Existence of Solution for Nonlinear Elliptic Equations with Unbounded Coefficients and 𝐿¹ Data

Abstract

An existence result of a renormalized solution for a class of nonlinear elliptic equations is established. The diffusion functions 𝑎(𝑥,𝑢,∇𝑢) may not be in (𝐿¹_loc(Ω))^𝑁 for a finite value of the unknown and the data belong to 𝐿¹(Ω).

1. Introduction

In this paper we investigate the problem of existence of a renormalized solutions for elliptic equations of the type

−div(𝑎(𝑥,𝑢,∇𝑢))=𝑓inΩ,𝑢=0on𝜕Ω, (1.1) where Ω is an open bounded subset of ℝ^𝑁,𝑁≥1, with the data 𝑓 in 𝐿¹(Ω). The operator −div(𝑎(𝑥,𝑢,∇𝑢)) is a Leray-Lions operator defined on the weighted sobolev spaces 𝑊₀^1,𝑝(Ω,𝑤), but which is not restricted by any growth condition with respect to 𝑢 (see assumptions (2.2), (2.4), and (2.5) of Section 3). The function 𝑎(𝑥,𝑠,𝜉) is controlled by a real function 𝑏∶]−∞,𝑚[→ℝ which blows up for a finite value 𝑚>0 (see (2.2), (2.3)).

There are mainly two types of difficulties that are studying Problem (1.1). One consists to give a sense to the flux 𝑎(𝑥,𝑢,∇𝑢) on the set {𝑥∈Ω;𝑢(𝑥)=𝑚}. The second one is that the data 𝑓 only belong to 𝐿¹, so that proving existence of a weak solution (i.e., in the distribution meaning) seems to be an arduous task. To overcome this difficulty we use in this paper the framework of renormalized solutions. This notion was introduced by DiPerna and Lions [1] for the study of Boltzmann equation (see also Lions [2] for a few applications to fluid mechanics models). This notion was then adapted to elliptic vesion of (1.1) in Boccardo et al. [3], and Murat [4, 5] (see also [6, 7] for nonlinear parabolic problems). At the same time the equivalent notion of entropy solutions has been developed independently by Bénilan et al. [8] for the study of nonlinear elliptic problems.

In the case where 𝑎(𝑥,𝑢,∇𝑢) is replaced by (𝑑(𝑢)+𝐴(𝑢))∇𝑢 (problems with diffusion matrices that have at least one diagonal coefficient that blows up for a finite value of the unknown) and 𝑓∈𝐿²(Ω), existence and uniqueness has been established in Blanchard and Redwane [9, 10].

As far as we have the stationary and evolution equations case (1.1), the existence and a partial uniqueness of renormalized solutions have been proved in Blanchard et al. [11] in the case where 𝑎(𝑥,𝑢,∇𝑢) is replaced by 𝐴(𝑥,𝑢)∇𝑢 (where 𝐴(𝑥,𝑠) is a Carathéodory symmetric matrices, such that 𝐴(𝑥,𝑠) blows up as 𝑠→𝑚⁻ uniformly with respect to 𝑥). It has also been applied to the study of linear and nonlinear elliptic and parabolic equations when the diffusion coefficient has a singularity for a finite value of the unknown (see García Vázquez and Ortegón Gallego [12, 13] and Orsina [14]).

The paper is organized as follows. In Section 2 we will precise some basic properties of weighted Sobolev spaces. Section 3 is devoted to specify the assumptions on 𝑎(𝑥,𝑠,𝜉),𝑏(𝑠), and 𝑓 needed in the present study and gives the definition of a renormalized solution of (1.1). In Section 4 (Theorem 4.1) we establish the existence of such a solution.

2. Preliminaries

Throughout the paper, we assume that the following assumptions hold true. Ω is a bounded open subset on ℝ^𝑁,𝑁≥1. Let us suppose that 1<𝑝<∞ is a real number, and 𝜔(𝑥)={𝜔_𝑖(𝑥)}_{{0≤𝑖≤𝑁}} is a vector of weight functions. Furthermore we suppose that every component 𝜔_𝑖(𝑥) is a measurable function which is strictly positive and satisfies

𝜔_𝑖∈𝐿¹_loc(Ω),𝜔_𝑖^{−1/(𝑝−1)}∈𝐿¹_loc(Ω). (2.1) We define the weighted Lebesgue space 𝐿^𝑝(Ω,𝜔₀) with weight 𝜔₀, as the space of all real-valued measurable functions 𝑢 for which

‖𝑢‖_𝑝,𝜔0=(_Ω|𝑢(𝑥)|^𝑝𝜔₀(𝑥)𝑑𝑥)^1/𝑝<+∞. (2.2) In order to define the weighted Sobolev space of 𝑊^1,𝑝(Ω,𝜔), as the space of all real-valued functions 𝑢∈𝐿^𝑝(Ω,𝜔₀) such that the derivaties in the sense of distributions satisfy 𝜕𝑢/𝜕𝑥_𝑖∈𝐿^𝑝(Ω,𝜔_𝑖) for all 𝑖=1,…,𝑁. This set of functions forms a Banach space under the norm ‖𝑢‖_1,𝑝,𝜔=(_Ω|𝑢(𝑥)|^𝑝𝜔₀(𝑥)𝑑𝑥+^𝑁_𝑖=1Ω|𝜕𝑢𝜕𝑥_𝑖|^𝑝𝜔_𝑖(𝑥)𝑑𝑥)^1/𝑝. (2.3) To deal with the Dirichlet problem, we use the space 𝑋=𝑊₀^1,𝑝(Ω,𝜔) defined as the closure of 𝐶^∞₀(Ω) with respect to the norm ‖⋅‖_1,𝑝,𝜔. Note that, 𝐶^∞₀(Ω) is dense in 𝑊₀^1,𝑝(Ω,𝜔) and (𝑊₀^1,𝑝(Ω,𝜔),‖⋅‖_1,𝑝,𝜔) is a reflexive Banach space. Note that the expression

‖𝑢‖_𝑋=(^𝑁_𝑖=1Ω|𝜕𝑢𝜕𝑥_𝑖|^𝑝𝜔_𝑖(𝑥)𝑑𝑥)^1/𝑝 (2.4) is a norm defined on 𝑋 and is equivalent to the norm (2.3). Moreover (𝑋,‖⋅‖_𝑋) is a reflexive Banach space, and there exist a weight function 𝜎 on Ω and a parameter 1<𝑞<∞ such that the Hardy inequality

(_Ω|𝑢|^𝑞𝜎(𝑥)𝑑𝑥)^1/𝑞≤𝐶(^𝑁_𝑖=1Ω|𝜕𝑢𝜕𝑥_𝑖|^𝑝𝜔_𝑖(𝑥)𝑑𝑥)^1/𝑝 (2.5) holds for every 𝑢∈𝑋 with a constant 𝐶>0 independent of 𝑢. Moreover, the imbedding 𝑋↪𝐿^𝑞(Ω,𝜎) is compact.

We recall that the dual of the weighted Sobolev spaces 𝑊₀^1,𝑝(Ω,𝜔) is equivalent to 𝑊^−1,𝑝′(Ω,𝜔^∗), where 𝜔^∗={𝜔^∗_𝑖=𝜔^1−𝑝′_𝑖;𝑖=1,…,𝑁} and 𝑝^′=𝑝/(𝑝−1) is the conjugate of 𝑝. For more details we refer the reader to [15] (see also [16]).

3. Assumptions on the Data and Definition of a Renormalized Solution

Throughout the paper, we assume that the following assumptions hold true. Ω is a bounded open set on ℝ^𝑁,𝑁≥1. Let 1<𝑝<∞, and let 𝜔(𝑥)={𝜔_𝑖(𝑥)}_{{0≤𝑖≤𝑁}} be a vector of weight functions.

Let now −div(𝑎(𝑥,𝑢,∇𝑢)) be a Leray-Lions operator defined on 𝑊₀^1,𝑝(Ω,𝜔) into 𝑊^−1,𝑝′(Ω,𝜔^∗) and where 𝑎∶Ω×ℝ×ℝ^𝑁→ℝ^𝑁isaCarathéodoryfunction,suchthat (3.1) there exists a positive function 𝑏∈𝐶⁰((−∞,𝑚)) which satisfies

lim_{𝑟→𝑚−}𝑏(𝑟)=+∞;^𝑚₀𝑏(𝑠)𝑑𝑠<+∞,𝑏(𝑟)≥𝛼>0∀𝑟∈]−∞,𝑚[,𝑎(𝑥,𝑠,𝜉)⋅𝜉≥𝑏(𝑠)^𝑁𝑝−1_𝑖=1𝜔_𝑖(𝑥)|𝜉_𝑖|^𝑝,𝑎(𝑥,𝑠,0)=0, (3.2) for almost every 𝑥∈Ω, for every 𝑠∈ℝ and 𝜉∈ℝ^𝑁.

For any 𝑖=1,…,𝑁, |𝑎_𝑖(𝑥,𝑠,𝜉)|≤𝜔_𝑖(𝑥)^1/𝑝[𝐿(𝑥)+𝜎(𝑥)^1/𝑝′|𝑠|^𝑞/𝑝′+𝑏(𝑠)^𝑁𝑝−1_𝑗=1𝜔^1/𝑝′_𝑗(𝑥)|𝜉_𝑗|^𝑝−1], (3.3) for almost every 𝑥∈Ω, for every 𝑠 and 𝜉, and where 𝐿(𝑥) is a positive function in 𝐿^𝑝′(Ω)

[𝑎(𝑥,𝑠,𝜉)−𝑎(𝑥,𝑠,𝜉^′)][𝜉−𝜉^′]≥0, (3.4) for any 𝜉,𝜉^′∈ℝ^𝑁, for any 𝑠∈ℝ and for almost every 𝑥∈Ω

𝑓isanelementof𝐿¹(Ω). (3.5)

Remark 3.1. As already mentioned in the introduction, problem (1.1) does not admit a weak solution under assumptions (3.1)–(3.5) since the growth of 𝑎(𝑥,𝑢,∇𝑢) is not controlled with respect to 𝑢, the field 𝑎(𝑥,𝑢,∇𝑢) is not, in general, defined as a distribution because the difficulty is defining the field 𝑎(𝑥,𝑢,∇𝑢) on the subset {𝑥∈Ω;𝑢(𝑥)=𝑚} of Ω, (since on this set, 𝑏(𝑢)=+∞).
The following notations will be used throughout the paper. For any 𝐾≥0, the truncation at height 𝐾 is defined by 𝑇_𝐾(𝑟)=max(−𝐾,min(𝑟,𝐾)), for any positive numbers 𝑙 and 𝐾, the functions 𝑇^𝐾_𝑙 are defined by
𝑇^𝐾_𝑙(𝑟)=−𝐾,if𝑟≤−𝐾,𝑟,if−𝐾≤𝑟≤𝑙,𝑙,if𝑟≥𝑙. (3.6) We define for 𝑛≥1 fixed 𝜃_𝑛(𝑟)=𝑇₁(𝑟−𝑇_𝑛(𝑟))=0,if|𝑟|≤𝑛,𝑟−𝑛𝑠𝑔(𝑠),if𝑛≤|𝑟|≤𝑛+1,𝑠𝑔(𝑠),if|𝑟|≥𝑛+1, (3.7) and 𝑆_𝑛(𝑟)=1−|𝜃_𝑛(𝑟)|, for all 𝑟∈ℝ.
The definition of a renormalized solution for Problem (1.1) can be stated as follows.

Definition 3.2. A measurable function 𝑢 defined on Ω is a renormalized solution of Problem (1.1) if 𝑇_𝐾(𝑢)∈𝑊₀^1,𝑝(Ω,𝜔)∀𝐾≥0,𝑢(𝑥)≤𝑚foralmostevery𝑥∈Ω,𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))𝜒_{𝑢<𝑚}∈^𝑁_𝑖=1𝐿^𝑝′(Ω,𝑤^1−𝑝′_𝑖),_{{−𝑛−1≤𝑢(𝑥)≤−𝑛}}𝑎(𝑥,𝑢,∇𝑢)∇𝑢𝑑𝑥→0as𝑛→+∞,1𝛿_{{𝑚−2𝛿≤𝑢(𝑥)≤𝑚−𝛿}}𝑎(𝑥,𝑢,∇𝑢)∇𝑢𝑑𝑥→_{𝑢=𝑚}𝑓𝑑𝑥as𝛿→0, (3.8)(3.9)(3.10)(3.11)(3.12) and if, for every function 𝑆 in 𝑊^1,∞(ℝ) such that supp(𝑆) is compact and 𝑆(𝑚)=0,𝑢 satisfies _Ω𝑎(𝑥,𝑢,∇𝑢)∇(𝑆(𝑢)𝜑)𝑑𝑥=_Ω𝑓𝑆(𝑢)𝜑𝑑𝑥,∀𝜑∈𝑊₀^1,𝑝(Ω,𝜔)∩𝐿^∞(Ω). (3.13)

The following remarks are concerned with Definition 3.2.

Remark 3.3. Notice that, thanks to our regularity assumptions (3.8), (3.9), (3.10) and the choice of 𝑆, all terms in (3.13) are well defined.
The following two identifications are made in (3.13):
(i)𝑎(𝑥,𝑢,∇𝑢)∇(𝑆(𝑢)𝜑) identifies with 𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))∇(𝑆(𝑢)𝜑) for almost every 𝑥∈Ω, where 𝐾>0 and supp(𝑆)⊂[−𝐾,𝐾]. As a consequence of (3.8), (3.9), and (3.10), and of 𝑆∈𝑊^1,∞(ℝ),𝜑∈𝑊₀^1,𝑝(Ω,𝜔)∩𝐿^∞(Ω), it follows that 𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))∇(𝑆^′(𝑢)𝜑)∈𝐿¹(Ω), (3.14)(ii)𝑓𝑆(𝑢)𝜑∈𝐿¹(Ω), because 𝑓∈𝐿¹(Ω) and 𝑆(𝑢)𝜑∈𝐿^∞(Ω).

4. Existence Result

This section is devoted to establish the following existence theorem.

Theorem 4.1. Under assumptions (3.1)–(3.5) there exists a renormalized solution 𝑢 of Problem (1.1).

Proof. The proof is divided into 7 steps. In Step 1, we introduce an approximate problem. Step 2 is devoted to establish a few a priori estimates, the limit 𝑢 of the approximate solutions 𝑢^𝜀 is introduced and it is shown that 𝑢 satisfies (3.8) and (3.9). Step 3 is devoted to prove an energy estimate (Lemma 4.2) which is a key point for the monotonicity arguments that are developed in Step 4. Step 5 is devoted to prove that 𝑢 satisfies (3.11). In Step 6 we prove that 𝑢 satisfies (3.12). Finally, Step 7 is devoted to prove that 𝑢 satisfies (3.13) of Definition 3.2.

Step 1. Let us introduce the following regularization of the data: 𝑏^𝜀(𝑟)=𝑏(𝑇^1/𝜀_𝑚−𝜀(𝑟))∀𝑟∈ℝfor𝜀>0,𝑎^𝜀(𝑥,𝑠,𝜉)=𝑎(𝑥,𝑇^1/𝜀_𝑚−𝜀(𝑠),𝜉)a.e.inΩ,∀𝑠∈ℝ,∀𝜉∈ℝ^𝑁,𝑓^𝜀∈𝐿^𝑝′(Ω);‖𝑓^𝜀‖_𝐿1(Ω)≤‖𝑓‖_𝐿1(Ω)∶𝑓^𝜀→𝑓stronglyin𝐿¹(Ω)as𝜀tendsto0. (4.1)(4.2)(4.3)
Let us now consider the following regularized problem:
−div(𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀))=𝑓^𝜀inΩ,𝑢^𝜀=0on𝜕Ω. (4.4)(4.5) In view of (3.3), (4.1), and (4.2), 𝑎^𝜀 satisfy. For 𝑖=1,…,𝑁|𝑎^𝜀_𝑖(𝑥,𝑠,𝜉)|≤𝜔_𝑖(𝑥)^1/𝑝[𝐿(𝑥)+𝜎(𝑥)^1/𝑝′|𝑇^1/𝜀_𝑚−𝜀(𝑠)|^𝑞/𝑝′+𝑏^𝜀(𝑠)^𝑁𝑝−1_𝑗=1𝜔^1/𝑝′_𝑗(𝑥)|𝜉_𝑗|^𝑝−1] (4.6) a.e. 𝑥∈Ω, for all 𝑠∈ℝ,𝜉∈ℝ^𝑁. And 𝛼≤𝑏^𝜀(𝑟)≤max_{{−1/𝜀≤𝑟≤𝑚−𝜀}}𝑏(𝑟)=𝐶_𝜀∀𝑟∈ℝ. (4.7)

As a consequence, proving existence of a weak solution 𝑢^𝜀∈𝑊₀^1,𝑝(Ω,𝜔) of (4.4) and (4.5) is an easy task (see, e.g., Theorem 2.1 and Remark 2.1 in Chapter 2 of [17] and see also [18]).

Step 2. A priori estimates and pointwise convergence of 𝑢^𝜀.
Using 𝑇_𝐾(𝑢^𝜀) as a test function in (4.4) leads to
_Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑇_𝐾(𝑢^𝜀)𝑑𝑥=_Ω𝑓^𝜀𝑇_𝐾(𝑢^𝜀)𝑑𝑥≤𝐾‖𝑓‖_𝐿1(Ω). (4.8) Since 𝑎^𝜀 satisfies (3.2), (4.2), and owing to (4.8) we have _Ω𝑏^𝜀(𝑢^𝜀)^𝑁𝑝−1_𝑖=1|𝜕𝑇_𝐾(𝑢^𝜀)𝜕𝑥_𝑖|^𝑝𝜔_𝑖(𝑥)𝑑𝑥≤𝐾‖𝑓‖_𝐿1(Ω),𝛼^𝑝−1_Ω^𝑁_𝑖=1|𝜕𝑇_𝐾(𝑢^𝜀)𝜕𝑥_𝑖|^𝑝𝜔_𝑖(𝑥)𝑑𝑥≤𝐾‖𝑓‖_𝐿1(Ω). (4.9)(4.10) From (4.10) we deduce with a classical argument (see, e.g., [18]) that, for a subsequence still indexed by 𝜀, 𝑢^𝜀→𝑢a.e.inΩ,𝑇_𝐾(𝑢^𝜀)→𝑇_𝐾(𝑢)weaklyin𝑊₀^1,𝑝(Ω,𝜔)andstronglyin𝐿^𝑞(Ω,𝜎), (4.11)(4.12) as 𝜀 tends to 0, where 𝑢 is a measurable function defined on Ω which is finite a.e. in Ω.
Taking now 𝑍^𝜀=∫^{𝑇𝐾𝑚(𝑢𝜀)}₀𝑏^𝜀(𝑠)𝑑𝑠 as a test function in (4.4) gives
_Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑍^𝜀𝑑𝑥=_Ω𝑓^𝜀𝑍^𝜀𝑑𝑥. (4.13) Since 𝑎^𝜀 satisfies (3.2) and 𝑏 satisfies (3.1), permit to deduce from (4.13) that _Ω^𝑁_𝑖=1|𝜕𝑍^𝜀𝜕𝑥_𝑖|^𝑝𝜔_𝑖(𝑥)𝑑𝑥≤𝐶_𝐾‖𝑓‖_𝐿1(Ω), (4.14) where |𝑍^𝜀|≤∫^𝑚_−𝐾𝑏(𝑠)𝑑𝑠=𝐶_𝐾 is a constant independent of 𝜀.
Now for a fixed 𝐾>0, assumption (3.3) gives for 𝑖=1,…,𝑁,
|𝑎^𝜀_𝑖(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))|≤𝜔_𝑖(𝑥)^1/𝑝[𝐿(𝑥)+𝜎(𝑥)^1/𝑝′max(𝐾,𝑚)^𝑞/𝑝′+^𝑁_𝑗=1𝜔^1/𝑝′_𝑗(𝑥)|𝜕𝑍^𝜀𝜕𝑥_𝑗|^𝑝−1]. (4.15) In view of (4.14) and (4.15), we deduce that 𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))isboundedin^𝑁_𝑖=1𝐿^𝑝′(Ω,𝑤^1−𝑝′_𝑖), (4.16) then there exists a function 𝑋_𝐾∈∏^𝑁_𝑖=1𝐿^𝑝′(Ω,𝑤^1−𝑝′_𝑖) such that 𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))⇀𝑋_𝐾weaklyin^𝑁_𝑖=1𝐿^𝑝′(Ω,𝑤^1−𝑝′_𝑖)as𝜀→0. (4.17) To prove that 𝑢 is less or equal to 𝑚 is an easy task which is performed exactly as in [10, 11]. Using 𝑇⁺_2𝑚(𝑢^𝜀)−𝑇⁺_𝑚(𝑢^𝜀) as a test function in (4.4) leads to _Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇(𝑇⁺_2𝑚(𝑢^𝜀)−𝑇⁺_𝑚(𝑢^𝜀))𝑑𝑥=_Ω𝑓^𝜀(𝑇⁺_2𝑚(𝑢^𝜀)−𝑇⁺_𝑚(𝑢^𝜀))𝑑𝑥, (4.18) which implies easily that _Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇(𝑇⁺_2𝑚(𝑢^𝜀)−𝑇⁺_𝑚(𝑢^𝜀)))∇(𝑇⁺_2𝑚(𝑢^𝜀)−𝑇⁺_𝑚(𝑢^𝜀))𝑑𝑥≤𝑚‖𝑓‖_𝐿1(Ω). (4.19)
Then (3.2), (4.1), and (4.2) yield
𝑏(𝑚−𝜀)^𝑝−1_Ω^𝑁_𝑖=1|𝜕(𝑇⁺_2𝑚(𝑢^𝜀)−𝑇⁺_𝑚(𝑢^𝜀))𝜕𝑥_𝑖|^𝑝𝜔_𝑖(𝑥)𝑑𝑥≤𝑚‖𝑓‖_𝐿1(Ω). (4.20) With the help of Poincaré's inequality, we have _Ω|𝑇⁺_2𝑚(𝑢^𝜀)−𝑇⁺_𝑚(𝑢^𝜀)|^𝑝𝜔₀(𝑥)𝑑𝑥≤𝐶𝑚𝑏(𝑚−𝜀)^𝑝−1‖𝑓‖_𝐿1(Ω), (4.21) where 𝐶 does not depend on 𝜀. Then in view of (3.1), (4.11), and 𝜔₀>0, we can pass to the limit in (4.21) as 𝜀 tends to 0, to deduce that 𝑇⁺_2𝑚(𝑢)−𝑇⁺_𝑚(𝑢)=0a.e.inΩ,𝑢≤𝑚a.e.inΩ. (4.22)
Let us now take 𝑇_𝐾(𝑣^𝜀) as a test function in (4.4), where 𝑣^𝜀=∫^𝑢𝜀₀𝑏^𝜀(𝑠)𝑑𝑠. We obtain
_Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑇_𝐾(𝑣^𝜀)𝑑𝑥=_Ω𝑓^𝜀𝑇_𝐾(𝑣^𝜀)𝑑𝑥≤𝐾‖𝑓‖_𝐿1(Ω). (4.23) Then (3.2) yields _Ω^𝑁_𝑖=1|𝜕𝑇_𝐾(𝑣^𝜀)𝜕𝑥_𝑖|^𝑝𝜔_𝑖(𝑥)𝑑𝑥≤𝐾‖𝑓‖_𝐿1(Ω). (4.24) We deduce with a classical argument that, for a subsequence still indexed by 𝜀, 𝑣^𝜀→𝑣a.e.inΩ,𝑇_𝐾(𝑣^𝜀)→𝑇_𝐾(𝑣)weaklyin𝑊₀^1,𝑝(Ω,𝜔), (4.25)(4.26) as 𝜀 tends to 0, where 𝑣 is a measurable function defined on Ω which is finite a.e. in Ω.
Using the admissible test function 𝜃_𝑛(𝑣^𝜀) in (4.4) leads to
_Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝜃_𝑛(𝑣^𝜀)𝑑𝑥=_Ω𝑓^𝜀𝜃_𝑛(𝑣^𝜀)𝑑𝑥. (4.27) As a consequence of the previous convergence results, we are in a position to pass to the limit as 𝜀 tends to 0 in (4.27) lim_𝜀→0Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝜃_𝑛(𝑣^𝜀)𝑑𝑥=_Ω𝑓𝜃_𝑛(𝑣)𝑑𝑥. (4.28) Using the pointwise convergence of 𝜃_𝑛(𝑢) to 0 as 𝑛 tends to +∞ and |𝜃_𝑛(𝑢)|≤1 a.e. in Ω independently of 𝑛, since 𝑓∈𝐿¹(Ω), Lebesgue's convergence theorem shows that ∫_Ω𝑓𝜃_𝑛(𝑣)𝑑𝑥→0, as 𝑛 tends to +∞. Passing to the limit in (4.28) we obtain lim_𝑛→+∞lim_{𝜀→0{𝑛≤|𝑣𝜀|≤𝑛+1}}𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑣^𝜀𝑑𝑥=0. (4.29)

Step 3. In this step we prove the following monotonicity estimate.Lemma 4.2. The subsequence of 𝑢^𝜀 defined in Step 1 satisfies for any 𝐾≥0lim_𝜀→0Ω[𝑎^𝜀(𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1−𝑎^𝜀(𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢))𝑏^𝜀(𝑢^𝜀)^𝑝−1][∇𝑇^𝐾_𝑚(𝑢^𝜀)−∇𝑇^𝐾_𝑚(𝑢)]𝑑𝑥=0. (4.30)Proof. Let 𝐾≥0 be fixed. Equality (4.30) is split into _Ω[𝑎^𝜀(𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1−𝑎^𝜀(𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢))𝑏^𝜀(𝑢^𝜀)^𝑝−1][∇𝑇^𝐾_𝑚(𝑢^𝜀)−∇𝑇^𝐾_𝑚(𝑢)]𝑑𝑥=𝐴^𝜀₁+𝐴^𝜀₂+𝐴^𝜀₃, (4.31) where 𝐴^𝜀₁=_Ω𝑎^𝜀(𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1∇𝑇^𝐾_𝑚(𝑢^𝜀𝐴)𝑑𝑥𝑑𝑠𝑑𝑡,_𝜀2=−_Ω𝑎^𝜀(𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1∇𝑇^𝐾_𝑚𝐴(𝑢)𝑑𝑥𝑑𝑠𝑑𝑡,_𝜀3=−_Ω𝑎^𝜀(𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢))𝑏^𝜀(𝑢^𝜀)^𝑝−1(∇𝑇^𝐾_𝑚(𝑢^𝜀)−∇𝑇^𝐾_𝑚(𝑢))𝑑𝑥𝑑𝑠𝑑𝑡. (4.32) In the sequel we pass to the limit in (4.31) when 𝜀 tends to 0.Limit of 𝐴^𝜀₁
Using the admissible test function 𝑆_𝑛(𝑣^𝜀)∫^𝑇𝐾𝑚₀^(𝑢)(1/𝑏(𝑠)^𝑝−1)𝑑𝑠 in (4.4) leads to _Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(x,𝑢^𝜀,∇𝑢^𝜀)∇𝑇^𝐾_𝑚(𝑢)𝑏(𝑢)^𝑝−1𝑑𝑥+_Ω𝑎^𝜀(𝑢^𝜀,∇𝑢^𝜀)∇𝑆_𝑛(𝑣^𝜀)⋅(^𝑇𝐾𝑚₀^(𝑢)1𝑏(𝑠)^𝑝−1=𝑑𝑠)𝑑𝑥_Ω𝑓^𝜀𝑆_𝑛(𝑣^𝜀)^𝑇𝐾𝑚₀^(𝑢)1𝑏(𝑠)^𝑝−1𝑑𝑠𝑑𝑥, (4.33) where 𝑣^𝜀=∫^𝑢𝜀₀𝑏^𝜀(𝑠)𝑑𝑠, pass to the limit as 𝜀 tends to 0 in (4.33).
Since supp(𝑆_𝑛)⊂[−(𝑛+1),𝑛+1] and {𝑥∈Ω;|𝑣^𝜀|≤𝑛+1}⊂{𝑥∈Ω;|𝑢^𝜀|≤(𝑛+1)/𝛼}, we have for 𝑖=1,…,𝑁 and 𝜀≤𝛼/(𝑛+1)
|𝑎^𝜀_𝑖(𝑥,𝑢^𝜀,∇𝑢^𝜀)𝑆_𝑛(𝑣^𝜀)|=|𝑎^𝜀_𝑖(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀),∇𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀))𝑆_𝑛(𝑣^𝜀)|≤‖𝑆_𝑛‖_𝐿∞(ℝ)𝜔_𝑖(𝑥)^1/𝑝[𝐿(𝑥)+𝜎(𝑥)^1/𝑝′|𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀)|^𝑞/𝑝′+^𝑁_𝑗=1𝜔^1/𝑝′_𝑗(𝑥)|𝜕𝑇_𝑛+1(𝑣^𝜀)𝜕𝑥_𝑗|^𝑝−1]. (4.34) In view of (4.24), (4.34) we deduce that for fixed 𝑛≥1: 𝑎^𝜀(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀),∇𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀))𝑆_𝑛(𝑣^𝜀)isboundedin^𝑁_𝑖=1𝐿^𝑝′(Ω,𝑤^1−𝑝′_𝑖), (4.35) independently of 𝜀≤𝛼/(𝑛+1). Then there exists a function 𝑌_𝑛∈∏^𝑁_𝑖=1𝐿^𝑝′(Ω,𝑤^1−𝑝′_𝑖) such that for fixed 𝑛≥1: 𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀),∇𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀))⇀𝑌_𝑛weaklyin^𝑁_𝑖=1𝐿^𝑝′(Ω,𝑤^1−𝑝′_𝑖)as𝜀→0. (4.36) Now for max(𝐾,𝑚)≤𝑛/𝛼, we have 𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀),∇𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀))𝜒_{{−𝐾<𝑢𝜀<𝑚}}=𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑇^𝑚_𝐾(𝑢^𝜀),∇𝑇^𝑚_𝐾(𝑢^𝜀))𝜒_{{−𝐾<𝑢𝜀<𝑚}} (4.37) a.e. in Ω, which implies that, through the use of (4.17), (4.25), and (4.36) and passing to the limit as 𝜀 tends to 0, 𝑌_𝑛𝜒_{{−𝐾<𝑢<𝑚}}=𝑆_𝑛(𝑣)𝑋_𝐾𝜒_{{−𝐾<𝑢<𝑚}} (4.38) a.e. in Ω−{{𝑢=−𝐾}∪{𝑢=𝑚}}formax(𝐾,𝑚)≤𝑛/𝛼. As a consequence of (4.38) we have for max(𝐾,𝑚)≤𝑛/𝛼𝑌_𝑛∇𝑇^𝐾_𝑚(𝑢)=𝑆_𝑛(𝑣)𝑋_𝐾∇^𝐾_𝑚(𝑢)a.e.inΩ. (4.39) We are now in a position to exploit (4.33), which gives together with (4.36) and (4.39) lim_𝜀→0Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑇^𝐾_𝑚(𝑢)𝑏(𝑢)^𝑝−1𝑑𝑥=lim_𝜀→0ΩS_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀),∇𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀))∇𝑇^𝐾_𝑚(𝑢)𝑏(𝑢)^𝑝−1=𝑑𝑥_Ω𝑌_𝑛∇𝑇^𝐾_𝑚(𝑢)𝑏(𝑢)^𝑝−1=𝑑𝑥_Ω𝑆_𝑛(𝑣)𝑋_𝐾∇𝑇^𝐾_𝑚(𝑢)𝑏(𝑢)^𝑝−1𝑑𝑥. (4.40) Passing to the limit as 𝑛 tends to +∞ in (4.40) leads to lim_𝑛→+∞lim_𝜀→0Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑇^𝐾_𝑚(𝑢)𝑏(𝑢)^𝑝−1𝑑𝑥=_Ω𝑋_𝐾∇𝑇^𝐾_𝑚(𝑢)𝑏(𝑢)^𝑝−1𝑑𝑥. (4.41) The second term of (4.33) |_Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑆_𝑛(𝑣^𝜀).(^𝑇𝐾𝑚₀^(𝑢)1𝑏(𝑠)^𝑝−1≤𝑑𝑠)𝑑𝑥|max(𝑚,𝐾)𝛼^𝑝−1_{{𝑛≤|𝑣}^𝜀_|≤𝑛+1}𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑣^𝜀𝑑𝑥. (4.42) Then (4.29) implies that lim_𝑛→+∞lim_𝜀→0Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑆_𝑛(𝑣^𝜀).(^𝑇𝐾𝑚₀^(𝑢)1𝑏(𝑠)^𝑝−1𝑑𝑠)𝑑𝑥=0. (4.43) In view (4.41) and (4.43), passing to the limit as 𝜀 tends to 0 and as 𝑛 tends to +∞ in (4.33) is an easy task and leads to _Ω𝑋_𝐾∇𝑇^𝐾_𝑚(𝑢)𝑏(𝑢)^𝑝−1𝑑𝑥=_Ω𝑓^𝑇𝐾𝑚₀^(𝑢)1𝑏(𝑠)^𝑝−1𝑑𝑠𝑑𝑥. (4.44) We are now in a position to exploit (4.44).
The use of the test function ∫^{𝑇𝐾𝑚(𝑢𝜀)}₀(1/𝑏^𝜀(𝑠)^𝑝−1)𝑑𝑠 in (4.4), yields
_Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑇^𝐾_𝑚(𝑢^𝜀)𝑏^𝜀(𝑢^𝜀)^𝑝−1𝑑𝑥=_Ω𝑓^{𝜀𝑇𝐾𝑚(𝑢𝜀)}₀1𝑏^𝜀(𝑠)^𝑝−1𝑑𝑠𝑑𝑥. (4.45) Passing to the limit as 𝜀 tends to 0 in (4.45), in view (4.44), we have lim_𝜀→0𝐴^𝜀₁=lim_𝜀→0Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑇^𝐾_𝑚(𝑢^𝜀)𝑏^𝜀(𝑢^𝜀)^𝑝−1𝑑𝑥=_Ω𝑋_𝐾∇𝑇^𝐾_𝑚(𝑢)𝑏(𝑢)^𝑝−1𝑑𝑥. (4.46)Limit of 𝐴^𝜀₂
In view of (4.12), (4.17) and since 1/𝑏^𝜀(𝑢^𝜀)^𝑝−1 converges to 1/𝑏(𝑢)^𝑝−1 a.e. in Ω and due to the bound 1/𝑏^𝜀(𝑢^𝜀)^𝑝−1≤1/𝛼^𝑝−1 a.e. in Ω, we have lim_𝜀→0𝐴^𝜀₂=−_Ω𝑋_𝐾∇𝑇^𝐾_𝑚(𝑢)𝑏(𝑢)^𝑝−1𝑑𝑥. (4.47)Limit of 𝐴^𝜀₃
Let us remark that (3.1), (4.1), and (4.11) imply that 𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢))𝑏^𝜀(𝑢^𝜀)^𝑝−1→𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),𝐷𝑇^𝐾_𝑚(𝑢))𝑏(𝑢)^𝑝−1a.e.inΩ, (4.48) as 𝜀 tends to 0, and that for 𝑖=1,…,𝑁|𝑎^𝜀_𝑖(𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢))𝑏^𝜀(𝑢^𝜀)^𝑝−1|≤𝑤_𝑖^1/𝑝1(𝑥)[𝛼^𝑝−1𝐿(𝑥)+max(𝐾,𝑚)^𝑞/𝑝′𝛼^𝑝−1𝜎^1/𝑝′(𝑥)+^𝑁_𝑗=1𝑤^1/𝑝′_𝑗|𝜕𝑇^𝐾_𝑚(𝑢)𝜕𝑥_𝑗|^𝑝−1] (4.49) a.e. in Ω, uniformly with respect to 𝜀.
It follows that when 𝜀 tends to 0
𝑎(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢))𝑏^𝜀(𝑢^𝜀)^𝑝−1→𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇_𝐾(𝑢))𝑏(𝑢)^𝑝−1stronglyin^𝑁_𝑖=1𝐿^𝑝′(Ω,𝑤^1−𝑝′_𝑖). (4.50) In view of (4.12), we conclude that (∇𝑇^𝐾_𝑚(𝑢^𝜀)−∇𝑇^𝐾_𝑚(𝑢))⇀0weaklyin^𝑁_𝑖=1𝐿^𝑝(Ω,𝑤_𝑖),as𝜀goesto0. (4.51) As a consequence of (4.50) and (4.51) we have for all 𝐾>0lim_𝜀→0𝐴^𝜀₃=0. (4.52) Equations (4.46), (4.46), (4.47), and (4.46) allow to pass to the limit as 𝜀 tends to zero in (4.31) and to obtain (4.30) of Lemma 4.2.

Step 4. In this step we identify the weak limit 𝑋_𝐾 and we prove the weak 𝐿¹ convergence of the “truncated” energy (𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))/𝑏^𝜀(𝑢^𝜀)^𝑝−1)∇𝑇^𝐾_𝑚(𝑢^𝜀) as 𝜀 tends to 0.

Lemma 4.3. For fixed 𝐾≥0, one has 𝑋_𝐾=𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))a.e.in{𝑥∈Ω;𝑢(𝑥)<𝑚}. (4.53) And as 𝜀 tends to 0 𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1∇𝑇^𝐾_𝑚(𝑢^𝜀)⇀𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))𝑏(𝑢)^𝑝−1∇𝑇^𝐾_𝑚(𝑢)weaklyin𝐿¹(Ω). (4.54)

Proof. Let 𝐾≥0 be fixed. From (4.11) and (4.50) together with (4.30) of Lemma 4.2, we obtain lim_𝜀→0Ω𝑎^𝜀(𝑥,𝑇^𝐾_m(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏(𝑇^𝐾_𝑚(𝑢^𝜀))^𝑝−1∇𝑇^𝐾_𝑚(𝑢^𝜀)𝑑𝑥=_Ω𝑋_𝐾𝑏(𝑇^𝐾_𝑚(𝑢))^𝑝−1∇𝑇_𝐾(𝑢)𝑑𝑥. (4.55) We remark the monotone character 𝑎 (with respect to 𝜉) and since 1/𝑏^𝜀(𝑢^𝜀)^𝑝−1 converges to 1/𝑏(𝑢)^𝑝−1 a.e. in Ω and due to the bound 1/𝑏^𝜀(𝑢^𝜀)^𝑝−1≤1/𝛼^𝑝−1 a.e. in Ω, we conclude that for all 𝜓∈∏^𝑁_𝑖=1𝐿^𝑝(Ω,𝑤_𝑖) we have 0≤lim_𝜀→0Ω[𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1−𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),𝜓)𝑏^𝜀(𝑢^𝜀)^𝑝−1][∇𝑇^𝐾_𝑚(𝑢^𝜀)−𝜓]𝑑𝑥=lim_𝜀→0Ω𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1[∇𝑇^𝐾_𝑚(𝑢^𝜀)−𝜓]𝑑𝑥−lim_𝜀→0Ω𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),𝜓)𝑏^𝜀(𝑢^𝜀)^𝑝−1[∇𝑇^𝐾_𝑚(𝑢^𝜀)−𝜓]𝑑𝑥=_Ω𝑋_𝐾𝑏(𝑢)^𝑝−1[∇𝑇^𝐾_𝑚(𝑢)−𝜓]𝑑𝑥−_Ω𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),𝜓)𝑏(𝑢)^𝑝−1[∇𝑇^𝐾_𝑚×(𝑢)−𝜓]𝑑𝑥_Ω[𝑋_𝐾𝑏(𝑢)^𝑝−1−𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),𝜓)𝑏(𝑢)^𝑝−1][∇𝑇^𝐾_𝑚(𝑢)−𝜓]𝑑𝑥. (4.56) The usual Minty's argument applies in view of (4.56). It follows that 𝑋_𝐾𝑏(𝑢)^𝑝−1=𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))𝑏(𝑢)^𝑝−1a.e.inΩ (4.57) which together with (4.20) yields (4.53) of Lemma 4.3.
In order to prove (4.54), we observe that the monotone character of 𝑎 (with respect to 𝜉) and (4.30) give
[𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1−𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢))𝑏^𝜀(𝑢^𝜀)^𝑝−1][∇𝑇^𝐾_𝑚(𝑢^𝜀)−∇𝑇^𝐾_𝑚(𝑢)]→0 (4.58) strongly in 𝐿¹(Ω) as 𝜀 tends to 0. Moreover (4.12), (4.17), (4.50), and (4.53) imply that 𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1∇𝑇^𝐾_𝑚(𝑢)⇀𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))𝑏(𝑢)^𝑝−1∇𝑇^𝐾_𝑚(𝑢) (4.59) weakly in 𝐿¹(Ω) as 𝜀 tends to 0 𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢))𝑏^𝜀(𝑢^𝜀)^𝑝−1∇𝑇^𝐾_𝑚(𝑢^𝜀)⇀𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))𝑏(𝑢)^𝑝−1∇𝑇^𝐾_𝑚(𝑢) (4.60) weakly in 𝐿¹(Ω) as 𝜀 tends to 0, and 𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢))𝑏^𝜀(𝑢^𝜀)^𝑝−1∇𝑇^𝐾_𝑚(𝑢)⇀𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))𝑏(𝑢)^𝑝−1∇𝑇^𝐾_𝑚(𝑢) (4.61) strongly in 𝐿¹(Ω) as 𝜀 tends to 0.
Using the above convergence results (4.59), (4.60), and (4.61) in (4.58) we obtain that for any 𝐾≥0
𝑎^𝜀(𝑥,𝑇^𝐾_𝑚(𝑢^𝜀),∇𝑇^𝐾_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1∇𝑇^𝐾_𝑚(𝑢^𝜀)⇀𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))𝑏(𝑢)^𝑝−1∇𝑇^𝐾_𝑚(𝑢) (4.62) weakly in 𝐿¹(Ω) as 𝜀 tends to 0.

Step 5. In this step we prove that 𝑢 satisfies (3.11).
Using (𝑇_𝑚^𝑛+1(𝑢^𝜀)−𝑇^𝑛_𝑚(𝑢^𝜀)) as a test function in (4.4) leads to
_Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇(𝑇_𝑚^𝑛+1(𝑢^𝜀)−𝑇^𝑛_𝑚(𝑢^𝜀))𝑑𝑥=_Ω𝑓^𝜀(𝑇_𝑚^𝑛+1(𝑢^𝜀)−𝑇^𝑛_𝑚(𝑢^𝜀))𝑑𝑥. (4.63) Since supp(𝑇_𝑚^𝑛+1(⋅)−𝑇^𝑛_𝑚(⋅))⊂[−(𝑛+1),−𝑛], we have _{{−𝑛−1≤𝑢𝜀(𝑥)≤−𝑛}}𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑢^𝜀=𝑑𝑥_Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇(𝑇_𝑚^𝑛+1(𝑢^𝜀)−𝑇^𝑛_𝑚(𝑢^𝜀=))𝑑𝑥_Ω𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)𝑏^𝜀(𝑢^𝜀)^𝑝−1∇(𝑇_𝑚^𝑛+1(𝑢^𝜀)−𝑇^𝑛_𝑚(𝑢^𝜀))𝑏(𝑇^𝑛+1_𝑚−1(𝑢^𝜀))^𝑝−1𝑑𝑥=_Ω𝑎^𝜀(𝑥,𝑇_𝑚^𝑛+1(𝑢^𝜀),∇𝑇_𝑚^𝑛+1(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1∇𝑇_𝑚^𝑛+1(𝑢^𝜀)𝑏(𝑇^𝑛+1_𝑚−1(𝑢^𝜀))^𝑝−1𝑑𝑥−_Ω𝑎^𝜀(𝑥,𝑇^𝑛_𝑚(𝑢^𝜀),∇𝑇^𝑛_𝑚(𝑢^𝜀))𝑏^𝜀(𝑢^𝜀)^𝑝−1∇𝑇^𝑛_𝑚(𝑢^𝜀)𝑏(𝑇^𝑛+1_𝑚−1(𝑢^𝜀))^𝑝−1𝑑𝑥. (4.64) In view of (4.54) of Lemma 4.3 and since 𝑏(𝑇^𝑛+1_𝑚−1(𝑢^𝜀))^𝑝−1 converges to 𝑏(𝑇^𝑛+1_𝑚−1(𝑢))^𝑝−1 a.e. in Ω and due to the bound 𝑏(𝑇^𝑛+1_𝑚−1(𝑢^𝜀))^𝑝−1≤max_{𝑠∈[−𝑛−1,𝑚−1]}𝑏(𝑠)^𝑝−1 a.e. in Ω, we can pass to the limit as 𝜀 tends to 0 for fixed 𝑛≥0 to obtain lim_{𝜀→0{−𝑛−1≤𝑢𝜀(𝑥)≤−𝑛}}𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑢^𝜀=𝑑𝑥_Ω𝑎(𝑥,𝑇_𝑚^𝑛+1(𝑢),∇𝑇_𝑚^𝑛+1(𝑢))𝑏(𝑢)^𝑝−1∇𝑇_𝑚^𝑛+1(𝑢)𝑏(𝑇^𝑛+1_𝑚−1(𝑢))^𝑝−1−𝑑𝑥_Ω𝑎(𝑥,𝑇^𝑛_𝑚(𝑢),∇𝑇^𝑛_𝑚(𝑢))𝑏(𝑢)^𝑝−1∇𝑇^𝑛_𝑚(𝑢)𝑏(𝑇^𝑛+1_𝑚−1(𝑢))^𝑝−1=𝑑𝑥_{{−𝑛−1≤𝑢(𝑥)≤−𝑛}}𝑎(𝑥,𝑢,∇𝑢)∇𝑢𝑑𝑥. (4.65) Taking the limit as 𝜀 tends to 0 and 𝑛 tends to +∞ in (4.63) and using the estimate (4.64) and (4.65) show that lim_{𝑛→+∞{−𝑛−1≤𝑢(𝑥)≤−𝑛}}𝑎(𝑥,𝑢,∇𝑢)∇𝑢𝑑𝑥≤lim_{𝑛→+∞{𝑢≤−𝑛}}|𝑓|𝑑𝑥=0. (4.66)

Step 6. In this step we prove that 𝑢 satisfies (3.12).
Using 𝑆_𝑛(𝑣^𝜀)(1/𝛿)(𝑇⁺_𝑚−𝛿(𝑢)−𝑇⁺_𝑚−2𝛿(𝑢)) as a test function in (4.4) leads to
1𝛿_Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇(𝑇⁺_𝑚−𝛿(𝑢)−𝑇⁺_𝑚−2𝛿=(𝑢))𝑑𝑥_Ω𝑆_𝑛(𝑣^𝜀)𝑓^𝜀(𝑇⁺_𝑚−𝛿(𝑢)−𝑇⁺_𝑚−2𝛿(𝑢))𝛿𝑑𝑥, (4.67) where 𝑣^𝜀=∫^𝑢𝜀₀𝑏^𝜀(𝑠)𝑑𝑠. Since supp(𝑆_𝑛)⊂[−(𝑛+1),𝑛+1] and {𝑥∈Ω;|𝑣^𝜀|≤𝑛+1}⊂{𝑥∈Ω;|𝑢^𝜀|≤(𝑛+1)/𝛼} we have 1𝛿_Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇(𝑇⁺_𝑚−𝛿(𝑢)−𝑇⁺_𝑚−2𝛿=1(𝑢))𝑑𝑥𝛿_Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀),∇𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀))∇(𝑇⁺_𝑚−𝛿(𝑢)−𝑇⁺_𝑚−2𝛿(𝑢))𝑑𝑥. (4.68) In view of (4.22), (4.36), (4.39), and (4.53), passing to the limit as 𝜀 tends to 0 and 𝑛 tends to +∞lim_𝑛→+∞lim_𝜀→01𝛿_Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀),∇𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀))∇(𝑇⁺_𝑚−𝛿(𝑢)−𝑇⁺_𝑚−2𝛿(𝑢))𝑑𝑥=lim_𝑛→+∞1𝛿_Ω𝑆_𝑛(𝑣)𝑎(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢),∇𝑇_{(𝑛+1)/𝛼}(𝑢))∇(𝑇⁺_𝑚−𝛿(𝑢)−𝑇⁺_𝑚−2𝛿(𝑢))𝑑𝑥=lim_𝑛→+∞1𝛿_Ω𝑆_𝑛(𝑣)𝑎(𝑥,𝑇_𝑚−𝛿(𝑢),∇𝑇_𝑚−𝛿(𝑢))∇(𝑇⁺_𝑚−𝛿(𝑢)−𝑇⁺_𝑚−2𝛿=1(𝑢))𝑑𝑥𝛿_Ω𝑎(𝑥,𝑇_𝑚−𝛿(𝑢),∇𝑇_𝑚−𝛿(𝑢))∇(𝑇⁺_𝑚−𝛿(𝑢)−𝑇⁺_𝑚−2𝛿=1(𝑢))𝑑𝑥𝛿_{{𝑚−2𝛿≤𝑢≤𝑚−𝛿}}𝑎(𝑥,𝑢,∇𝑢)∇𝑢𝑑𝑥. (4.69) Taking the limit as 𝜀 tends to 0, 𝑛 tends to +∞ and 𝛿 tends to 0 in (4.67) and using the estimate (4.68) and (4.69) show that lim_𝛿→01𝛿_{{𝑚−2𝛿≤𝑢≤𝑚−𝛿}}𝑎(𝑥,𝑢,∇𝑢)∇𝑢𝑑𝑥=lim_𝛿→0Ω𝑓(𝑇⁺_𝑚−𝛿(𝑢)−𝑇⁺_𝑚−2𝛿(𝑢))𝛿=𝑑𝑥_{𝑢=𝑚}𝑓(𝑥)𝑑𝑥. (4.70)

Step 7. In this step, 𝑢 is shown to satisfy (3.13). Let 𝜑∈𝑊₀^1,𝑝(Ω,𝜔)∩𝐿^∞(Ω) and let 𝑆 be a function in 𝑊^1,∞(ℝ) such that 𝑆 has a compact support and 𝑆(𝑚)=0. Let 𝐾 be a positive real number such that supp(𝑆)⊂[−𝐾,𝐾] and 𝑣^𝜀=∫^𝑢𝜀₀𝑏^𝜀(𝑠)𝑑𝑠. Using 𝑆(𝑢)𝑆_𝑛(𝑣^𝜀)𝜑 as a test function in (4.4) leads to _Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇(𝑆(𝑢)𝜑)𝑑𝑥+_Ω𝑆(𝑢)𝜑𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑆_𝑛(𝑣^𝜀=)𝑑𝑥_Ω𝑓^𝜀𝑆_𝑛(𝑣^𝜀)𝑆(𝑢)𝜑𝑑𝑥. (4.71)
In what follows we pass to the limit as 𝜀 tends to 0 and 𝑛 tends to +∞ in each term of (4.71).

Limit of First Term in (4.71)
Since supp 𝑆_𝑛⊂[−(𝑛+1),𝑛+1] and {𝑥∈Ω;|𝑣^𝜀|≤𝑛+1}⊂{𝑥∈Ω;|𝑢^𝜀|≤(𝑛+1)/𝛼}, we have 𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)=𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀),∇𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀))a.e.inΩ. (4.72) In view of (4.22), (4.36), (4.39), and (4.53), passing to the limit as 𝜀 tends to 0 lim_𝜀→0Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇(𝑆(𝑢)𝜑)𝑑𝑥=lim_𝜀→0Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀),∇𝑇_{(𝑛+1)/𝛼}(𝑢^𝜀))∇(𝑆(𝑢)𝜑)𝑑𝑥=_Ω𝑆_𝑛(𝑣)𝑎(𝑥,𝑇_{(𝑛+1)/𝛼}(𝑢),∇𝑇_{(𝑛+1)/𝛼}=(𝑢))∇(𝑆(𝑢)𝜑)𝑑𝑥_Ω𝑆_𝑛(𝑣)𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))∇(𝑆(𝑢)𝜑)𝑑𝑥,lim_𝑛→+∞lim_𝜀→0Ω𝑆_𝑛(𝑣^𝜀)𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇(𝑆(𝑢)𝜑)𝑑𝑥=lim_{𝑛→+∞Ω}𝑆_𝑛(𝑣)𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚=(𝑢))∇(𝑆(𝑢)𝜑)𝑑𝑥_Ω𝑎(𝑥,𝑇^𝐾_𝑚(𝑢),∇𝑇^𝐾_𝑚(𝑢))∇(𝑆(𝑢)𝜑)𝑑𝑥=_Ω𝑎(𝑥,𝑢,∇𝑢)∇(𝑆(𝑢)𝜑)𝑑𝑥. (4.73)

Limit of Second Term in (4.71)
Since supp(𝑆^′_𝑛)⊂[−(𝑛+1),−𝑛]∪[𝑛+1,𝑛] for any 𝑛≥1. As a consequence |_Ω𝑆(𝑢)𝜑𝑎^𝜀(𝑥,𝑢^𝜀,∇𝑢^𝜀)∇𝑆_𝑛(𝑣^𝜀)𝑑𝑥|≤‖𝑆‖_𝐿^∞_(Ω)‖𝜑‖_{𝐿∞(Ω){𝑛≤|𝑣𝜀|≤𝑛+1}}𝑎^𝜀(𝑥,𝑢^𝜀,∇