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

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

Volume 6, Issue 2, Article 54, 2005

STRONGLY NONLINEAR ELLIPTIC UNILATERAL PROBLEMS IN ORLICZ SPACE AND L¹ DATA

L. AHAROUCH AND M. RHOUDAF

DÉPARTEMENT DEMATHÉMATIQUES ETINFORMATIQUE

FACULTÉ DESSCIENCESDHAR-MAHRAZ

B.P. 1796 ATLAS, FÈS, MAROC. l_aharouch@yahoo.fr rhoudaf_mohamed@yahoo.fr

Received 21 December, 2004; accepted 06 April, 2005 Communicated by A. Fiorenza

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,

whereAis a Leray-Lions operator from its domain D(A) ⊂ W₀¹LM(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 side fbelongs toL¹(Ω).

Key words and phrases: Orlicz Sobolev spaces, Boundary value problems, Truncations, Unilateral problems.

2000 Mathematics Subject Classification. 35J60.

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) ⊂ W₀¹LM(Ω), with M an N-function and whereg 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.

ISSN (electronic): 1443-5756

c 2005 Victoria University. All rights reserved.

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

250-04

In the case where f ∈ W⁻¹E_M(Ω), an existence theorem has been proved in [14] with the nonlinearitiesg depends only onxandu, and in [4] whereg 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 ong 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.

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^p case where there is no restriction can be found in [17], and in [12, 9, 8] in the case of obstacle problems, see also [18].

2. PRELIMINARIES

Let M : R⁺ → R⁺ be an N-function, i.e. M is continuous, convex, with M(t) > 0for t >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, witha(0) = 0, a(t)>0fort >0anda(t)tends to∞ast→

∞.

TheN-function M conjugate to M is defined byM = Rt

0 ¯a(s)ds, where ¯a : R⁺ → R⁺ is given by¯a(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 for t ≥ some t₀ > 0 then 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.

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

kukM,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 byEM(Ω).

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

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

kuk_1,M = X

|α|≤1

kD^αuk_M.

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 spaceD(Ω)inW¹E_M(Ω) and the spaceW₀¹L_M(Ω)as theσ(Q

L_M,Q

E_M)closure ofD(Ω)inW¹L_M(Ω).

Let W⁻¹L_M(Ω) (resp. W⁻¹E_M(Ω)) denote the space of distributions on Ωwhich can be written as sums of derivatives of order ≤ 1 of functions in L_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. LetF :R→Rbe uniformly Lipschitzian, withF(0) = 0. LetMbe anN-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 ofF⁰is 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 assume 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).

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 andQbeN-functions such thatQP, and letf : Ω×R→Rbe a Carathéodory function such that, for a.e.x∈Ω and 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_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).

We define T₀^1,M(Ω) to be the set of measurable functions u : Ω → R such that T_k(u) ∈ W₀¹L_M(Ω), whereT_k(s) = max(−k,min(k, s))fors ∈ Randk ≥ 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 functionv : Ω−→R^N such that

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

We will define the gradient ofuas the functionv, and we will denote it byv =∇u.

Lemma 2.5. Let λ ∈ R and let u and v be two measurable functions defined on Ω which are finite almost everywhere, and which are such that T_k(u), T_k(v)andT_k(u+λv)belong to W₀¹L_M(Ω)for everyk >0then

∇(u+λv) = ∇(u) +λ∇(v) a.e. inΩ

where∇(u), ∇(v)and∇(u+λv)are the gradients ofu,v andu+λv introduced in Lemma 2.4.

The proof of this lemma is similar to the proof of Lemma 2.12 in [10] for theL^pcase.

Below, we will use the following technical lemma.

Lemma 2.6. Let(fn), 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).

3. MAINRESULTS

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 twoN-functionsM andP withP M, functionc(x)inE_M(Ω),constants k₁, k₂, k₃, k₄such that, for a.e.xinΩand for alls∈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 element

v₀inK_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.

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

Furthermore letg : Ω×R×R^N → Rbe a Carathéodory function such that for a.e. x ∈ Ω and 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, andhis a given nonegative function inL¹(Ω).

Consider the following Dirichlet problem:

(3.3) A(u) +g(x, u,∇u) = f in Ω.

Remark 3.1. The condition(A₅)holds if one of the following conditions is verified.

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

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

We shall prove the following existence theorem.

Theorem 3.2. Assume that(A₁)–(A₅),(G₁)and(G₂)hold and f ∈L¹(Ω).Then there exists at least one solution of the following unilateral problem,

(P)



















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 ∈Kψ∩L^∞(Ω), ∀k > 0.

4. PROOF OFTHEOREM3.2 To prove the existence theorem, we proceed by steps.

STEP 1: Approximate unilateral problems.

Let us define

g_n(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 ∈K_ψ.

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

From Gossez and Mustonen ([15, Proposition 5]), the problem (P_n) 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]).

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

Ω

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

Z

Ω

gn(x, un,∇un)ϕk(Tl(un−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, un,∇un)∇(un−v0)ϕ⁰_k(Tl(un−v0))dx

≤ Z

{|un|≤k}

|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.

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 +

Z

Ω

δ(x)ϕ⁰_k(T_l(u_n−v₀))dx+ Z

Ω

f_nϕ_k(T_l(u_n−v₀))dx.

Since

{x∈Ω,|un(x)| ≤k} ⊆ {x∈Ω :|un−v0| ≤l}

and the fact thath, δ ∈L¹(Ω), furtherfnis 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, whereckis 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(|∇Tk(un)|)dx≤2ck.

SinceT_k(u_n)is bounded inW₀¹L_M(Ω), there exists somev_k∈W₀¹L_M(Ω)such that (4.3) Tk(un)* vk weakly in W₀¹LM(Ω) for σ(Q

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

STEP 3: Convergence in measure of u_n

Letk₀ ≥ kv₀k∞andk > k₀, takingv =u_n−T_k(u_n−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

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

Z

Ω

a(x, un,∇un)∇Tk(un−v0)dx≤k Z

{|un|≤k₀}

|gn(x, un,∇un)|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(|∇Tk0(un)|)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].

By(A₄), we obtain,

Z

{|un−v₀|≤k}

M(|∇un|)dx≤kc1, wherec₁is independent ofk, sincek is 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 that u_n converges to some function u in measure (and therefore, we can always assume that the convergence is a.e. after passing to a suitable subsequence). We shall show thatu_nis a Cauchy sequence in measure.

Letk > 0large enough, by Lemma 5.7 of [13], there exist two positive constantsc₃ andc₄ such 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{|u_n|> k}= Z

{|un|>k}

M(c₃T_k(u_n))dx ≤c₅k, hence

(4.8) meas(|u_n|> k)≤ c5k

M(c₃k) ∀n,∀k.

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

For everyλ >0, we have

(4.9) meas({|u_n−u_m|> λ})≤meas({|u_n|> k}) + meas({|u_m|> k})

+ meas({|T_k(u_n)−T_k(u_m)|> λ}).

Consequently, by (4.3) we can assume thatT_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), T_k(u_n)→T_k(u) strongly in E_M(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, which implies that

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

(4.11) Z

{|u_n−v₀|≤k}

a(x, u_n,∇u_n)(w− ∇v₀)dx

≤ Z

{|un−v0|≤k}

a(x, u_n,∇u_n)(∇u_n− ∇v₀)dx +

Z

{|un−v0|≤k}

a(x, u_n, w)(w− ∇u_n)dx.

We claim that, (4.12)

Z

{|u_n−v₀|≤k}

a(x, u_n,∇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, un,∇un)(∇un− ∇v0)dx+ Z

Ω

gn(x, un,∇un)Tk(un−v0)dx

≤ Z

Ω

f_nT_k(u_n−v₀)dx.

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∞(un)

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

{|un−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,∇u_n))χ{|u_n−v₀|≤k})_n remains bounded in(L_M(Ω))^N. Sincekis arbitrary, we deduce that(a(x, T_k(u_n),∇T_k(u_n)))_n is also bounded in(L_M(Ω))^N, which implies that, for allk > 0there exists a function h_k ∈ (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 byχ_rthe characteristic function ofΩ_r. Clearly,Ω_r⊂Ω_r+1andmeas(Ω\Ω_r)−→0asr −→ ∞.

Fixrand lets≥r, we have, 0≤

Z

Ωr

[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 (4.17)

≤ Z

Ωs

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

By(A₅)there exists a sequencev_j ∈K_ψ∩W₀¹E_M(Ω)∩L^∞(Ω)which converges toT_k(u)for the modular converge inW₀¹L_M(Ω).

Here, we define

w_n,j^h =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^h_n,j as test function in (P_n), we obtain,

A(un), ηϕk w^h_n,j +

Z

Ω

gn(x, un,∇un)ηϕk w_n,j^h dx≤

Z

Ω

fnηϕk 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.

It follows that (4.20)

Z

Ω

a(x, un,∇un)∇w^h_n,jϕ⁰_k w^h_n,j dx+

Z

Ω

gn(x, un,∇un)ϕk w_n,j^h dx

≤ Z

Ω

f_nϕ_k w_n,j^h dx.

Note that, ∇w_n,j^h = 0 on the set where |un| > 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 amongn, j andh, 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 onn, 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_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, In view of (4.10), we haveϕ_k w_n,j^h

→ϕ_k(w^h_j)weakly^∗ asn →+∞inL^∞(Ω), and then Z

Ω

f_nϕ_k w_n,j^h dx→

Z

Ω

f ϕ_k(w_j^h)dx as n →+∞.

Again tendsjto infinity, we get Z

Ω

f ϕ_k(w^h_j)dx→ Z

Ω

f ϕ_k(w^h)dx as j →+∞, finally lettinghthe infinity, we deduce by using the Lebesgue TheoremR

Ωf ϕ_k(w^h)dx→0.

So that

Z

Ω

f_nϕ_k w_n,j^h

dx=(n, j, h).

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

≥0, we deduce from (4.20) that

(4.21)

Z

Ω

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

Z

{|u_n|≤k}

g_n(x, u_n,∇u_n)ϕ_k w_n,j^h

dx≤(n, j, h).

Splitting the first integral on the left hand side of (4.21) where|u_n| ≤ k and|u_n| > k, we can write,

(4.22) Z

Ω

a(x, Tm(un),∇Tm(un))∇w^h_n,jϕ⁰_k w^h_n,j 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

{|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

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

{|u_n|>k}

|a(x, Tk(un),0)||∇Tk(vj)|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).

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

Z

{|u_n|>k}

a(x, T_m(u_n),∇T_m(u_n))∇w_n,j^h ϕ⁰_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 inL_M(Ω), we have, for a subsequence

|a(x, T_m(u_n),∇T_m(u_n))|* l_m weakly inL_M(Ω)inσ(L_M, E_M)asntends to infinity, and since

|∇T_k(v_j)|χ_{|un|>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}

lm|∇Tk(vj)|dx → −ϕ⁰_k(2k) Z

{|u|>k}

lm|∇Tk(u)|dx= 0 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

{|u_n−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^h_n,jϕ⁰_k w^h_n,j 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, T_m(u_n),∇T_m(u_n))∇w^h_n,jϕ⁰_k w^h_n,j dx

≥ Z

Ω

a(x, Tk(un),∇Tk(un))−a x, Tk(un),∇Tk(vj)χ^j_s

×

∇T_k(u_n)− ∇T_k(v_j)χ^j_s

ϕ⁰_k w^h_n,j dx +

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_n,j^h dx

− Z

Ω\Ω^j_s

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

+(n, h) +(n, j) +_h(n, j), whereχ^j_sdenotes the characteristic function of the subsetΩ^j_s ={x∈Ω :|∇T_k(v_j)| ≤s}.

By (4.16) and the fact that ∇Tk(vj)χ_Ω\Ω^j

sϕ⁰_k w^h_n,j

tends to ∇Tk(vj)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^js

h_k∇T_k(v_j)ϕ⁰_k w_j^h 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^h_j dx →

Z

Ω\Ω_s

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

Finally

(4.29) −

Z

Ω\Ω^j_s

a(x, Tk(un),∇Tk(un))∇Tk(vj)ϕ⁰_k w^h_n,j dx

=− Z

Ω\Ω_s

h_k∇T_k(u)ϕ⁰_k(w^h)dx+_h(n, j).

Concerning the second term on the right hand side of (4.28) we can write (4.30)

Z

Ω

a x, Tk(un),∇Tk(vj)χ^j_s ∇Tk(un)− ∇Tk(vj)χ^j_s

ϕ⁰_k w^h_n,j dx

= Z

Ω

a x, T_k(u_n),∇T_k(v_j)χ^j_s

∇T_k(u_n)ϕ⁰_k(T_k(u_n)−T_k(v_j))dx

− Z

Ω

a(x, T_k(u_n),∇T_k(v_j)χ^j_s)∇T_k(v_j)χ^j_sϕ⁰_k w_n,j^h dx.

The first term on the right hand side of (4.30) tends to the quantity Z

Ω

a x, T_k(u),∇T_k(v_j)χ^j_s

∇T_k(u)ϕ⁰_k(T_k(u)−T_k(v_j))dx as n → ∞ since

a x, T_k(u_n),∇T_k(v_j)χ^j_s

ϕ⁰_k(T_k(u_n)−T_k(v_j))

→a x, T_k(u),∇T_k(v_j)χ^j_s

ϕ⁰_k(T_k(u)−T_k(v_j))

strongly in (E_M(Ω))^N by Lemma 2.3 and ∇T_k(u_n) * ∇T_k(u) weakly in (L_M(Ω))^N for σ(Q

L_M,Q E_M).

For the second term on the right hand side of (4.30) it is easy to see that

(4.31) Z

Ω

a x, Tk(un),∇Tk(vj)χ^j_s

∇Tk(vj)χ^j_sϕ⁰_k w^h_n,j dx

−→

Z

Ω

a x, T_k(u),∇T_k(v_j)χ^j_s

∇T_k(v_j)χ^j_sϕ⁰_k w^h_j dx.

as n→ ∞.

Consequently, we have

(4.32) 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

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

ϕ⁰_k w^h_j

dx+_j,h(n) since

∇Tk(vj)χ^j_sϕ⁰_k(w_j^h)→ ∇Tk(u)χsϕ⁰_k(w^h) strongly in(E_M(Ω))^N asj →+∞, it is easy to see that

Z

Ω

a x, Tk(u),∇Tk(vj)χ^j_s ∇Tk(u)− ∇Tk(vj)χ^j_s)

ϕ⁰_k w_j^h dx

−→

Z

Ω\Ωs

a(x, T_k(u),0)∇T_k(u)ϕ⁰_k(w^h)dx

asj →+∞, thus

(4.33) 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_s

a(x, Tk(u),0)∇Tk(u)ϕ⁰_k(0)dx+(n, j).

Combining (4.28), (4.29) and (4.32), we get

(4.34) Z

Ω

a(x, Tm(un),∇Tm(un))∇w^h_n,jϕ⁰_k w^h_n,j 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

− Z

Ω\Ω_s

hk∇Tk(u)ϕ⁰_k(0)dx+ Z

Ω\Ω_s

a(x, Tk(u),0)∇Tk(u)ϕ⁰_k(0)dx+(n, j, h).

We now, turn to the second term on the left hand side of (4.21), we have Z

{|un|≤k}

g_n(x, u_n,∇u_n)ϕ_k w_n,j^h dx (4.35)

≤b(k) Z

Ω

(h(x) +M(|∇T_k(u_n)|))

ϕ_k w^h_n,j dx

≤b(k) Z

Ω

h(x)|ϕ_k w_n,j^h

|dx+ b(k) α

Z

Ω

δ(x)

ϕ_k w_n,j^h dx + b(k)

α Z

Ω

a(x, T_k(u_n),∇T_k(u_n))∇T_k(u_n)

ϕ_k w_n,j^h dx

− b(k) α

Z

Ω

a(x, T_k(u_n),∇T_k(u_n))∇v₀

ϕ_k w^h_n,j dx

≤(n, j, h) + b(k) α

Z

Ω

a(x, T_k(u_n),∇T_k(u_n))∇T_k(u_n)

ϕ_k w^h_n,j dx.

The last term on the last side of this inequality reads as (4.36) b(k)

α 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 ϕ_k w^h_n,j dx + b(k)

α 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) ϕ_k w^h_n,j dx + b(k)

α Z

Ω

a(x, Tk(un),∇Tk(un))∇Tk(vj)χ^j_s

ϕk w_n,j^h dx and reasoning as above, it is easy to see that

b(k) α

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 ϕ_k w_n,j^h

dx=(n, j) and

−b(k) α

Z

Ω

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

ϕ_k w^h_n,j

dx=(n, j, h).

So that (4.37)

Z

{|un|≤k}

g_n(x, u_n,∇u_n)ϕ_k w^h_n,j dx

≤ b(k) α

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 ϕ_k w_n,j^h

dx+(n, j, h).

Combining (4.21), (4.34) and (4.37), we obtain (4.38)

Z

Ω

a(x, Tk(un),∇Tk(un))−a x, Tk(un),∇Tk(vj)χ^j_s

×

∇T_k(u_n)− ∇T_k(v_j)χ^j_s

ϕ⁰_k w_n,j^h

− b(k) α

ϕ_k w^h_n,j

dx

≤ Z

Ω\Ω_s

hk∇Tk(u)ϕ⁰_k(0)dx+ Z

Ω\Ω_s

a(x, Tk(u),0)∇Tk(u)ϕ⁰_k(0)dx+(n, j, h),