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volume 6, issue 2, article 54, 2005.

Received 21 December, 2004;

accepted 06 April, 2005.

Communicated by:A. Fiorenza

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Journal of Inequalities in Pure and Applied Mathematics

STRONGLY NONLINEAR ELLIPTIC UNILATERAL PROBLEMS IN ORLICZ SPACE ANDL¹ DATA

L. AHAROUCH AND M. RHOUDAF

Département de Mathématiques et Informatique Faculté des Sciences Dhar-Mahraz

B.P. 1796 Atlas, Fès, Maroc.

EMail:l_aharouch@yahoo.fr EMail:rhoudaf_mohamed@yahoo.fr

c

2000Victoria University ISSN (electronic): 1443-5756 250-04

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Strongly Nonlinear Elliptic Unilateral Problems in Orlicz

Space andL¹Data L. Aharouch and M. Rhoudaf

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J. Ineq. Pure and Appl. Math. 6(2) Art. 54, 2005

http://jipam.vu.edu.au

Abstract

In this paper, we shall be concerned with the existence result of Unilateral problem associated to the equations of the form,

Au+g(x, u,∇u) =f,

whereA is a Leray-Lions operator from its domain D(A) ⊂ W₀¹L_M(Ω) into W⁻¹E_M(Ω). On the nonlinear lower order termg(x, u,∇u), we assume that it is a Carathéodory function having natural growth with respect to|∇u|, and satisfies the sign condition. The right hand sidefbelongs toL¹(Ω).

2000 Mathematics Subject Classification:35J60.

Key words: Orlicz Sobolev spaces, Boundary value problems, Truncations, Unilat- eral problems.

The authors would like to thank the referee for his comments.

1. Introduction

LetΩbe an open bounded subset ofR^N, N ≥2, with segment property. Let us consider the following nonlinear Dirichlet problem

(1.1) −div(a(x, u,∇u)) +g(x, u,∇u) =f,

where f ∈ L¹(Ω), Au = −diva(x, u,∇u) is a Leray-Lions operator defined on its domain D(A) ⊂ W₀¹L_M(Ω), with M an N-function and where g is a nonlinearity with the "natural" growth condition:

|g(x, s, ξ)| ≤b(|s|)(h(x) +M(|ξ|)) and which satisfies the classical sign condition

g(x, s, ξ)·s≥0.

In the case where f ∈ W⁻¹E_M(Ω), an existence theorem has been proved in [14] with the nonlinearities g depends only on x and u, and in [4] where g depends also the∇u.

For the case wheref ∈ L¹(Ω), the authors in [5] studied the problem (1.1), with the added assumption of exact natural growth

|g(x, s, ξ)| ≥βM(|ξ|) for |s| ≥µ

and in [6] no coercivity condition is assumed on g but the result is restricted to theN-function,M satisfying a∆₂-condition, while in [11] the authors were concerned about the above problem without assuming a∆2-condition onM.

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The purpose of this paper is to prove an existence result for unilateral problems associated to (1.1) without assuming the∆2-condition in the setting of the Orlicz-Sobolev space.

Further work for the equation (1.1) in theL^pcase where there is no restriction can be found in [17], and in [12,9,8] in the case of obstacle problems, see also [18].

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2. Preliminaries

Let M : R⁺ → R⁺ be an N-function, i.e. M is continuous, convex, with M(t)>0fort >0, ^M_t^(t) →0ast →0and ^M(t)_t → ∞ast → ∞.

Equivalently, M admits the representation: M(t) = Rt

0 a(s)ds where a : R⁺ → R⁺ is nondecreasing, right continuous, with a(0) = 0, a(t) > 0for t >0anda(t)tends to∞ast → ∞.

The N-function M conjugate to M is defined by M = Rt

0 ¯a(s)ds, where

¯

a :R⁺ →R⁺is given bya(t) = sup{s¯ :a(s)≤t}.

TheN-functionM is said to satisfy the∆₂-condition if, for somek

(2.1) M(2t)≤kM(t), ∀t≥0.

When (2.1) holds only fort ≥ some t₀ > 0then M is said to satisfy the ∆₂- condition near infinity.

We will extend theseN-functions to even functions on allR, i.e. M(t) = M(|t|)ift≤0.

Moreover, we have the following Young’s inequality

∀s, t ≥0, st≤M(t) +M(s).

LetP andQbe twoN-functions. P Qmeans thatP grows essentially less rapidly thanQ, i.e., for each >0,_Q(t)^P^(t) →0ast → ∞.This is the case if and only iflimt→∞ Q⁻¹(t)

P⁻¹(t) = 0.

Let Ω be an open subset of R^N. The Orlicz class KM(Ω) (resp. the Or- licz space L_M(Ω) is defined as the set of (equivalence classes of) real valued

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measurable functionsuonΩsuch that Z

Ω

M(u(x))dx <+∞

resp.

Z

Ω

M

u(x) λ

dx <+∞for someλ >0

. L_M(Ω)is a Banach space under the norm

kuk_M,Ω = inf

λ >0, Z

Ω

M

u(x) λ

dx≤1

andK_M(Ω)is a convex subset ofL_M(Ω).

The closure inL_M(Ω)of the set of bounded measurable functions with com- pact support inΩis denoted byE_M(Ω).

The dual ofE_M(Ω) can be identified with L_M(Ω) by means of the pairing R

Ωuv dx, and the dual norm ofL_M(Ω)is equivalent tok · k_{M ,Ω}.

We now turn to the Orlicz-Sobolev space,W¹L_M(Ω)(resp. W¹E_M(Ω)) is the space of all functionsusuch thatuand its distributional derivatives of order 1 lie inL_M(Ω)(resp. E_M(Ω)). It is a Banach space under the norm

kuk1,M = X

|α|≤1

kD^αukM.

Thus, W¹L_M(Ω)andW¹E_M(Ω)can be identified with subspaces of the product ofN+ 1copies ofL_M(Ω). Denoting this product byQ

L_M, we will use the weak topologiesσ(Q

L_M,Q

E_M)andσ(Q

L_M,Q L_M).

The spaceW₀¹E_M(Ω)is defined as the (norm) closure of the Schwartz space D(Ω) inW¹E_M(Ω) and the spaceW₀¹L_M(Ω) as theσ(Q

L_M,Q

E_M)closure ofD(Ω)inW¹LM(Ω).

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LetW⁻¹L_M(Ω)(resp. W⁻¹E_M(Ω)) denote the space of distributions onΩ which can be written as sums of derivatives of order≤1of functions inL_M(Ω) (resp. E_M(Ω)). It is a Banach space under the usual quotient norm (for more details see [1]).

We recall some lemmas introduced in [4] which will be used later.

Lemma 2.1. Let F : R → Rbe uniformly Lipschitzian, with F(0) = 0. Let M be anN-function and letu ∈ W¹L_M(Ω) (resp. W¹E_M(Ω)). ThenF(u)∈ W¹L_M(Ω)( resp. W¹E_M(Ω)). Moreover, if the setDof discontinuity points of F⁰ is finite, then

∂

∂x_iF(u) =

( F⁰(u)_∂x^∂

iu a.e. in {x∈Ω :u(x)∈/D}, 0 a.e. in {x∈Ω :u(x)∈D}.

Lemma 2.2. LetF :R→Rbe uniformly Lipschitzian, withF(0) = 0. We as- sume that the set of discontinuity points ofF⁰is finite. LetM be anN-function, then the mappingF :W¹L_M(Ω) →W¹L_M(Ω)is sequentially continuous with respect to the weak* topologyσ(Q

L_M,Q E_M).

We give now the following lemma which concerns operators of Nemytskii type in Orlicz spaces (see [4]).

Lemma 2.3. LetΩbe an open subset ofR^N with finite measure. LetM, P and QbeN-functions such thatQP, and letf : Ω×R→Rbe a Carathéodory function such that, for a.e.x∈Ωand alls∈R:

|f(x, s)| ≤c(x) +k₁P⁻¹M(k₂|s|),

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wherek₁, k₂are real constants andc(x)∈E_Q(Ω). Then the Nemytskii operator Nf defined byNf(u)(x) = f(x, u(x))is strongly continuous from

P

E_M(Ω), 1 k2

=

u∈L_M(Ω) :d(u, E_M(Ω))< 1 k2

intoE_Q(Ω).

We defineT₀^1,M(Ω) to be the set of measurable functionsu : Ω → R such that T_k(u) ∈ W₀¹L_M(Ω), where T_k(s) = max(−k,min(k, s))for s ∈ R and k ≥0.We give the following lemma which is a generalization of Lemma 2.1 of [2] in Orlicz spaces.

Lemma 2.4. For everyu∈ T₀^1,M(Ω), there exists a unique measurable function v : Ω−→R^N such that

∇T_k(u) = vχ{|u|<k}, almost everywhere in Ω for every k > 0.

We will define the gradient of u as the function v, and we will denote it by v =∇u.

Lemma 2.5. Let λ ∈ R and let uand v be two measurable functions defined onΩwhich are finite almost everywhere, and which are such thatT_k(u), T_k(v) andTk(u+λv)belong toW₀¹LM(Ω)for everyk > 0then

∇(u+λv) =∇(u) +λ∇(v) a.e. inΩ

where∇(u),∇(v)and∇(u+λv)are the gradients ofu,vandu+λvintroduced in Lemma2.4.

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The proof of this lemma is similar to the proof of Lemma 2.12 in [10] for the L^p case.

Below, we will use the following technical lemma.

Lemma 2.6. Let(f_n), f ∈L¹(Ω)such that (i) f_n ≥0a.e. inΩ

(ii) f_n →f a.e. inΩ (iii) R

Ωf_n(x)dx→R

Ωf(x)dx thenf_n→f strongly inL¹(Ω).

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3. Main Results

LetΩbe an open bounded subset ofR^N,N ≥2, with the segment property.

Given an obstacle functionψ : Ω→R,we consider (3.1) K_ψ ={u∈W₀¹L_M(Ω); u≥ψ a.e. in Ω},

this convex set is sequentially σ(ΠL_M,ΠE_M)closed inW₀¹L_M(Ω) (see [15]).

We now state conditions on the differential operator

(3.2) Au =−div(a(x, u,∇u))

(A₁) a(x, s, ξ) : Ω×R×R^N →R^N is a Carathéodory function.

(A₂) There exist two N-functions M and P with P M, function c(x) in E_M(Ω), constantsk₁, k₂, k₃, k₄ such that, for a.e. xin Ωand for all s ∈ R, ζ ∈R^N

|a(x, s, ζ)| ≤c(x) +k₁P⁻¹M(k₂|s|) +k₃M⁻¹M(k₄|ζ|).

(A3) [a(x, s, ζ)−a(x, s, ζ⁰)](ζ−ζ⁰)>0 for a.e.xinΩ,sinRandζ, ζ⁰ inR^N, withζ 6=ζ⁰.

(A4) There existδ(x)inL¹(Ω), strictly positive constantαsuch that, for some fixed elementv₀inK_ψ∩W₀¹E_M(Ω)∩L^∞(Ω),

a(x, s, ζ)(ζ−Dv₀)≥αM(|ζ|)−δ(x) for a.e. xinΩ, and alls ∈R, ζ ∈R^N.

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(A₅) For eachv ∈K_ψ∩L^∞(Ω)there exists a sequencev_n∈K_ψ∩W₀¹E_M(Ω)∩ L^∞(Ω)such thatvn →ufor the modular convergence.

Furthermore letg : Ω×R×R^N →Rbe a Carathéodory function such that for a.e. x∈Ωand for alls ∈R, ζ∈R^N

(G₁) g(x, s, ζ)s ≥0

(G₂) |g(x, s, ζ)| ≤b(|s|) (h(x) +M(|ζ|)),

whereb : R+ → R+ is a continuous non decreasing function, and his a given nonegative function inL¹(Ω).

Consider the following Dirichlet problem:

(3.3) A(u) +g(x, u,∇u) = f in Ω.

Remark 1. The condition(A₅)holds if one of the following conditions is veri- fied.

1. There existψ ∈ K_ψ such thatψ −ψ is continuous inΩ, (see [15, Propo- sition 9]).

2. ψ ∈W₀¹E_M(Ω), (see [15, Proposition 10]).

We shall prove the following existence theorem.

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Theorem 3.1. Assume that(A₁)–(A₅),(G₁)and(G₂)hold and f ∈ L¹(Ω).

Then there exists at least one solution of the following unilateral problem,

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









u∈ T₀^1,M(Ω), u≥ψ a.e. in Ω, g(x, u,∇u)∈L¹(Ω)

R

Ωa(x, u,∇u)∇T_k(u−v)dx+R

Ωg(x, u,∇u)T_k(u−v)dx

≤R

Ωf T_k(u−v)dx,

∀ v ∈K_ψ∩L^∞(Ω), ∀k >0.

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4. Proof of Theorem 3.1

To prove the existence theorem, we proceed by steps.

STEP 1: Approximate unilateral problems.

Let us define

gn(x, s, ξ) = g(x, s, ξ) 1 + _n¹|g(x, s, ξ)|

and let us consider the approximate unilateral problems:

(P_n)











u_n ∈K_ψ∩D(A), hAu_n, u_n−vi+R

Ωg_n(x, u_n,∇u_n)(u_n−v)dx

≤R

Ωf_n(u_n−v)dx,

∀v ∈Kψ.

wheref_nis a regular function such thatf_nstrongly converges tof inL¹(Ω).

From Gossez and Mustonen ([15, Proposition 5]), the problem (Pn) has at least one solution.

STEP 2: A priori estimates.

Letk ≥ kv0k∞and letϕk(s) = se^γs², where γ = b(k)

α

2

. It is well known that

(4.1) ϕ⁰_k(s)−b(k)

α |ϕ_k(s)| ≥ 1

2, ∀s∈R (see [9]).

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Taking u_n−ηϕ_k(T_l(u_n−v₀))as test function in (P_n), wherel = k+kv₀k∞, we obtain,

Z

Ω

a(x, un,∇un)∇Tl(un−v0)ϕ⁰_k(Tl(un−v0))dx +

Z

Ω

g_n(x, u_n,∇u_n)ϕ_k(T_l(u_n−v₀))dx

≤ Z

Ω

f_nϕ_k(T_l(u_n−v₀))dx.

Since

g_n(x, u_n,∇u_n)ϕ_k(T_l(u_n−v₀))≥0 on the subset{x∈Ω :|u_n(x)|> k},then

Z

{|un−v0|≤l}

a(x, u_n,∇u_n)∇(u_n−v₀)ϕ⁰_k(T_l(u_n−v₀))dx

≤ Z

{|u_n|≤k}

|gn(x, un,∇un)||ϕk(Tl(un−v0))|dx +

Z

Ω

By using(A₄)and(G₁), we have α

Z

{|un−v0|≤l}

M(|∇u_n|)ϕ⁰_k(T_l(u_n−v₀))dx

≤b(|k|) Z

Ω

(h(x) +M(∇T_k(u_n)))|ϕ_k(T_l(u_n−v₀))|dx

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+ Z

Ω

δ(x)ϕ⁰_k(T_l(u_n−v₀))dx+ Z

Ω

Since

{x∈Ω,|u_n(x)| ≤k} ⊆ {x∈Ω :|u_n−v₀| ≤l}

and the fact thath, δ∈L¹(Ω), furtherf_nis bounded inL¹(Ω), then Z

Ω

M(|∇T_k(u_n)|)ϕ⁰_k(T_l(u_n−v₀))dx

≤ b(k) α

Z

Ω

M(|∇T_k(u_n)|)|ϕ_k(T_l(u_n−v₀))|dx+c_k, wherec_kis a positive constant depending onk, which implies that

Z

Ω

M(|∇T_k(u_n)|)

ϕ⁰_k(T_l(u_n−v₀))− b(k)

α |ϕ_k(T_l(u_n−v₀))|

dx≤c_k. By using (4.1), we deduce,

(4.2)

Z

Ω

M(|∇T_k(u_n)|)dx≤2c_k.

SinceT_k(u_n)is bounded inW₀¹L_M(Ω), there exists somev_k ∈W₀¹L_M(Ω)such that

(4.3) T_k(u_n)* v_k weakly in W₀¹L_M(Ω) for σ(Q

L_M,Q E_M), T_k(u_n)→v_k strongly in E_M(Ω) and a.e. in Ω.

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STEP 3: Convergence in measure of u_n

Letk0 ≥ kv0k∞andk > k0, takingv =un−Tk(un−v0)as test function in (P_n) gives,

(4.4) Z

Ω

a(x, u_n,∇u_n)∇T_k(u_n−v₀)dx +

Z

Ω

g_n(x, u_n,∇u_n)T_k(u_n−v₀)dx

≤ Z

Ω

f_nT_k(u_n−v₀)dx, since g_n(x, u_n,∇u_n)T_k(u_n −v₀) ≥ 0 on the subset {x ∈ Ω,|u_n(x)| > k₀}, hence (4.4) implies that,

Z

Ω

a(x, u_n,∇u_n)∇T_k(u_n−v₀)dx≤k Z

{|u_n|≤k₀}

|g_n(x, u_n,∇u_n)|dx+kkfk_L¹_(Ω) which gives, by using(G₁),

(4.5) Z

Ω

a(x, u_n,∇u_n)∇T_k(u_n−v₀)dx

≤kb(k₀) Z

Ω

|h(x)|dx+ Z

Ω

M(|∇T_k₀(u_n)|)dx

+kc.

Combining (4.2) and (4.5), we have, Z

Ω

a(x, u_n,∇u_n)∇T_k(u_n−v₀)dx ≤k[c_k₀ +c].

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By(A₄), we obtain, Z

{|un−v0|≤k}

M(|∇u_n|)dx≤kc₁, wherec₁ is independent ofk, sincekis arbitrary, we have

Z

{|un|≤k}

M(|∇u_n|)dx≤ Z

{|un−v0|≤k+kv0k∞}

M(|∇u_n|)dx≤kc₂, i.e.,

(4.6)

Z

Ω

M(|∇T_k(u_n)|)dx ≤kc₂.

Now, we prove thatu_nconverges to some functionuin measure (and therefore, we can always assume that the convergence is a.e. after passing to a suit- able subsequence). We shall show thatu_nis a Cauchy sequence in measure.

Let k > 0 large enough, by Lemma 5.7 of [13], there exist two positive constantsc3 andc4such that

(4.7)

Z

Ω

M(c₃T_k(u_n))dx≤c₄ Z

Ω

M(|∇T_k(u_n)|)dx≤kc₅, then, we deduce, by using (4.7) that

M(c3k) meas{|un|> k}= Z

{|u_n|>k}

M(c3Tk(un))dx≤c5k,

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hence

(4.8) meas(|u_n|> k)≤ c₅k

M(c₃k) ∀n,∀k.

Letting k to infinity, we deduce that,meas{|u_n| > k}tends to 0 ask tends to infinity.

For everyλ >0, we have

(4.9) meas({|un−um|> λ})≤meas({|un|> k}) + meas({|um|> k}) + meas({|T_k(u_n)−T_k(u_m)|> λ}).

Consequently, by (4.3) we can assume that T_k(u_n) is a Cauchy sequence in measure inΩ.

Let > 0then, by (4.9) there exists somek() > 0such thatmeas({|u_n− u_m|> λ})< for alln, m≥h₀(k(), λ). This proves that(u_n)is a Cauchy sequence in measure inΩ, thus converges almost everywhere to some measurable functionu. Then

(4.10) T_k(u_n)* T_k(u) weakly in W₀¹L_M(Ω) for σ(Q

L_M,Q E_M), Tk(un)→Tk(u) strongly in EM(Ω) and a.e. in Ω.

Step 4: Boundedness of (a(x, T_k(u_n),∇T_k(u_n))_nin(L_M(Ω))^N. Letw∈(E_M(Ω))^N be arbitrary, by(A₃)we have

(a(x, u_n,∇u_n)−a(x, u_n, w))(∇u_n−w)≥0,

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which implies that

a(x, un,∇un)(w−∇v0)≤a(x, un,∇un)(∇un−∇v0)−a(x, un, w)(∇un−w) and integrating on the subset{x∈Ω,|u_n−v₀| ≤k}, we obtain,

(4.11) Z

{|un−v0|≤k}

a(x, u_n,∇u_n)(w− ∇v₀)dx

≤ Z

{|u_n−v₀|≤k}

a(x, un,∇un)(∇un− ∇v0)dx +

Z

{|un−v0|≤k}

a(x, u_n, w)(w− ∇u_n)dx.

We claim that, (4.12)

Z

{|un−v₀|≤k}

a(x, un,∇u_n)(∇u_n−v0)dx≤c10, wherec10is a positive constant depending onk.

Indeed, if we takev =u_n−T_k(u_n−v₀)as test function in (P_n), we get, Z

{|un−v0|≤k}

a(x, u_n,∇u_n)(∇u_n− ∇v₀)dx +

Z

Ω

g_n(x, u_n,∇u_n)T_k(u_n−v₀)dx

≤ Z

Ω

f_nT_k(u_n−v₀)dx.

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Sinceg_n(x, u_n,∇u_n)T_k(u_n−v₀) ≥ 0on the subset {x ∈ Ω, |u_n| ≥ kv₀k∞}, which implies

(4.13) Z

{|u_n−v₀|≤k}

a(x, u_n,∇u_n)(∇u_n− ∇v₀)dx

≤b(kv₀k∞) Z

Ω

h(x)dx+ Z

Ω

M(∇Tkv₀k∞(u_n)

dx+kkfk_L¹_(Ω). Combining (4.2) and (4.13), we deduce (4.12).

On the other hand, forλlarge enough, we have by using(A₂) (4.14)

Z

{|un−v0|≤k}

M

a(x, u_n, w) λ

dx

≤M

c(x) λ

+ k₃

λM(k₄|w|) +c≤c₁₁, hence, |a(x, u_n, w)|bounded inL_M(Ω), which implies that the second term of the right hand side of (4.11) is bounded

Consequently, we obtain, (4.15)

Z

{|u_n−v₀|≤k}

a(x, u_n,∇u_n)(w− ∇v₀)dx≤c₁₂, withc₁₂is positive constant depending ofk.

Hence, by the Theorem of Banach–Steinhaus, the sequence (a(x, u_n,

∇un))χ{|un−v0|≤k})n remains bounded in (L_M(Ω))^N. Since k is arbitrary, we

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deduce that(a(x, T_k(u_n),∇T_k(u_n)))_nis also bounded in(L_M(Ω))^N, which implies that, for allk > 0there exists a functionhk ∈(L_M(Ω))^N, such that, (4.16) a(x, T_k(u_n),∇T_k(u_n))* h_k weakly in (L_M(Ω))^N

for σ(ΠL_M(Ω),ΠE_M(Ω)).

STEP 5: Almost everywhere convergence of the gradient.

We fixk > kv₀k_∞. LetΩ_r ={x ∈ Ω,|∇T_k(u(x))| ≤ r}and denote by χ_r the characteristic function ofΩ_r. Clearly,Ω_r ⊂ Ω_r+1 andmeas(Ω\Ω_r) −→ 0 asr−→ ∞.

Fixrand lets≥r, we have, 0≤

Z

Ωr

[a(x, Tk(un),∇Tk(un))−a(x, Tk(un),∇Tk(u))]

(4.17)

×[∇T_k(u_n)− ∇T_k(u)]dx

≤ Z

Ωs

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u))]

×[∇T_k(u_n)− ∇T_k(u)]dx

= Z

Ωs

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u)χ_s)]

×[∇T_k(u_n)− ∇T_k(u)χ_s]dx

≤ Z

Ω

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u)χ_s)]

×[∇T_k(u_n)− ∇T_k(u)χ_s]dx.

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By(A₅)there exists a sequencev_j ∈K_ψ∩W₀¹E_M(Ω)∩L^∞(Ω)which converges toTk(u)for the modular converge inW₀¹LM(Ω).

Here, we define

w^h_n,j =T_2k(u_n−v₀−T_h(u_n−v₀) +T_k(u_n)−T_k(v_j)), w_j^h =T_2k(u−v₀−T_h(u−v₀) +T_k(u)−T_k(v_j)) and

w^h =T_2k(u−v₀−T_h(u−v₀)), whereh >2k > 0.

Forη= exp(−4γk²), we defined the following function as (4.18) v_n,j^h =u_n−ηϕ_k w_n,j^h

. We takev_n,j^h as test function in (P_n), we obtain, A(u_n), ηϕ_k w_n,j^h

+ Z

Ω

g_n(x, u_n,∇u_n)ηϕ_k w_n,j^h dx≤

Z

Ω

f_nηϕ_k w_n,j^h dx.

Which, implies that (4.19)

A(u_n), ϕ_k w_n,j^h +

Z

Ω

g_n(x, u_n,∇u_n)ϕ_k w_n,j^h dx

≤ Z

Ω

f_nϕ_k w_n,j^h dx.

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It follows that (4.20)

Z

Ω

a(x, u_n,∇u_n)∇w_n,j^h ϕ⁰_k w_n,j^h dx +

Z

Ω

g_n(x, u_n,∇u_n)ϕ_k w^h_n,j dx≤

Z

Ω

f_nϕ_k w_n,j^h dx.

Note that,∇w^h_n,j = 0on the set where |u_n| > h+ 5k, therefore, setting m = 5k+h, and denoting by(n, j, h)any quantity such that

h→+∞lim lim

j→+∞ lim

n→+∞(n, j, h) = 0.

If the quantity we consider does not depend on one parameter among n, j and h, we will omit the dependence on the corresponding parameter: as an example, (n, h)is any quantity such that

h→+∞lim lim

n→+∞(n, h) = 0.

Finally, we will denote (for example) by _h(n, j) a quantity that depends on n, j, hand is such that

j→+∞lim lim

n→+∞_h(n, j) = 0 for any fixed value ofh.

We get, by (4.20), Z

Ω

a(x, T_m(u_n),∇T_m(u_n))∇w^h_n,jϕ⁰_k w^h_n,j dx +

Z

Ω

g_n(x, u_n,∇u_n)ϕ_k w_n,j^h dx≤

Z

Ω

f_nϕ_k w_n,j^h dx,

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In view of (4.10), we have ϕ_k w_n,j^h

→ ϕ_k(w^h_j) weakly^∗ as n → +∞ in L^∞(Ω), and then

Z

Ω

f_nϕ_k w^h_n,j dx→

Z

Ω

f ϕ_k(w^h_j)dx as n →+∞.

Again tendsj to infinity, we get Z

Ω

f ϕ_k(w^h_j)dx→ Z

Ω

f ϕ_k(w^h)dx as j →+∞,

finally letting h the infinity, we deduce by using the Lebesgue Theorem that R

Ωf ϕ_k(w^h)dx→0.

So that

Z

Ω

fnϕk w^h_n,j

dx=(n, j, h).

Since in the set{x∈Ω,|u_n(x)|> k}, we haveg(x, u_n,∇u_n)ϕ_k w_n,j^h

≥0, we deduce from (4.20) that

(4.21) Z

Ω

a(x, T_m(u_n),∇T_m(u_n))∇w_n,j^h ϕ⁰_k w_n,j^h dx +

Z

{|un|≤k}

gn(x, un,∇un)ϕk w_n,j^h

dx≤(n, j, h).

Splitting the first integral on the left hand side of (4.21) where |un| ≤ k and

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|u_n|> k, we can write, (4.22)

Z

Ω

a(x, T_m(u_n),∇T_m(u_n))∇w_n,j^h ϕ⁰_k w_n,j^h dx

= Z

{|un|≤k}

a(x, T_m(u_n),∇T_m(u_n))[∇T_k(u_n)− ∇T_k(v_j)]ϕ⁰_k w_n,j^h dx +

Z

{|un|>k

a(x, T_m(u_n),∇T_m(u_n))∇w^h_n,jϕ⁰_k w_n,j^h dx.

The first term of the right hand side of the last inequality can write as (4.23)

Z

{|u_n|≤k}

a(x, T_m(u_n),∇T_m(u_n))[∇T_k(u_n)− ∇T_k(v_j)]ϕ⁰_k w_n,j^h dx

≥ Z

Ω

a(x, T_k(u_n),∇T_k(u_n))[∇T_k(u_n)− ∇T_k(v_j)]ϕ⁰_k w^h_n,j dx

−ϕ⁰_k(2k) Z

{|un|>k}

|a(x, T_k(u_n),0)||∇T_k(v_j)|dx.

Recalling that, |a(x, T_k(u_n),0)|χ{|un|>k} converges to |a(x, T_k(u),0)|χ{|u|>k}

strongly inL_M(Ω), moreover, since|∇T_k(v_j)|modular converges to|∇T_k(u)|, then

−ϕ⁰_k(2k) Z

{|un|>k}

|a(x, T_k(u_n),0)||∇T_k(v_j)|dx=(n, j).

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For the second term of the right hand side of(4.14)we can write, using(A₃) (4.24)

Z

{|un|>k}

a(x, T_m(u_n),∇T_m(u_n))∇w^h_n,jϕ⁰_k w_n,j^h dx

≥ −ϕ⁰_k(2k) Z

{|un|>k}

|a(x, T_m(u_n),∇T_m(u_n))|∇T_k(v_j)|dx

−ϕ⁰(2k) Z

{|un−v₀|>h

δ(x)dx.

Since |a(x, T_m(u_n),∇T_m(u_n))| is bounded in L_M(Ω), we have, for a subsequence

|a(x, Tm(un),∇Tm(un))|* lm

weakly inL_M(Ω)inσ(L_M, E_M)asntends to infinity, and since

|∇T_k(v_j)|χ_{|u_n_|>k} → |∇T_k(v_j)|χ_{|u|>k}

strongly inE_M(Ω)asntends to infinity, we have

−ϕ⁰_k(2k) Z

{|un|>k}

|a(x, T_m(u_n),∇T_m(u_n))|∇T_k(v_j)|dx

→ −ϕ⁰_k(2k) Z

{|u|>k}

l_m|∇T_k(v_j)|dx asntends to infinity.

Using now, the modular convergence of(v_j), we get

−ϕ⁰_k(2k) Z

{|u|>k}

l_m|∇T_k(v_j)|dx→ −ϕ⁰_k(2k) Z

{|u|>k}

l_m|∇T_k(u)|dx= 0

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asj tends to infinity.

Finally

(4.25) −ϕ⁰_k(2k) Z

{|u_n|>k}

|a(x, T_m(u_n),∇T_m(u_n))|∇T_k(v_j)|dx=_h(n, j).

On the other hand, sinceδ∈L¹(Ω)it is easy to see that

(4.26) −ϕ⁰_k(2k)

Z

{|un−v0|>h

δ(x)dx=(n, h).

Combining (4.23) – (4.26), we deduce (4.27)

Z

Ω

a(x, T_m(u_n),∇T_m(u_n))∇w_n,j^h ϕ⁰_k w_n,j^h dx

≥ Z

Ω

a(x, T_k(u_n),∇T_k(u_n))[∇T_k(u_n)− ∇T_k(v_j)]ϕ⁰_k w^h_n,j dx +(n, h) +(n, j) +_h(n, j), which implies that

(4.28) Z

Ω

a(x, Tm(un),∇Tm(un))∇w_n,j^h ϕ⁰_k w_n,j^h dx

≥ Z

Ω

a(x, T_k(u_n),∇T_k(u_n))−a x, T_k(u_n),∇T_k(v_j)χ^j_s

×

∇T_k(u_n)− ∇T_k(v_j)χ^j_s

ϕ⁰_k w_n,j^h dx

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+ Z

Ω

a x, T_k(u_n),∇T_k(v_j)χ^j_s ∇T_k(u_n)− ∇T_k(v_j)χ^j_s

ϕ⁰_k w^h_n,j dx

− Z

Ω\Ω^js

a(x, T_k(u_n),∇T_k(u_n))∇T_k(v_j)ϕ⁰_k w_n,j^h dx

+(n, h) +(n, j) +_h(n, j), where χ^j_s denotes the characteristic function of the subset Ω^j_s = {x ∈ Ω :

|∇Tk(vj)| ≤s}.

By (4.16) and the fact that∇T_k(v_j)χ_Ω\Ω^j

sϕ⁰_k w^h_n,j

tends to∇T_k(v_j)χ_Ω\Ω^j

sϕ⁰_k(w^h_j) strongly in(E_M(Ω))^N, the third term on the right hand side of(4.28)tends to the quantity

Z

Ω\Ω^j_s

h_k∇T_k(v_j)ϕ⁰_k w^h_j dx asntends to infinity.

Letting nowj tend to infinity, by using the modular convergence ofv_j, we have

Z

Ω

h_k∇T_k(v_j)χ_Ω\Ω^j

sϕ⁰_k w_j^h dx→

Z

Ω\Ωs

h_k∇T_k(u)ϕ⁰_k(w^h)dx asj tends to infinity.