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volume 4, issue 5, article 98, 2003.

Received 26 March, 2003;

accepted 05 August, 2003.

Communicated by:A. Fiorenza

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Journal of Inequalities in Pure and Applied Mathematics

ASYMPTOTIC BEHAVIOUR OF SOME EQUATIONS IN ORLICZ SPACES

D. MESKINE AND A. ELMAHI

Département de Mathématiques et Informatique Faculté des Sciences Dhar-Mahraz

B.P 1796 Atlas-Fès,Fès Maroc.

EMail:meskinedriss@hotmail.com C.P.R de Fès, B.P 49, Fès, Maroc.

c

2000Victoria University ISSN (electronic): 1443-5756 040-03

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Asymptotic Behaviour of Some Equations in Orlicz Spaces

D. Meskine and A. Elmahi

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J. Ineq. Pure and Appl. Math. 4(5) Art. 98, 2003

http://jipam.vu.edu.au

Abstract

In this paper, we prove an existence and uniqueness result for solutions of some bilateral problems of the form







hAu, v−ui ≥ hf, v−ui, ∀v∈K u∈K

whereAis a standard Leray-Lions operator defined onW₀¹LM(Ω), withMan N-function which satisfies the∆₂-condition, and whereK is a convex subset ofW₀¹L_M(Ω)with obstacles depending on some Carathéodory functiong(x, u).

We consider first, the casef ∈W⁻¹E_M(Ω)and secondly wheref∈L¹(Ω). Our method deals with the study of the limit of the sequence of solutionsu_nof some approximate problem with nonlinearity term of the form|g(x, u_n)|ⁿ⁻¹g(x, u_n)× M(|∇u_n|).

2000 Mathematics Subject Classification:35J25, 35J60.

Key words: Strongly nonlinear elliptic equations, Natural growth, Truncations, Varia- tional inequalities, Bilateral problems.

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1. Introduction

Let Ωbe an open bounded subset of R^N, N ≥ 2, with the segment property.

Consider the following obstacle problem:

(P)







hAu, v−ui ≥ hf, v−ui, ∀v ∈K, u∈K,

whereA(u) =−div(a(x, u,∇u))is a Leray-Lions operator defined onW₀¹L_M(Ω), withM being an N-function which satisfies the∆₂-condition and whereK is a convex subset ofW₀¹L_M(Ω).

In the variational case (i.e. wheref ∈ W⁻¹E_M(Ω)), it is well known that problem (P) has been already studied by Gossez and Mustonen in [10].

In this paper, we consider a recent approach of penalization in order to prove an existence theorem for solutions of some bilateral problems of (P) type.

We recall that L. Boccardo and F. Murat, see [6], have approximated the model variational inequality:







h−∆pu, v−ui ≥ hf, v−ui, ∀v ∈K

u∈K ={v ∈W₀^1,p(Ω) :|v(x)| ≤1a.e. inΩ},

withf ∈W^−1,p⁰(Ω)and−∆_pu=−div(|∇u|^p−2∇u), by the sequence of problems:







−∆_pu_n+|u_n|ⁿ⁻¹u_n =f inD⁰(Ω) u_n ∈W₀^1,p(Ω)∩Lⁿ(Ω).

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In [7], A. Dall’aglio and L. Orsina generalized this result by taking increasing powers depending also on some Carathéodory function g satisfying the sign condition and some hypothesis of integrability. Following this idea, we have studied in [5] the sequence of problems:







−∆_pu_n+|g(x, u_n)|ⁿ⁻¹g(x, u_n)|∇u_n|^p =f inD⁰(Ω) u_n∈W₀^1,p(Ω), |g(x, u_n)|ⁿ|∇u_n|^p ∈L¹(Ω)

Here, we introduce the general sequence of equations in the setting of Orlicz- Sobolev spaces







Au_n+|g(x, u_n)|ⁿ⁻¹g(x, u_n)M(|∇u_n|) =f inD⁰(Ω) u_n∈W₀¹L_M(Ω), |g(x, u_n)|ⁿM(|∇u_n|)∈L¹(Ω).

We are interested throughout the paper in studying the limit of the sequenceu_n. We prove that this limit satisfies some bilateral problem of the (P) form under some conditions ong. In the first we takef ∈W⁻¹E_M(Ω)and next inL¹(Ω).

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2. Preliminaries

2.1. N −Functions

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

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

TheN-functionM conjugate toM is defined byM(t) = Rt

0a(s)ds, where¯ a :R⁺ →R⁺is given bya(t) = sup{s¯ :a(s)≤t}(see [1]).

TheN-function is said to satisfy the∆₂ condition, denoted by M ∈ ∆₂, if for somek >0:

(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 into even functions on allR.

LetP andQbe two N-functions. P Q means 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.

2.2. Orlicz spaces

LetΩbe an open subset ofR^N. The Orlicz classK_M(Ω)(resp. the Orlicz space LM(Ω)) is defined as the set of (equivalence classes of) real valued measurable

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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 equalityE_M(Ω) =L_M(Ω)holds if only ifM satisfies the∆₂condition, for alltor fortlarge according to whetherΩhas infinite measure or not.

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

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

The space LM(Ω) is reflexive if and only ifM and M satisfy the ∆2 condition, for all t or fort large, according to whether Ωhas infinite measure or not.

2.3. Orlicz-Sobolev spaces

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)). It is a Banach space

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under the norm

kuk1,M = X

|α|≤1

kD^αukM.

Thus,W¹LM(Ω)andW¹EM(Ω)can be identified with subspaces of product of N+ 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 Schwarz space D(Ω) inW¹E_M(Ω)and the spaceW₀¹L_M(Ω)as theσ(Q

L_M,Q

E_M)closure ofD(Ω)inW¹L_M(Ω).

We say thatunconverges toufor the modular convergence inW¹LM(Ω)if for someλ >0

Z

Ω

M

D^αu_n−D^αu λ

dx→0 for all |α| ≤1.

This implies convergence forσ(Q

L_M,Q L_M).

IfM satisfies the∆₂-condition onR⁺, then modular convergence coincides with norm convergence.

2.4. The spaces W

⁻¹

L

M¯

(Ω) and W

⁻¹

E

M¯

(Ω)

Let W⁻¹L_M(Ω) (resp. W⁻¹E_M(Ω)) denote the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of functions in L_M (resp. E_M(Ω)). It is a Banach space under the usual quotient norm.

If the open setΩhas the segment property then the spaceD(Ω)is dense in W₀¹L_M(Ω)for the modular convergence and thus for the topologyσ(Q

L_M,Q L_M)

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(cf. [8,9]). Consequently, the action of a distribution inW⁻¹L_M(Ω)on an ele- ment ofW₀¹LM(Ω)is well defined.

2.5. Lemmas related to the Nemytskii operators in Orlicz spaces

We recall some lemmas introduced in [3] which will be used in this paper.

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 sup- pose that the set of discontinuity points ofF⁰ is 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).

2.6. Abstract lemma applied to the truncation operators

We now give the following lemma which concerns operators of the Nemytskii type in Orlicz spaces (see [3]).

Lemma 2.3. LetΩbe an open subset ofR^N with finite measure.

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LetM, P andQbeN-functions such thatQ P, and letf : Ω×R →R be a Carathéodory function such that a.e. x∈Ωand alls∈R:

|f(x, s)| ≤c(x) +k₁P⁻¹M(k₂|s|), wherek₁, k₂ are real constants andc(x)∈E_Q(Ω).

Then the Nemytskii operatorN_fdefined byN_f(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

intoEQ(Ω).

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3. The Main Result

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

LetM be anN-function satisfying the∆₂-condition near infinity.

LetA(u) = −div(a(x,∇u))be a Leray-Lions operator defined onW₀¹L_M(Ω) into

W⁻¹L_M(Ω), where a : Ω× R^N → R^N is a Carathéodory function satisfying for a.e. x∈Ωand for allζ, ζ⁰ ∈R^N,(ζ 6=ζ⁰) :

(3.1) |a(x, ζ)| ≤h(x) +M⁻¹M(k1|ζ|)

(3.2) (a(x, ζ)−a(x, ζ⁰))(ζ−ζ⁰)>0

(3.3) a(x, ζ)ζ ≥αM

|ζ|

λ

withα, λ >0, k₁ ≥0, h∈E_M(Ω).

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

(3.4) g(x, s)s≥0

(3.5) |g(x, s)| ≤b(|s|)

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(3.6)











for almostx∈Ω\Ω^∞₊ there exists=(x)>0such that:

g(x, s)>1, ∀s∈]q₊(x), q₊(x) +[;

for almost x∈Ω\Ω^∞₋ there exists =(x)>0such that:

g(x, s)<−1, ∀s∈]q−(x)−, q−(x)[,

whereb:R⁺→R⁺is a continuous and nondecreasing function, withb(0) = 0 and where

q₊(x) = inf{s >0 :g(x, s)≥1}

q−(x) = sup{s <0 :g(x, s)≤ −1}

Ω^∞₊ ={x∈Ω :q+(x) = +∞}

Ω^∞₋ ={x∈Ω :q−(x) = −∞}.

We define forsandkinR, k ≥0, T_k(s) = max(−k,min(k, s)).

Theorem 3.1. Let f ∈ W⁻¹E_M(Ω). Assume that (3.1) – (3.6) hold true and that the functions → g(x, s)is nondecreasing for a.e. x ∈ Ω. Then, for any real numberµ > 0, the problem

(Pn)







A(u_n) +|g(x, u_n)|ⁿ⁻¹g(x, u_n)M_|∇u

n| µ

=f inD⁰(Ω) u_n ∈W₀¹L_M(Ω),|g(x, u_n)|ⁿM_|∇u

n| µ

∈L¹(Ω)

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admits at least one solutionu_nsuch that:

(3.7) ∀k >0 T_k(u_n)→T_k(u)for modular convergence inW₀¹L_M(Ω) whereuis the unique solution of the following bilateral problem

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





hAu, v−ui ≥ hf, v−ui, ∀v ∈K

u∈K ={v ∈W₀¹LM(Ω) :q−≤v ≤q+ a.e.},

Remark 3.1. If the functions→g(x, s)is strictly nondecreasing for a.e. x∈Ω then the assumption (3.6) holds true.

Proof.

Step 1: A priori estimates.

The existence ofu_nis given by Theorem 3.1 of [3]. Choosingv =u_nas a test function in (P_n), and using the sign condition (3.4), we get

hAu_n, u_ni ≤ hf, u_ni.

By Proposition 5 of [11] one has:

(3.8) Z

Ω

M

|∇u_n| λ

dx≤C, and Z

Ω

a(x, u_n,∇u_n)∇u_ndx≤C,

(3.9) (a(x, un,∇un))is bounded in(L_M(Ω))^N,

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(3.10)

Z

Ω

|g(x, u_n)|ⁿ⁻¹g(x, u_n)M

|∇un| µ

u_ndx≤C.

We then deduce Z

{|un|>k}

|g(x, un)|ⁿM

|∇u_n| µ

dx ≤C, for all k >0.

Sincebis continuous and sinceb(0) = 0there existsδ >0such that b(|s|)≤1for all|s| ≤δ.

On the other hand, by the∆₂ condition there exist two positive constants KandK⁰ such that

M t

µ

≤KM t

λ

+K⁰ for all t≥0, which implies

Z

{|un|≤δ}

|g(x, un)|ⁿM

|∇u_n| µ

dx

≤ Z

{|u_n|≤δ}

K⁰+KM

|∇un| λ

dx.

Consequently from (3.8) (3.11)

Z

Ω

|g(x, un)|ⁿM

|∇u_n| µ

dx≤C, for all n.

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Step 2: Almost everywhere convergence of the gradients.

Since (un) is a bounded sequence in W₀¹LM(Ω) there exist some u ∈ W₀¹L_M(Ω)such that (for a subsequence still denoted byu_n)

u_n * uweakly inW₀¹L_M(Ω)forσY

L_M,Y E_M

, (3.12)

strongly inEM(Ω) and a.e. inΩ.

Furthermore, if we have

Au_n =f− |g(x, u_n)|ⁿ⁻¹g(x, u_n)M

|∇u_n| µ

with|g(x, u_n)|ⁿ⁻¹g(x, u_n)M_|∇u

n| µ

being bounded in L¹(Ω)then as in [2], one can show that

(3.13) ∇u_n→ ∇ua.e. inΩ.

Step 3: u∈K ={v ∈W₀¹L_M(Ω) :q−≤v ≤q₊ a.e. inΩ}.

Sinces→g(x, s)is nondecreasing, then in view of (3.6), we have:

{s∈R:|g(x, s)| ≤1a.e. inΩ}={s∈R:q− ≤s≤q₊ a.e. inΩ}.

It suffices to verify that|g(x, u)| ≤1a.e.

We have

Z

Ω

|g(x, u_n)|ⁿM

|∇u_n| µ

dx≤C,

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which gives Z

{|g(x,un)|>k}

|g(x, u_n)|ⁿM

|∇u_n| µ

|dx≤C

and Z

{|g(x,u_n)|>k}

M

|∇un| µ

dx ≤ C kⁿ

wherek > 1. Lettingn → +∞fork fixed, we deduce by using Fatou’s lemma

Z

{|g(x,u)|>k}

M

|∇u_n| µ

dx= 0 and so that,

|g(x, u)| ≤1a.e. inΩ.

Step 4: Strong convergence of the truncations.

Letφ(s) =sexp(γs²),whereγ is chosen such thatγ ≥ _α¹2

.

It is well known thatφ⁰(s)−^2K_α |φ(s)| ≥ ¹₂,∀s∈R,whereKis a constant which will be used later. The use of the test functionv_n = φ(z_n)in (P_n) wherez_n =T_k(u_n)−T_k(u)gives

hAu_n, φ(z_n)i+

Z

Ω

|∇un| µ

φ(z_n)dx=hf, φ(z_n)i which implies, by using the fact that g(x, un)φ(zn) ≥ 0 on {x ∈ Ω :

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|u_n|> k},

hAu_n, φ(z_n)i+ Z

{0≤u_n≤T_k(u)}∩{|u_n|≤k}

|g(x, u_n)|ⁿ⁻¹

×g(x, u_n)M

|∇u_n| µ

φ(z_n)dx +

Z

{T_k(u)≤un≤0}∩{|un|≤k}

|g(x, u_n)|ⁿ⁻¹

×g(x, u_n)M

|∇u_n| µ

φ(z_n)dx ≤ hf, φ(z_n)i.

The second and the third terms of the last inequality will be denoted respectively byI_n,k¹ andI_n,k² and_i(n)denote various sequences of real num- bers which tend to0asn→+∞.

On the one hand we have I_n,k¹

≤ Z

{0≤u_n≤T_k(u)}∩{|u_n|≤k}

|g(x, u_n)|ⁿM

|∇un| µ

|φ(z_n)|dx

≤ Z

{0≤u_n≤u}∩{|u_n|≤k}

|g(x, u_n)|ⁿM

|∇u_n| µ

|φ(z_n)|dx, but since|g(x, u_n)| ≤1on{x∈Ω : 0≤u_n≤u}, then we have

I_n,k¹ ≤

Z

{|u_n|≤k}

M

|∇u_n| µ

|φ(z_n)|dx.

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By using the fact that M

|∇u_n| µ

≤K⁰+KM

|∇u_n| λ

we obtain I_n,k¹

≤ Z

Ω

K⁰|φ(z_n)|dx+K α

Z

Ω

a(x,∇T_k(u_n))∇T_k(u_n)|φ(z_n)|dx, which gives

(3.14)

I_n,k¹

≤₁(n) + K α

Z

Ω

a(x,∇T_k(u_n))∇T_k(u_n)|φ(z_n)|dx.

Similarly, I_n,k²

≤ Z

{|un|≤k}

M

|∇u_n| µ

|φ(z_n)|dx (3.15)

≤₁(n) + K α

Z

Ω

a(x,∇T_k(u_n))∇T_k(u_n)|φ(z_n)|dx.

The first term on the left hand side of the last inequality can be written as:

(3.16) Z

Ω

a(x,∇u_n)[∇T_k(u_n)− ∇T_k(u)]φ⁰(z_n)dx

= Z

{|u_n|≤k}

− Z

{|un|>k}

a(x,∇u_n)∇T_k(u)φ⁰(z_n)dx.

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For the second term on the right hand side of the last equality, we have

Z

{|u_n|>k}

a(x,∇u_n)∇T_k(u)φ⁰(z_n)dx

≤C_k Z

Ω

|a(x,∇u_n)||∇T_k(u)|χ_{|u_n_|>k}dx.

The right hand side of the last inequality tends to 0 as n tends to infinity. Indeed, the sequence (a(x,∇u_n))_n is bounded in (L_M(Ω))^N while

∇T_k(u)χ_{|u_n_|>k}tends to 0 strongly in(E_M(Ω))^N.

We define for every s > 0,Ω_s = {x ∈ Ω : |∇T_k(u(x))| ≤ s} and we denote byχ_sits characteristic function. For the first term of the right hand side of (3.16), we can write

(3.17) Z

{|un|≤k}

= Z

Ω

[a(x,∇T_k(u_n))−a(x,∇T_k(u)χ_s)][∇T_k(u_n)− ∇T_k(u)χ_s]φ⁰(z_n)dx +

Z

Ω

a(x,∇T_k(u)χ_s)[∇T_k(u_n)− ∇T_k(u)χ_s]φ⁰(z_n)dx

− Z

Ω

a(x,∇Tk(un))∇Tk(u)χΩ\Ωsφ⁰(zn)dx.

The second term of the right hand side of (3.17) tends to 0 since a(x,∇T_k(u_n)χ_s)φ⁰(z_n)→a(x,∇T_k(u)χ_s)strongly in(E_M(Ω))^N

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by Lemma2.3and

∇T_k(u_n)*∇T_k(u)weakly in(L_M(Ω))^N forσY

L_M(Ω),Y

E_M(Ω) . The third term of (3.17) tends to −R

Ωa(x,∇T_k(u))∇T_k(u)χ_Ω\Ω_sdx as n→ ∞since

a(x,∇Tk(un))* a(x,∇Tk(u))weakly forσY

E_M(Ω),Y

LM(Ω)

. Consequently, from (3.16) we have

(3.18) Z

Ω

= Z

Ω

[a(x,∇Tk(un))−a(x,∇Tk(u)χs)]

×[∇T_k(u_n)− ∇T_k(u)χ_s]φ⁰(z_n)dx+₂(n).

We deduce that, in view of (3.17) and (3.18), Z

Ω

[a(x,∇Tk(un))−a(x,∇Tk(u)χs)]

×[∇T_k(u_n)− ∇T_k(u)χ_s]

φ⁰(z_n)−2K

α |φ(z_n)|

dx

≤₃(n) + Z

Ω

a(x,∇T_k(u))∇T_k(u)χΩ\Ω_sdx,

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and so Z

Ω

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

≤2₃(n) + 2 Z

Ω

a(x,∇T_k(u))∇T_k(u)χ_Ω\Ω_sdx.

Hence Z

Ω

a(x,∇T_k(u_n))∇T_k(u_n)dx

≤ Z

Ω

a(x,∇T_k(u_n))∇T_k(u)χ_sdx +

Z

Ω

a(x,∇T_k(u)χ_s)[∇T_k(u_n)− ∇T_k(u)χ_s]dx + 23(n) + 2

Z

Ω

a(x,∇Tk(u))∇Tk(u)χΩ\Ωsdx.

Now considering the limit sup overn, one has (3.19) lim sup

n→+∞

Z

Ω

≤lim sup

n→+∞

Z

Ω

a(x,∇T_k(u_n))∇T_k(u)χ_sdx + lim sup

n→+∞

Z

Ω

a(x,∇T_k(u)χ_s)[∇T_k(u_n)− ∇T_k(u)χ_s]dx + 2

Z

Ω

a(x,∇T_k(u))∇T_k(u)χ_Ω\Ω_sdx.

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The second term of the right hand side of the inequality (3.19) tends to 0, since

a(x,∇T_k(u_n)χ_s)→a(x,∇T_k(u)χ_s)strongly inE_M(Ω), while∇T_k(u_n)tends weakly to∇T_k(u).

The first term of the right hand side of (3.19) tends to R

Ωa(x,∇T_k(u))

∇T_k(u)χ_sdxsince

a(x,∇T_k(u_n))* a(x,∇T_k(u))weakly in(L_M(Ω))^N forσ(Q

L_M,Q

E_M)while∇T_k(u)χ_s ∈E_M(Ω). We deduce then lim sup

n→+∞

Z

Ω

≤ Z

Ω

a(x,∇T_k(u))∇T_k(u)χ_sdx + 2

Z

Ω

a(x,∇Tk(u))∇Tk(u)χΩ\Ωsdx, by using the fact thata(x,∇T_k(u))∇T_k(u) ∈ L¹(Ω) and letting s → ∞ we get, sincemeas(Ω\Ω_s)→0

lim sup

n→+∞

Z

Ω

a(x,∇T_k(u_n))∇T_k(u_n)dx≤ Z

Ω

a(x,∇T_k(u))∇T_k(u)dx

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which gives, by using Fatou’s lemma, (3.20) lim

n→+∞

Z

Ω

= Z

Ω

a(x,∇T_k(u))∇T_k(u)dx.

On the other hand, we have M

|∇Tk(un)|

µ

≤K⁰+ K α

Z

Ω

a(x,∇T_k(u_n))∇T_k(u_n)dx, then by using (3.20) and Vitali’s theorem, one easily has

(3.21) M

|∇T_k(u_n)|

µ

→M

|∇T_k(u)|

µ

strongly inL¹(Ω).

By writing

(3.22) M

|∇T_k(u_n)− ∇T_k(u)|

2µ

≤

M_|∇T

k(un)|

µ

2 +

M_|∇T

k(un)|

µ

2 one has, by (3.21) and Vitali’s theorem again,

(3.23) T_k(u_n)→T_k(u)for modular convergence inW₀¹L_M(Ω).

Step 5: uis the solution of the variational inequality (P).

Choosingw= Tk(un−θTm(v))as a test function in (Pn), wherev ∈ K

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and0< θ <1, gives hAu_n, T_k(u_n−θT_m(v))i

+ Z

Ω

|g(x, un)|ⁿ⁻¹g(x, un)M

|∇u_n| µ

Tk(un−θTm(v))dx

=hf, T_k(u_n−θT_m(v))i, sinceg(x, u_n)T_k(u_n−θT_m(v))≥0on

{x∈Ω :u_n ≥0andu_n ≥θT_m(v)}∪{x∈Ω :u_n≤0andu_n≤θT_m(v)}

we have Z

Ω

|∇u_n| µ

T_k(u_n−θT_m(v))dx

≥ Z

{0≤un≤θTm(v)}

|∇u_n| µ

T_k(u_n−θT_m(v))dx +

Z

{θTm(v)≤un≤0}

|∇u_n| µ

T_k(u_n−θT_m(v))dx.

The first and the second terms in the right hand side of the last inequality will be denoted respectively byJ_n,m¹ andJ_n,m² .

Defining

δ_1,m(x) = sup

0≤s≤θTm(v)

g(x, s) we get0≤δ_1,m(x)<1a.e. and

|J_n,m¹ | ≤k Z

{0≤un≤θTm(v)}

(δ_1,m(x))ⁿM

|∇u_n| µ

dx.

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Since

(δ_1,m(x))ⁿM

|∇u_n| µ

χ_{|u_n_|≤m}

≤M

|∇T_m(u_n)|

µ

, we have then by using (3.23) and Lebesgue’s theorem

J_n,m¹ −→0asn →+∞.

Similarly

|J_n,m² | ≤k Z

{|u_n|≤m}

|δ2,m(x)|ⁿM

|∇Tm(un)|

µ

dx→0asn →+∞, where

δ_2,m(x) = inf

θTm(v)≤s≤0g(x, s).

On the other hand , by using Fatou’s lemma and the fact that

a(x,∇u_n)→a(x,∇u)weakly in(L_M(Ω))^N forσ(ΠL_M,ΠE_M), one easily has

lim inf

n→+∞hAu_n, T_k(u_n−θT_m(v))i ≤ hAu, T_k(u−θT_m(v))i.

Consequently

hAu, T_k(u−θT_m(v))i ≤ hf, T_k(u−θT_m(v))i,

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this implies that by lettingk→+∞, sinceT_k(u−θT_m(v))→u−θT_m(v) for modular convergence inW₀¹LM(Ω),

hAu, u−θT_m(v)i ≤ hf, u−θT_m(v)i,

in which we can easily pass to the limit asθ→1andm→+∞to obtain hAu, u−vii ≤ hf, u−vi.

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4. The L

¹

Case

In this section, we study the same problems as before but we assume that q₋ andq₊are bounded.

Theorem 4.1. Let f ∈ L¹(Ω). Assume that the hypotheses are as in Theorem 3.1, q− andq₊ belong toL^∞(Ω). Then the problem (P_n)admits at least one solutionu_nsuch that:

u_n →ufor modular convergence inW₀¹L_M(Ω), whereuis the unique solution of the bilateral problem:

(Q)







hAu, v−ui ≥ Z

Ω

f(v−u)dx,∀v ∈K

u∈K ={v ∈W₀¹L_M(Ω) :q− ≤v ≤q₊a.e.}.

Proof. We sketch the proof since the steps are similar to those in Section3.

The existence ofu_n is given by Theorem 1 of [4]. Indeed, it is easy to see that

|g(x, s)| ≥ 1 on {|s| ≥ γ},where γ = max{supessq+,−infessq−} and so that

|g(x, s)|ⁿM |ζ|

µ

≥M |ζ|

µ

for|s| ≥γ.

Step 1: A priori estimates.

Choosingv =T_γ(u_n), as a test function in (P_n), and using the sign condition (3.4), we obtain

(4.1) α

Z

Ω

M

|∇T_γ(u_n)|

λ

dx≤γkfk₁

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and Z

{|un|>γ}

|g(x, un)|ⁿM

|∇u_n| µ

dx≤ kfk1, which gives

Z

{|u_n|>γ}

M

|∇u_n| µ

dx≤C and finally

(4.2)

Z

Ω

M

|∇un| max{λ, µ}

dx≤C.

On the other hand, as in Section3, we have (4.3)

Z

Ω

|g(x, u_n)|ⁿM

|∇u_n| µ

dx≤C.

Step 2: Almost everywhere convergence of the gradients.

Due to (4.2), there exists some u ∈ W₀¹L_M(Ω) such that (for a subsequence)

u_n * uweakly inW₀¹L_M(Ω)forσ(ΠL_M,ΠE_M).

Write

Au_n =f− |g(x, u_n)|ⁿ⁻¹g(x, u_n)M

|∇u_n| µ

and remark that, by (4.2), the second hand side is uniformly bounded in L¹(Ω). Then as in Section3

∇u_n→ ∇ua.e. inΩ.

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Step 3: u∈K ={v ∈W₀¹L_M(Ω) :q−≤v ≤q₊a.e. inΩ}.

Similarly, as in the proof of Theorem3.1, one can prove this step with the aid of property (4.3).

Step 4: Strong convergence of the truncations.

It is easy to see that the proof is the same as in Section3.

Step 5: uis the solution of the bilateral problem(Q).

Letv ∈ K and0 < θ < 1. Taking v_n = T_k(u_n −θv), k > 0 as a test function in (P_n), one can see that the proof is the same by replacingT_m(v) withv in Section3. We remark thatK ⊂L^∞(Ω).

Step 6: u_n→ufor modular convergence inW₀¹L_M(Ω).

We shall prove that∇u_n → ∇uin(L_M(Ω))^N for the modular convergence by using Vitali’s theorem.

LetE be a measurable subset ofΩ, we have for anyk >0 Z

E

M

|∇u_n| µ

dx

≤ Z

E∩{|un|≤k}

M

|∇u_n| µ

dx+

Z

E∩{|un|>k}

M

|∇u_n| µ

dx.

Let > 0. By virtue of the modular convergence of the truncates, there exists someη(, k)such that for anyEmeasurable

(4.4) |E|< η(, k)⇒ Z

E∩{|un|≤k}

M

|∇u_n| µ

dx <

2, ∀n.

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ChoosingT₁(u_n−T_k(u_n)), withk >0a test function in (P_n) we obtain:

hAu_n, T₁(u_n−T_k(u_n))i +

Z

Ω

|∇un| µ

T1(un−Tk(un))dx

= Z

Ω

f T₁(u_n−T_k(u_n))dx, which implies

Z

{|un|>k+1}

|g(x, un)|ⁿM

|∇u_n| µ

dx≤

Z

{|un|>k}

|f|dx.

Note thatmeas{x∈Ω :|u_n(x)|> k} →0uniformly onnwhenk → ∞.

We deduce then that there existsk =k()such that Z

{|un|>k}

|f|dx <

2, ∀n, which gives

Z

{|un|>k+1}

|g(x, u_n)|ⁿM

|∇u_n| µ

dx <

2, ∀n.

By settingt() = max{k+ 1, γ}we obtain (4.5)

Z

{|un|>t()}

M

|∇u_n| µ

dx <

2, ∀n.

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Combining (4.4) and (4.5) we deduce that there existsη >0such that Z

E

M

|∇u_n| µ

< , ∀nwhen|E|< η, E measurable, which shows the equi-integrability ofM_|∇u

n| µ

inL¹(Ω), and therefore we have

M

|∇u_n| µ

→M

|∇u|

µ

strongly inL¹(Ω).

By remarking that M

|∇u_n− ∇u|

2µ

≤ 1 2

M

|∇u_n| µ

+M

|∇u|

µ

one easily has, by using the Lebesgue theorem Z

Ω

M

|∇u_n− ∇u|

2µ

dx→0asn→+∞, which completes the proof.

Remark 4.1. The conditionb(0) = 0is not necessary. Indeed, takingθ_h(u_n), h >

0,as a test function in(Pn)with θ_h(s) =







hs if |s| ≤ _h¹ sgn(s) if |s| ≥ _h¹,

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we obtain Z

Ω

|∇u_n| µ

θh(un)dx≤ Z

Ω

f θh(un)dx.

and then, by lettingh→+∞, Z

Ω

|g(x, un)|ⁿM

|∇u_n| µ

dx≤C.

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References

[1] R. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.

[2] A. BENKIRANE AND A. ELMAHI, Almost everywhere convergence of the gradients of solutions to elliptic equations in Orlicz spaces and appli- cation, Nonlinear Anal. T.M.A., 28 (11) (1997), 1769–1784.

[3] A. BENKIRANE ANDA. ELMAHI, An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces, Nonlinear Anal. T.M.A., 36 (1999), 11–24.

[4] A. BENKIRANE AND A. ELMAHI, A strongly nonlinear elliptic equa- tion having natural growth terms andL¹data, Nonlinear Anal. T.M.A., 39 (2000), 403–411.

[5] A. BENKIRANE, A. ELMAHIANDD. MESKINE, On the limit of some nonlinear elliptic problems, Archives of Inequalities and Applications, 1 (2003), 207–220.

[6] L. BOCCARDO AND F. MURAT, Increase of power leads to bilateral problems, in Composite Media and Homogenization Theory, G. Dal Maso and G. F. Dell’Antonio (Eds.), World Scientific, Singapore, 1995, pp. 113–

123.

[7] A. DALL’AGLIOANDL. ORSINA, On the limit of some nonlinear elliptic equations involving increasing powers, Asympt. Anal., 14 (1997), 49–71.