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

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http://jipam.vu.edu.au/

Volume 4, Issue 5, Article 98, 2003

ASYMPTOTIC BEHAVIOUR OF SOME EQUATIONS IN ORLICZ SPACES

D. MESKINE AND A. ELMAHI

DÉPARTEMENT DEMATHÉMATIQUES ETINFORMATIQUE

FACULTÉ DESSCIENCESDHAR-MAHRAZ

B.P 1796 ATLAS-FÈS,FÈSMAROC. [email protected] C.P.RDEFÈS, B.P 49, FÈS, MAROC.

Received 26 March, 2003; accepted 05 August, 2003 Communicated by A. Fiorenza

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

whereA is a standard Leray-Lions operator defined on W₀¹LM(Ω), with M an N-function which satisfies the ∆2-condition, and where K is a convex subset of W₀¹LM(Ω) with ob- stacles 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 solutionsunof some approximate problem with nonlinearity term of the form

|g(x, un)|ⁿ⁻¹g(x, un)×M(|∇un|).

Key words and phrases: Strongly nonlinear elliptic equations, Natural growth, Truncations, Variational inequalities, Bilateral problems.

2000 Mathematics Subject Classification. 35J25, 35J60.

1. INTRODUCTION

LetΩbe an open bounded subset ofR^N, N ≥ 2, with the segment property. Consider the following obstacle problem:

(P)







hAu, v−ui ≥ hf, v−ui, ∀v ∈K, u∈K,

where A(u) = −div(a(x, u,∇u)) is a Leray-Lions operator defined on W₀¹L_M(Ω), with M being an N-function which satisfies the ∆₂-condition and where K is a convex subset of W₀¹L_M(Ω).

ISSN (electronic): 1443-5756

040-03

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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) un∈W₀^1,p(Ω)∩Lⁿ(Ω).

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







Aun+|g(x, un)|ⁿ⁻¹g(x, un)M(|∇un|) =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 on g. In the first we takef ∈W⁻¹E_M(Ω)and next inL¹(Ω).

2. PRELIMINARIES

2.1. N−Functions. LetM :R⁺ →R⁺be anN-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) > 0 fort > 0 anda(t)tends to ∞as t→ ∞.

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

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

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

(2.1) M(2t)≤kM(t) ∀t≥0;

when (2.1) holds only for t ≥ some t₀ > 0 then M is said to satisfy the ∆₂ condition near infinity.

We will extend theseN-functions into even functions on allR.

LetP andQbe twoN-functions. P Qmeans thatP grows essentially less rapidly than Q, 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.

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2.2. Orlicz spaces. LetΩbe an open subset ofR^N. The Orlicz classK_M(Ω)(resp. the Orlicz spaceL_M(Ω)) is defined as the set of (equivalence classes of) real valued measurable functions uonΩsuch 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 compact support in Ωis denoted byE_M(Ω).

The equalityEM(Ω) =LM(Ω)holds if only ifM satisfies the∆2condition, for alltor fort large according to whetherΩhas infinite measure or not.

The dual ofE_M(Ω) can be identified withL_M(Ω)by means of the pairingR

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

The spaceL_M(Ω)is reflexive if and only if M andM satisfy the∆₂ condition, for all tor fortlarge, according to whetherΩhas infinite measure or not.

2.3. Orlicz-Sobolev spaces. We now turn to the Orlicz-Sobolev space, W¹L_M(Ω) (resp.

W¹EM(Ω)) is the space of all functionsu such that u and its distributional derivatives up to order 1 lie inLM(Ω)(resp.EM(Ω)). It is a Banach space under the norm

kuk_1,M = X

|α|≤1

kD^αuk_M.

Thus,W¹L_M(Ω)andW¹E_M(Ω) can be identified with subspaces of 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

LM,Q L_M).

The spaceW₀¹EM(Ω)is defined as the (norm) closure of the Schwarz spaceD(Ω)inW¹EM(Ω) and the spaceW₀¹L_M(Ω)as theσ(Q

L_M,Q

E_M)closure ofD(Ω)inW¹L_M(Ω).

We say thatu_nconverges toufor the modular convergence inW¹L_M(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 spacesW⁻¹LM¯ (Ω)andW⁻¹EM¯ (Ω). LetW⁻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 inL_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 inW₀¹L_M(Ω)for the modular convergence and thus for the topology σ(Q

L_M,Q

L_M) (cf. [8, 9]). Consequently, the action of a distribution inW⁻¹L_M(Ω)on an element ofW₀¹L_M(Ω)is well defined.

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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 → R be uniformly Lipschitzian, with F(0) = 0. Let M be an N−function and let u ∈ W¹L_M(Ω) (resp. W¹E_M(Ω)). Then F(u) ∈ W¹L_M(Ω) (resp.

W¹E_M(Ω)). Moreover, if the setDof discontinuity points ofF⁰ 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. Let F : R → R be uniformly Lipschitzian, with F(0) = 0. We suppose that the set of discontinuity points of F⁰ is finite. Let M be an N-function, then the mapping F : W¹L_M(Ω) → W¹L_M(Ω) is sequentially continuous with respect to the weak* topology σ(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.

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

|f(x, s)| ≤c(x) +k₁P⁻¹M(k₂|s|), wherek₁, k₂are real constants andc(x)∈E_Q(Ω).

Then the Nemytskii operator N_f defined by N_f(u)(x) = f(x, u(x))is strongly continuous from

P

EM(Ω), 1 k₂

=

u∈LM(Ω) :d(u, EM(Ω))< 1 k₂

intoE_Q(Ω).

3. THEMAINRESULT

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

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

Let A(u) = −div(a(x,∇u)) be a Leray-Lions operator defined on W₀¹LM(Ω) into W⁻¹L_M(Ω), where a : Ω×R^N → R^N is a Carathéodory function satisfying for a.e. x ∈ Ω and 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

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(3.5) |g(x, s)| ≤b(|s|)

(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) = 0and 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, Tk(s) = max(−k,min(k, s)).

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







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) 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

(P)







hAu, v−ui ≥ hf, v−ui, ∀v ∈K

u∈K ={v ∈W₀¹L_M(Ω) :q− ≤v ≤q₊ a.e.},

Remark 3.2. If the function s → 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 (Pn), 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

|∇un| λ

dx≤C, and Z

Ω

a(x, u_n,∇u_n)∇u_ndx ≤C, (3.9) (a(x, u_n,∇u_n))is bounded in(L_M(Ω))^N,

(3.10)

Z

Ω

|g(x, u_n)|ⁿ⁻¹g(x, u_n)M

|∇un| µ

u_ndx≤C.

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We then deduce Z

{|un|>k}

|g(x, u_n)|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∆₂condition there exist two positive constantsKandK⁰such that

M t

µ

≤KM t

λ

+K⁰ for all t≥0, which implies

Z

{|un|≤δ}

|g(x, u_n)|ⁿM

|∇u_n| µ

dx≤

Z

{|un|≤δ}

K⁰ +KM

|∇u_n| λ

dx.

Consequently from (3.8) (3.11)

Z

Ω

|g(x, u_n)|ⁿM

|∇un| µ

dx≤C, for all n.

Step 2: Almost everywhere convergence of the gradients.

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

(3.12) un * uweakly inW₀¹LM(Ω)forσY

LM,Y E_M

, 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) ∇un→ ∇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, which gives

Z

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

|g(x, u_n)|ⁿM

|∇u_n| µ

|dx≤C and

Z

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

M

|∇u_n| µ

dx≤ C kⁿ

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wherek >1. Lettingn →+∞forkfixed, 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) where z_n=T_k(u_n)−T_k(u) gives

hAu_n, φ(z_n)i+ Z

Ω

|∇u_n| µ

φ(z_n)dx=hf, φ(z_n)i which implies, by using the fact thatg(x, u_n)φ(z_n)≥0on{x∈Ω :|u_n|> k}, hAun, φ(zn)i+

Z

{0≤un≤T_k(u)}∩{|un|≤k}

|g(x, un)|ⁿ⁻¹g(x, un)M

|∇u_n| µ

φ(zn)dx +

Z

{T_k(u)≤un≤0}∩{|un|≤k}

|∇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 by I_n,k¹ and I_n,k² and _i(n) denote various sequences of real numbers which tend to 0 as n →+∞.

On the one hand we have I_n,k¹

≤ Z

{0≤un≤T_k(u)}∩{|un|≤k}

|g(x, un)|ⁿM

|∇u_n| µ

|φ(zn)|dx

≤ Z

{0≤un≤u}∩{|un|≤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

{|un|≤k}

M

|∇u_n| µ

|φ(z_n)|dx.

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,∇Tk(un))∇Tk(un)|φ(zn)|dx.

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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

{|un|≤k}

− Z

{|u_n|>k}

a(x,∇un)∇Tk(u)φ⁰(zn)dx.

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{|un|>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))_nis bounded in(L_M(Ω))^N while∇T_k(u)χ{|un|>k} tends to 0 strongly in(E_M(Ω))^N.

We define for everys > 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,∇T_k(u_n))∇T_k(u)χΩ\Ω_sφ⁰(z_n)dx.

The second term of the right hand side of (3.17) tends to 0 since

a(x,∇Tk(un)χs)φ⁰(zn)→a(x,∇Tk(u)χs)strongly in(E_M(Ω))^N by Lemma 2.3 and

∇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_sdxasn→ ∞since a(x,∇T_k(u_n))* a(x,∇T_k(u))weakly forσY

E_M(Ω),Y

L_M(Ω) .

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Consequently, from (3.16) we have (3.18)

Z

Ω

= 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+₂(n).

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

Ω

[a(x,∇Tk(un))−a(x,∇Tk(u)χs)]

×[∇Tk(un)− ∇Tk(u)χs]

φ⁰(zn)− 2K

α |φ(zn)|

dx

≤₃(n) + Z

Ω

a(x,∇T_k(u))∇T_k(u)χΩ\Ωsdx, 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)χΩ\Ωsdx.

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,∇T_k(u))∇T_k(u)χ_Ω\Ω_sdx.

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

n→+∞

Z

Ω

a(x,∇T_k(u_n))∇T_k(u_n)dx

≤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)χΩ\Ωsdx.

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∇Tk(un)tends weakly to∇Tk(u).

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

Ωa(x,∇Tk(u))∇Tk(u)χsdx since

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

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forσ(Q

L_M,Q

E_M)while∇T_k(u)χ_s ∈E_M(Ω). We deduce then 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)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 lettings → ∞we get, since meas(Ω\Ωs)→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 which gives, by using Fatou’s lemma,

(3.20) lim

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.

On the other hand, we have M

|∇T_k(u_n)|

µ

≤K⁰+K α

Z

Ω

a(x,∇Tk(un))∇Tk(un)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

|∇Tk(un)− ∇Tk(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) Tk(un)→Tk(u)for modular convergence inW₀¹LM(Ω).

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

Choosingw=Tk(un−θTm(v))as a test function in (Pn), wherev ∈Kand0< θ <1, gives

hAu_n, T_k(u_n−θT_m(v))i+ Z

Ω

|∇u_n| µ

T_k(u_n−θT_m(v))dx

=hf, Tk(un−θTm(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≤u_n≤θT_m(v)}

|∇un| µ

T_k(u_n−θT_m(v))dx +

Z

{θT_m(v)≤u_n≤0}

|∇u_n| µ

Tk(un−θTm(v))dx.

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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

δ1,m(x) = sup

0≤s≤θT_m(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.

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}

|δ_2,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,

this implies that by lettingk→+∞, sinceTk(u−θTm(v))→u−θTm(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.

4. THEL¹ CASE

In this section, we study the same problems as before but we assume that q− and q+ 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),

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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 Section 3.

The existence ofu_nis given by Theorem 1 of [4]. Indeed, it is easy to see that|g(x, s)| ≥1on {|s| ≥γ},whereγ = max{supessq₊,−infessq−}and so that

|g(x, s)|ⁿM |ζ|

µ

≥M |ζ|

µ

for|s| ≥γ.

Step 1: A priori estimates.

Choosingv =Tγ(un), as a test function in (Pn), and using the sign condition (3.4), we obtain

(4.1) α

Z

Ω

M

|∇T_γ(u_n)|

λ

dx ≤γkfk1

and

Z

{|un|>γ}

|g(x, u_n)|ⁿM

|∇u_n| µ

dx ≤ kfk₁, which gives

Z

{|un|>γ}

M

|∇u_n| µ

dx≤C and finally

(4.2)

Z

Ω

M

|∇u_n| max{λ, µ}

dx≤C.

On the other hand, as in Section 3, 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 someu∈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 inL¹(Ω). Then as in Section 3

∇u_n→ ∇ua.e. inΩ.

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

Similarly, as in the proof of Theorem 3.1, 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 Section 3.

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

Let v ∈ K and 0 < 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)withvin Section 3. 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.

LetEbe 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 some η(, k)such that for anyEmeasurable

(4.4) |E|< η(, k)⇒ Z

E∩{|u_n|≤k}

M

|∇un| µ

dx <

2, ∀n.

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

hAun, T1(un−Tk(un))i+ Z

Ω

|∇u_n| 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.

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

|∇un| µ

→M

|∇u|

µ

strongly inL¹(Ω).

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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

|∇un− ∇u|

2µ

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

Remark 4.2. 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, we obtain

Z

Ω

|∇u_n| µ

θh(un)dx≤ Z

Ω

f θh(un)dx.

and then, by lettingh→+∞, Z

Ω

|g(x, u_n)|ⁿM

|∇u_n| µ

dx≤C.

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

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