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

Authors: Mohammed Kbiri Alaoui
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 1. Introduction

We deal with boundary value problems(Pμ)A(u)=g(u)M(|∇u|)+μinΩ,u=0on∂Ω,
where
(1.1)A(u)=-div(a(·,u,∇u)),Ω is a bounded domain of RN, with the segment property. a:Ω×R×RN→RN is a Carathéodory function (i.e., measurable with respect to x in Ω for every (s,ξ) in R×RN, and continuous with respect to (s,ξ) in R×RN for almost every x in Ω) such that there exist two N-functions P≪M and for all ξ,ξ*∈RN, ξ≠ξ*, the following hypotheses are true
(1.2)a(x,s,ξ)ξ≥αM(|ξ|λ),(1.3)[a(x,s,ξ)-a(x,s,ξ*)][ξ-ξ*]>0,(1.4)|a(x,s,ξ)|≤c(x)+k1P¯-1M(k2|s|)+k3M¯-1M(k4|ξ|),
where c(x) belongs to EM¯(Ω), c≥0, ki(i=1,2,3,4) to R+, and α, λ to R*+. 						(1.5)M(t)≪tN,∫·+∞‍M¯(t)t1+N′dt=+∞,(1.6)g:R+→R+anintegrablefunctiononR+,(1.7)μ∈Mb+(Ω).
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(1.8)-Δpu=g(u)|∇u|p+μ.
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
(1.9)-Δpv=λf(x)(1+b(v))p-1,
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 W1LM(Ω) when the operator does not satisfy the classical polynomial growth.


## 2. Preliminaries

LetM: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}.LetP and Q be two N-functions. P≪Q means that P grows essentially less rapidly than Q; that is, for each ε>0,
(2.1)P(t)Q(εt)→0ast→∞.
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.LetΩ be an open subset of RN. The Orlicz class ℒM(Ω) (resp., the Orlicz space LM(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 thatLM(Ω) is a Banach space under the norm ∥u∥M,Ω=inf{λ>0:∫Ω‍M(u(x)/λ)dx≤1} and ℒM(Ω) is a convex subset of LM(Ω). The closure in LM(Ω) of the set of bounded measurable functions with compact support in Ω¯ is denoted by EM(Ω). In general EM(Ω)≠LM(Ω) and the dual of EM(Ω) can be identified with LM¯(Ω) by means of the pairing ∫Ω‍u(x)v(x)dx, and the dual norm on LM¯(Ω) is equivalent to ∥·∥M¯,Ω.We now turn to the Orlicz-Sobolev space.W1LM(Ω) (resp. W1EM(Ω)) is the space of all functions u such that u and its distributional derivatives up to order 1 lie in LM(Ω) (resp. EM(Ω)). This is a Banach space under the norm ∥u∥1,M,Ω=∑|α|≤1‍∥Dαu∥M,Ω. Thus W1LM(Ω) and W1EM(Ω) can be identified with subspaces of the product of N+1 copies of LM(Ω). Denoting this product by ΠLM, we will use the weak topologies σ(ΠLM,ΠEM¯) and σ(ΠLM,ΠLM¯). The space W01EM(Ω) is defined as the (norm) closure of the Schwartz space 𝒟(Ω) in W1EM(Ω) and the space W01LM(Ω) as the σ(ΠLM,ΠEM¯) closure of 𝒟(Ω) in W1LM(Ω). We say that un converges to u for the modular convergence in W1LM(Ω) if for some λ>0, ∫Ω‍M((Dαun-Dαu)/λ)dx→0 for all |α|≤1. This implies convergence for σ(ΠLM,ΠLM¯). 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].Fork>0, we define the truncation at height k,Tk:R→R by: Tk(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.Thep-capacity Cp(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
(3.1)Cp(K,Ω)=inf{∫Ω‍|∇φ|pdx:φ∈D(Ω),φ≥χK},
where χK 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
(3.2)Cp(U,Ω)=sup{Cp(K,Ω),KcompactK⊂Ω}.
Finally, the p-capacity of any subset B⊂Ω is defined by
(3.3)Cp(B,Ω)=inf{Cp(U,Ω),UopenB⊂U}.Definition 3.1.
We say thatu is a weak solution of the problem (Pμ) if
(3.4)uismeasurable,Tk(u)∈W01LM(Ω),∫Ω‍a(·,u,∇u)∇vdx=∫Ω‍g(u)M(|∇u|)vdx+∫Ω‍vdμ∀v∈D(Ω).
We define Mb(Ω) as the space of all Radon measures on Ω with bounded total variation, and Cb(Ω) as the space of all bounded, continuous functions on Ω, so that ∫Ω‍φdμ is defined for φ∈Cb(Ω) and μ∈Mb(Ω).We say that a sequence(μn) of measures in Mb(Ω) converges to a measure μ in Mb(Ω) if
(3.5)limn→+∞∫Ω‍φdμn=∫Ω‍φdμn
for every φ∈Cb(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 Mb(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
(Pf)A(u)=g(u)M(|∇u|)+finΩ,u=0on∂Ω.For the proof see [12].LetM 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)
∫0·‍DoB-1(1/r1-1/N)dr<+∞,B(t)=M(t)/t,(iii)
there exists anN-function H such that HoM¯-1oM≤Dand H¯≤Dnear infinity.Lemma 3.3.
Let(un) be a sequence of solutions of the problem
(3.6)A(un)=Fn,un∈W01LM(Ω),
where (Fn) is a sequence of functions bounded in L1(Ω). Then (un) is bounded in W01LD(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 by𝒳N=:NCN1/N, CN the measure of the unit ball of RN, and μ(θ)=|{|u|>θ}|.
Letφ be a truncation defined by
(3.7)φ(ξ)={00≤ξ≤θ1h(ξ-t)θ<ξ<θ+h1ξ≥θ+h-φ(-ξ)ξ<0
for all θ,h>0.
Usingv=φ(un) as a test function, we obtain after tending h to zero
(3.8)-ddθ∫{|un|>θ}‍M(|∇un|)dx≤C∫{|un|≥θ}‍|Fn|dx.
Following the same way as in [14], we have for D∈K,
(3.9)-ddθ∫{|un|>θ}‍D(|∇un|)dx≤(-μ′(θ))DoB-1((-1𝒳Nμ(θ)1-1/Nddθ∫{|un|>θ}‍M(|∇un|)dx)).
We obtain
(3.10)-ddθ∫{|un|>θ}‍D(|∇un|)dx≤(-μ′(θ))DoB-1(-C𝒳Nμ(θ)1-1/N).
So,
(3.11)∫Ω‍D(|∇un|)dx=∫0+∞‍(-ddθ∫{|un|>θ}‍D(|∇un|)dx)dt≤1C′∫0C′|Ω|‍DoB-1((Cs1-1/N))ds.
Then the sequence (un) is bounded in W01LD(Q).Lemma 3.4.
Letμ be a nonnegative Radon measure which is concentrated in a set E of zero p-capacity. Then there exists a sequence (ψδ) of D(Ω) functions such that
(3.12)limδ→∞∫Ω‍|∇ψδ|pdx=0,0≤ψδ≤1,limδ→∞∫Ω‍(1-ψδ)dμ=0.For the proof see [15].Remark 3.5.
By the above lemma, we have that(ψδ) converge to zero both strongly in W01,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 setK is nonempty.Theorem 3.6.
Letx0∈Ω, under the hypotheses (1.2)–(1.6) and N≥2, there exists at least one positive weak solution of the problem
(Pδx0)A(u)=g(u)M(|∇u|)+δx0inΩ,u=0on∂Ω.Remark 3.7.
(
1
) The condition N≥2 is very important to ensure the existence of cut functions for the measure δx0. The case N=1 is easily treated, since we come back to the variational case.
(1.9) The condition ∫·+∞‍(M¯(t)/t1+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(Pδx0) and improve the one given by Porretta in [5]. If M(t)=tp, one can find a solution u such that ∫Ω‍|∇u|N(p-1)/(N-1)/Logσ(e+|∇u|)<+∞, for some σ>1.Remark 3.9 (see [5]).
The conditiong∈L1 is optimal in the sense introduced by Porretta in [5].
Indeed, ifμ 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 (Pμ) 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 writeC 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
(3.13)limm→∞lims→∞limδ→∞limj→∞limn→∞ϵ(n,j,δ,s,m)=0,
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(Pfn)A(un)=g(un)M(|∇un|)+fninΩ,un=0in∂Ω,
where (fn) is a smooth sequence of functions such that ∥fn∥1≤C and fn⇀δx0 in Mb(Ω).The existence of solutions of the above problemun∈W01LM(Ω) was ensured by Lemma 3.2.


#### 3.2.2. A Priori Estimates

Lemma 3.10.
There exists a subsequence of(un) (also denoted (un)); there exists a measurable function u such that Tk(u)∈W01LM(Ω) and Tk(un)⇀Tk(u) weakly in W01LM(Ω), strongly in EM(Ω), and a.e. in Ω.Proof.
Letv=Tk(un)exp(∫0un‍g(s)ds) as test function in (Pfn). One has
(3.14)∫Ω‍a(x,un,∇un)∇Tk(un)exp(∫0un‍g(s)ds)22222+∫Ω‍a(x,un,∇un)∇ung(un)Tk(un)exp(∫0un‍g(s)ds)2222=∫Ω‍M(|∇un|)g(un)Tk(un)exp(∫0un‍g(s)ds)+∫Ω‍fnTk(un)exp(∫0un‍g(s)ds),
then
(3.15)∫Ω‍M(|∇Tk(un)|)exp(∫0un‍g(s)ds)dx≤∫Ω‍fnTk(un)exp(∫0un‍g(s)ds)≤Ck,
so (Tk(un)) is bounded in W01LM(Ω).
There exist a subsequence also denoted(un) and a measurable function ωk such that
(3.16)Tk(un)⇀ωk,weaklyinW01LM(Ω),a.e.inΩ,andstronglyinEM(Ω).
By an easy argument we can see that, there exists a measurable function u such that
(3.17)Tk(u)=ωk.


#### 3.2.3. Almost Everywhere Convergence of Gradients

Lemma 3.11.
The subsequence(un) obtained in Lemma 3.10 satisfies
(3.18)∇un→∇ua.e.inΩ.Proof.
Let us recall that since theN-capacity CN({x0},Ω)=0, there exists a sequence (ψδ) satisfying the Lemma 3.4 with μ=δx0 and p=N.Step 1.
In this step we will show the following:(i)
limn→+∞∫Ωr‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))(1-ψδ)dx=0,(ii)
limn→+∞∫Ωr‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))ψδdx=0.
Letm>0, k>0 such that m>k. Let ρm be a truncation defined by
(3.19)ρm(s)={1|s|≤m,m+1-|s|m<|s|<m+1,0|s|≥m+1.
Let s>0, Ωs={x∈Ω:|∇Tk(u)|≤s}, Ωjs={x∈Ω:|∇Tk(vj)|≤s}.
We denote, respectively, byχs, χjs the indicator function of Ωs,Ωjs.
Letvj∈D(Ω) such that vj→Tk(u) with the modular convergence in W01LM(Ω) (see [11]).
Letzn,mj,δ=(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)ρm(un), as test function in the approximate problem. Then
(3.20)〈A(un),zn,mj,δ〉=∫Ω‍g(un)M(|∇un|)zn,mj,δdx+∫Ω‍fnzn,mj,δdx:=J1+J2.
We have
(3.21)∫Ω‍a(·,un,∇un)∇zn,mj,δdx=∫Ω‍a(·,un,∇un)ρm(un)(∇Tk(un)-∇Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx-∫Ω‍a(·,un,∇un)ρm(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)∇ψδdx+∫Ω‍a(·,un,∇un)∇unρm′(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,un,∇un)∇unρm(un)(Tk(un)-Tk(vj))g(un)exp(∫0un‍g(s)ds)(1-ψδ)dx=∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(vj)χjs))(∇Tk(un)-∇Tk(vj)χjs)ρm(un)×exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,Tk(un),∇Tk(vj)χjs)(∇Tk(un)-∇Tk(vj)χjs)ρm(un)×exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(un)ρm(un)exp(∫0un‍g(s)ds)(1-ψδ)dx-∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(vj)ρm(un)exp(∫0un‍g(s)ds)(1-ψδ)dx-∫|un|>k‍a(·,un,∇un)∇Tk(vj)ρm(un)exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,un,∇un)∇unρm′(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,un,∇un)∇unρm(un)g(un)exp(∫0un‍g(s)ds)(1-ψδ)dx=I1+I2+I3+I4+I5+I6+I7.
Then,
(3.22)〈A(un),zn,mj,δ〉=∑i=17‍Ii=J1+J2.AboutI2: it is obvious since
(3.23)a(·,Tk(un),∇Tk(vj)χjs)→a(·,Tk(u),∇Tk(vj)χjs)strongly,
that is
(3.24)∫Ω‍a(·,Tk(un),∇Tk(vj)χjs)(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx=ϵ(n,j).AboutI4: since (Tk(un)) is bounded in W01LM(Ω) and ∇Tk(vj)χΩ-Ωjs∈(EM(Ω))N, there exists a measurable function hk such that (up to a subsequence also denoted Tk(un))
(3.25)a(·,Tk(un),∇Tk(un))⇀hkweaklyin(LM¯(Ω))N.
Then -∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(vj)(1-ψδ)dx=-∫Ω-Ωjs‍hk∇Tk(vj)(1-ψδ)dx+ϵ(n).Sincevj→Tk(u) in modular convergence, then
(3.26)-∫Ω-Ωjs‍hk∇Tk(vj)(1-ψδ)dx=-∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+ϵ(j).
So,
(3.27)-∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(vj)(1-ψδ)dx=-∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+ϵ(n,j).AboutI5: since ρ(un)=0 on the set {|un|≥m+1}, then
(3.28)-∫|un|>k‍a(·,un,∇un)∇Tk(vj)ρm(un)(1-ψδ)dx2221=-∫|un|>k‍a(·,Tm+1(un),∇Tm+1(un))∇Tk(vj)ρm(un)(1-ψδ)dx
and as for I4, one has
(3.29)-∫|un|>k‍a(·,un,∇un)∇Tk(vj)ρm(un)(1-ψδ)dx2221=-∫|u|>k‍hm+1∇Tk(u)ρm(u)(1-ψδ)dx+ϵ(n,j)=ϵ(n,j)
since ∇Tk(u)=0 when |u|≥k.About
I
6: let v=ρm(un)(1-ψδ)exp(∫0un‍g(s)ds) as test function, in one hand we obtain
(3.30)∫{m≤un≤m+1}‍a(·,un,∇un)∇unexp(∫0un‍g(s)ds)(1-ψδ)dx22221+∫Ω‍a(·,un,∇un)∇ung(un)exp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx22221-∫Ω‍a(·,un,∇un)∇ψδexp(∫0un‍g(s)ds)ρm(un)dx22=∫Ω‍g(un)M(|∇un|)exp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx22221+∫Ω‍fnexp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx.
So,
(3.31)∫{m≤un≤m+1}‍a(·,un,∇un)∇unexp(∫0un‍g(s)ds)(1-ψδ)dx222≤∫Ω‍a(·,un,∇un)∇ψδexp(∫0un‍g(s)ds)ρm(un)dx22222+∫Ω‍fnexp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx222≤C∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx.
On the other hand
(3.32)I6=∫Ω‍a(·,un,∇un)∇unρm′(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx=∫{m≤un≤m+1}‍a(·,un,∇un)∇un(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx≤2k[∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx]=ϵ(n,δ,m).
Let us come back to our main estimation (i). We have
(3.33)∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)χs))(∇Tk(un)-∇Tk(u)χs)ρm(un)(1-ψδ)dx=∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(vj)χjs))×(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx+∫Ω‍a(·,Tk(un),∇Tk(un))(∇Tk(vj)χjs-∇Tk(u)χs)ρm(un)(1-ψδ)dx(=ϵ(n,j))-∫Ω‍a(·,Tk(un),∇Tk(u)χs)(∇Tk(un)-∇Tk(u)χs)ρm(un)(1-ψδ)dx(=ϵ(n,j))+∫Ω‍a(·,Tk(un),∇Tk(vj)χjs)(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx(=ϵ(n,j)).
Then we deduce that
(3.34)∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)χs))(∇Tk(un)-∇Tk(u)χs)ρm(un)(1-ψδ)dx=∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(vj)χjs))×(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx+ϵ(n,j)≤ϵ(n,j)+C∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+C∫|un|≥m‍fn(1-ψδ)dx.
Since we have ∑i=17‍Ii=J1+J2, I3≥0 and using (1.2), we deduce that
(3.35)I1≤J2-I2-I3-I4-I5-I6≤∫Ω‍fnzn,mj,δ+C∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+Ck∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx.
For r≤s, one has
(3.36)0≤I1(r)≤I1(s)≤C∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+Ck∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx+ϵ(n,j).
We have (Tm+1(un)) is bounded in W01LM(Ω),(a(·,Tm+1(un),∇Tm+1(un))) in (LM¯(Ω))N) and so in (LN′(Ω))N since M(t)≪tN.Since we have(3.37)limδ→+∞∫Ω‍|∇ψδ|Ndx=0,limδ→+∞∫Ω‍(1-ψδ)dδx0=0,lims→+∞|Ω-Ωs|=0,limm→+∞|{|u|≥m}|=0,
we easily deduce (i).For (ii) we proceed as in (i) by using the fact that∫|un|≥m‍fnψδdx=ϵ(n,δ,m).Then we conclude that(3.38)limn→+∞∫Ωr‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))dx=0.Step 2.
In this step we prove that∇un→∇u a.e. in Ω.
Since(Tk(un)) is bounded in W01LM(Ω), then (|∇Tk(un)|) is finite a.e. in Ω.
Hence there exists a measurable setE such that |E|=0 and |∇Tk(un)|<+∞ in Ω-E.
Letη(x)=limn(∇Tk(un)), then by using Fatou lemma, we get
(3.39)∫Ω‍|η(x)|dx≤lim_∫Ω‍|∇Tk(un)|dx≤C.
So, |η(x)|<+∞ a.e. in Ω and ∇Tk(un)→η a.e. in Ω, and for a subsequence still denoted by Tk(un), we obtain from (3.38) that
(3.40)0=limn→∞(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))inΩ=(a(·,u,η)-a(·,u,∇Tk(u)))(η-∇Tk(u)).
Combining with (1.3), we get
(3.41)∇Tk(u)=ηa.e.inΩ.
Therefore, we have
(3.42)∇Tk(un)→∇Tk(u)a.e.inΩ.Lemma 3.12.
For allk>0,
(3.43)∇Tk(un)→∇Tk(u)forthemodularconvergencein(LM(Ω))N.For the proof we can adopt the same way as in [13].


#### 3.2.4. The Convergence of the Problems(Pfn) and the Completion of the Proof of Theorem 3.6

In one hand by using as test functionφ=∫0un‍g(s)χ{s>h}exp(∫0un‍g(s)ds) in (Pfn), we can deduce as in [5] that g(un)M(|∇un|) converge strongly in L1(Ω) to g(u)M(|∇u|).In the other hand, by Lemma3.3, we deduce that (un) is bounded in W01LD(Ω) for every D∈K. Then (a(·,un,∇un)) is bounded in LH(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=∑i=1J‍αjδxj(xj∈Ω) and ∑i=1J‍αj=1) and the problem (Pμ) admits at least one weak solution.The proof is a simple adaptation of the one of Theorem3.6 by taking in zn,mj,δ,∑i=1J‍αj(1-ψδxj) in the place of (1-ψδxj), where ψδxj is the cut functions corresponding to the measure δxj.Remark 3.14.
The technique used in the proof of Theorem3.6 allows us to prove that the problem
(3.44)A(u)=g(u)a(·,u,∇u)∇u+δx0inΩ,u=0on∂Ω
has a weak solution.


### 3.3. General Case

Before giving the general case of Theorem3.6, 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 byμn the sequence defined by μn=∑i=1n‍αiδxi for some xi∈Ω, and αi∈R+ with ∑i=1n‍αi=1.Theorem 3.15.
Let(μn) be a sequence of Radon measures convergent to a Radon measure μ, and let (un), a sequence of weak solutions of (Pμn). Then there exists M>0 such that
(3.45)∫Ω‍M(|∇Tk(un)|)dx≤Mk
for every n and every k>0. Moreover, there exists a measurable function u such that Tk(u)∈W01LM(Ω) and u is a weak solution of (Pμ).Proof.
A priori estimate: Let Jη be a sequence of mollifiers functions. Let v=Tk(un)exp(∫0un‍g(s)ds)*Jη as test function in (Pμn)(3.46)∫Ω‍a(x,un,∇un)[∇Tk(un)exp(∫0un‍g(s)ds)*Jη]+∫Ω‍a(x,un,∇un)[∇ung(un)Tk(un)exp(∫0un‍g(s)ds)*Jη]=∫Ω‍M(|∇un|)g(un)Tk(un)exp(∫0un‍g(s)ds)*Jη+∫Ω‍Tk(un)exp(∫0un‍g(s)ds)*Jηdμn,
then, by using an easy argument, the properties of convolution, and tending η to zero, we get
(3.47)∫Ω‍M(|∇Tk(un)|)exp(∫0un‍g(s)ds)dx≤Ck.
So (Tk(un)) is bounded in W01LM(Ω) and as above there exists a measurable function u such that
(3.48)Tk(un)⇀Tk(u),weaklyinW01LM(Ω),a.e.inΩ,andstronglyinEM(Ω).
Almost everywhere convergence of gradients: Let us consider zn,mj,δ*Jη as test function in (Pμn), we have
(3.49)〈A(un),zn,mj,δ*Jη〉=∫Ω‍g(un)M(|∇un|)zn,mj,δ*Jηdx+∫Ω‍zn,mj,δ*Jηdμn.
We prove easily, since zn,mj,δ∈W01LM(Ω)∩L∞(Ω), that
(3.50)〈A(un),zn,mj,δ*Jη〉=〈A(un),zn,mj,δ〉+ϵ(η),∫Ω‍g(un)M(|∇un|)zn,mj,δ*Jηdx=∫Ω‍g(un)M(|∇un|)zn,mj,δdx+ϵ(η),∫Ω‍zn,mj,δ*Jηdμn=∑i=1n‍αizn,mj,δ(xi)+ϵ(η),∑i=1n‍αizn,mj,δ(xi)=ϵ(η,n,δ,m),
Also, ∫Ω‍ρm(un)(1-ψδ)exp(∫0un‍g(s)ds)*Jηdμn=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 (Pμ) admits at least one weak solution.SinceΩ 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 theN-functions M and M¯ satisfy the Δ2 condition, we can prove with the same way as in Theorem 3.6 that the problem (Pμ) 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(3.51)-div(a(·,u,∇u))=g(u)|∇u|p(x)+μinΩ,u=0on∂Ω,
where a satisfies, for almost every x in Ω, for every s in R, for every ξ and ξ* in RN,
(3.52)a(x,s,ξ)ξ≥|ξ|p(x),[a(x,s,ξ)-a(x,s,ξ*)][ξ-ξ*]>0,ξ≠ξ*,|a(x,s,ξ)|≤C(k(x)+|s|p(x)-1+|ξ|p(x)-1).
Here p:Ω→]1,+∞[ is a measurable function such that
(3.53)1<essinfx∈Ωp(x)≤esssupx∈Ωp(x)<N,
where k is a nonnegative function in Lp′(·)(Ω), p′(x)=p(x)/(p(x)-1), and C is a positive real.


## 3.1. Useful Results

First, we give the following definitions and results which will be used in our main result.Thep-capacity Cp(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
(3.1)Cp(K,Ω)=inf{∫Ω‍|∇φ|pdx:φ∈D(Ω),φ≥χK},
where χK 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
(3.2)Cp(U,Ω)=sup{Cp(K,Ω),KcompactK⊂Ω}.
Finally, the p-capacity of any subset B⊂Ω is defined by
(3.3)Cp(B,Ω)=inf{Cp(U,Ω),UopenB⊂U}.Definition 3.1.
We say thatu is a weak solution of the problem (Pμ) if
(3.4)uismeasurable,Tk(u)∈W01LM(Ω),∫Ω‍a(·,u,∇u)∇vdx=∫Ω‍g(u)M(|∇u|)vdx+∫Ω‍vdμ∀v∈D(Ω).
We define Mb(Ω) as the space of all Radon measures on Ω with bounded total variation, and Cb(Ω) as the space of all bounded, continuous functions on Ω, so that ∫Ω‍φdμ is defined for φ∈Cb(Ω) and μ∈Mb(Ω).We say that a sequence(μn) of measures in Mb(Ω) converges to a measure μ in Mb(Ω) if
(3.5)limn→+∞∫Ω‍φdμn=∫Ω‍φdμn
for every φ∈Cb(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 Mb(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
(Pf)A(u)=g(u)M(|∇u|)+finΩ,u=0on∂Ω.For the proof see [12].LetM 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)
∫0·‍DoB-1(1/r1-1/N)dr<+∞,B(t)=M(t)/t,(iii)
there exists anN-function H such that HoM¯-1oM≤Dand H¯≤Dnear infinity.Lemma 3.3.
Let(un) be a sequence of solutions of the problem
(3.6)A(un)=Fn,un∈W01LM(Ω),
where (Fn) is a sequence of functions bounded in L1(Ω). Then (un) is bounded in W01LD(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 by𝒳N=:NCN1/N, CN the measure of the unit ball of RN, and μ(θ)=|{|u|>θ}|.
Letφ be a truncation defined by
(3.7)φ(ξ)={00≤ξ≤θ1h(ξ-t)θ<ξ<θ+h1ξ≥θ+h-φ(-ξ)ξ<0
for all θ,h>0.
Usingv=φ(un) as a test function, we obtain after tending h to zero
(3.8)-ddθ∫{|un|>θ}‍M(|∇un|)dx≤C∫{|un|≥θ}‍|Fn|dx.
Following the same way as in [14], we have for D∈K,
(3.9)-ddθ∫{|un|>θ}‍D(|∇un|)dx≤(-μ′(θ))DoB-1((-1𝒳Nμ(θ)1-1/Nddθ∫{|un|>θ}‍M(|∇un|)dx)).
We obtain
(3.10)-ddθ∫{|un|>θ}‍D(|∇un|)dx≤(-μ′(θ))DoB-1(-C𝒳Nμ(θ)1-1/N).
So,
(3.11)∫Ω‍D(|∇un|)dx=∫0+∞‍(-ddθ∫{|un|>θ}‍D(|∇un|)dx)dt≤1C′∫0C′|Ω|‍DoB-1((Cs1-1/N))ds.
Then the sequence (un) is bounded in W01LD(Q).Lemma 3.4.
Letμ be a nonnegative Radon measure which is concentrated in a set E of zero p-capacity. Then there exists a sequence (ψδ) of D(Ω) functions such that
(3.12)limδ→∞∫Ω‍|∇ψδ|pdx=0,0≤ψδ≤1,limδ→∞∫Ω‍(1-ψδ)dμ=0.For the proof see [15].Remark 3.5.
By the above lemma, we have that(ψδ) converge to zero both strongly in W01,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 setK is nonempty.Theorem 3.6.
Letx0∈Ω, under the hypotheses (1.2)–(1.6) and N≥2, there exists at least one positive weak solution of the problem
(Pδx0)A(u)=g(u)M(|∇u|)+δx0inΩ,u=0on∂Ω.Remark 3.7.
(
1
) The condition N≥2 is very important to ensure the existence of cut functions for the measure δx0. The case N=1 is easily treated, since we come back to the variational case.
(1.9) The condition ∫·+∞‍(M¯(t)/t1+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(Pδx0) and improve the one given by Porretta in [5]. If M(t)=tp, one can find a solution u such that ∫Ω‍|∇u|N(p-1)/(N-1)/Logσ(e+|∇u|)<+∞, for some σ>1.Remark 3.9 (see [5]).
The conditiong∈L1 is optimal in the sense introduced by Porretta in [5].
Indeed, ifμ 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 (Pμ) 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 writeC 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
(3.13)limm→∞lims→∞limδ→∞limj→∞limn→∞ϵ(n,j,δ,s,m)=0,
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(Pfn)A(un)=g(un)M(|∇un|)+fninΩ,un=0in∂Ω,
where (fn) is a smooth sequence of functions such that ∥fn∥1≤C and fn⇀δx0 in Mb(Ω).The existence of solutions of the above problemun∈W01LM(Ω) was ensured by Lemma 3.2.


### 3.2.2. A Priori Estimates

Lemma 3.10.
There exists a subsequence of(un) (also denoted (un)); there exists a measurable function u such that Tk(u)∈W01LM(Ω) and Tk(un)⇀Tk(u) weakly in W01LM(Ω), strongly in EM(Ω), and a.e. in Ω.Proof.
Letv=Tk(un)exp(∫0un‍g(s)ds) as test function in (Pfn). One has
(3.14)∫Ω‍a(x,un,∇un)∇Tk(un)exp(∫0un‍g(s)ds)22222+∫Ω‍a(x,un,∇un)∇ung(un)Tk(un)exp(∫0un‍g(s)ds)2222=∫Ω‍M(|∇un|)g(un)Tk(un)exp(∫0un‍g(s)ds)+∫Ω‍fnTk(un)exp(∫0un‍g(s)ds),
then
(3.15)∫Ω‍M(|∇Tk(un)|)exp(∫0un‍g(s)ds)dx≤∫Ω‍fnTk(un)exp(∫0un‍g(s)ds)≤Ck,
so (Tk(un)) is bounded in W01LM(Ω).
There exist a subsequence also denoted(un) and a measurable function ωk such that
(3.16)Tk(un)⇀ωk,weaklyinW01LM(Ω),a.e.inΩ,andstronglyinEM(Ω).
By an easy argument we can see that, there exists a measurable function u such that
(3.17)Tk(u)=ωk.


### 3.2.3. Almost Everywhere Convergence of Gradients

Lemma 3.11.
The subsequence(un) obtained in Lemma 3.10 satisfies
(3.18)∇un→∇ua.e.inΩ.Proof.
Let us recall that since theN-capacity CN({x0},Ω)=0, there exists a sequence (ψδ) satisfying the Lemma 3.4 with μ=δx0 and p=N.Step 1.
In this step we will show the following:(i)
limn→+∞∫Ωr‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))(1-ψδ)dx=0,(ii)
limn→+∞∫Ωr‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))ψδdx=0.
Letm>0, k>0 such that m>k. Let ρm be a truncation defined by
(3.19)ρm(s)={1|s|≤m,m+1-|s|m<|s|<m+1,0|s|≥m+1.
Let s>0, Ωs={x∈Ω:|∇Tk(u)|≤s}, Ωjs={x∈Ω:|∇Tk(vj)|≤s}.
We denote, respectively, byχs, χjs the indicator function of Ωs,Ωjs.
Letvj∈D(Ω) such that vj→Tk(u) with the modular convergence in W01LM(Ω) (see [11]).
Letzn,mj,δ=(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)ρm(un), as test function in the approximate problem. Then
(3.20)〈A(un),zn,mj,δ〉=∫Ω‍g(un)M(|∇un|)zn,mj,δdx+∫Ω‍fnzn,mj,δdx:=J1+J2.
We have
(3.21)∫Ω‍a(·,un,∇un)∇zn,mj,δdx=∫Ω‍a(·,un,∇un)ρm(un)(∇Tk(un)-∇Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx-∫Ω‍a(·,un,∇un)ρm(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)∇ψδdx+∫Ω‍a(·,un,∇un)∇unρm′(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,un,∇un)∇unρm(un)(Tk(un)-Tk(vj))g(un)exp(∫0un‍g(s)ds)(1-ψδ)dx=∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(vj)χjs))(∇Tk(un)-∇Tk(vj)χjs)ρm(un)×exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,Tk(un),∇Tk(vj)χjs)(∇Tk(un)-∇Tk(vj)χjs)ρm(un)×exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(un)ρm(un)exp(∫0un‍g(s)ds)(1-ψδ)dx-∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(vj)ρm(un)exp(∫0un‍g(s)ds)(1-ψδ)dx-∫|un|>k‍a(·,un,∇un)∇Tk(vj)ρm(un)exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,un,∇un)∇unρm′(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,un,∇un)∇unρm(un)g(un)exp(∫0un‍g(s)ds)(1-ψδ)dx=I1+I2+I3+I4+I5+I6+I7.
Then,
(3.22)〈A(un),zn,mj,δ〉=∑i=17‍Ii=J1+J2.AboutI2: it is obvious since
(3.23)a(·,Tk(un),∇Tk(vj)χjs)→a(·,Tk(u),∇Tk(vj)χjs)strongly,
that is
(3.24)∫Ω‍a(·,Tk(un),∇Tk(vj)χjs)(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx=ϵ(n,j).AboutI4: since (Tk(un)) is bounded in W01LM(Ω) and ∇Tk(vj)χΩ-Ωjs∈(EM(Ω))N, there exists a measurable function hk such that (up to a subsequence also denoted Tk(un))
(3.25)a(·,Tk(un),∇Tk(un))⇀hkweaklyin(LM¯(Ω))N.
Then -∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(vj)(1-ψδ)dx=-∫Ω-Ωjs‍hk∇Tk(vj)(1-ψδ)dx+ϵ(n).Sincevj→Tk(u) in modular convergence, then
(3.26)-∫Ω-Ωjs‍hk∇Tk(vj)(1-ψδ)dx=-∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+ϵ(j).
So,
(3.27)-∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(vj)(1-ψδ)dx=-∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+ϵ(n,j).AboutI5: since ρ(un)=0 on the set {|un|≥m+1}, then
(3.28)-∫|un|>k‍a(·,un,∇un)∇Tk(vj)ρm(un)(1-ψδ)dx2221=-∫|un|>k‍a(·,Tm+1(un),∇Tm+1(un))∇Tk(vj)ρm(un)(1-ψδ)dx
and as for I4, one has
(3.29)-∫|un|>k‍a(·,un,∇un)∇Tk(vj)ρm(un)(1-ψδ)dx2221=-∫|u|>k‍hm+1∇Tk(u)ρm(u)(1-ψδ)dx+ϵ(n,j)=ϵ(n,j)
since ∇Tk(u)=0 when |u|≥k.About
I
6: let v=ρm(un)(1-ψδ)exp(∫0un‍g(s)ds) as test function, in one hand we obtain
(3.30)∫{m≤un≤m+1}‍a(·,un,∇un)∇unexp(∫0un‍g(s)ds)(1-ψδ)dx22221+∫Ω‍a(·,un,∇un)∇ung(un)exp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx22221-∫Ω‍a(·,un,∇un)∇ψδexp(∫0un‍g(s)ds)ρm(un)dx22=∫Ω‍g(un)M(|∇un|)exp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx22221+∫Ω‍fnexp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx.
So,
(3.31)∫{m≤un≤m+1}‍a(·,un,∇un)∇unexp(∫0un‍g(s)ds)(1-ψδ)dx222≤∫Ω‍a(·,un,∇un)∇ψδexp(∫0un‍g(s)ds)ρm(un)dx22222+∫Ω‍fnexp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx222≤C∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx.
On the other hand
(3.32)I6=∫Ω‍a(·,un,∇un)∇unρm′(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx=∫{m≤un≤m+1}‍a(·,un,∇un)∇un(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx≤2k[∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx]=ϵ(n,δ,m).
Let us come back to our main estimation (i). We have
(3.33)∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)χs))(∇Tk(un)-∇Tk(u)χs)ρm(un)(1-ψδ)dx=∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(vj)χjs))×(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx+∫Ω‍a(·,Tk(un),∇Tk(un))(∇Tk(vj)χjs-∇Tk(u)χs)ρm(un)(1-ψδ)dx(=ϵ(n,j))-∫Ω‍a(·,Tk(un),∇Tk(u)χs)(∇Tk(un)-∇Tk(u)χs)ρm(un)(1-ψδ)dx(=ϵ(n,j))+∫Ω‍a(·,Tk(un),∇Tk(vj)χjs)(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx(=ϵ(n,j)).
Then we deduce that
(3.34)∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)χs))(∇Tk(un)-∇Tk(u)χs)ρm(un)(1-ψδ)dx=∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(vj)χjs))×(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx+ϵ(n,j)≤ϵ(n,j)+C∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+C∫|un|≥m‍fn(1-ψδ)dx.
Since we have ∑i=17‍Ii=J1+J2, I3≥0 and using (1.2), we deduce that
(3.35)I1≤J2-I2-I3-I4-I5-I6≤∫Ω‍fnzn,mj,δ+C∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+Ck∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx.
For r≤s, one has
(3.36)0≤I1(r)≤I1(s)≤C∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+Ck∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx+ϵ(n,j).
We have (Tm+1(un)) is bounded in W01LM(Ω),(a(·,Tm+1(un),∇Tm+1(un))) in (LM¯(Ω))N) and so in (LN′(Ω))N since M(t)≪tN.Since we have(3.37)limδ→+∞∫Ω‍|∇ψδ|Ndx=0,limδ→+∞∫Ω‍(1-ψδ)dδx0=0,lims→+∞|Ω-Ωs|=0,limm→+∞|{|u|≥m}|=0,
we easily deduce (i).For (ii) we proceed as in (i) by using the fact that∫|un|≥m‍fnψδdx=ϵ(n,δ,m).Then we conclude that(3.38)limn→+∞∫Ωr‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))dx=0.Step 2.
In this step we prove that∇un→∇u a.e. in Ω.
Since(Tk(un)) is bounded in W01LM(Ω), then (|∇Tk(un)|) is finite a.e. in Ω.
Hence there exists a measurable setE such that |E|=0 and |∇Tk(un)|<+∞ in Ω-E.
Letη(x)=limn(∇Tk(un)), then by using Fatou lemma, we get
(3.39)∫Ω‍|η(x)|dx≤lim_∫Ω‍|∇Tk(un)|dx≤C.
So, |η(x)|<+∞ a.e. in Ω and ∇Tk(un)→η a.e. in Ω, and for a subsequence still denoted by Tk(un), we obtain from (3.38) that
(3.40)0=limn→∞(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))inΩ=(a(·,u,η)-a(·,u,∇Tk(u)))(η-∇Tk(u)).
Combining with (1.3), we get
(3.41)∇Tk(u)=ηa.e.inΩ.
Therefore, we have
(3.42)∇Tk(un)→∇Tk(u)a.e.inΩ.Lemma 3.12.
For allk>0,
(3.43)∇Tk(un)→∇Tk(u)forthemodularconvergencein(LM(Ω))N.For the proof we can adopt the same way as in [13].


### 3.2.4. The Convergence of the Problems(Pfn) and the Completion of the Proof of Theorem 3.6

In one hand by using as test functionφ=∫0un‍g(s)χ{s>h}exp(∫0un‍g(s)ds) in (Pfn), we can deduce as in [5] that g(un)M(|∇un|) converge strongly in L1(Ω) to g(u)M(|∇u|).In the other hand, by Lemma3.3, we deduce that (un) is bounded in W01LD(Ω) for every D∈K. Then (a(·,un,∇un)) is bounded in LH(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=∑i=1J‍αjδxj(xj∈Ω) and ∑i=1J‍αj=1) and the problem (Pμ) admits at least one weak solution.The proof is a simple adaptation of the one of Theorem3.6 by taking in zn,mj,δ,∑i=1J‍αj(1-ψδxj) in the place of (1-ψδxj), where ψδxj is the cut functions corresponding to the measure δxj.Remark 3.14.
The technique used in the proof of Theorem3.6 allows us to prove that the problem
(3.44)A(u)=g(u)a(·,u,∇u)∇u+δx0inΩ,u=0on∂Ω
has a weak solution.


## 3.2.1. A Sequence of Approximating Problems

Consider the approximate problem(Pfn)A(un)=g(un)M(|∇un|)+fninΩ,un=0in∂Ω,
where (fn) is a smooth sequence of functions such that ∥fn∥1≤C and fn⇀δx0 in Mb(Ω).The existence of solutions of the above problemun∈W01LM(Ω) was ensured by Lemma 3.2.


## 3.2.2. A Priori Estimates

Lemma 3.10.
There exists a subsequence of(un) (also denoted (un)); there exists a measurable function u such that Tk(u)∈W01LM(Ω) and Tk(un)⇀Tk(u) weakly in W01LM(Ω), strongly in EM(Ω), and a.e. in Ω.Proof.
Letv=Tk(un)exp(∫0un‍g(s)ds) as test function in (Pfn). One has
(3.14)∫Ω‍a(x,un,∇un)∇Tk(un)exp(∫0un‍g(s)ds)22222+∫Ω‍a(x,un,∇un)∇ung(un)Tk(un)exp(∫0un‍g(s)ds)2222=∫Ω‍M(|∇un|)g(un)Tk(un)exp(∫0un‍g(s)ds)+∫Ω‍fnTk(un)exp(∫0un‍g(s)ds),
then
(3.15)∫Ω‍M(|∇Tk(un)|)exp(∫0un‍g(s)ds)dx≤∫Ω‍fnTk(un)exp(∫0un‍g(s)ds)≤Ck,
so (Tk(un)) is bounded in W01LM(Ω).
There exist a subsequence also denoted(un) and a measurable function ωk such that
(3.16)Tk(un)⇀ωk,weaklyinW01LM(Ω),a.e.inΩ,andstronglyinEM(Ω).
By an easy argument we can see that, there exists a measurable function u such that
(3.17)Tk(u)=ωk.


## 3.2.3. Almost Everywhere Convergence of Gradients

Lemma 3.11.
The subsequence(un) obtained in Lemma 3.10 satisfies
(3.18)∇un→∇ua.e.inΩ.Proof.
Let us recall that since theN-capacity CN({x0},Ω)=0, there exists a sequence (ψδ) satisfying the Lemma 3.4 with μ=δx0 and p=N.Step 1.
In this step we will show the following:(i)
limn→+∞∫Ωr‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))(1-ψδ)dx=0,(ii)
limn→+∞∫Ωr‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))ψδdx=0.
Letm>0, k>0 such that m>k. Let ρm be a truncation defined by
(3.19)ρm(s)={1|s|≤m,m+1-|s|m<|s|<m+1,0|s|≥m+1.
Let s>0, Ωs={x∈Ω:|∇Tk(u)|≤s}, Ωjs={x∈Ω:|∇Tk(vj)|≤s}.
We denote, respectively, byχs, χjs the indicator function of Ωs,Ωjs.
Letvj∈D(Ω) such that vj→Tk(u) with the modular convergence in W01LM(Ω) (see [11]).
Letzn,mj,δ=(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)ρm(un), as test function in the approximate problem. Then
(3.20)〈A(un),zn,mj,δ〉=∫Ω‍g(un)M(|∇un|)zn,mj,δdx+∫Ω‍fnzn,mj,δdx:=J1+J2.
We have
(3.21)∫Ω‍a(·,un,∇un)∇zn,mj,δdx=∫Ω‍a(·,un,∇un)ρm(un)(∇Tk(un)-∇Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx-∫Ω‍a(·,un,∇un)ρm(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)∇ψδdx+∫Ω‍a(·,un,∇un)∇unρm′(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,un,∇un)∇unρm(un)(Tk(un)-Tk(vj))g(un)exp(∫0un‍g(s)ds)(1-ψδ)dx=∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(vj)χjs))(∇Tk(un)-∇Tk(vj)χjs)ρm(un)×exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,Tk(un),∇Tk(vj)χjs)(∇Tk(un)-∇Tk(vj)χjs)ρm(un)×exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(un)ρm(un)exp(∫0un‍g(s)ds)(1-ψδ)dx-∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(vj)ρm(un)exp(∫0un‍g(s)ds)(1-ψδ)dx-∫|un|>k‍a(·,un,∇un)∇Tk(vj)ρm(un)exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,un,∇un)∇unρm′(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx+∫Ω‍a(·,un,∇un)∇unρm(un)g(un)exp(∫0un‍g(s)ds)(1-ψδ)dx=I1+I2+I3+I4+I5+I6+I7.
Then,
(3.22)〈A(un),zn,mj,δ〉=∑i=17‍Ii=J1+J2.AboutI2: it is obvious since
(3.23)a(·,Tk(un),∇Tk(vj)χjs)→a(·,Tk(u),∇Tk(vj)χjs)strongly,
that is
(3.24)∫Ω‍a(·,Tk(un),∇Tk(vj)χjs)(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx=ϵ(n,j).AboutI4: since (Tk(un)) is bounded in W01LM(Ω) and ∇Tk(vj)χΩ-Ωjs∈(EM(Ω))N, there exists a measurable function hk such that (up to a subsequence also denoted Tk(un))
(3.25)a(·,Tk(un),∇Tk(un))⇀hkweaklyin(LM¯(Ω))N.
Then -∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(vj)(1-ψδ)dx=-∫Ω-Ωjs‍hk∇Tk(vj)(1-ψδ)dx+ϵ(n).Sincevj→Tk(u) in modular convergence, then
(3.26)-∫Ω-Ωjs‍hk∇Tk(vj)(1-ψδ)dx=-∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+ϵ(j).
So,
(3.27)-∫Ω-Ωjs‍a(·,Tk(un),∇Tk(un))∇Tk(vj)(1-ψδ)dx=-∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+ϵ(n,j).AboutI5: since ρ(un)=0 on the set {|un|≥m+1}, then
(3.28)-∫|un|>k‍a(·,un,∇un)∇Tk(vj)ρm(un)(1-ψδ)dx2221=-∫|un|>k‍a(·,Tm+1(un),∇Tm+1(un))∇Tk(vj)ρm(un)(1-ψδ)dx
and as for I4, one has
(3.29)-∫|un|>k‍a(·,un,∇un)∇Tk(vj)ρm(un)(1-ψδ)dx2221=-∫|u|>k‍hm+1∇Tk(u)ρm(u)(1-ψδ)dx+ϵ(n,j)=ϵ(n,j)
since ∇Tk(u)=0 when |u|≥k.About
I
6: let v=ρm(un)(1-ψδ)exp(∫0un‍g(s)ds) as test function, in one hand we obtain
(3.30)∫{m≤un≤m+1}‍a(·,un,∇un)∇unexp(∫0un‍g(s)ds)(1-ψδ)dx22221+∫Ω‍a(·,un,∇un)∇ung(un)exp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx22221-∫Ω‍a(·,un,∇un)∇ψδexp(∫0un‍g(s)ds)ρm(un)dx22=∫Ω‍g(un)M(|∇un|)exp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx22221+∫Ω‍fnexp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx.
So,
(3.31)∫{m≤un≤m+1}‍a(·,un,∇un)∇unexp(∫0un‍g(s)ds)(1-ψδ)dx222≤∫Ω‍a(·,un,∇un)∇ψδexp(∫0un‍g(s)ds)ρm(un)dx22222+∫Ω‍fnexp(∫0un‍g(s)ds)ρm(un)(1-ψδ)dx222≤C∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx.
On the other hand
(3.32)I6=∫Ω‍a(·,un,∇un)∇unρm′(un)(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx=∫{m≤un≤m+1}‍a(·,un,∇un)∇un(Tk(un)-Tk(vj))exp(∫0un‍g(s)ds)(1-ψδ)dx≤2k[∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx]=ϵ(n,δ,m).
Let us come back to our main estimation (i). We have
(3.33)∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)χs))(∇Tk(un)-∇Tk(u)χs)ρm(un)(1-ψδ)dx=∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(vj)χjs))×(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx+∫Ω‍a(·,Tk(un),∇Tk(un))(∇Tk(vj)χjs-∇Tk(u)χs)ρm(un)(1-ψδ)dx(=ϵ(n,j))-∫Ω‍a(·,Tk(un),∇Tk(u)χs)(∇Tk(un)-∇Tk(u)χs)ρm(un)(1-ψδ)dx(=ϵ(n,j))+∫Ω‍a(·,Tk(un),∇Tk(vj)χjs)(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx(=ϵ(n,j)).
Then we deduce that
(3.34)∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)χs))(∇Tk(un)-∇Tk(u)χs)ρm(un)(1-ψδ)dx=∫Ω‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(vj)χjs))×(∇Tk(un)-∇Tk(vj)χjs)ρm(un)(1-ψδ)dx+ϵ(n,j)≤ϵ(n,j)+C∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+C∫|un|≥m‍fn(1-ψδ)dx.
Since we have ∑i=17‍Ii=J1+J2, I3≥0 and using (1.2), we deduce that
(3.35)I1≤J2-I2-I3-I4-I5-I6≤∫Ω‍fnzn,mj,δ+C∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+Ck∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx.
For r≤s, one has
(3.36)0≤I1(r)≤I1(s)≤C∫Ω-Ωs‍hk∇Tk(u)(1-ψδ)dx+Ck∥∇ψδ∥N∥a(·,Tm+1(un),∇Tm+1(un))∥N′+C∫|un|≥m‍fn(1-ψδ)dx+ϵ(n,j).
We have (Tm+1(un)) is bounded in W01LM(Ω),(a(·,Tm+1(un),∇Tm+1(un))) in (LM¯(Ω))N) and so in (LN′(Ω))N since M(t)≪tN.Since we have(3.37)limδ→+∞∫Ω‍|∇ψδ|Ndx=0,limδ→+∞∫Ω‍(1-ψδ)dδx0=0,lims→+∞|Ω-Ωs|=0,limm→+∞|{|u|≥m}|=0,
we easily deduce (i).For (ii) we proceed as in (i) by using the fact that∫|un|≥m‍fnψδdx=ϵ(n,δ,m).Then we conclude that(3.38)limn→+∞∫Ωr‍(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))dx=0.Step 2.
In this step we prove that∇un→∇u a.e. in Ω.
Since(Tk(un)) is bounded in W01LM(Ω), then (|∇Tk(un)|) is finite a.e. in Ω.
Hence there exists a measurable setE such that |E|=0 and |∇Tk(un)|<+∞ in Ω-E.
Letη(x)=limn(∇Tk(un)), then by using Fatou lemma, we get
(3.39)∫Ω‍|η(x)|dx≤lim_∫Ω‍|∇Tk(un)|dx≤C.
So, |η(x)|<+∞ a.e. in Ω and ∇Tk(un)→η a.e. in Ω, and for a subsequence still denoted by Tk(un), we obtain from (3.38) that
(3.40)0=limn→∞(a(·,Tk(un),∇Tk(un))-a(·,Tk(un),∇Tk(u)))(∇Tk(un)-∇Tk(u))inΩ=(a(·,u,η)-a(·,u,∇Tk(u)))(η-∇Tk(u)).
Combining with (1.3), we get
(3.41)∇Tk(u)=ηa.e.inΩ.
Therefore, we have
(3.42)∇Tk(un)→∇Tk(u)a.e.inΩ.Lemma 3.12.
For allk>0,
(3.43)∇Tk(un)→∇Tk(u)forthemodularconvergencein(LM(Ω))N.For the proof we can adopt the same way as in [13].


## 3.2.4. The Convergence of the Problems(Pfn) and the Completion of the Proof of Theorem 3.6

In one hand by using as test functionφ=∫0un‍g(s)χ{s>h}exp(∫0un‍g(s)ds) in (Pfn), we can deduce as in [5] that g(un)M(|∇un|) converge strongly in L1(Ω) to g(u)M(|∇u|).In the other hand, by Lemma3.3, we deduce that (un) is bounded in W01LD(Ω) for every D∈K. Then (a(·,un,∇un)) is bounded in LH(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=∑i=1J‍αjδxj(xj∈Ω) and ∑i=1J‍αj=1) and the problem (Pμ) admits at least one weak solution.The proof is a simple adaptation of the one of Theorem3.6 by taking in zn,mj,δ,∑i=1J‍αj(1-ψδxj) in the place of (1-ψδxj), where ψδxj is the cut functions corresponding to the measure δxj.Remark 3.14.
The technique used in the proof of Theorem3.6 allows us to prove that the problem
(3.44)A(u)=g(u)a(·,u,∇u)∇u+δx0inΩ,u=0on∂Ω
has a weak solution.


## 3.3. General Case

Before giving the general case of Theorem3.6, 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 byμn the sequence defined by μn=∑i=1n‍αiδxi for some xi∈Ω, and αi∈R+ with ∑i=1n‍αi=1.Theorem 3.15.
Let(μn) be a sequence of Radon measures convergent to a Radon measure μ, and let (un), a sequence of weak solutions of (Pμn). Then there exists M>0 such that
(3.45)∫Ω‍M(|∇Tk(un)|)dx≤Mk
for every n and every k>0. Moreover, there exists a measurable function u such that Tk(u)∈W01LM(Ω) and u is a weak solution of (Pμ).Proof.
A priori estimate: Let Jη be a sequence of mollifiers functions. Let v=Tk(un)exp(∫0un‍g(s)ds)*Jη as test function in (Pμn)(3.46)∫Ω‍a(x,un,∇un)[∇Tk(un)exp(∫0un‍g(s)ds)*Jη]+∫Ω‍a(x,un,∇un)[∇ung(un)Tk(un)exp(∫0un‍g(s)ds)*Jη]=∫Ω‍M(|∇un|)g(un)Tk(un)exp(∫0un‍g(s)ds)*Jη+∫Ω‍Tk(un)exp(∫0un‍g(s)ds)*Jηdμn,
then, by using an easy argument, the properties of convolution, and tending η to zero, we get
(3.47)∫Ω‍M(|∇Tk(un)|)exp(∫0un‍g(s)ds)dx≤Ck.
So (Tk(un)) is bounded in W01LM(Ω) and as above there exists a measurable function u such that
(3.48)Tk(un)⇀Tk(u),weaklyinW01LM(Ω),a.e.inΩ,andstronglyinEM(Ω).
Almost everywhere convergence of gradients: Let us consider zn,mj,δ*Jη as test function in (Pμn), we have
(3.49)〈A(un),zn,mj,δ*Jη〉=∫Ω‍g(un)M(|∇un|)zn,mj,δ*Jηdx+∫Ω‍zn,mj,δ*Jηdμn.
We prove easily, since zn,mj,δ∈W01LM(Ω)∩L∞(Ω), that
(3.50)〈A(un),zn,mj,δ*Jη〉=〈A(un),zn,mj,δ〉+ϵ(η),∫Ω‍g(un)M(|∇un|)zn,mj,δ*Jηdx=∫Ω‍g(un)M(|∇un|)zn,mj,δdx+ϵ(η),∫Ω‍zn,mj,δ*Jηdμn=∑i=1n‍αizn,mj,δ(xi)+ϵ(η),∑i=1n‍αizn,mj,δ(xi)=ϵ(η,n,δ,m),
Also, ∫Ω‍ρm(un)(1-ψδ)exp(∫0un‍g(s)ds)*Jηdμn=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 (Pμ) admits at least one weak solution.SinceΩ 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 theN-functions M and M¯ satisfy the Δ2 condition, we can prove with the same way as in Theorem 3.6 that the problem (Pμ) 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(3.51)-div(a(·,u,∇u))=g(u)|∇u|p(x)+μinΩ,u=0on∂Ω,
where a satisfies, for almost every x in Ω, for every s in R, for every ξ and ξ* in RN,
(3.52)a(x,s,ξ)ξ≥|ξ|p(x),[a(x,s,ξ)-a(x,s,ξ*)][ξ-ξ*]>0,ξ≠ξ*,|a(x,s,ξ)|≤C(k(x)+|s|p(x)-1+|ξ|p(x)-1).
Here p:Ω→]1,+∞[ is a measurable function such that
(3.53)1<essinfx∈Ωp(x)≤esssupx∈Ωp(x)<N,
where k is a nonnegative function in Lp′(·)(Ω), p′(x)=p(x)/(p(x)-1), and C is a positive real.



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*Source: 615816-2012-10-15.xml*