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

Volume 6, Issue 2, Article 54, 2005

**STRONGLY NONLINEAR ELLIPTIC UNILATERAL PROBLEMS IN ORLICZ**
**SPACE AND** L^{1} **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) ⊂ W_{0}^{1}LM(Ω) into W^{−1}E_{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 toL^{1}(Ω).

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

*2000 Mathematics Subject Classification. 35J60.*

**1. I****NTRODUCTION**

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 ∈ L^{1}(Ω), Au = −diva(x, u,∇u) is a Leray-Lions operator defined on its domain
D(A) ⊂ W_{0}^{1}LM(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^{−1}E_{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 ∈ L^{1}(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. P****RELIMINARIES**

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 t_{0} > 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,W^{1}L_{M}(Ω) (resp. W^{1}E_{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^{α}uk_{M}.

Thus,W^{1}L_{M}(Ω)andW^{1}E_{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 spaceW_{0}^{1}E_{M}(Ω)is defined as the (norm) closure of the Schwartz spaceD(Ω)inW^{1}E_{M}(Ω)
and the spaceW_{0}^{1}L_{M}(Ω)as theσ(Q

L_{M},Q

E_{M})closure ofD(Ω)inW^{1}L_{M}(Ω).

Let W^{−1}L_{M}(Ω) (resp. W^{−1}E_{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. Let*F :R→R

*be uniformly Lipschitzian, with*F(0) = 0. LetM

*be an*N

*-function*

*and let*u ∈ W

^{1}L

_{M}(Ω)

*(resp.*W

^{1}E

_{M}(Ω)). Then F(u) ∈ W

^{1}L

_{M}(Ω)

*( resp.*W

^{1}E

_{M}(Ω)).

*Moreover, if the set*D*of discontinuity points of*F^{0}*is finite, then*

∂

∂x_{i}F(u) =

( F^{0}(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*F

^{0}

*is finite. Let*M

*be an*N-function, then the mapping F : W

^{1}L

_{M}(Ω) → W

^{1}L

_{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. Let*Ω

*be an open subset of*R

^{N}

*with finite measure. Let*M, P

*and*Q

*be*N

*-functions*

*such that*QP

*, and let*f : Ω×R→R

*be a Carathéodory function such that, for a.e.*x∈Ω

*and all*s∈R

*:*

|f(x, s)| ≤c(x) +k_{1}P^{−1}M(k_{2}|s|),

*where*k_{1}, k_{2} *are real constants and*c(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
k_{2}

=

u∈LM(Ω) :d(u, EM(Ω))< 1
k_{2}

*into*E_{Q}(Ω).

We define T_{0}^{1,M}(Ω) to be the set of measurable functions u : Ω → R such that T_{k}(u) ∈
W_{0}^{1}L_{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 every*u∈ T

_{0}

^{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 of*u*as the function*v, 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)

*and*T

_{k}(u+λv)

*belong to*W

_{0}

^{1}L

_{M}(Ω)

*for every*k >0

*then*

∇(u+λv) = ∇(u) +λ∇(v) *a.e. in*Ω

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

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

Below, we will use the following technical lemma.

* Lemma 2.6. Let*(fn), f ∈L

^{1}(Ω)

*such that*(i) f

_{n}≥0

*a.e. in*Ω

(ii) f_{n}→f *a.e. in*Ω
(iii) R

Ωf_{n}(x)dx→R

Ωf(x)dx
*then*f_{n} →f *strongly in*L^{1}(Ω).

**3. M****AIN****R****ESULTS**

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∈W_{0}^{1}L_{M}(Ω); u≥ψ a.e. in Ω},

this convex set is sequentially σ(ΠL_{M},ΠE_{M}) closed inW_{0}^{1}L_{M}(Ω) (see [15]). We now state
conditions on the differential operator

(3.2) Au=−div(a(x, u,∇u))

(A_{1}) a(x, s, ξ) : Ω×R×R^{N} →R^{N} is a Carathéodory function.

(A_{2}) There exist twoN-functionsM andP withP M, functionc(x)inE_{M}(Ω),constants
k_{1}, k_{2}, k_{3}, k_{4}such that, for a.e.xinΩand for alls∈R, ζ ∈R^{N}

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

(A_{3}) [a(x, s, ζ)−a(x, s, ζ^{0})](ζ−ζ^{0})>0 for a.e. xinΩ,sinRandζ, ζ^{0} inR^{N}, withζ 6=ζ^{0}.
(A_{4}) There existδ(x)inL^{1}(Ω), strictly positive constantαsuch that, for some fixed element

v_{0}inK_{ψ}∩W_{0}^{1}E_{M}(Ω)∩L^{∞}(Ω),

a(x, s, ζ)(ζ−Dv_{0})≥αM(|ζ|)−δ(x)
for a.e. xinΩ, and alls∈R, ζ ∈R^{N}.

(A_{5}) For eachv ∈K_{ψ}∩L^{∞}(Ω)there exists a sequencev_{n}∈K_{ψ}∩W_{0}^{1}E_{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}

(G_{1}) g(x, s, ζ)s≥0

(G_{2}) |g(x, s, ζ)| ≤b(|s|) (h(x) +M(|ζ|)),

whereb:R+→R+is a continuous non decreasing function, andhis a given nonegative
function inL^{1}(Ω).

Consider the following Dirichlet problem:

(3.3) A(u) +g(x, u,∇u) = f in Ω.

**Remark 3.1. The condition**(A_{5})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 ∈W_{0}^{1}E_{M}(Ω), (see [15, Proposition 10]).

We shall prove the following existence theorem.

* Theorem 3.2. Assume that*(A

_{1})

*–*(A

_{5}),(G

_{1})

*and*(G

_{2})

*hold and*f ∈L

^{1}(Ω).

*Then there exists*

*at least one solution of the following unilateral problem,*

(P)

u∈ T_{0}^{1,M}(Ω), u≥ψ a.e. *in* Ω,
g(x, u,∇u)∈L^{1}(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. P****ROOF OF****T****HEOREM****3.2**
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 + _{n}^{1}|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_{n}is a regular function such thatf_{n}strongly converges tof inL^{1}(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^{γs}^{2}, where γ =
b(k)

α

2

. It is well known that

(4.1) ϕ^{0}_{k}(s)− b(k)

α |ϕ_{k}(s)| ≥ 1

2, ∀s ∈R (see [9]).

Takingu_{n}−ηϕ_{k}(T_{l}(u_{n}−v_{0}))as test function in (P_{n}), wherel =k+kv_{0}k∞, we obtain,
Z

Ω

a(x, u_{n},∇u_{n})∇T_{l}(u_{n}−v_{0})ϕ^{0}_{k}(T_{l}(u_{n}−v_{0}))dx
+

Z

Ω

gn(x, un,∇un)ϕk(Tl(un−v0))dx

≤ Z

Ω

f_{n}ϕ_{k}(T_{l}(u_{n}−v_{0}))dx.

Since

g_{n}(x, u_{n},∇u_{n})ϕ_{k}(T_{l}(u_{n}−v_{0}))≥0

on the subset{x∈Ω :|u_{n}(x)|> k},then
Z

{|un−v_{0}|≤l}

a(x, un,∇un)∇(un−v0)ϕ^{0}_{k}(Tl(un−v0))dx

≤ Z

{|un|≤k}

|g_{n}(x, u_{n},∇u_{n})||ϕ_{k}(T_{l}(u_{n}−v_{0}))|dx+
Z

Ω

f_{n}ϕ_{k}(T_{l}(u_{n}−v_{0}))dx.

By using(A_{4})and(G_{1}), we have
α

Z

{|un−v0|≤l}

M(|∇u_{n}|)ϕ^{0}_{k}(T_{l}(u_{n}−v_{0}))dx

≤b(|k|) Z

Ω

(h(x) +M(∇T_{k}(u_{n})))|ϕ_{k}(T_{l}(u_{n}−v_{0}))|dx
+

Z

Ω

δ(x)ϕ^{0}_{k}(T_{l}(u_{n}−v_{0}))dx+
Z

Ω

f_{n}ϕ_{k}(T_{l}(u_{n}−v_{0}))dx.

Since

{x∈Ω,|un(x)| ≤k} ⊆ {x∈Ω :|un−v0| ≤l}

and the fact thath, δ ∈L^{1}(Ω), furtherfnis bounded inL^{1}(Ω), then
Z

Ω

M(|∇T_{k}(u_{n})|)ϕ^{0}_{k}(T_{l}(u_{n}−v_{0}))dx≤ b(k)
α

Z

Ω

M(|∇T_{k}(u_{n})|)|ϕ_{k}(T_{l}(u_{n}−v_{0}))|dx+c_{k},
whereckis a positive constant depending onk, which implies that

Z

Ω

M(|∇T_{k}(u_{n})|)

ϕ^{0}_{k}(T_{l}(u_{n}−v_{0}))− b(k)

α |ϕ_{k}(T_{l}(u_{n}−v_{0}))|

dx≤c_{k}.
By using (4.1), we deduce,

(4.2)

Z

Ω

M(|∇Tk(un)|)dx≤2ck.

SinceT_{k}(u_{n})is bounded inW_{0}^{1}L_{M}(Ω), there exists somev_{k}∈W_{0}^{1}L_{M}(Ω)such that
(4.3) Tk(un)* vk weakly in W_{0}^{1}LM(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}

Letk_{0} ≥ kv_{0}k∞andk > k_{0}, takingv =u_{n}−T_{k}(u_{n}−v_{0})as test function in (P_{n}) gives,
(4.4)

Z

Ω

a(x, u_{n},∇u_{n})∇T_{k}(u_{n}−v_{0})dx+
Z

Ω

g_{n}(x, u_{n},∇u_{n})T_{k}(u_{n}−v_{0})dx

≤ Z

Ω

fnTk(un−v0)dx,
sinceg_{n}(x, u_{n},∇u_{n})T_{k}(u_{n}−v_{0})≥0on the subset{x∈Ω,|u_{n}(x)|> k_{0}},hence (4.4) implies
that,

Z

Ω

a(x, un,∇un)∇Tk(un−v0)dx≤k Z

{|un|≤k_{0}}

|gn(x, un,∇un)|dx+kkfk_{L}^{1}_{(Ω)}

which gives, by using(G_{1}),
(4.5)

Z

Ω

a(x, u_{n},∇u_{n})∇T_{k}(u_{n}−v_{0})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}−v_{0})dx≤k[c_{k}_{0}+c].

By(A_{4}), we obtain,

Z

{|un−v_{0}|≤k}

M(|∇un|)dx≤kc1,
wherec_{1}is independent ofk, sincek is arbitrary, we have

Z

{|un|≤k}

M(|∇u_{n}|)dx≤
Z

{|un−v_{0}|≤k+kv_{0}k∞}

M(|∇u_{n}|)dx≤kc_{2},
i.e.,

(4.6)

Z

Ω

M(|∇T_{k}(u_{n})|)dx≤kc_{2}.

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_{n}is a Cauchy sequence in measure.

Letk > 0large enough, by Lemma 5.7 of [13], there exist two positive constantsc_{3} andc_{4}
such that

(4.7)

Z

Ω

M(c_{3}T_{k}(u_{n}))dx≤c_{4}
Z

Ω

M(|∇T_{k}(u_{n})|)dx≤kc_{5},
then, we deduce, by using (4.7) that

M(c_{3}k) meas{|u_{n}|> k}=
Z

{|un|>k}

M(c_{3}T_{k}(u_{n}))dx ≤c_{5}k,
hence

(4.8) meas(|u_{n}|> k)≤ c5k

M(c_{3}k) ∀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 ≥ h_{0}(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 W_{0}^{1}L_{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}))

_{n}

*in*(L

_{M}(Ω))

^{N}.

Letw∈(E_{M}(Ω))^{N} be arbitrary, by(A_{3})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− ∇v_{0})≤a(x, u_{n},∇u_{n})(∇u_{n}− ∇v_{0})−a(x, u_{n}, w)(∇u_{n}−w)
and integrating on the subset{x∈Ω,|u_{n}−v_{0}| ≤k}, we obtain,

(4.11) Z

{|u_{n}−v_{0}|≤k}

a(x, u_{n},∇u_{n})(w− ∇v_{0})dx

≤ Z

{|un−v0|≤k}

a(x, u_{n},∇u_{n})(∇u_{n}− ∇v_{0})dx
+

Z

{|un−v0|≤k}

a(x, u_{n}, w)(w− ∇u_{n})dx.

We claim that, (4.12)

Z

{|u_{n}−v_{0}|≤k}

a(x, u_{n},∇u_{n})(∇u_{n}−v_{0})dx ≤c_{10},
wherec_{10}is a positive constant depending onk.

Indeed, if we takev =u_{n}−T_{k}(u_{n}−v_{0})as test function in (P_{n}), we get,
Z

{|un−v_{0}|≤k}

a(x, un,∇un)(∇un− ∇v0)dx+ Z

Ω

gn(x, un,∇un)Tk(un−v0)dx

≤ Z

Ω

f_{n}T_{k}(u_{n}−v_{0})dx.

Sinceg_{n}(x, u_{n},∇u_{n})T_{k}(u_{n}−v_{0})≥0on the subset{x∈Ω, |u_{n}| ≥ kv_{0}k∞},which implies
(4.13)

Z

{|u_{n}−v_{0}|≤k}

a(x, u_{n},∇u_{n})(∇u_{n}− ∇v_{0})dx

≤b(kv0k∞) Z

Ω

h(x)dx+ Z

Ω

M(∇Tkv0k∞(un)

dx+kkfk_{L}^{1}_{(Ω)}.
Combining (4.2) and (4.13), we deduce (4.12).

On the other hand, forλlarge enough, we have by using(A_{2})
(4.14)

Z

{|un−v0|≤k}

M

a(x, u_{n}, w)
λ

dx≤M

c(x) λ

+k_{3}

λM(k_{4}|w|) +c≤c_{11},
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− ∇v_{0})dx≤c_{12},
withc_{12}is positive constant depending ofk.

Hence, by the Theorem of Banach-Steinhaus, the sequence (a(x, u_{n},∇u_{n}))χ{|u_{n}−v_{0}|≤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 >kv_{0}k∞. LetΩ_{r} ={x∈ Ω,|∇T_{k}(u(x))| ≤ r}and denote byχ_{r}the characteristic
function ofΩ_{r}. Clearly,Ω_{r}⊂Ω_{r+1}andmeas(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(A_{5})there exists a sequencev_{j} ∈K_{ψ}∩W_{0}^{1}E_{M}(Ω)∩L^{∞}(Ω)which converges toT_{k}(u)for
the modular converge inW_{0}^{1}L_{M}(Ω).

Here, we define

w_{n,j}^{h} =T_{2k}(u_{n}−v_{0}−T_{h}(u_{n}−v_{0}) +T_{k}(u_{n})−T_{k}(v_{j})),
w_{j}^{h} =T_{2k}(u−v_{0}−T_{h}(u−v_{0}) +T_{k}(u)−T_{k}(v_{j}))
and

w^{h} =T_{2k}(u−v_{0}−T_{h}(u−v_{0})),
whereh >2k >0.

Forη= exp(−4γk^{2}), 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_{n}ϕ_{k} w_{n,j}^{h}
dx.

It follows that (4.20)

Z

Ω

a(x, un,∇un)∇w^{h}_{n,j}ϕ^{0}_{k} w^{h}_{n,j}
dx+

Z

Ω

gn(x, un,∇un)ϕk w_{n,j}^{h}
dx

≤ Z

Ω

f_{n}ϕ_{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,j}ϕ^{0}_{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_{n}ϕ_{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_{n}ϕ_{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_{n}ϕ_{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,j}ϕ^{0}_{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,j}ϕ^{0}_{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})]ϕ^{0}_{k} w_{n,j}^{h}
dx
+

Z

{|un|>k

a(x, T_{m}(u_{n}),∇T_{m}(u_{n}))∇w^{h}_{n,j}ϕ^{0}_{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})]ϕ^{0}_{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})]ϕ^{0}_{k} w^{h}_{n,j}
dx

−ϕ^{0}_{k}(2k)
Z

{|u_{n}|>k}

|a(x, Tk(un),0)||∇Tk(vj)|dx.

Recalling that,|a(x, T_{k}(u_{n}),0)|χ_{{|u}_{n}_{|>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

−ϕ^{0}_{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(A_{3})
(4.24)

Z

{|u_{n}|>k}

a(x, T_{m}(u_{n}),∇T_{m}(u_{n}))∇w_{n,j}^{h} ϕ^{0}_{k} w_{n,j}^{h}
dx

≥ −ϕ^{0}_{k}(2k)
Z

{|un|>k}

|a(x, T_{m}(u_{n}),∇T_{m}(u_{n}))|∇T_{k}(v_{j})|dx

−ϕ^{0}(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_{{|u}_{n}_{|>k}} → |∇T_{k}(v_{j})|χ_{{|u|>k}}

strongly inE_{M}(Ω)asntends to infinity, we have

−ϕ^{0}_{k}(2k)
Z

{|un|>k}

|a(x, T_{m}(u_{n}),∇T_{m}(u_{n}))|∇T_{k}(v_{j})|dx→ −ϕ^{0}_{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

−ϕ^{0}_{k}(2k)
Z

{|u|>k}

lm|∇Tk(vj)|dx → −ϕ^{0}_{k}(2k)
Z

{|u|>k}

lm|∇Tk(u)|dx= 0 asj tends to infinity.

Finally

(4.25) −ϕ^{0}_{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 ∈L^{1}(Ω)it is easy to see that

(4.26) −ϕ^{0}_{k}(2k)

Z

{|u_{n}−v_{0}|>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,j}ϕ^{0}_{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})]ϕ^{0}_{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,j}ϕ^{0}_{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}

ϕ^{0}_{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}

ϕ^{0}_{k} w_{n,j}^{h}
dx

− Z

Ω\Ω^{j}_{s}

a(x, Tk(un),∇Tk(un))∇Tk(vj)ϕ^{0}_{k} w_{n,j}^{h}
dx

+(n, h) +(n, j) +_{h}(n, j),
whereχ^{j}_{s}denotes 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}

sϕ^{0}_{k} w^{h}_{n,j}

tends to ∇Tk(vj)χ_{Ω\Ω}^{j}

sϕ^{0}_{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})ϕ^{0}_{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}

sϕ^{0}_{k} w^{h}_{j}
dx →

Z

Ω\Ω_{s}

h_{k}∇T_{k}(u)ϕ^{0}_{k}(w^{h})dx
asj tends to infinity.

Finally

(4.29) −

Z

Ω\Ω^{j}_{s}

a(x, Tk(un),∇Tk(un))∇Tk(vj)ϕ^{0}_{k} w^{h}_{n,j}
dx

=− Z

Ω\Ω_{s}

h_{k}∇T_{k}(u)ϕ^{0}_{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}

ϕ^{0}_{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})ϕ^{0}_{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}_{s}ϕ^{0}_{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)ϕ^{0}_{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}

ϕ^{0}_{k}(T_{k}(u_{n})−T_{k}(v_{j}))

→a x, T_{k}(u),∇T_{k}(v_{j})χ^{j}_{s}

ϕ^{0}_{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}_{s}ϕ^{0}_{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}_{s}ϕ^{0}_{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})

ϕ^{0}_{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})

ϕ^{0}_{k} w^{h}_{j}

dx+_{j,h}(n)
since

∇Tk(vj)χ^{j}_{s}ϕ^{0}_{k}(w_{j}^{h})→ ∇Tk(u)χsϕ^{0}_{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})

ϕ^{0}_{k} w_{j}^{h}
dx

−→

Z

Ω\Ωs

a(x, T_{k}(u),0)∇T_{k}(u)ϕ^{0}_{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}

ϕ^{0}_{k} w^{h}_{n,j}
dx

= Z

Ω\Ω_{s}

a(x, Tk(u),0)∇Tk(u)ϕ^{0}_{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,j}ϕ^{0}_{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}

ϕ^{0}_{k} w_{n,j}^{h}
dx

− Z

Ω\Ω_{s}

hk∇Tk(u)ϕ^{0}_{k}(0)dx+
Z

Ω\Ω_{s}

a(x, Tk(u),0)∇Tk(u)ϕ^{0}_{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}))∇v_{0}

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}

ϕ^{0}_{k} w_{n,j}^{h}

− b(k) α

ϕ_{k} w^{h}_{n,j}

dx

≤ Z

Ω\Ω_{s}

hk∇Tk(u)ϕ^{0}_{k}(0)dx+
Z

Ω\Ω_{s}

a(x, Tk(u),0)∇Tk(u)ϕ^{0}_{k}(0)dx+(n, j, h),