Journal of Inequalities in Pure and
Applied Mathematics
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
D ÉPARTEMENT DE M ATHÉMATIQUES ET I NFORMATIQUE
FACULTÉ DES S CIENCES D HAR -M AHRAZ
B.P 1796 ATLAS -F ÈS ,F ÈS M AROC .
meskinedriss@hotmail.com
C.P.R DE F ÈS , B.P 49, F ÈS , M AROC .
Received 26 March, 2003; accepted 05 August, 2003
Communicated by A. Fiorenza
A BSTRACT. 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
where A is a standard Leray-Lions operator defined on W01 LM (Ω), with M an N-function
which satisfies the ∆2 -condition, and where K is a convex subset of W01 LM (Ω) with obstacles depending on some Carathéodory function g(x, u). We consider first, the case f ∈
W −1 EM (Ω) and secondly where f ∈ L1 (Ω). Our method deals with the study of the limit
of the sequence of solutions un of some approximate problem with nonlinearity term of the form
|g(x, un )|n−1 g(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. I NTRODUCTION
Let Ω be an open bounded subset of RN , N ≥ 2, with the segment property. Consider the
following obstacle problem:

 hAu, v − ui ≥ hf, v − ui, ∀v ∈ K,
(P)
 u ∈ K,
where A(u) = −div(a(x, u, ∇u)) is a Leray-Lions operator defined on W01 LM (Ω), with M
being an N -function which satisfies the ∆2 -condition and where K is a convex subset of
W01 LM (Ω).
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
040-03
2
D. M ESKINE AND A. E LMAHI
In the variational case (i.e. where f ∈ W −1 EM (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−∆p u, v − ui ≥ hf, v − ui, ∀v ∈ K
 u ∈ K = {v ∈ W 1,p (Ω) : |v(x)| ≤ 1 a.e. in Ω},
0
0
with f ∈ W −1,p (Ω) and −∆p u = −div(|∇u|p−2 ∇u), by the sequence of problems:

 −∆p un + |un |n−1 un = f in D0 (Ω)
 u ∈ W 1,p (Ω) ∩ Ln (Ω).
n
0
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:

 −∆p un + |g(x, un )|n−1 g(x, un )|∇un |p = f in D0 (Ω)

un ∈ W01,p (Ω), |g(x, un )|n |∇un |p ∈ L1 (Ω)
Here, we introduce the general sequence of equations in the setting of Orlicz-Sobolev spaces

 Aun + |g(x, un )|n−1 g(x, un )M (|∇un |) = f in D0 (Ω)
 u ∈ W 1 L (Ω), |g(x, u )|n M (|∇u |) ∈ L1 (Ω).
n
n
n
0 M
We are interested throughout the paper in studying the limit of the sequence un . We prove that
this limit satisfies some bilateral problem of the (P) form under some conditions on g. In the
first we take f ∈ W −1 EM (Ω) and next in L1 (Ω).
2. P RELIMINARIES
2.1. N −Functions. Let M : R+ → R+ be an N -function, i.e. M is continuous, convex, with
M (t) > 0 for t > 0, Mt(t) → 0 as t → 0 and Mt(t) → ∞ as t → ∞.
Rt
Equivalently, M admits the representation: M (t) = 0 a(s)ds, where a : R+ → R+ is
nondecreasing, right continuous, with a(0) = 0, a(t) > 0 for t > 0 and a(t) tends to ∞ as
t → ∞.
Rt
The N -function M conjugate to M is defined by M (t) = 0 ā(s)ds, where a : R+ → R+ is
given by ā(t) = sup{s : a(s) ≤ t} (see [1]).
The N -function is said to satisfy the ∆2 condition, denoted by M ∈ ∆2 , if for some k > 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 these N -functions into even functions on all R.
Let P and Q be two N -functions. P Q means that P grows essentially less rapidly than
−1 (t)
P (t)
Q, i.e. for each > 0, Q(t)
→ 0 as t → ∞. This is the case if and only if limt→∞ Q
= 0.
P −1 (t)
J. Inequal. Pure and Appl. Math., 4(5) Art. 98, 2003
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A SYMPTOTIC B EHAVIOUR OF S OME E QUATIONS IN O RLICZ S PACES
3
2.2. Orlicz spaces. Let Ω be an open subset of RN . The Orlicz class KM (Ω) (resp. the Orlicz
space LM (Ω)) is defined as the set of (equivalence classes of) real valued measurable functions
u on Ω such that:
Z
Z
u(x)
M (u(x))dx < +∞ resp.
M(
)dx < +∞ for some λ > 0 .
λ
Ω
Ω
LM (Ω) is a Banach space under the norm
Z
u(x)
kukM,Ω = inf λ > 0 :
M(
)dx ≤ 1
λ
Ω
and KM (Ω) is a convex subset of LM (Ω).
The closure in LM (Ω) of the set of bounded measurable functions with compact support in
Ω is denoted by EM (Ω).
The equality EM (Ω) = LM (Ω) holds if only if M satisfies the ∆2 condition, for all t or for t
large according to whether Ω has infinite measure or not.
R
The dual of EM (Ω) can be identified with LM (Ω) by means of the pairing Ω uvdx, and the
dual norm of LM (Ω) is equivalent to k · kM ,Ω .
The space LM (Ω) is reflexive if and only if M and M satisfy the ∆2 condition, for all t or
for t large, according to whether Ω has infinite measure or not.
2.3. Orlicz-Sobolev spaces. We now turn to the Orlicz-Sobolev space, W 1 LM (Ω) (resp.
W 1 EM (Ω)) is the space of all functions u such that u and its distributional derivatives up to
order 1 lie in LM (Ω) (resp. EM (Ω)). It is a Banach space under the norm
X
kuk1,M =
kDα ukM .
|α|≤1
Thus, W 1 LM (Ω) and W 1 EM (Ω) canQbe identified with subspaces of product ofQN + 1Qcopies
of LM (Ω).
this product by LM , we will use the weak topologies σ ( LM , EM )
Q Denoting
Q
and σ ( LM , LM ).
The space W01 EM (Ω) is defined as
the (norm)
closure of the Schwarz space D(Ω) in W 1 EM (Ω)
Q
Q
and the space W01 LM (Ω) as the σ ( LM , EM ) closure of D(Ω) in W 1 LM (Ω).
We say that un converges to u for the modular convergence in W 1 LM (Ω) if for some λ > 0
α
Z
D un − D α u
M
dx → 0 for all |α| ≤ 1.
λ
Ω
Q
Q
This implies convergence for σ ( LM , LM ).
If M satisfies the ∆2 -condition on R+ , then modular convergence coincides with norm convergence.
2.4. The spaces W −1 LM̄ (Ω) and W −1 EM̄ (Ω). Let W −1 LM (Ω) (resp. W −1 EM (Ω)) denote
the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of
functions in LM (resp. EM (Ω)). It is a Banach space under the usual quotient norm.
If the open set Ω has the segment property then theQspace Q
D(Ω) is dense in W01 LM (Ω) for the
modular convergence and thus for the topology σ ( LM , LM ) (cf. [8, 9]). Consequently,
the action of a distribution in W −1 LM (Ω) on an element of W01 LM (Ω) is well defined.
J. Inequal. Pure and Appl. Math., 4(5) Art. 98, 2003
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4
D. M ESKINE AND A. E LMAHI
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 ∈ W 1 LM (Ω) (resp. W 1 EM (Ω)). Then F (u) ∈ W 1 LM (Ω) (resp.
W 1 EM (Ω)). Moreover, if the set D of discontinuity points of F 0 is finite, then
 0
/ D},
 F (u) ∂x∂ i u a.e. in {x ∈ Ω : u(x) ∈
∂
F (u) =

∂xi
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 F 0 is finite. Let M be an N -function, then the mapping
1
F Q
: W 1 LM
Q(Ω) → W LM (Ω) is sequentially continuous with respect to the weak* topology
σ ( LM , EM ).
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. Let Ω be an open subset of RN with finite measure.
Let M, P and Q be N -functions such that Q P , and let f : Ω×R → R be a Carathéodory
function such that a.e. x ∈ Ω and all s ∈ R:
|f (x, s)| ≤ c(x) + k1 P −1 M (k2 |s|),
where k1 , k2 are real constants and c(x) ∈ EQ (Ω).
Then the Nemytskii operator Nf defined by Nf (u)(x) = f (x, u(x)) is strongly continuous
from
1
1
P EM (Ω),
= u ∈ LM (Ω) : d(u, EM (Ω)) <
k2
k2
into EQ (Ω).
3. T HE M AIN R ESULT
Let Ω be an open bounded subset of RN , N ≥ 2, with the segment property.
Let M be an N -function satisfying the ∆2 -condition near infinity.
Let A(u) = −div(a(x, ∇u)) be a Leray-Lions operator defined on W01 LM (Ω) into
W −1 LM (Ω), where a : Ω × RN → RN is a Carathéodory function satisfying for a.e. x ∈ Ω
and for all ζ, ζ 0 ∈ RN , (ζ 6= ζ 0 ) :
(3.1)
(3.2)
|a(x, ζ)| ≤ h(x) + M
−1
M (k1 |ζ|)
0
(a(x, ζ) − a(x, ζ ))(ζ − ζ 0 ) > 0
(3.3)
a(x, ζ)ζ ≥ αM
|ζ|
λ
with α, λ > 0, k1 ≥ 0, h ∈ EM (Ω).
Furthermore, let g : Ω × R → R be a Carathéodory function such that for a.e. x ∈ Ω and for
all s ∈ R:
(3.4)
J. Inequal. Pure and Appl. Math., 4(5) Art. 98, 2003
g(x, s)s ≥ 0
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A SYMPTOTIC B EHAVIOUR OF S OME E QUATIONS IN O RLICZ S PACES
|g(x, s)| ≤ b(|s|)
(3.5)
(3.6)
5

for almost x ∈ Ω\Ω∞

+ there exists = (x) > 0 such that:





 g(x, s) > 1, ∀s ∈]q+ (x), q+ (x) + [;


for almost x ∈ Ω\Ω∞

− there exists = (x) > 0 such that:




g(x, s) < −1, ∀s ∈]q− (x) − , q− (x)[,
where b : R+ → R+ is a continuous and nondecreasing function, with b(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 for s and k in R, k ≥ 0, Tk (s) = max(−k, min(k, s)).
Theorem 3.1. Let f ∈ W −1 EM (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 number µ > 0, the problem

n−1
|∇un |

g(x, un )M
= f in D0 (Ω)
 A(un ) + |g(x, un )|
µ
(Pn )

 un ∈ W 1 LM (Ω), |g(x, un )|n M |∇un | ∈ L1 (Ω)
0
µ
admits at least one solution un such that:
(3.7)
Tk (un ) → Tk (u) for modular convergence in W01 LM (Ω)
∀k > 0
where u is the unique solution of the following bilateral problem

 hAu, v − ui ≥ hf, v − ui, ∀v ∈ K
(P )
 u ∈ K = {v ∈ W 1 L (Ω) : q ≤ v ≤ q a.e.},
−
+
0 M
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 of un is given by Theorem 3.1 of [3]. Choosing v = un as a test function
in (Pn ), and using the sign condition (3.4), we get
hAun , un i ≤ hf, un i.
(3.8)
By Proposition 5 of [11] one has:
Z
Z
|∇un |
M
dx ≤ C, and
a(x, un , ∇un )∇un dx ≤ C,
λ
Ω
Ω
(a(x, un , ∇un )) is bounded in (LM (Ω))N ,
(3.9)
Z
n−1
|g(x, un )|
(3.10)
Ω
J. Inequal. Pure and Appl. Math., 4(5) Art. 98, 2003
g(x, un )M
|∇un |
µ
un dx ≤ C.
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6
D. M ESKINE AND A. E LMAHI
We then deduce
Z
n
|g(x, un )| M
{|un |>k}
|∇un |
µ
dx ≤ C, for all k > 0.
Since b is continuous and since b(0) = 0 there exists δ > 0 such that
b(|s|) ≤ 1 for all |s| ≤ δ.
On the other hand, by the ∆2 condition there exist two positive constants K and K 0 such
that
t
t
≤ KM
+ K 0 for all t ≥ 0,
M
µ
λ
which implies
Z
Z
|∇un |
|∇un |
n
0
|g(x, un )| M
dx ≤
K + KM
dx.
µ
λ
{|un |≤δ}
{|un |≤δ}
Consequently from (3.8)
Z
|∇un |
n
(3.11)
|g(x, un )| M
dx ≤ C, for all n.
µ
Ω
Step 2: Almost everywhere convergence of the gradients.
Since (un ) is a bounded sequence in W01 LM (Ω) there exist some u ∈ W01 LM (Ω) such
that (for a subsequence still denoted by un )
Y
Y
1
(3.12)
un * u weakly in W0 LM (Ω) for σ
LM ,
EM , strongly in EM (Ω),
and a.e. in Ω.
Furthermore, if we have
n−1
Aun = f − |g(x, un )|
with |g(x, un )|n−1 g(x, un )M
show that
|∇un |
µ
g(x, un )M
|∇un |
µ
being bounded in L1 (Ω) then as in [2], one can
∇un → ∇u a.e. in Ω.
(3.13)
Step 3: u ∈ K = {v ∈ W01 LM (Ω) : q− ≤ v ≤ q+ a.e. in Ω}.
Since s → g(x, s) is nondecreasing, then in view of (3.6), we have:
{s ∈ R : |g(x, s)| ≤ 1 a.e. in Ω} = {s ∈ R : q− ≤ s ≤ q+ a.e. in Ω}.
It suffices to verify that |g(x, u)| ≤ 1 a.e.
We have
Z
n
|g(x, un )| M
Ω
|∇un |
µ
dx ≤ C,
which gives
Z
n
|g(x, un )| M
{|g(x,un )|>k}
|∇un |
µ
|dx ≤ C
and
Z
M
{|g(x,un )|>k}
J. Inequal. Pure and Appl. Math., 4(5) Art. 98, 2003
|∇un |
µ
dx ≤
C
kn
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A SYMPTOTIC B EHAVIOUR OF S OME E QUATIONS IN O RLICZ S PACES
7
where k > 1. Letting n → +∞ for k fixed, we deduce by using Fatou’s lemma
Z
|∇un |
M
dx = 0
µ
{|g(x,u)|>k}
and so that,
|g(x, u)| ≤ 1 a.e. in Ω.
Step 4: Strong convergence of the truncations.
2
Let φ(s) = s exp(γs2 ), where γ is chosen such that γ ≥ α1 .
0
It is well known that φ (s)− 2K
|φ(s)| ≥ 12 , ∀s ∈ R, where K is a constant which will be
α
used later. The use of the test function vn = φ(zn ) in (Pn ) where zn = Tk (un ) − Tk (u)
gives
Z
|∇un |
n−1
hAun , φ(zn )i + |g(x, un )|
g(x, un )M
φ(zn )dx = hf, φ(zn )i
µ
Ω
which implies, by using the fact that g(x, un )φ(zn ) ≥ 0 on {x ∈ Ω : |un | > k},
Z
|∇un |
n−1
hAun , φ(zn )i +
|g(x, un )|
g(x, un )M
φ(zn )dx
µ
{0≤un ≤Tk (u)}∩{|un |≤k}
Z
|∇un |
n−1
φ(zn )dx ≤ hf, φ(zn )i.
+
|g(x, un )|
g(x, un )M
µ
{Tk (u)≤un ≤0}∩{|un |≤k}
The second and the third terms of the last inequality will be denoted respectively by
1
2
In,k
and In,k
and i (n) denote various sequences of real numbers which tend to 0 as
n → +∞.
On the one hand we have
Z
1 |∇u
|
n
n
In,k ≤
|g(x, un )| M
|φ(zn )|dx
µ
{0≤un ≤Tk (u)}∩{|un |≤k}
Z
|∇un |
n
≤
|g(x, un )| M
|φ(zn )|dx,
µ
{0≤un ≤u}∩{|un |≤k}
but since |g(x, un )| ≤ 1 on {x ∈ Ω : 0 ≤ un ≤ u}, then we have
Z
1 |∇un |
In,k ≤
M
|φ(zn )|dx.
µ
{|un |≤k}
By using the fact that
M
|∇un |
µ
0
≤ K + KM
|∇un |
λ
we obtain
Z
Z
1 K
0
In,k ≤
K |φ(zn )|dx +
a(x, ∇Tk (un ))∇Tk (un )|φ(zn )|dx,
α Ω
Ω
which gives
(3.14)
1 In,k ≤ 1 (n) + K
α
J. Inequal. Pure and Appl. Math., 4(5) Art. 98, 2003
Z
a(x, ∇Tk (un ))∇Tk (un )|φ(zn )|dx.
Ω
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8
D. M ESKINE AND A. E LMAHI
Similarly,
2 In,k ≤
(3.15)
Z
M
{|un |≤k}
K
≤ 1 (n) +
α
|∇un |
µ
|φ(zn )|dx
Z
a(x, ∇Tk (un ))∇Tk (un )|φ(zn )|dx.
Ω
The first term on the left hand side of the last inequality can be written as:
Z
(3.16)
0
a(x, ∇un )[∇Tk (un ) − ∇Tk (u)]φ (zn )dx
Ω
Z
0
=
a(x, ∇un )[∇Tk (un ) − ∇Tk (u)]φ (zn )dx
{|un |≤k}
Z
0
−
a(x, ∇un )∇Tk (u)φ (zn )dx.
{|un |>k}
For the second term on the right hand side of the last equality, we have
Z
Z
0
a(x, ∇un )∇Tk (u)φ (zn )dx ≤ Ck
|a(x, ∇un )||∇Tk (u)|χ{|un |>k} dx.
{|un |>k}
Ω
The right hand side of the last inequality tends to 0 as n tends to infinity. Indeed,
the sequence (a(x, ∇un ))n is bounded in (LM (Ω))N while ∇Tk (u)χ{|un |>k} tends to 0
strongly in (EM (Ω))N .
We define for every s > 0, Ωs = {x ∈ Ω : |∇Tk (u(x))| ≤ s} and we denote by χs its
characteristic function. For the first term of the right hand side of (3.16), we can write
Z
0
a(x, ∇un )[∇Tk (un ) − ∇Tk (u)]φ (zn )dx
(3.17)
{|un |≤k}
Z
=
0
[a(x, ∇Tk (un )) − a(x, ∇Tk (u)χs )][∇Tk (un ) − ∇Tk (u)χs ]φ (zn )dx
Ω
Z
0
+ a(x, ∇Tk (u)χs )[∇Tk (un ) − ∇Tk (u)χs ]φ (zn )dx
Ω
Z
0
− a(x, ∇Tk (un ))∇Tk (u)χΩ\Ωs φ (zn )dx.
Ω
The second term of the right hand side of (3.17) tends to 0 since
0
a(x, ∇Tk (un )χs )φ (zn ) → a(x, ∇Tk (u)χs ) strongly in (EM (Ω))N
by Lemma 2.3 and
∇Tk (un ) * ∇Tk (u) weakly in (LM (Ω))N for σ
The third term of (3.17) tends to −
Y
LM (Ω),
Y
EM (Ω) .
R
a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx as n → ∞ since
Y
Y
a(x, ∇Tk (un )) * a(x, ∇Tk (u)) weakly for σ
EM (Ω),
LM (Ω) .
J. Inequal. Pure and Appl. Math., 4(5) Art. 98, 2003
Ω
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A SYMPTOTIC B EHAVIOUR OF S OME E QUATIONS IN O RLICZ S PACES
9
Consequently, from (3.16) we have
Z
0
(3.18)
a(x, ∇un )[∇Tk (un ) − ∇Tk (u)]φ (zn )dx
Ω
Z
= [a(x, ∇Tk (un )) − a(x, ∇Tk (u)χs )]
Ω
0
× [∇Tk (un ) − ∇Tk (u)χs ]φ (zn )dx + 2 (n).
We deduce that, in view of (3.17) and (3.18),
Z
[a(x, ∇Tk (un )) − a(x, ∇Tk (u)χs )]
Ω
2K
|φ(zn )| dx
× [∇Tk (un ) − ∇Tk (u)χs ] φ (zn ) −
α
Z
≤ 3 (n) + a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx,
0
Ω
and so
Z
Ω
[a(x, ∇Tk (un )) − a(x, ∇Tk (u)χs )][∇Tk (un ) − ∇Tk (u)χs ]dx
Z
≤ 23 (n) + 2 a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx.
Ω
Hence
Z
Ω
a(x, ∇Tk (un ))∇Tk (un )dx
Z
Z
≤
a(x, ∇Tk (un ))∇Tk (u)χs dx + a(x, ∇Tk (u)χs )[∇Tk (un ) − ∇Tk (u)χs ]dx
Ω
Ω
Z
+ 23 (n) + 2 a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx.
Ω
Now considering the limit sup over n, one has
Z
(3.19) lim sup a(x, ∇Tk (un ))∇Tk (un )dx
n→+∞
Ω
Z
Z
≤ lim sup a(x, ∇Tk (un ))∇Tk (u)χs dx + lim sup a(x, ∇Tk (u)χs )
n→+∞
n→+∞
Ω
Ω
Z
× [∇Tk (un ) − ∇Tk (u)χs ]dx + 2 a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx.
Ω
The second term of the right hand side of the inequality (3.19) tends to 0, since
a(x, ∇Tk (un )χs ) → a(x, ∇Tk (u)χs ) strongly in EM (Ω),
while ∇Tk (un ) tends weakly to ∇Tk (u).
R
The first term of the right hand side of (3.19) tends to Ω a(x, ∇Tk (u))∇Tk (u)χs dx
since
a(x, ∇Tk (un )) * a(x, ∇Tk (u)) weakly in (LM (Ω))N
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10
D. M ESKINE AND A. E LMAHI
Q
Q
for σ ( LM , EM ) while ∇Tk (u)χs ∈ EM (Ω). We deduce then
Z
Z
lim sup a(x, ∇Tk (un ))∇Tk (un )dx ≤
a(x, ∇Tk (u))∇Tk (u)χs dx
n→+∞
Ω
Ω
Z
+ 2 a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx,
Ω
1
by using the fact that a(x, ∇Tk (u))∇Tk (u) ∈ L (Ω) and letting s → ∞ we get, since
meas(Ω\Ωs ) → 0
Z
Z
lim sup a(x, ∇Tk (un ))∇Tk (un )dx ≤
a(x, ∇Tk (u))∇Tk (u)dx
n→+∞
Ω
Ω
which gives, by using Fatou’s lemma,
Z
Z
(3.20)
lim
a(x, ∇Tk (un ))∇Tk (un )dx =
a(x, ∇Tk (u))∇Tk (u)dx.
n→+∞
Ω
Ω
On the other hand, we have
Z
|∇Tk (un )|
K
0
M
≤K +
a(x, ∇Tk (un ))∇Tk (un )dx,
µ
α Ω
then by using (3.20) and Vitali’s theorem, one easily has
|∇Tk (un )|
|∇Tk (u)|
(3.21)
M
→M
strongly in L1 (Ω).
µ
µ
By writing
(3.22)
M
|∇Tk (un ) − ∇Tk (u)|
2µ
M
≤
|∇Tk (un )|
µ
2
M
+
|∇Tk (un )|
µ
2
one has, by (3.21) and Vitali’s theorem again,
Tk (un ) → Tk (u) for modular convergence in W01 LM (Ω).
(3.23)
Step 5: u is the solution of the variational inequality (P ).
Choosing w = Tk (un − θTm (v)) as a test function in (Pn ), where v ∈ K and 0 < θ < 1,
gives
Z
|∇un |
n−1
hAun , Tk (un − θTm (v))i + |g(x, un )|
g(x, un )M
Tk (un − θTm (v))dx
µ
Ω
= hf, Tk (un − θTm (v))i,
since g(x, un )Tk (un − θTm (v)) ≥ 0 on
{x ∈ Ω : un ≥ 0 and un ≥ θTm (v)} ∪ {x ∈ Ω : un ≤ 0 and un ≤ θTm (v)}
we have
Z
Ω
n−1
|g(x, un )|
Z
≥
|∇un |
µ
Tk (un − θTm (v))dx
|∇un |
n−1
|g(x, un )|
g(x, un )M
Tk (un − θTm (v))dx
µ
{0≤un ≤θTm (v)}
Z
|∇un |
n−1
+
|g(x, un )|
g(x, un )M
Tk (un − θTm (v))dx.
µ
{θTm (v)≤un ≤0}
g(x, un )M
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A SYMPTOTIC B EHAVIOUR OF S OME E QUATIONS IN O RLICZ S PACES
11
The first and the second terms in the right hand side of the last inequality will be denoted
1
2
respectively by Jn,m
and Jn,m
.
Defining
δ1,m (x) =
sup g(x, s)
0≤s≤θTm (v)
we get 0 ≤ δ1,m (x) < 1 a.e. and
Z
1
|Jn,m | ≤ k
n
(δ1,m (x)) M
{0≤un ≤θTm (v)}
|∇un |
µ
dx.
Since
(δ1,m (x))n M |∇un | χ{|un |≤m} ≤ M |∇Tm (un )| ,
µ
µ
we have then by using (3.23) and Lebesgue’s theorem
1
Jn,m
−→ 0 as n → +∞.
Similarly
2
|Jn,m
|
|δ2,m (x)| M
δ2,m (x) =
inf
n
≤k
|∇Tm (un )|
µ
Z
{|un |≤m}
dx → 0 as n → +∞,
where
g(x, s).
θTm (v)≤s≤0
On the other hand , by using Fatou’s lemma and the fact that
a(x, ∇un ) → a(x, ∇u) weakly in (LM (Ω))N for σ(ΠLM , ΠEM ),
one easily has
lim inf hAun , Tk (un − θTm (v))i ≤ hAu, Tk (u − θTm (v))i.
n→+∞
Consequently
hAu, Tk (u − θTm (v))i ≤ hf, Tk (u − θTm (v))i,
this implies that by letting k → +∞, since Tk (u − θTm (v)) → u − θTm (v) for modular
convergence in W01 LM (Ω),
hAu, u − θTm (v)i ≤ hf, u − θTm (v)i,
in which we can easily pass to the limit as θ → 1 and m → +∞ to obtain
hAu, u − vii ≤ hf, u − vi.
4. T HE L1 C ASE
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− and q+
belong to L∞ (Ω). Then the problem (Pn ) admits at least one solution un such that:
un → u for modular convergence in W01 LM (Ω),
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12
D. M ESKINE AND A. E LMAHI
where u is the unique solution of the bilateral problem:
Z

 hAu, v − ui ≥
f (v − u)dx, ∀v ∈ K
(Q)
Ω

u ∈ K = {v ∈ W01 LM (Ω) : q− ≤ v ≤ q+ a.e.}.
Proof. We sketch the proof since the steps are similar to those in Section 3.
The existence of un is given by Theorem 1 of [4]. Indeed, it is easy to see that |g(x, s)| ≥ 1 on
{|s| ≥ γ}, where γ = max {supess q+ , − infess q− } and so that
|ζ|
|ζ|
n
|g(x, s)| M
≥M
for |s| ≥ γ.
µ
µ
Step 1: A priori estimates.
Choosing v = Tγ (un ), as a test function in (Pn ), and using the sign condition (3.4), we
obtain
Z
|∇Tγ (un )|
(4.1)
α M
dx ≤ γkf k1
λ
Ω
and
Z
|∇un |
n
|g(x, un )| M
dx ≤ kf k1 ,
µ
{|un |>γ}
which gives
Z
|∇un |
M
dx ≤ C
µ
{|un |>γ}
and finally
Z
|∇un |
(4.2)
M
dx ≤ C.
max{λ, µ}
Ω
On the other hand, as in Section 3, we have
Z
|∇un |
n
dx ≤ C.
(4.3)
|g(x, un )| M
µ
Ω
Step 2: Almost everywhere convergence of the gradients.
Due to (4.2), there exists some u ∈ W01 LM (Ω) such that (for a subsequence)
un * u weakly in W01 LM (Ω) for σ(ΠLM , ΠEM ).
Write
|∇un |
Aun = f − |g(x, un )|
g(x, un )M
µ
and remark that, by (4.2), the second hand side is uniformly bounded in L1 (Ω). Then as
in Section 3
∇un → ∇u a.e. in Ω.
Step 3: u ∈ K = {v ∈ W01 LM (Ω) : q− ≤ v ≤ q+ a.e. in Ω}.
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.
n−1
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A SYMPTOTIC B EHAVIOUR OF S OME E QUATIONS IN O RLICZ S PACES
13
Step 5: u is the solution of the bilateral problem (Q).
Let v ∈ K and 0 < θ < 1. Taking vn = Tk (un − θv), k > 0 as a test function in
(Pn ), one can see that the proof is the same by replacing Tm (v) with v in Section 3. We
remark that K ⊂ L∞ (Ω).
Step 6: un → u for modular convergence in W01 LM (Ω).
We shall prove that ∇un → ∇u in (LM (Ω))N for the modular convergence by using
Vitali’s theorem.
Let E be a measurable subset of Ω, we have for any k > 0
Z
Z
Z
|∇un |
|∇un |
|∇un |
M
dx ≤
M
dx +
M
dx.
µ
µ
µ
E∩{|un |≤k}
E∩{|un |>k}
E
(4.4)
Let > 0. By virtue of the modular convergence of the truncates, there exists some
η(, k) such that for any E measurable
Z
|∇un |
dx < , ∀n.
|E| < η(, k) ⇒
M
µ
2
E∩{|un |≤k}
Choosing T1 (un − Tk (un )), with k > 0 a test function in (Pn ) we obtain:
Z
|∇un |
n−1
hAun , T1 (un − Tk (un ))i + |g(x, un )|
g(x, un )M
T1 (un − Tk (un ))dx
µ
Ω
Z
=
f T1 (un − Tk (un ))dx,
Ω
which implies
Z
n
|g(x, un )| M
{|un |>k+1}
|∇un |
µ
Z
dx ≤
|f |dx.
{|un |>k}
Note that meas{x ∈ Ω : |un (x)| > k} → 0 uniformly on n when k → ∞. We deduce
then that there exists k = k() such that
Z
|f |dx < , ∀n,
2
{|un |>k}
which gives
Z
n
|g(x, un )| M
{|un |>k+1}
(4.5)
|∇un |
µ
dx < , ∀n.
2
By setting t() = max {k + 1, γ} we obtain
Z
|∇un |
M
dx < , ∀n.
µ
2
{|un |>t()}
Combining (4.4) and (4.5) we deduce that there exists η > 0 such that
Z
|∇un |
M
< , ∀n when |E| < η, E measurable,
µ
E
which shows the equi-integrability of M |∇uµ n | in L1 (Ω), and therefore we have
|∇un |
|∇u|
M
→M
strongly in L1 (Ω).
µ
µ
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14
D. M ESKINE AND A. E LMAHI
By remarking that
|∇un − ∇u|
|∇un |
|∇u|
1
M
≤
M
+M
2µ
2
µ
µ
one easily has, by using the Lebesgue theorem
Z
|∇un − ∇u|
dx → 0 as n → +∞,
M
2µ
Ω
which completes the proof.
Remark 4.2. The condition b(0) = 0 is not necessary. Indeed, taking θh (un ), h > 0, as a test
function in (Pn ) with

 hs
if |s| ≤ h1
θh (s) =
 sgn(s) if |s| ≥ 1 ,
h
we obtain
Z
n−1
|g(x, un )|
g(x, un )M
Ω
and then, by letting h → +∞,
Z
n
|∇un |
µ
|g(x, un )| M
Ω
Z
θh (un )dx ≤
|∇un |
µ
f θh (un )dx.
Ω
dx ≤ C.
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