On Elliptic Equations in Orlicz Spaces Involving Natural Growth Term and Measure Data

Abstract

The paper deals with the existence of solutions of elliptic equations in the framework of Orlicz spaces with right-hand side measure and natural growth term.

1. Introduction

We deal with boundary value problems 𝐴(𝑢)=𝑔(𝑢)𝑀(|∇𝑢|)+𝜇inΩ,𝑢=0on𝜕Ω, 𝑃_𝜇 where 𝐴(𝑢)=−div(𝑎(⋅,𝑢,∇𝑢)), (1.1)Ω is a bounded domain of 𝐑^𝑁, with the segment property. 𝑎∶Ω×𝐑×𝐑^𝑁→𝐑^𝑁 is a Carathéodory function (i.e., measurable with respect to 𝑥 in Ω for every (𝑠,𝜉) in 𝐑×𝐑^𝑁, and continuous with respect to (𝑠,𝜉) in 𝐑×𝐑^𝑁 for almost every 𝑥 in Ω) such that there exist two 𝑁-functions 𝑃≪𝑀 and for all 𝜉,𝜉^∗∈𝐑^𝑁, 𝜉≠𝜉^∗, the following hypotheses are true 𝑎()𝜉𝑥,𝑠,𝜉𝜉≥𝛼𝑀𝜆,𝑎(𝑥,𝑠,𝜉)−𝑎𝑥,𝑠,𝜉^∗𝜉−𝜉^∗>0,𝑎(𝑥,𝑠,𝜉)≤𝑐(𝑥)+𝑘₁𝑃⁻¹𝑀𝑘₂|𝑠|+𝑘₃𝑀⁻¹𝑀𝑘₄𝜉, (1.2)(1.3)(1.4) where 𝑐(𝑥) belongs to 𝐸_𝑀(Ω), 𝑐≥0, 𝑘_𝑖(𝑖=1,2,3,4) to 𝐑⁺, and 𝛼, 𝜆 to 𝐑⁺_∗. 𝑀(𝑡)≪𝑡^𝑁,_⋅^+∞𝑀(𝑡)𝑡^1+𝑁′𝑑𝑡=+∞,𝑔∶𝐑⁺→𝐑⁺anintegrablefunctionon𝐑⁺,𝜇∈𝑀⁺_𝑏(Ω). (1.5)(1.6)(1.7) For the sake of simplicity, we suppose in (1.2) that 𝛼=𝜆=1.

This paper is devoted to study the Dirichlet problem for some nonlinear elliptic equations whose simplest model is −Δ_𝑝𝑢=𝑔(𝑢)|∇𝑢|^𝑝+𝜇. (1.8) This kind of problems has been widely studied. Many authors have proved results for second-order elliptic problems with lower-order terms depending on the gradient; these works include, for instance, [1–6]. After the classical example by Kazdan and Kramer (see [7]), which shows that (1.8) cannot always have solutions, two different kind of questions have been considered. On the one hand, in some papers, the existence of solutions when the source 𝑓 is small in a suitable norm are proved. On the other hand, conditions on which the function 𝑔 have been considered in order to get a solution for all 𝑓 in a given Lebesgue space. This is the way chosen in [1, 5, 6] under the hypothesis 𝑔∈𝐿¹.

In [8], the authors present some results concerning existence, nonexistence, multiplicity, and regularity of positive solutions for two elliptic quasilinear problems with Dirichlet data in a bounded domain. The first problem is similar to (1.8) and the other is −Δ_𝑝𝑣=𝜆𝑓(𝑥)(1+𝑏(𝑣))^𝑝−1, (1.9) where 𝜆, 𝑓, 𝑏, 𝑔 required some specified conditions. The first one, of unknown 𝑢, involves a gradient term with natural growth. The second one, of unknown 𝑣, presents a source term of order 0. They gave and established a precise connection between problems in 𝑢 and 𝑣. Also, they proved a result of existence for the problem in 𝑣 with general bounded Radon measures data and obtained some results for the problem in 𝑢 by using the connection between these two problems. Other authors have established this connection between the two problems (1.8) and (1.9); one can see, for example, [9].

Many researchers have investigated the possibility to find solutions of (1.8) under the sign condition 𝑔(𝑠)𝑠≥0, in which case the term 𝑔(𝑢)|∇𝑢|^𝑝 is said to be an absorption term. In [5], the author treated the problem (1.8) without using the above sign condition but by supposing the summability of the function 𝑔. The principal tools used is the Lebesgue decomposition theorem (see [10]) and the cut functions with respect to the measure 𝜇. The result given is optimal.

In the spirit of the work [5], our purpose in this paper is to prove existence results in the setting of the Orlicz Sobolev space 𝑊¹𝐿_𝑀(Ω) when the operator does not satisfy the classical polynomial growth.

2. Preliminaries

Let 𝑀 ∶𝐑⁺→𝐑⁺ be an 𝑁-function, that is, 𝑀 is continuous, convex, with 𝑀(𝑡)>0 for 𝑡>0, 𝑀(𝑡)/𝑡→0 as 𝑡→0 and 𝑀(𝑡)/𝑡→∞ as 𝑡→∞. The 𝑁-function 𝑀 conjugate to 𝑀 is defined by 𝑀(𝑡)=sup{𝑠𝑡−𝑀(𝑠)∶𝑠>0}.

Let 𝑃 and 𝑄 be two 𝑁-functions. 𝑃≪𝑄 means that 𝑃 grows essentially less rapidly than 𝑄; that is, for each 𝜀>0, 𝑃(𝑡)𝑄(𝜀𝑡)→0as𝑡→∞. (2.1) The 𝑁-function 𝑀 is said to satisfy the Δ₂ condition if for some 𝑘>0: 𝑀(2𝑡)≤𝑘𝑀(𝑡) for all 𝑡≥0; when this inequality holds only for 𝑡≥𝑡₀>0, 𝑀 is said to satisfy the Δ₂ condition near infinity.

Let Ω be an open subset of 𝐑^𝑁. The Orlicz class ℒ_𝑀(Ω) (resp., the Orlicz space 𝐿_𝑀(Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions 𝑢 on Ω such that ∫_Ω𝑀(𝑢(𝑥))𝑑𝑥<+∞ (resp. ∫_Ω𝑀(𝑢(𝑥)/𝜆)𝑑𝑥<+∞for some 𝜆>0).

Note that 𝐿_𝑀(Ω) is a Banach space under the norm ‖𝑢‖_𝑀,Ω=inf{𝜆>0∶∫_Ω𝑀(𝑢(𝑥)/𝜆)𝑑𝑥≤1} and ℒ_𝑀(Ω) is a convex subset of 𝐿_𝑀(Ω). The closure in 𝐿_𝑀(Ω) of the set of bounded measurable functions with compact support in Ω is denoted by 𝐸_𝑀(Ω). In general 𝐸_𝑀(Ω)≠𝐿_𝑀(Ω) and the dual of 𝐸_𝑀(Ω) can be identified with 𝐿_𝑀(Ω) by means of the pairing ∫_Ω𝑢(𝑥)𝑣(𝑥)𝑑𝑥, and the dual norm on 𝐿_𝑀(Ω) is equivalent to ‖⋅‖_𝑀,Ω.

We now turn to the Orlicz-Sobolev space. 𝑊¹𝐿_𝑀(Ω) (resp. 𝑊¹𝐸_𝑀(Ω)) is the space of all functions 𝑢 such that 𝑢 and its distributional derivatives up to order 1 lie in 𝐿_𝑀(Ω) (resp. 𝐸_𝑀(Ω)). This is a Banach space under the norm ‖𝑢‖_1,𝑀,Ω=∑_|𝛼|≤1‖𝐷^𝛼𝑢‖_𝑀,Ω. Thus 𝑊¹𝐿_𝑀(Ω) and 𝑊¹𝐸_𝑀(Ω) can be identified with subspaces of the product of 𝑁+1 copies of 𝐿_𝑀(Ω). Denoting this product by Π𝐿_𝑀, we will use the weak topologies 𝜎(Π𝐿_𝑀,Π𝐸_𝑀) and 𝜎(Π𝐿_𝑀,Π𝐿_𝑀). The space 𝑊¹₀𝐸_𝑀(Ω) is defined as the (norm) closure of the Schwartz space 𝒟(Ω) in 𝑊¹𝐸_𝑀(Ω) and the space 𝑊¹₀𝐿_𝑀(Ω) as the 𝜎(Π𝐿_𝑀,Π𝐸_𝑀) closure of 𝒟(Ω) in 𝑊¹𝐿_𝑀(Ω). We say that 𝑢_𝑛 converges to 𝑢 for the modular convergence in 𝑊¹𝐿_𝑀(Ω) if for some 𝜆>0, ∫_Ω𝑀((𝐷^𝛼𝑢_𝑛−𝐷^𝛼𝑢)/𝜆)𝑑𝑥→0 for all |𝛼|≤1. This implies convergence for 𝜎(Π𝐿_𝑀,Π𝐿_𝑀). If 𝑀 satisfies the Δ₂ condition on 𝐑⁺ (near infinity only when Ω has finite measure), then modular convergence coincides with norm convergence.

Fore more details about the Orlicz spaces and their properties one can see [11, 12].

For 𝑘>0, we define the truncation at height 𝑘,𝑇_𝑘∶𝐑→𝐑 by: 𝑇_𝑘(𝑠)=max(−𝑘,min(𝑘,𝑠)).

3. Main Result

3.1. Useful Results

First, we give the following definitions and results which will be used in our main result.

The 𝑝-capacity 𝐶_𝑝(𝐵,Ω) of any set 𝐵⊂Ω with respect to Ω is defined in the following classical way. The 𝑝-capacity of any compact set 𝐾⊂Ω is first defined as 𝐶_𝑝(𝐾,Ω)=inf_Ω∇𝜑^𝑝𝑑𝑥∶𝜑∈𝐷(Ω),𝜑≥𝜒_𝐾, (3.1) where 𝜒_𝐾 is the characteristic function of 𝐾; we will use the convention that inf𝜙=+∞. The 𝑝-capacity of any open subset 𝑈⊂Ω is then defined by 𝐶_𝑝(𝑈,Ω)=sup𝐶_𝑝(𝐾,Ω),𝐾compact𝐾⊂Ω. (3.2) Finally, the 𝑝-capacity of any subset 𝐵⊂Ω is defined by 𝐶_𝑝(𝐵,Ω)=inf𝐶_𝑝(𝑈,Ω),𝑈open𝐵⊂𝑈. (3.3)

Definition 3.1. We say that 𝑢 is a weak solution of the problem 𝑃_𝜇 if 𝑢ismeasurable,𝑇_𝑘(𝑢)∈𝑊¹₀𝐿_𝑀(Ω),_Ω𝑎(⋅,𝑢,∇𝑢)∇𝑣𝑑𝑥=_Ω𝑔(𝑢)𝑀(|∇𝑢|)𝑣𝑑𝑥+_Ω𝑣𝑑𝜇∀𝑣∈𝐷(Ω). (3.4) We define 𝑀_𝑏(Ω) as the space of all Radon measures on Ω with bounded total variation, and 𝐶_𝑏(Ω) as the space of all bounded, continuous functions on Ω, so that ∫_Ω𝜑𝑑𝜇 is defined for 𝜑∈𝐶_𝑏(Ω) and 𝜇∈𝑀_𝑏(Ω).

We say that a sequence (𝜇_𝑛) of measures in 𝑀_𝑏(Ω) converges to a measure 𝜇 in 𝑀_𝑏(Ω) if lim_{𝑛→+∞Ω}𝜑𝑑𝜇_𝑛=_Ω𝜑𝑑𝜇_𝑛 (3.5) for every 𝜑∈𝐶_𝑏(Ω). If this convergence holds only for all the continuous functions 𝜑 with compact support in Ω, then we have the usual weak ∗ convergence in 𝑀_𝑏(Ω).

Lemma 3.2. Under the hypotheses (1.2)–(1.6), 𝑓∈𝐿^∞(Ω) and 𝑓≥0, there exists at least one positive weak solution of the problem 𝐴(𝑢)=𝑔(𝑢)𝑀(|∇𝑢|)+𝑓inΩ,𝑢=0on𝜕Ω. 𝑃_𝑓

For the proof see [12].

Let 𝑀 be a fixed 𝑁-function, we define 𝐾 as the set of 𝑁 functions 𝐷 satisfying the following conditions:(i)𝑀(𝐷⁻¹(𝑠)) is a convex function,(ii)∫^⋅₀𝐷𝑜𝐵⁻¹(1/𝑟^1−1/𝑁)𝑑𝑟<+∞,𝐵(𝑡)=𝑀(𝑡)/𝑡, (iii)there exists an 𝑁-function 𝐻 such that 𝐻𝑜𝑀⁻¹𝑜𝑀≤𝐷and 𝐻≤𝐷near infinity.

Lemma 3.3. Let (𝑢_𝑛) be a sequence of solutions of the problem 𝐴𝑢_𝑛=𝐹_𝑛,𝑢_𝑛∈𝑊¹₀𝐿_𝑀(Ω), (3.6) where (𝐹_𝑛) is a sequence of functions bounded in 𝐿¹(Ω). Then (𝑢_𝑛) is bounded in 𝑊¹₀𝐿_𝐷(Ω) for all 𝑁-functions 𝐷∈𝐾.

For the proof one can see that the technique used in [13] and adapted to the elliptic case gives the result, but for the simplicity we give a sketched proof.

Proof. Let denote by 𝒳_𝑁=∶𝑁𝐶_𝑁^1/𝑁, 𝐶_𝑁 the measure of the unit ball of 𝐑^𝑁, and 𝜇(𝜃)=|{|𝑢|>𝜃}|.
Let 𝜑 be a truncation defined by 𝜑(𝜉)=100≤𝜉≤𝜃ℎ()()𝜉−𝑡𝜃<𝜉<𝜃+ℎ1𝜉≥𝜃+ℎ−𝜑−𝜉𝜉<0 (3.7) for all 𝜃,ℎ>0.
Using 𝑣=𝜑(𝑢_𝑛) as a test function, we obtain after tending ℎ to zero −𝑑𝑑𝜃_{{|𝑢𝑛|>𝜃}}𝑀∇𝑢_𝑛𝑑𝑥≤𝐶_{{|𝑢𝑛|≥𝜃}}𝐹_𝑛𝑑𝑥. (3.8) Following the same way as in [14], we have for 𝐷∈𝐾, −𝑑𝑑𝜃_{{|𝑢𝑛|>𝜃}}𝐷∇𝑢_𝑛𝑑𝑥≤−𝜇^′(𝜃)𝐷𝑜𝐵⁻¹−1𝒳_𝑁𝜇(𝜃)^1−1/𝑁𝑑𝑑𝜃_{{|𝑢𝑛|>𝜃}}𝑀∇𝑢_𝑛𝑑𝑥. (3.9) We obtain −𝑑𝑑𝜃_{{|𝑢𝑛|>𝜃}}𝐷∇𝑢_𝑛𝑑𝑥≤−𝜇^′(𝜃)𝐷𝑜𝐵⁻¹−𝐶𝒳_𝑁𝜇(𝜃)^1−1/𝑁. (3.10) So, _Ω𝐷∇𝑢_𝑛𝑑𝑥=₀^+∞−𝑑𝑑𝜃_{{|𝑢𝑛|>𝜃}}𝐷∇𝑢_𝑛≤1𝑑𝑥𝑑𝑡𝐶^{′𝐶′0|Ω|}𝐷𝑜𝐵⁻¹𝐶𝑠^1−1/𝑁𝑑𝑠. (3.11) Then the sequence (𝑢_𝑛) is bounded in 𝑊¹₀𝐿_𝐷(𝑄).

Lemma 3.4. Let 𝜇 be a nonnegative Radon measure which is concentrated in a set 𝐸 of zero p-capacity. Then there exists a sequence (𝜓_𝛿) of 𝐷(Ω) functions such that lim_𝛿→∞Ω∇𝜓_𝛿^𝑝𝑑𝑥=0,0≤𝜓_𝛿≤1,lim_𝛿→∞Ω1−𝜓_𝛿𝑑𝜇=0. (3.12)

For the proof see [15].

Remark 3.5. By the above lemma, we have that (𝜓_𝛿) converge to zero both strongly in 𝑊₀^1,𝑝(Ω), a.e. in Ω, and in the weak * topology of 𝐿^∞(Ω).

3.2. Existence Result

In what follows, we suppose that the set 𝐾 is nonempty.

Theorem 3.6. Let 𝑥₀∈Ω, under the hypotheses (1.2)–(1.6) and 𝑁≥2, there exists at least one positive weak solution of the problem 𝐴(𝑢)=𝑔(𝑢)𝑀(|∇𝑢|)+𝛿_𝑥0inΩ,𝑢=0on𝜕Ω. 𝑃_𝛿𝑥0

Remark 3.7. (1) The condition 𝑁≥2 is very important to ensure the existence of cut functions for the measure 𝛿_𝑥0. The case 𝑁=1 is easily treated, since we come back to the variational case.
(1.9) The condition ∫_⋅^+∞(𝑀(𝑡)/𝑡^1+𝑁′)𝑑𝑡=+∞ is supposed to guarantee that we are not in the variational case and the study has a sense (see [16]).

Remark 3.8. The conditions (i), (ii), and (iii) permit us to determine the regularity of the solutions of 𝑃_𝛿𝑥0 and improve the one given by Porretta in [5]. If 𝑀(𝑡)=𝑡^𝑝, one can find a solution 𝑢 such that ∫_Ω|∇𝑢|^{𝑁(𝑝−1)/(𝑁−1)}/𝐿𝑜𝑔^𝜎(𝑒+|∇𝑢|)<+∞, for some 𝜎>1.

Remark 3.9 (see [5]). The condition 𝑔∈𝐿¹ is optimal in the sense introduced by Porretta in [5].
Indeed, if 𝜇 is the Dirac mass, there is no solution which can be obtained by approximation. In particular, in the reaction case (𝑔(𝑠)𝑠≤0), if 𝜇 is approximated by a sequence of smooth functions, the sequence of approximating solutions converges to a solution of 𝑃_𝜇 if 𝑔∈𝐿¹, while it blows up everywhere if 𝑔∉𝐿¹.
Finally, let us say a few words on how positive constant will be denoted hereafter. If no otherwise specified, we will write 𝐶 to denote any positive constant (possibly different) which only depends on the data, that is on quantities that are fixed in the assumptions (𝑁, Ω, and so on…); in any case such constants never depend on the different indexes having a limit. In the sequel and throughout the paper, we will omit for simplicity the dependence on 𝑥 in the function 𝑎(𝑥,𝑠,𝜉) and denote 𝜖(𝑛,𝑗,𝛿,𝑠,𝑚) all quantities (possibly different) such that lim_𝑚→∞lim_𝑠→∞lim_𝛿→∞lim_𝑗→∞lim_𝑛→∞𝜖𝑛,𝑗,𝛿,𝑠,𝑚=0, (3.13) and this will be in the order in which the parameters we use will tend to infinity, that is, first 𝑛, then 𝑗,𝛿,𝑠, and finally 𝑚. Similarly, we will write only 𝜖(𝑛), or 𝜖(𝑛,𝑗),…to mean that the limits are made only on the specified parameters. Moreover, for the sake of simplicity, in what follows, the convergence, even if not explicitly stressed, may be understood to be taken possibly up to a suitable subsequence extraction.

3.2.1. A Sequence of Approximating Problems

Consider the approximate problem 𝐴𝑢_𝑛=𝑔𝑢_𝑛𝑀∇𝑢_𝑛+𝑓_𝑛𝑢inΩ,_𝑛=0in𝜕Ω, 𝑃_𝑓𝑛 where (𝑓_𝑛) is a smooth sequence of functions such that ‖𝑓_𝑛‖₁≤𝐶 and 𝑓_𝑛⇀0𝑥00086𝛿_𝑥0 in 𝑀_𝑏(Ω).

The existence of solutions of the above problem 𝑢_𝑛∈𝑊¹₀𝐿_𝑀(Ω) was ensured by Lemma 3.2.

3.2.2. A Priori Estimates

Lemma 3.10. There exists a subsequence of (𝑢_𝑛) (also denoted (𝑢_𝑛)); there exists a measurable function 𝑢 such that 𝑇_𝑘(𝑢)∈𝑊¹₀𝐿_𝑀(Ω) and 𝑇_𝑘(𝑢_𝑛)⇀𝑇_𝑘(𝑢) weakly in 𝑊¹₀𝐿_𝑀(Ω), strongly in 𝐸_𝑀(Ω), and a.e. in Ω.

Proof. Let 𝑣=𝑇_𝑘(𝑢_𝑛)exp(∫^𝑢𝑛₀𝑔(𝑠)𝑑𝑠) as test function in 𝑃_𝑓𝑛. One has _Ω𝑎𝑥,𝑢_𝑛,∇𝑢_𝑛∇𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)+𝑑𝑠_Ω𝑎𝑥,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛𝑔𝑢_𝑛𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)=𝑑𝑠_Ω𝑀∇𝑢_𝑛𝑔𝑢_𝑛𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠+_Ω𝑓_𝑛𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠, (3.14) then _Ω𝑀∇𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠𝑑𝑥≤_Ω𝑓_𝑛𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠≤𝐶𝑘, (3.15) so (𝑇_𝑘(𝑢_𝑛)) is bounded in 𝑊¹₀𝐿_𝑀(Ω).
There exist a subsequence also denoted (𝑢_𝑛) and a measurable function 𝜔_𝑘 such that 𝑇_𝑘𝑢_𝑛⇀𝜔_𝑘,weaklyin𝑊¹₀𝐿_𝑀(Ω),a.e.inΩ,andstronglyin𝐸_𝑀(Ω). (3.16) By an easy argument we can see that, there exists a measurable function 𝑢 such that 𝑇_𝑘(𝑢)=𝜔_𝑘. (3.17)

3.2.3. Almost Everywhere Convergence of Gradients

Lemma 3.11. The subsequence (𝑢_𝑛) obtained in Lemma 3.10 satisfies ∇𝑢_𝑛→∇𝑢a.e.inΩ. (3.18)

Proof. Let us recall that since the 𝑁-capacity 𝐶_𝑁({𝑥₀},Ω)=0, there exists a sequence (𝜓_𝛿) satisfying the Lemma 3.4 with 𝜇=𝛿_𝑥0 and 𝑝=𝑁.

Step 1. In this step we will show the following: (i)lim_𝑛→+∞∫_Ω𝑟(𝑎(⋅,𝑇_𝑘(𝑢_𝑛),∇𝑇_𝑘(𝑢_𝑛))−𝑎(⋅,𝑇_𝑘(𝑢_𝑛),∇𝑇_𝑘(𝑢)))(∇𝑇_𝑘(𝑢_𝑛)−∇𝑇_𝑘(𝑢))(1−𝜓_𝛿)𝑑𝑥=0, (ii)lim_𝑛→+∞∫_Ω𝑟(𝑎(⋅,𝑇_𝑘(𝑢_𝑛),∇𝑇_𝑘(𝑢_𝑛))−𝑎(⋅,𝑇_𝑘(𝑢_𝑛),∇𝑇_𝑘(𝑢)))(∇𝑇_𝑘(𝑢_𝑛)−∇𝑇_𝑘(𝑢))𝜓_𝛿𝑑𝑥=0.
Let 𝑚>0, 𝑘>0 such that 𝑚>𝑘. Let 𝜌_𝑚 be a truncation defined by 𝜌_𝑚(𝑠)=1|𝑠||𝑠||𝑠|0|𝑠|≤𝑚,𝑚+1−𝑚<<𝑚+1,≥𝑚+1. (3.19) Let 𝑠>0, Ω^𝑠={𝑥∈Ω∶|∇𝑇_𝑘(𝑢)|≤𝑠}, Ω^𝑠_𝑗={𝑥∈Ω∶|∇𝑇_𝑘(𝑣_𝑗)|≤𝑠}.
We denote, respectively, by 𝜒^𝑠, 𝜒^𝑠_𝑗 the indicator function of Ω^𝑠,Ω^𝑠_𝑗.
Let 𝑣_𝑗∈𝐷(Ω) such that 𝑣_𝑗→𝑇_𝑘(𝑢) with the modular convergence in 𝑊¹₀𝐿_𝑀(Ω) (see [11]).
Let 𝑧^𝑗,𝛿_𝑛,𝑚=(𝑇_𝑘(𝑢_𝑛)−𝑇_𝑘(𝑣_𝑗))exp(∫^𝑢𝑛₀𝑔(𝑠)𝑑𝑠)(1−𝜓_𝛿)𝜌_𝑚(𝑢_𝑛), as test function in the approximate problem. Then 𝐴𝑢_𝑛,𝑧^𝑗,𝛿_𝑛,𝑚=_Ω𝑔𝑢_𝑛𝑀∇𝑢_𝑛𝑧^𝑗,𝛿_𝑛,𝑚𝑑𝑥+_Ω𝑓_𝑛𝑧^𝑗,𝛿_𝑛,𝑚𝑑𝑥∶=𝐽₁+𝐽₂. (3.20) We have _Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑧^𝑗,𝛿_𝑛,𝑚=𝑑𝑥_Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛𝜌_𝑚𝑢_𝑛∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘𝑣_𝑗exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿−𝑑𝑥_Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛𝜌_𝑚𝑢_𝑛𝑇_𝑘𝑢_𝑛−𝑇_𝑘𝑣_𝑗exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠∇𝜓_𝛿+𝑑𝑥_Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛𝜌^′_𝑚𝑢_𝑛𝑇_𝑘𝑢_𝑛−𝑇_𝑘𝑣_𝑗exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿+𝑑𝑥_Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛𝜌_𝑚𝑢_𝑛𝑇_𝑘𝑢_𝑛−𝑇_𝑘𝑣_𝑗𝑔𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿=𝑑𝑥_Ω𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛−𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗𝜌_𝑚𝑢_𝑛×exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿+𝑑𝑥_Ω𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗𝜌_𝑚𝑢_𝑛×exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿+𝑑𝑥_{Ω−Ω𝑠𝑗}𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛∇𝑇_𝑘𝑢_𝑛𝜌_𝑚𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿−𝑑𝑥_{Ω−Ω𝑠𝑗}𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛∇𝑇_𝑘𝑣_𝑗𝜌_𝑚𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿𝑑𝑥−_{|𝑢𝑛|>𝑘}𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑇_𝑘𝑣_𝑗𝜌_𝑚𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿𝑑𝑥+_Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛𝜌^′_𝑚𝑢_𝑛𝑇_𝑘𝑢_𝑛−𝑇_𝑘𝑣_𝑗exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿+𝑑𝑥_Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛𝜌_𝑚𝑢_𝑛𝑔𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿𝑑𝑥=𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅+𝐼₆+𝐼₇. (3.21) Then, 𝐴𝑢_𝑛,𝑧^𝑗,𝛿_𝑛,𝑚=⁷_𝑖=1𝐼_𝑖=𝐽₁+𝐽₂. (3.22)About 𝐼₂: it is obvious since 𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗→𝑎⋅,𝑇_𝑘(𝑢),∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗strongly, (3.23) that is _Ω𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗𝜌_𝑚𝑢_𝑛1−𝜓_𝛿𝑑𝑥=𝜖𝑛,𝑗. (3.24)About 𝐼₄: since (𝑇_𝑘(𝑢_𝑛)) is bounded in 𝑊¹₀𝐿_𝑀(Ω) and ∇𝑇_𝑘(𝑣_𝑗)𝜒_{Ω−Ω𝑠𝑗}∈(𝐸_𝑀(Ω))^𝑁, there exists a measurable function ℎ_𝑘 such that (up to a subsequence also denoted 𝑇_𝑘(𝑢_𝑛)) 𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛⇀ℎ_𝑘weaklyin𝐿_𝑀(Ω)^𝑁. (3.25) Then −∫_{Ω−Ω𝑠𝑗}𝑎(⋅,𝑇_𝑘(𝑢_𝑛),∇𝑇_𝑘(𝑢_𝑛))∇𝑇_𝑘(𝑣_𝑗)(1−𝜓_𝛿)𝑑𝑥=−∫_{Ω−Ω𝑠𝑗}ℎ_𝑘∇𝑇_𝑘(𝑣_𝑗)(1−𝜓_𝛿)𝑑𝑥+𝜖(𝑛).

Since 𝑣_𝑗→𝑇_𝑘(𝑢) in modular convergence, then −_{Ω−Ω𝑠𝑗}ℎ_𝑘∇𝑇_𝑘𝑣_𝑗1−𝜓_𝛿𝑑𝑥=−_Ω−Ω𝑠ℎ_𝑘∇𝑇_𝑘(𝑢)1−𝜓_𝛿𝑑𝑥+𝜖𝑗. (3.26) So, −_{Ω−Ω𝑠𝑗}𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛∇𝑇_𝑘𝑣_𝑗1−𝜓_𝛿𝑑𝑥=−_Ω−Ω𝑠ℎ_𝑘∇𝑇_𝑘(𝑢)1−𝜓_𝛿𝑑𝑥+𝜖𝑛,𝑗. (3.27)About 𝐼₅: since 𝜌(𝑢_𝑛)=0 on the set {|𝑢_𝑛|≥𝑚+1}, then −_{|𝑢𝑛|>𝑘}𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑇_𝑘𝑣_𝑗𝜌_𝑚𝑢_𝑛1−𝜓_𝛿𝑑𝑥=−_{|𝑢𝑛|>𝑘}𝑎⋅,𝑇_𝑚+1𝑢_𝑛,∇𝑇_𝑚+1𝑢_𝑛∇𝑇_𝑘𝑣_𝑗𝜌_𝑚𝑢_𝑛1−𝜓_𝛿𝑑𝑥 (3.28) and as for 𝐼₄, one has −_{|𝑢𝑛|>𝑘}𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑇_𝑘𝑣_𝑗𝜌_𝑚𝑢_𝑛1−𝜓_𝛿𝑑𝑥=−_|𝑢|>𝑘ℎ_𝑚+1∇𝑇_𝑘(𝑢)𝜌_𝑚(𝑢)1−𝜓_𝛿𝑑𝑥+𝜖𝑛,𝑗=𝜖𝑛,𝑗 (3.29) since ∇𝑇_𝑘(𝑢)=0 when |𝑢|≥𝑘.

About 𝐼₆: let 𝑣=𝜌_𝑚(𝑢_𝑛)(1−𝜓_𝛿)exp(∫^𝑢𝑛₀𝑔(𝑠)𝑑𝑠) as test function, in one hand we obtain _{{𝑚≤𝑢𝑛≤𝑚+1}}𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿+𝑑𝑥_Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛𝑔𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠𝜌_𝑚𝑢_𝑛1−𝜓_𝛿−𝑑𝑥_Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝜓_𝛿exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠𝜌_𝑚𝑢_𝑛=𝑑𝑥_Ω𝑔𝑢_𝑛𝑀∇𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠𝜌_𝑚𝑢_𝑛1−𝜓_𝛿+𝑑𝑥_Ω𝑓_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠𝜌_𝑚𝑢_𝑛1−𝜓_𝛿𝑑𝑥. (3.30) So, _{{𝑚≤𝑢𝑛≤𝑚+1}}𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿≤𝑑𝑥_Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝜓_𝛿exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠𝜌_𝑚𝑢_𝑛+𝑑𝑥_Ω𝑓_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠𝜌_𝑚𝑢_𝑛1−𝜓_𝛿𝑑𝑥≤𝐶∇𝜓_𝛿𝑁𝑎⋅,𝑇_𝑚+1𝑢_𝑛,∇𝑇_𝑚+1𝑢_𝑛𝑁′+𝐶_{|𝑢𝑛|≥𝑚}𝑓_𝑛1−𝜓_𝛿𝑑𝑥. (3.31) On the other hand 𝐼₆=_Ω𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛𝜌^′_𝑚𝑢_𝑛𝑇_𝑘𝑢_𝑛−𝑇_𝑘𝑣_𝑗exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿=𝑑𝑥_{{𝑚≤𝑢𝑛≤𝑚+1}}𝑎⋅,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛𝑇_𝑘𝑢_𝑛−𝑇_𝑘𝑣_𝑗exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠1−𝜓_𝛿𝑑𝑥≤2𝑘∇𝜓_𝛿𝑁𝑎⋅,𝑇_𝑚+1𝑢_𝑛,∇𝑇_𝑚+1𝑢_𝑛𝑁′+𝐶_{|𝑢𝑛|≥𝑚}𝑓_𝑛1−𝜓_𝛿().𝑑𝑥=𝜖𝑛,𝛿,𝑚 (3.32) Let us come back to our main estimation (i). We have _Ω𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛−𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘(𝑢)𝜒^𝑠∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘(𝑢)𝜒^𝑠𝜌_𝑚𝑢_𝑛1−𝜓_𝛿=𝑑𝑥_Ω𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛−𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗×∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗𝜌_𝑚𝑢_𝑛1−𝜓_𝛿+𝑑𝑥_Ω𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗−∇𝑇_𝑘(𝑢)𝜒^𝑠𝜌_𝑚𝑢_𝑛1−𝜓_𝛿−𝑑𝑥=𝜖𝑛,𝑗_Ω𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘(𝑢)𝜒^𝑠∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘(𝑢)𝜒^𝑠𝜌_𝑚𝑢_𝑛1−𝜓_𝛿+𝑑𝑥=𝜖𝑛,𝑗_Ω𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗𝜌_𝑚𝑢_𝑛1−𝜓_𝛿𝑑𝑥=𝜖𝑛,𝑗. (3.33) Then we deduce that _Ω𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛−𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘(𝑢)𝜒^𝑠∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘(𝑢)𝜒^𝑠𝜌_𝑚𝑢_𝑛1−𝜓_𝛿=𝑑𝑥_Ω𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛−𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗×∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘𝑣_𝑗𝜒^𝑠_𝑗𝜌_𝑚𝑢_𝑛1−𝜓_𝛿𝑑𝑥+𝜖𝑛,𝑗≤𝜖𝑛,𝑗+𝐶_Ω−Ω𝑠ℎ_𝑘∇𝑇_𝑘(𝑢)1−𝜓_𝛿𝑑𝑥+𝐶_{|𝑢𝑛|≥𝑚}𝑓_𝑛1−𝜓_𝛿𝑑𝑥. (3.34) Since we have ∑⁷_𝑖=1𝐼_𝑖=𝐽₁+𝐽₂, 𝐼₃≥0 and using (1.2), we deduce that 𝐼₁≤𝐽₂−𝐼₂−𝐼₃−𝐼₄−𝐼₅−𝐼₆≤_Ω𝑓_𝑛𝑧^𝑗,𝛿_𝑛,𝑚+𝐶_Ω−Ω𝑠ℎ_𝑘∇𝑇_𝑘(𝑢)1−𝜓_𝛿𝑑𝑥+𝐶𝑘∇𝜓_𝛿𝑁𝑎⋅,𝑇_𝑚+1𝑢_𝑛,∇𝑇_𝑚+1𝑢_𝑛𝑁′+𝐶_{|𝑢𝑛|≥𝑚}𝑓_𝑛1−𝜓_𝛿𝑑𝑥. (3.35) For 𝑟≤𝑠, one has 0≤𝐼₁(𝑟)≤𝐼₁(𝑠)≤𝐶_Ω−Ω𝑠ℎ_𝑘∇𝑇_𝑘(𝑢)1−𝜓_𝛿𝑑𝑥+𝐶𝑘∇𝜓_𝛿𝑁𝑎⋅,𝑇_𝑚+1𝑢_𝑛,∇𝑇_𝑚+1𝑢_𝑛𝑁′+𝐶_{|𝑢𝑛|≥𝑚}𝑓_𝑛1−𝜓_𝛿𝑑𝑥+𝜖𝑛,𝑗. (3.36) We have (𝑇_𝑚+1(𝑢_𝑛)) is bounded in 𝑊¹₀𝐿_𝑀(Ω),(𝑎(⋅,𝑇_𝑚+1(𝑢_𝑛),∇𝑇_𝑚+1(𝑢_𝑛))) in (𝐿_𝑀(Ω))^𝑁) and so in (𝐿^𝑁′(Ω))^𝑁 since 𝑀(𝑡)≪𝑡^𝑁.

Since we have lim_{𝛿→+∞Ω}∇𝜓_𝛿^𝑁𝑑𝑥=0,lim_{𝛿→+∞Ω}1−𝜓_𝛿𝑑𝛿_𝑥0=0,lim_𝑠→+∞Ω−Ω^𝑠=0,lim_𝑚→+∞|{|𝑢|≥𝑚}|=0, (3.37) we easily deduce (i).

For (ii) we proceed as in (i) by using the fact that ∫_{|𝑢𝑛|≥𝑚}𝑓_𝑛𝜓_𝛿𝑑𝑥=𝜖(𝑛,𝛿,𝑚).

Then we conclude that lim_{𝑛→+∞Ω𝑟}𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛−𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘(𝑢)∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘(𝑢)𝑑𝑥=0. (3.38)

Step 2. In this step we prove that ∇𝑢_𝑛→∇𝑢 a.e. in Ω.
Since (𝑇_𝑘(𝑢_𝑛)) is bounded in 𝑊¹₀𝐿_𝑀(Ω), then (|∇𝑇_𝑘(𝑢_𝑛)|) is finite a.e. in Ω.
Hence there exists a measurable set 𝐸 such that |𝐸|=0 and |∇𝑇_𝑘(𝑢_𝑛)|<+∞ in Ω−𝐸.
Let 𝜂(𝑥)=lim_𝑛(∇𝑇_𝑘(𝑢_𝑛)), then by using Fatou lemma, we get _Ω𝜂(𝑥)𝑑𝑥≤lim_Ω∇𝑇_𝑘𝑢_𝑛𝑑𝑥≤𝐶. (3.39) So, |𝜂(𝑥)|<+∞ a.e. in Ω and ∇𝑇_𝑘(𝑢_𝑛)→𝜂 a.e. in Ω, and for a subsequence still denoted by 𝑇_𝑘(𝑢_𝑛), we obtain from (3.38) that 0=lim_𝑛→∞𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘𝑢_𝑛−𝑎⋅,𝑇_𝑘𝑢_𝑛,∇𝑇_𝑘(𝑢)∇𝑇_𝑘𝑢_𝑛−∇𝑇_𝑘(𝑢)=𝑎inΩ⋅,𝑢,𝜂−𝑎⋅,𝑢,∇𝑇_𝑘(𝑢)𝜂−∇𝑇_𝑘(𝑢). (3.40) Combining with (1.3), we get ∇𝑇_𝑘(𝑢)=𝜂a.e.inΩ. (3.41) Therefore, we have ∇𝑇_𝑘𝑢_𝑛→∇𝑇_𝑘(𝑢)a.e.inΩ. (3.42)

Lemma 3.12. For all 𝑘>0, ∇𝑇_𝑘𝑢_𝑛→∇𝑇_𝑘(𝑢)forthemodularconvergencein𝐿_𝑀(Ω)^𝑁. (3.43)

For the proof we can adopt the same way as in [13].

3.2.4. The Convergence of the Problems 𝑃_𝑓𝑛 and the Completion of the Proof of Theorem 3.6

In one hand by using as test function 𝜑=∫^𝑢𝑛₀𝑔(𝑠)𝜒_{𝑠>ℎ}exp(∫^𝑢𝑛₀𝑔(𝑠)𝑑𝑠) in 𝑃_𝑓𝑛, we can deduce as in [5] that 𝑔(𝑢_𝑛)𝑀(|∇𝑢_𝑛|) converge strongly in 𝐿¹(Ω) to 𝑔(𝑢)𝑀(|∇𝑢|).

In the other hand, by Lemma 3.3, we deduce that (𝑢_𝑛) is bounded in 𝑊¹₀𝐿_𝐷(Ω) for every 𝐷∈𝐾. Then (𝑎(⋅,𝑢_𝑛,∇𝑢_𝑛)) is bounded in 𝐿_𝐻(Ω) and the passage to the limit is an easy task.

Corollary 3.13. Under the hypotheses (1.2)–(1.7), 𝜇 is a convex combination of Dirac measures (𝜇=∑^𝐽_𝑖=1𝛼_𝑗𝛿_𝑥𝑗(𝑥_𝑗∈Ω) and ∑^𝐽_𝑖=1𝛼_𝑗=1) and the problem 𝑃_𝜇 admits at least one weak solution.

The proof is a simple adaptation of the one of Theorem 3.6 by taking in 𝑧^𝑗,𝛿_𝑛,𝑚,∑^𝐽_𝑖=1𝛼_𝑗(1−𝜓_𝛿𝑥𝑗) in the place of (1−𝜓_𝛿𝑥𝑗), where 𝜓_𝛿𝑥𝑗 is the cut functions corresponding to the measure 𝛿_𝑥𝑗.

Remark 3.14. The technique used in the proof of Theorem 3.6 allows us to prove that the problem 𝐴(𝑢)=𝑔(𝑢)𝑎(⋅,𝑢,∇𝑢)∇𝑢+𝛿_𝑥0inΩ,𝑢=0on𝜕Ω (3.44) has a weak solution.

3.3. General Case

Before giving the general case of Theorem 3.6, let's recall that the set of finite convex combination of Dirac measures is dense in the set of measures probability.

In the following theorem we denote by 𝜇_𝑛 the sequence defined by 𝜇_𝑛=∑^𝑛_𝑖=1𝛼_𝑖𝛿_𝑥𝑖 for some 𝑥_𝑖∈Ω, and 𝛼_𝑖∈𝐑⁺ with ∑^𝑛_𝑖=1𝛼_𝑖=1.

Theorem 3.15. Let (𝜇_𝑛) be a sequence of Radon measures convergent to a Radon measure 𝜇, and let (𝑢_𝑛), a sequence of weak solutions of (𝑃_𝜇𝑛). Then there exists 𝑀>0 such that _Ω𝑀∇𝑇_𝑘𝑢_𝑛𝑑𝑥≤𝑀𝑘 (3.45) for every 𝑛 and every 𝑘>0. Moreover, there exists a measurable function 𝑢 such that 𝑇_𝑘(𝑢)∈𝑊¹₀𝐿_𝑀(Ω) and 𝑢 is a weak solution of 𝑃_𝜇.

Proof. A priori estimate: Let 𝐽_𝜂 be a sequence of mollifiers functions. Let 𝑣=𝑇_𝑘(𝑢_𝑛)exp(∫^𝑢𝑛₀𝑔(𝑠)𝑑𝑠)∗𝐽_𝜂 as test function in (𝑃_𝜇𝑛)_Ω𝑎𝑥,𝑢_𝑛,∇𝑢_𝑛∇𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠∗𝐽_𝜂+_Ω𝑎𝑥,𝑢_𝑛,∇𝑢_𝑛∇𝑢_𝑛𝑔𝑢_𝑛𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠∗𝐽_𝜂=_Ω𝑀∇𝑢_𝑛𝑔𝑢_𝑛𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠∗𝐽_𝜂+_Ω𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠∗𝐽_𝜂𝑑𝜇_𝑛, (3.46) then, by using an easy argument, the properties of convolution, and tending 𝜂 to zero, we get _Ω𝑀∇𝑇_𝑘𝑢_𝑛exp^𝑢𝑛₀𝑔(𝑠)𝑑𝑠𝑑𝑥≤𝐶𝑘. (3.47) So (𝑇_𝑘(𝑢_𝑛)) is bounded in 𝑊¹₀𝐿_𝑀(Ω) and as above there exists a measurable function 𝑢 such that 𝑇_𝑘𝑢_𝑛⇀𝑇_𝑘(𝑢),weaklyin𝑊¹₀𝐿_𝑀(Ω),a.e.inΩ,andstronglyin𝐸_𝑀(Ω). (3.48)
Almost everywhere convergence of gradients: Let us consider 𝑧^𝑗,𝛿_𝑛,𝑚∗𝐽_𝜂 as test function in (𝑃_𝜇𝑛), we have 𝐴𝑢_𝑛,𝑧^𝑗,𝛿_𝑛,𝑚∗𝐽_𝜂=_Ω𝑔𝑢_𝑛𝑀∇𝑢_𝑛𝑧^𝑗,𝛿_𝑛,𝑚∗𝐽_𝜂𝑑𝑥+_Ω𝑧^𝑗,𝛿_𝑛,𝑚∗𝐽_𝜂𝑑𝜇_𝑛. (3.49) We prove easily, since 𝑧^𝑗,𝛿_𝑛,𝑚∈𝑊¹₀𝐿_𝑀(Ω)∩𝐿^∞(Ω), that 𝐴𝑢_𝑛,𝑧^𝑗,𝛿_𝑛,𝑚∗𝐽_𝜂=𝐴𝑢_𝑛,𝑧^𝑗,𝛿_𝑛,𝑚+𝜖_Ω𝜂,𝑔𝑢_𝑛𝑀∇𝑢_𝑛𝑧^𝑗,𝛿_𝑛,𝑚∗𝐽_𝜂𝑑𝑥=_Ω𝑔𝑢_𝑛𝑀∇𝑢_𝑛𝑧^𝑗,𝛿_𝑛,𝑚𝑑𝑥+𝜖_Ω𝜂,𝑧^𝑗,𝛿_𝑛,𝑚∗𝐽_𝜂𝑑𝜇_𝑛=^𝑛_𝑖=1𝛼_𝑖𝑧^𝑗,𝛿_𝑛,𝑚𝑥_𝑖+𝜖_𝑛𝑖=1𝜂,𝛼_𝑖𝑧^𝑗,𝛿_𝑛,𝑚𝑥_𝑖=𝜖𝜂,𝑛,𝛿,𝑚, (3.50) Also, ∫_Ω𝜌_𝑚(𝑢_𝑛)(1−𝜓_𝛿)exp(∫^𝑢𝑛₀𝑔(𝑠)𝑑𝑠)∗𝐽_𝜂𝑑𝜇_𝑛=𝜖(𝜂,𝑛,𝛿,𝑚).
By using the above estimation, we obtain a similar equation to (3.20) and we follow the same technique used in the step of almost everywhere convergence of gradients in Theorem 3.6 to prove the existence result.

Corollary 3.16. Under the hypotheses (1.2)–(1.7), 𝜇 is positive Radon measure and the problem 𝑃_𝜇 admits at least one weak solution.

Since Ω is bounded and by using the approximation of 𝜇/𝜇(Ω) which is a measure of probability, the existence of solutions can be obtained as consequence of the last theorem.

Remark 3.17. Let us recall that if we suppose that the 𝑁-functions 𝑀 and 𝑀 satisfy the Δ₂ condition, we can prove with the same way as in Theorem 3.6 that the problem 𝑃_𝜇 has a weak solution for all singular measure 𝜇 in the sense that it is concentrated in some Borel set with zero 𝑀-capacity.

Remark 3.18. One can see that the technique used in this paper can be adopted to prove the existence of solutions of the following problem −div(𝑎(⋅,𝑢,∇𝑢))=𝑔(𝑢)|∇𝑢|^𝑝(𝑥)+𝜇inΩ,𝑢=0on𝜕Ω, (3.51) where 𝑎 satisfies, for almost every 𝑥 in Ω, for every 𝑠 in 𝐑, for every 𝜉 and 𝜉^∗ in 𝐑^𝑁, 𝑎(𝑥,𝑠,𝜉)𝜉≥𝜉^𝑝(𝑥),𝑎(𝑥,𝑠,𝜉)−𝑎𝑥,𝑠,𝜉^∗𝜉−𝜉^∗>0,𝜉≠𝜉^∗,𝑎(𝑥,𝑠,𝜉)≤𝐶𝑘(𝑥)+|𝑠|^{𝑝(𝑥)−1}+𝜉^{𝑝(𝑥)−1}. (3.52) Here 𝑝∶Ω→]1,+∞[ is a measurable function such that 1<essinf_𝑥∈Ω𝑝(𝑥)≤esssup_𝑥∈Ω𝑝(𝑥)<𝑁, (3.53) where 𝑘 is a nonnegative function in 𝐿^𝑝′(⋅)(Ω), 𝑝^′(𝑥)=𝑝(𝑥)/(𝑝(𝑥)−1), and 𝐶 is a positive real.

Acknowledgments

This research is supported by the Deanship of Research, King Khalid University, Project no. KKU-SCI-11-011. The author would like to thank Dr. Meskine for his fruitful discussion.