¹₀(𝐷⧵Γ^𝑗). Then, (2.34) implies, for 𝑗=1,2,…, ^𝑛_{𝑖,𝑘=1𝐷⧵Γ′}𝜔∇𝑢^{(𝑗)𝑝−2}𝑏^(𝑗)_𝑖𝑘𝑢_𝑥^(𝑗)_𝑘𝜑_𝑥𝑖𝜉𝑑𝑥+^𝑛_𝑖,𝑘=11𝜏_𝐷⧵Γ^′𝜔∇𝑢^{(𝑗)𝑝−2}𝑏^(𝑗)_𝑖𝑘𝑢_𝑥^(𝑗)_𝑘𝑑_𝑥𝑖𝜉^′𝑑(𝑥)𝜏𝜑𝑑𝑥=𝛿_𝑗. (2.36) By virtue of the majorant Lebesgue theorem, for 𝜏→0, the first summand in (2.36) tends to the limit ∑^𝑛_𝑖,𝑘=1∫_{𝐷⧵Γ(𝑗)}𝜔|∇𝑢^(𝑗)|^𝑝−2𝑏^(𝑗)_𝑖𝑘𝑢_𝑥^(𝑗)_𝑘𝜑_𝑥𝑖𝑑𝑥. Let us now find the limit of the second summand. Applying the Federer formula, we have ^𝑛_𝑖,𝑘=11𝜏_𝐷⧵Γ^′𝜉^′𝑑(𝑥)𝜏𝜔∇𝑢^{(𝑗)𝑝−2}𝑏^(𝑗)_𝑖𝑘𝑢_𝑥^(𝑗)_𝑘𝜑_𝑥𝑖=𝑑𝑥_{𝑛𝑖,𝑘=1}1𝜏^𝜏_{0𝑑(𝑥)=𝑡∩𝐷⧵Γ}^′𝜑𝜔∇𝑢^{(𝑗)𝑝−2}𝑏^(𝑗)_𝑖𝑘𝑢^𝑗_𝑥𝑘𝑑_𝑥𝑖|∇𝑑|𝑑𝑠_𝑡𝜉^′𝑡𝜏𝑑𝑡. (2.37) Applying the mean value theorem, for some 𝑡₀∈(0,𝜏), we obtain =^𝑛_{𝑖,𝑘=1𝑑(𝑥)=𝑡0}𝜑𝜔∇𝑢^𝑗𝑝−2𝑏^(𝑗)_𝑖𝑘𝑢_𝑥𝑘𝑑_𝑥𝑖|∇𝑑|𝑑𝑠_𝑡01𝜏^𝜏₀𝜉^′𝑡𝜏⟶𝑑𝑡_{𝑛𝑖,𝑘=1𝑑(𝑥)=𝑡0}𝜑𝜔∇𝑢^{(𝑗)𝑝−2}𝑏^(𝑗)_𝑖𝑘𝑢_𝑥^(𝑗)_𝑘𝑑_𝑥𝑖|∇𝑑|𝑑𝑠_𝑡0⟶_𝜕Γ(𝑗)𝜑𝜔∇𝑢^{(𝑗)𝑝−2}𝜕𝑢^(𝑗)𝜕𝑣𝑑𝑠as𝜏⟶0. (2.38) Taking this limit relation into account, from (2.36), we obtain, as 𝜏→0, the following equality which is Green’ formula for approximation function 𝑢^𝑗: ^𝑛_{𝑖,𝑘=1𝐷⧵Γ(𝑗)}𝜔∇𝑢^{(𝑗)𝑝−2}𝑏^(𝑗)_𝑖𝑘𝑢_𝑥^(𝑗)_𝑘𝜑_𝑥𝑖𝑑𝑥=_𝜕Γ(𝑗)𝜑𝜔∇𝑢^{(𝑗)𝑝−2}𝜕𝑢^(𝑗)𝜕𝑣𝑑𝑠+𝛿_𝑗. (2.39) Whence, in view of the inequality |𝜑(𝑥)|≤‖𝜑‖_𝐶(𝐷), 𝑥∈𝐷, we have ^𝑛_{𝑖,𝑘=1𝐷⧵Γ(𝑗)}𝜔∇𝑢^{(𝑗)𝑝−2}𝑏^(𝑗)_𝑖𝑘𝑢_𝑥^(𝑗)_𝑘𝜑_𝑥𝑖𝑑𝑥≤𝜑_{𝐶(𝐷)𝜕Γ(𝑗)}𝜔∇𝑢^{(𝑗)𝑝−2}𝜕𝑢^(𝑗)𝜕𝑣𝑑𝑠+𝛿_𝑗≤𝜑_{𝐶(𝐷)𝜈𝛾𝜈}𝜔∇𝑢^{(𝑗)𝑝−2}𝜕𝑢^(𝑗)𝜕𝑣𝑑𝑠+𝛿_𝑗. (2.40)
Using the convergence ‖𝑢^(𝑗)−𝑢‖_{𝑊1𝑝𝜔(𝐺⧵Γ′)}→0  (𝑗→∞), Lemma 1.6, conditions (1.2), and Hölder inequality, we have the estimate for 1<𝑝<2: ^𝑛_{𝑖,𝑘=1𝐷⧵Γ(𝑗)}𝜔𝑏^(𝑗)_𝑖𝑘∇𝑢^{(𝑗)𝑝−2}𝑢_𝑥^(𝑗)_𝑖−|∇𝑢|^𝑝−2𝑢_𝑥𝑖𝜑_𝑥𝑘𝑑𝑥≤𝐶_𝐷⧵Γ′𝜔∇𝑢^(𝑗)−𝑢^𝑝−1∇𝜑𝑑𝑥≤𝐶_𝐷⧵Γ′𝜔∇𝑢^(𝑗)−𝑢^𝑝𝑑𝑥^1/𝑝′_𝐷⧵Γ′𝜔∇𝜑^𝑝𝑑𝑥^1/𝑝=𝛿_𝑗⟶0as𝑗⟶∞. (2.41) An analogous estimate for 2≤𝑝<∞ has the form ^𝑛_{𝑖,𝑘=1𝐷⧵Γ(𝑗)}𝜔𝑏^(𝑗)_𝑖𝑘∇𝑢^{(𝑗)𝑝−2}𝑢_𝑥^(𝑗)_𝑖−|∇𝑢|^𝑝−2𝑢_𝑥𝑖𝜑_𝑥𝑘𝑑𝑥≤𝐶_𝐷⧵Γ′𝜔∇𝑢^(𝑗)−𝑢|∇𝑢|^𝑝−2+∇𝑢^{(𝑗)𝑝−2}∇𝜑𝑑𝑥≤𝐶_𝐷⧵Γ′𝜔∇𝑢^(𝑗)−𝑢^𝑝𝑑𝑥^1/𝑝_𝐷⧵Γ′𝜔|∇𝑢|^𝑝+∇𝑢^(𝑗)𝑝𝑑𝑥^{(𝑝−2)/𝑝}×_𝐷𝜔∇𝜑^𝑝𝑑𝑥^1/(𝑝−2)≤2𝐶∇𝜑^𝑝′_{𝐿𝑝𝜔(𝐷)}‖∇𝑢‖_𝐿^𝑝−2_{𝑝𝜔(𝐷⧵Γ′)}∇𝑢^(𝑗)−𝑢_{𝐿𝑝𝜔(𝐷⧵Γ′)}=𝛿_𝑗⟶0as𝑗⟶∞. (2.42)
The Belongness𝑢∈𝑊¹_𝑝𝜔(𝐷)
Taking into account (2.41) and (2.42), the estimate (2.32), and the uniform convergence 𝑢^(𝑗)→𝑢 in 𝐺, convergence a.e. 𝑏^(𝑗)_𝑖𝑘→𝑏_𝑖𝑘, we find ^𝑛_{𝑖,𝑘=1𝐷⧵Γ(𝑗)}𝜔|∇𝑢|^𝑝−2𝑏_𝑖𝑘𝑢_𝑥𝑘𝜑_𝑥𝑖𝑑𝑥≤𝐶𝜑_{𝐶(𝐷)𝜈}𝑟_𝜈^−𝑝osc_2𝐵𝜈𝑢^{(𝑗)𝑝−1}𝜔𝐵_𝜈+𝛿_𝑗≤𝐶𝜑_{𝐶(𝐷)𝜈}𝑟_𝜈^{−𝑝+(𝑝−1)𝛼}𝜔𝐵_𝜈+𝛿_𝑗. (2.43) Therefore, ^𝑛_{𝑖,𝑘=1𝐷⧵Γ𝑗}𝜔|∇𝑢|^𝑝−2𝑏_𝑖𝑘𝑢_𝑥𝑘𝜑_𝑥𝑖𝑑𝑥=𝑂(𝜀)+𝛿_𝑗. (2.44) Taking into account the density of the class of functions 𝐶¹₀(𝐷) in the space 𝑊̇¹_𝑝𝜔(𝐷) and the fact that 𝑢∈𝑊¹_𝑝𝜔,loc(𝐷⧵𝐸), we also come to the same equality (2.44) for any function 𝜑∈𝑊̇¹_𝑝𝜔(𝐷). Assuming now that, in (2.44) 𝜑=𝑢𝜉^𝑝, where 𝜉∈𝐶^∞₀(𝐷) is a positive function equal to one in 𝐺, since Γ^𝑗⊂Γ^′′ and the integrand is positive, we obtain 𝜆_{𝐺⧵Γ′′}𝜔|∇𝑢|^𝑝𝑑𝑥≤^𝑛_{𝑖,𝑘=1𝐷⧵Γ𝑗}𝜔𝜉|∇𝑢|^𝑝−2𝑏_𝑖𝑘𝑢_𝑥𝑖𝑢_𝑥𝑘𝑑𝑥≤_{𝐷⧵Γ𝑗}𝜔𝜉^𝑝−1|∇𝑢|^𝑝−1∇𝜉|𝑢|𝑑𝑥+𝑂(𝜀)+𝛽_𝑗. (2.45) Whence, by means of Young’s inequality, we derive _{𝐷⧵Γ′′}𝜔|∇𝑢|^𝑝𝑑𝑥≤𝐶_𝐷𝜔∇𝜉^𝑝|𝑢|^𝑝𝑑𝑥=𝑂(1). (2.46) Then, by virtue of the arbitrariness of 𝜀,𝜂 (mes_𝑛𝐸=0), we obtain 𝑢∈𝑊¹_𝑝𝜔(𝐷).
proof of that 𝑢(𝑥) is a solution in 𝐷. Let us return to relation (2.44), from which, in view of 𝑢∈𝑊¹_𝑝𝜔(𝐷) and Γ^′′⊂𝐸^′, we have ^𝑛_{𝑖,𝑘=1𝐷⧵𝐸′}𝜔|∇𝑢|^𝑝−2𝑏_𝑖𝑘𝑢_𝑥𝑘𝜑_𝑥𝑖𝑑𝑥=𝑂(𝜀) (2.47) for any 𝜑∈𝑊̇¹_𝑝𝜔(𝐷). By virtue of the arbitrariness of 𝜀,𝜂, we find ^𝑛_{𝑖,𝑘=1𝐷}𝜔|∇𝑢|^𝑝−2𝑏_𝑖𝑘𝑢_𝑥𝑘𝜑_𝑥𝑖𝑑𝑥=0, (2.48) that is, the function 𝑢∈𝑊¹_𝑝𝜔(𝐷) is a solution of (1.5) throughout the domain 𝐷 and thereby of (1.1), too.
Theorem 1.7 is proved.

Proof of Theorem 1.8. Let Λ_𝜔^{−𝑝+(𝑝−1)𝛼}(𝐸)>0 for some compact set 𝐸⊂𝐷. Let us use the recent results for a Frosman type lemma with measure [11, 17] and follow the reasoning of the original paper [3]. We come to the following conclusion. There exists a Radon measure 𝜇 with a support on the set 𝐸, such that 𝜇(𝐸)>0 and for any ball 𝐵=𝐵(𝑥,𝑟) we have 𝜇(𝐵)≤𝐶𝑟^{−𝑝+(𝑝−1)𝛼}𝜔(𝐵). (2.49)
Let 𝑢∈𝑊̇¹_𝑝𝜔 be a solution of the equation div(𝐴(𝑥,𝑢,∇𝑢))=𝜇 (2.50) in the domain 𝐷=𝐵(0,𝑅), where 𝐵 is a sufficiently large ball. The solution of (2.50) is understood in the sense as follows: the integral identity _𝐷𝐴(𝑥,𝑢,∇𝑢)⋅∇𝜑𝑑𝑥=_𝐷𝜑𝑑𝜇 (2.51) is fulfilled for any test function 𝜑∈𝑊̇¹_𝑝𝜔(𝐷). Such a solution exists by virtue of 𝜇∈(𝑊¹_𝑝𝜔(𝐷))^∗. Let us show the latter inclusion.
By virtue of 𝜔∈𝐴_𝑝 for 𝑞>𝑝 and the fact that 𝑞 is sufficiently close to 𝑝, inequality (2.49) implies for 0<𝜌<𝑟: 𝑟^1−𝑛𝜇𝐵𝑥,𝜌^1/𝑞𝜎𝐵𝑥,𝜌^1/𝑝′≤𝐶𝜌^{𝛼(𝑝−1)/𝑞−(𝑛−1)(1−𝑝/𝑞)}𝜎𝐵𝑥,𝜌^{(𝑝−1)(1/𝑝−1/𝑞)}, (37^′) where 𝜎(𝐵(𝑥,𝜌))=∫_{𝐵(𝑥,𝜌)}𝜔^{−1/(𝑝−1)}(𝑦)𝑑𝑦. This inequality defines the constant in the Adams inequality _𝐵(0,𝑟)|𝑢|^𝑞𝑑𝜇^1/𝑞≤𝐶_𝑝,𝑞(𝑟)_𝐵()0,𝑟𝜔|∇𝑢|^𝑝𝑑𝑥^1/𝑝, (2.52) as 𝐶_𝑝,𝑞(𝑟)=sup_{()𝑥∈𝐵0,𝑟,0<𝜌<4𝑟}𝜌^1−𝑛𝜇𝐵𝑥,𝜌^1/𝑞𝜎𝐵𝑥,𝜌^1/𝑝′. (2.53) Then, by virtue of (37^′), we have 𝐶_𝑝,𝑞(𝑟)≤𝐶𝑟^{𝛼(𝑝−1)/𝑞−(𝑛−1)(1−𝑝/𝑞)}(𝜎(𝐵(0,𝑟)))^{(𝑝−1)(1/𝑝−1/𝑞)}. (37^′′) Whence, by virtue of Hölder inequality, we obtain _𝐵(0,𝑟)|𝑢|𝑑𝜇≤𝐶_𝑝,𝑞(𝑟)𝜇(𝐵(0,𝑟))^1/𝑞′_𝐵(0,𝑟)𝜔|∇𝑢|^𝑝𝑑𝑥^1/𝑝. (2.54) Taking into account inequalities (2.49), (37^′′) and the 𝐴_𝑝-condition, for any function 𝑢∈𝐶¹₀(𝐵(0,𝑟)), we have _𝐵(0,𝑟)|𝑢|𝑑𝜇≤𝐶(𝜔(𝐵(0,𝑟)))^1/𝑝′𝑟^{1−𝑝+(𝑝−1)𝛼}_𝐵(0,𝑟)𝜔|∇𝑢|^𝑝𝑑𝑥^1/𝑝, (38^′) whence it follows that 𝜇∈(𝑊¹_𝑝𝜔(𝐷))^∗.
Let us, following ideas of [10], show that 𝑢(𝑥)∈𝐶^𝛼(𝐷). Let ℎ∈𝑊¹_𝑝𝜔(𝐵(𝑥₀,𝑟)) be a solution of the equation div(𝐴(𝑥,ℎ,∇ℎ))=0 (2.55) with the condition ℎ−𝑢∈𝑊̇¹_𝑝𝜔(𝐵(𝑥₀,𝑟)). Then, for it, we have the integral identity _{𝐵(𝑥0,𝑟)}(𝐴(𝑥,𝑢,∇𝑢)−𝐴(𝑥,ℎ,∇ℎ))⋅∇𝑣𝑑𝑥=_{𝐵(𝑥0,𝑟)}𝑣𝑑𝜇, (2.56) where 𝑣=𝑢−ℎ. The integrand in the left-hand part of (2.56) is positive. Therefore, for 𝜌<𝑟/2, we have _{𝐵(𝑥0,𝜌)}(𝐴(𝑥,𝑢,∇𝑢)⋅∇𝑢+𝐴(𝑥,ℎ,∇ℎ)⋅∇ℎ)𝑑𝑥=_{𝐵(𝑥0,𝜌)}(𝐴(𝑥,ℎ,∇ℎ)∇𝑢+𝐴(𝑥,𝑢,∇𝑢)∇ℎ)𝑑𝑥+_{𝐵(𝑥0,𝑟)}𝑣𝑑𝜇. (2.57) By virtue of Young’s inequality and conditions (1.2), (1.3), from (2.57), we find _{𝐵(𝑥0,𝜌)}𝜔|∇𝑢|^𝑝𝑑𝑥≤𝐶_{𝐵(𝑥0,𝜌)}𝜔|∇ℎ|^𝑝𝑑𝑥+_{𝐵(𝑥0,𝑟)}𝑣𝑑𝜇;𝐶=𝐶𝑛,𝑝,𝜆. (2.58) From (2.55), we obtain _{𝐵(𝑥0,𝑟)}𝐴(𝑥,ℎ,∇ℎ)⋅∇𝑣𝑑𝑥=0, (2.59) whence by virtue of Hölder inequality, we find _{𝐵(𝑥0,𝑟)}𝜔|∇ℎ|^𝑝𝑑𝑥≤𝐶_{𝐵(𝑥0,𝑟)}𝜔|∇𝑢|^𝑝𝑑𝑥;𝐶=𝐶𝑛,𝑝,𝜆. (41^′) For the first summand of (2.58), we have the following estimates. According to [14], there exists a positive number 𝜅=𝜅(𝑛,𝑝,𝜆,𝐶_𝑝)∈(0,1) such that for the solution of (1.1) the inequality osc_{𝐵(𝑥0,𝑟1)}𝑟ℎ≤𝐶¹𝑟₂^𝜅osc_{𝐵(𝑥0,𝑟2)}ℎ, (2.60) where 𝐶=𝐶(𝑛,𝑝,𝐶_𝑝,𝜆,𝜅), is fulfilled for any 𝑟₁<𝑟₂. If we take into account the Caccioppoli type estimate (see [14]) _{𝐵(𝑥0,𝑟1)}𝜔|∇ℎ|^𝑝𝐶𝑑𝑥≤𝑟₂−𝑟₁^𝑝osc_{𝐵(𝑥0,𝑟2)}ℎ^𝑝𝜔𝐵𝑥₀,𝑟₂, (2.61) then, by virtue of Moser’s inequality, we obtain sup_{𝐵(𝑥0,𝑟2)}ℎ^𝑝≤𝐶𝜔𝐵𝑥₀,2𝑟_{2𝐵(𝑥0,2𝑟2)}𝜔ℎ−ℎ⁻_𝑟2^𝑝𝑑𝑥. (2.62) From (2.60), we derive the estimate _{𝐵(𝑥0,𝜌)}𝜔|∇ℎ|^𝑝𝜌𝑑𝑥≤𝐶𝑟^{−𝑝+𝑝𝜅}𝜔𝐵𝑥₀,𝜌𝜔𝐵𝑥₀,𝑟_{𝐵(𝑥0,𝑟)}𝜔|∇ℎ|^𝑝𝑑𝑥. (2.63) Indeed, _{𝐵(𝑥0,𝜌)}𝜔|∇ℎ|^𝑝𝐶𝑑𝑥≤𝜌_{𝐵(𝑥0,𝜌)}𝜔|∇ℎ|^𝑝−1ℎ−ℎ⁻_𝜌^𝑝𝐶𝑑𝑥≤𝜌^𝑝_{𝐵(𝑥0,𝜌)}𝜔ℎ−ℎ⁻_𝜌^𝑝≤𝐶𝑑𝑥𝜌^𝑝osc_{𝐵(𝑥0,𝜌)}ℎ^𝑝𝜔𝐵𝑥₀≤𝐶,𝜌𝜌^𝑝𝜌𝑟^𝜅𝑝osc_{𝐵(𝑥0,𝑟/2)}ℎ^𝑝𝜔𝐵𝑥₀≤𝐶,𝜌𝜌^𝑝𝜌𝑟^𝜅𝑝𝜔𝐵𝑥₀1,𝜌𝜔𝐵𝑥₀,𝑟_{𝐵(𝑥0,𝑟)}ℎ−ℎ⁻_𝑟^𝑝≤𝐶𝑑𝑥𝜌^𝑝𝜌𝑟^{−𝑝+𝑝𝜅}𝜔𝐵𝑥₀,𝜌𝜔𝐵𝑥₀,𝑟_{𝐵(𝑥0,𝑟)}𝜔|∇ℎ|^𝑝𝑑𝑥, (2.64) where ℎ⁻_𝑟 is the lower bound of the function ℎ in the ball 𝐵(𝑥₀,𝑟). Inequality (2.63) is proved.
Using the estimate (2.63) in (2.58), by virtue of (41^′), we have for 0<𝜌<𝑟/2_{𝐵(𝑥0,𝜌)}𝜔|∇𝑢|^𝑝𝜌𝑑𝑥≤𝐶𝑟^{−𝑝+𝑝𝜅}𝜔𝐵𝑥₀,𝜌𝜔𝐵𝑥₀,𝑟_{𝐵(𝑥0,𝑟)}𝜔|∇𝑢|^𝑝𝑑𝑥+_{𝐵(𝑥0,𝑟)}𝑣𝑑𝜇. (2.65)
Now, let us derive an estimate for the last summand in (2.65). To this end, we use inequality (38^′) to obtain _{𝐵(𝑥0,𝑟)}𝑣𝑑𝜇≤𝐶𝑝,𝑛,𝐶_{𝑝𝐵𝑥0,𝑟}𝜔|∇𝑣|^𝑝𝑑𝑥^1/𝑝, (2.66) where for the constant we have the estimate 𝐶𝑝,𝑛,𝐶_𝑝≤𝐶𝑟^{1−𝑝+(𝑝−1)𝛼}𝜔𝐵𝑥₀,𝑟^1/𝑝′ (2.67) By virtue of (2.65) and (2.67), we find _{𝐵(𝑥0,𝜌)}𝜔|∇𝑢|^𝑝𝜌𝑑𝑥≤𝐶𝑟^{−𝑝+𝑝𝜅}𝜔𝐵𝑥₀,𝜌𝜔𝐵𝑥₀,𝑟_{𝐵(𝑥0,𝑟)}𝜔|∇𝑢|^𝑝𝜔𝐵𝑥𝑑𝑥+𝐶⁰,𝑟𝑟^{(1−𝛼)𝑝1/𝑝′}_{𝐵(𝑥0,𝑟)}𝜔|∇𝑢|^𝑝𝑑𝑥^1/𝑝, (2.68) whence, by virtue of Young’s inequality, we obtain _{𝐵(𝑥0,𝜌)}𝜔|∇𝑢|^𝑝𝜌𝑑𝑥≤𝐶𝑟^{−𝑝+𝑝𝜅}𝜔𝐵𝑥₀,𝜌𝜔𝐵𝑥₀,𝑟+𝜀_{𝐵(𝑥0,𝑟)}𝜔|∇𝑢|^𝑝𝑑𝑥+𝐶_𝜀𝜔𝐵𝑥₀,𝑟𝑟^{−𝑝+𝑝𝛼}. (2.69) Assuming that 0<𝛼<𝜅, from (2.69) and Lemma 1.5, we obtain the estimate _{𝐵(𝑥0,𝜌)}𝜔|∇𝑢|^𝑝𝜌𝑑𝑥≤𝐶𝑟^{−𝑝+𝑝𝛼}𝜔𝐵𝑥₀,𝜌𝜔𝐵𝑥₀,𝑟_{𝐵(𝑥0,𝑟)}𝜔|∇𝑢|^𝑝𝑑𝑥+𝐶𝜔𝐵𝑥₀,𝜌𝜌^{−𝑝+𝑝𝛼}. (2.70) This inequality implies _{𝐵(𝑥0,𝜌)}|∇𝑢|𝑑𝑥≤_{𝐵(𝑥0,𝜌)}𝜔|∇𝑢|^𝑝𝑑𝑥^1/𝑝_{𝐵(𝑥0,𝜌)}𝜔^{−(1/(𝑝−1))}𝑑𝑥^1/𝑝′≤𝐶_{𝐵(𝑥0,𝜌)}𝜔𝑑𝑥^1/𝑝_{𝐵(𝑥0,𝜌)}𝜔^{−(1/(𝑝−1))}𝑑𝑥^1/𝑝′𝜌^−1+𝛼≤𝐶𝜌^{𝑛−1+𝛼}, (2.71) whence, by virtue of the Poincaré inequality, we obtain 𝑢−(_{𝐵(𝑥0,𝜌)}𝑢)_𝜌^𝑝𝑑𝑥≤𝐶𝜌^𝑛+𝛼, (2.72) where (𝑢)_𝜌 is average of the function 𝑢 with respect to the ball 𝐵(𝑥₀,𝜌).
By (2.72) and Campanato’s Lemma 1.4, we find 𝑢∈𝐶^𝛼.
Theorem 1.8 is proved.