Journal of Inequalities in Pure and
Applied Mathematics
ASYMPTOTIC BEHAVIOUR OF SOME EQUATIONS IN
ORLICZ SPACES
volume 4, issue 5, article 98,
2003.
D. MESKINE AND A. ELMAHI
Département de Mathématiques et Informatique
Faculté des Sciences Dhar-Mahraz
B.P 1796 Atlas-Fès,Fès Maroc.
EMail: meskinedriss@hotmail.com
Received 26 March, 2003;
accepted 05 August, 2003.
Communicated by: A. Fiorenza
C.P.R de Fès, B.P 49, Fès, Maroc.
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
040-03
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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
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 |).
2000 Mathematics Subject Classification: 35J25, 35J60.
Key words: Strongly nonlinear elliptic equations, Natural growth, Truncations, Variational inequalities, Bilateral problems.
Asymptotic Behaviour of Some
Equations in Orlicz Spaces
D. Meskine and A. Elmahi
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Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1
N −Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2
Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3
Orlicz-Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4
The spaces W −1 LM̄ (Ω) and W −1 EM̄ (Ω) . . . . . . . . 8
2.5
Lemmas related to the Nemytskii operators in Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.6
Abstract lemma applied to the truncation operators . . 9
3
The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4
The L1 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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1.
Introduction
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).
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)
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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 u ∈ W 1,p (Ω) ∩ Ln (Ω).
n
0
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In [7], A. Dall’aglio and L. Orsina generalized this result by taking increasing
powers depending also on some Carathéodory function g satisfying the sign
condition and some hypothesis of integrability. Following this idea, we have
studied in [5] the sequence of problems:

 −∆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 OrliczSobolev 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).
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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2.
2.1.
Preliminaries
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:
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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M (2t) ≤ kM (t) ∀t ≥ 0;
(2.1)
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
P (t)
less rapidly than Q, i.e. for each > 0, Q(t)
→ 0 as t → ∞. This is the case if
and only if limt→∞
2.2.
Q−1 (t)
P −1 (t)
= 0.
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
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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)
M(
)dx ≤ 1
kukM,Ω = inf λ > 0 :
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
uvdx, and the dual norm of LM (Ω) is equivalent to k · kM ,Ω .
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.
Asymptotic Behaviour of Some
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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
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under the norm
kuk1,M =
X
kDα ukM .
|α|≤1
Thus, W 1 LM (Ω) and W 1 EM (Ω) can be identified with
Q subspaces of product of
N + 1 copies of
Q product
Q by LM , we will use the weak
Q LM (Ω).
Q Denoting this
topologies σ ( LM , EM ) and σ ( LM , LM ).
The space W01 EM (Ω) is defined as the (norm) closureQof theQ
Schwarz space
1
1
D(Ω) in W EM (Ω) and the space W0 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.
Asymptotic Behaviour of Some
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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 the space D(Ω) Q
is denseQin
1
W0 LM (Ω) for the modular convergence and thus for the topology σ ( LM , LM )
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(cf. [8, 9]). Consequently, the action of a distribution in W −1 LM (Ω) on an element of W01 LM (Ω) is well defined.
2.5.
Lemmas related to the Nemytskii operators in Orlicz spaces
We recall some lemmas introduced in [3] which will be used in this paper.
Lemma 2.1. Let F : R → 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 F : W 1 LM (Ω) Q
→ W 1L
QM (Ω) is sequentially continuous with
respect to the weak* topology σ ( LM , EM ).
Asymptotic Behaviour of Some
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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.
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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
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into EQ (Ω).
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3.
The Main Result
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)
|a(x, ζ)| ≤ h(x) + M
−1
M (k1 |ζ|)
Asymptotic Behaviour of Some
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(3.2)
0
(a(x, ζ) − a(x, ζ ))(ζ − ζ 0 ) > 0
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(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:
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(3.4)
g(x, s)s ≥ 0
(3.5)
|g(x, s)| ≤ b(|s|)
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(3.6)

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

g(x, un )M |∇uµ n | = f in D0 (Ω)
 A(un ) + |g(x, un )|
(Pn )

 un ∈ W 1 LM (Ω), |g(x, un )|n M |∇un | ∈ L1 (Ω)
0
µ
Asymptotic Behaviour of Some
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admits at least one solution un such that:
∀k > 0
(3.7)
Tk (un ) → Tk (u) for modular convergence in W01 LM (Ω)
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 3.1. If the function s → g(x, s) is strictly nondecreasing for a.e. x ∈ Ω
then the assumption (3.6) holds true.
Proof.
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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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.
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By Proposition 5 of [11] one has:
Z
Z
|∇un |
(3.8)
M
dx ≤ C, and
a(x, un , ∇un )∇un dx ≤ C,
λ
Ω
Ω
(3.9)
(a(x, un , ∇un )) is bounded in (LM (Ω))N ,
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Z
n−1
|g(x, un )|
(3.10)
g(x, un )M
Ω
We then deduce
Z
n
|g(x, un )| M
{|un |>k}
|∇un |
µ
|∇un |
µ
un dx ≤ C.
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.
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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
|∇un |
n
|g(x, un )| M
dx
µ
{|un |≤δ}
Z
|∇un |
0
dx.
≤
K + KM
λ
{|un |≤δ}
Consequently from (3.8)
Z
|∇un |
n
(3.11)
|g(x, un )| M
dx ≤ C, for all n.
µ
Ω
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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
un * u weakly in W0 LM (Ω) for σ
LM ,
EM ,
(3.12)
strongly in EM (Ω) and a.e. in Ω.
Furthermore, if we have
Aun = f − |g(x, un )|n−1 g(x, un )M
with |g(x, un )|n−1 g(x, un )M
[2], one can show that
(3.13)
|∇un |
µ
|∇un |
µ
being bounded in L1 (Ω) then as in
∇un → ∇u a.e. in Ω.
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}.
Asymptotic Behaviour of Some
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It suffices to verify that |g(x, u)| ≤ 1 a.e.
We have
Z
|∇un |
n
|g(x, un )| M
dx ≤ C,
µ
Ω
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which gives
Z
n
|g(x, un )| M
{|g(x,un )|>k}
and
Z
M
{|g(x,un )|>k}
|∇un |
µ
|∇un |
µ
dx ≤
|dx ≤ C
C
kn
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
α
which will be 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
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which implies, by using the fact that g(x, un )φ(zn ) ≥ 0 on {x ∈ Ω :
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|un | > k},
Z
|g(x, un )|n−1
hAun , φ(zn )i +
{0≤un ≤Tk (u)}∩{|un |≤k}
× g(x, un )M
Z
+
|∇un |
µ
φ(zn )dx
|g(x, un )|n−1
{Tk (u)≤un ≤0}∩{|un |≤k}
|∇un |
× g(x, un )M
φ(zn )dx ≤ hf, φ(zn )i.
µ
The second and the third terms of the last inequality will be denoted re1
2
spectively by 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
|φ(zn )|dx,
≤
|g(x, un )| M
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}
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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By using the fact that
|∇un |
|∇un |
0
M
≤ K + KM
µ
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
Z
1 K
(3.14)
In,k ≤ 1 (n) +
a(x, ∇Tk (un ))∇Tk (un )|φ(zn )|dx.
α Ω
Asymptotic Behaviour of Some
Equations in Orlicz Spaces
D. Meskine and A. Elmahi
Similarly,
(3.15)
2 In,k ≤
Z
M
{|un |≤k}
≤ 1 (n) +
K
α
|∇un |
µ
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|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
0
(3.16)
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}
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For the second term on the right hand side of the last equality, we have
Z
a(x, ∇un )∇Tk (u)φ (zn )dx
{|un |>k}
Z
≤ Ck
|a(x, ∇un )||∇Tk (u)|χ{|un |>k} dx.
0
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
(3.17)
a(x, ∇un )[∇Tk (un ) − ∇Tk (u)]φ (zn )dx
{|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.
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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Ω
Page 19 of 34
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
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by Lemma 2.3 and
∇Tk (un ) * ∇Tk (u) weakly in (LM (Ω))N for σ
The third term of (3.17) tends to −
n → ∞ since
R
Ω
Y
LM (Ω),
Y
EM (Ω) .
a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx as
a(x, ∇Tk (un )) * a(x, ∇Tk (u)) weakly for σ
Y
EM (Ω),
Y
LM (Ω) .
Asymptotic Behaviour of Some
Equations in Orlicz Spaces
Consequently, from (3.16) we have
D. Meskine and A. Elmahi
Z
(3.18)
Ω
0
a(x, ∇un )[∇Tk (un ) − ∇Tk (u)]φ (zn )dx
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),
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Z
Ω
[a(x, ∇Tk (un )) − a(x, ∇Tk (u)χs )]
2K
0
× [∇Tk (un ) − ∇Tk (u)χs ] φ (zn ) −
|φ(zn )| dx
α
Z
≤ 3 (n) + a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx,
Ω
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and so
Z
[a(x, ∇Tk (un )) − a(x, ∇Tk (u)χs )][∇Tk (un ) − ∇Tk (u)χs ]dx
Ω
Z
≤ 23 (n) + 2 a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx.
Ω
Hence
Z
a(x, ∇Tk (un ))∇Tk (un )dx
Ω
Z
≤
a(x, ∇Tk (un ))∇Tk (u)χs dx
ΩZ
+ a(x, ∇Tk (u)χs )[∇Tk (un ) − ∇Tk (u)χs ]dx
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
≤ lim sup a(x, ∇Tk (un ))∇Tk (u)χs dx
n→+∞
Ω
Z
+ lim sup a(x, ∇Tk (u)χs )[∇Tk (un ) − ∇Tk (u)χs ]dx
n→+∞
Ω
Z
+ 2 a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx.
Ω
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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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
Q
Q
for σ ( LM , EM ) while ∇Tk (u)χs ∈ EM (Ω). We deduce then
Z
lim sup
n→+∞
a(x, ∇Tk (un ))∇Tk (un )dx
Ω
Z
≤
a(x, ∇Tk (u))∇Tk (u)χs dx
Ω
Z
+ 2 a(x, ∇Tk (u))∇Tk (u)χΩ\Ωs dx,
Ω
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
Ω
D. Meskine and A. Elmahi
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n→+∞
Asymptotic Behaviour of Some
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which gives, by using Fatou’s lemma,
Z
(3.20)
lim
a(x, ∇Tk (un ))∇Tk (un )dx
n→+∞ Ω
Z
=
a(x, ∇Tk (u))∇Tk (u)dx.
Ω
On the other hand, we have
Z
|∇Tk (un )|
K
0
M
≤K +
a(x, ∇Tk (un ))∇Tk (un )dx,
µ
α Ω
Asymptotic Behaviour of Some
Equations in Orlicz Spaces
D. Meskine and A. Elmahi
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
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By writing
(3.22) M
|∇Tk (un ) − ∇Tk (u)|
2µ
M
≤
|∇Tk (un )|
µ
2
M
+
|∇Tk (un )|
µ
2
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one has, by (3.21) and Vitali’s theorem again,
(3.23)
Tk (un ) → Tk (u) for modular convergence in
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W01 LM (Ω).
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
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and 0 < θ < 1, gives
hAun , Tk (un − θTm (v))i
Z
|∇un |
n−1
+ |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
|∇un |
n−1
|g(x, un )|
g(x, un )M
Tk (un − θTm (v))dx
µ
Ω
Z
|∇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}
The first and the second terms in the right hand side of the last inequality
1
2
will be denoted respectively by Jn,m
and Jn,m
.
Defining
δ1,m (x) =
sup g(x, s)
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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0≤s≤θTm (v)
we get 0 ≤ δ1,m (x) < 1 a.e. and
Z
1
|Jn,m | ≤ k
{0≤un ≤θTm (v)}
Page 24 of 34
n
(δ1,m (x)) M
|∇un |
µ
dx.
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Since
|∇T
(u
)|
|∇u
|
m
n
n
n
(δ1,m (x)) M
χ{|un |≤m} ≤ M
,
µ
µ
we have then by using (3.23) and Lebesgue’s theorem
1
Jn,m
−→ 0 as n → +∞.
Similarly
2
|Jn,m
|
Asymptotic Behaviour of Some
Equations in Orlicz Spaces
Z
≤k
n
|δ2,m (x)| M
{|un |≤m}
|∇Tm (un )|
µ
dx → 0 as n → +∞,
D. Meskine and A. Elmahi
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where
δ2,m (x) =
inf
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 ),
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one easily has
lim inf hAun , Tk (un − θTm (v))i ≤ hAu, Tk (u − θTm (v))i.
n→+∞
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Consequently
hAu, Tk (u − θTm (v))i ≤ hf, Tk (u − θTm (v))i,
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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.
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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4.
The L1 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− 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),
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
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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and
Z
n
|g(x, un )| M
{|un |>γ}
|∇un |
µ
dx ≤ kf k1 ,
which gives
Z
M
{|un |>γ}
|∇un |
µ
dx ≤ C
and finally
Z
M
(4.2)
Ω
|∇un |
max{λ, µ}
dx ≤ C.
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
n−1
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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∇un → ∇u a.e. in Ω.
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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.
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
|∇un |
M
µ
E
Z
≤
Z
D. Meskine and A. Elmahi
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dx
E∩{|un |≤k}
Asymptotic Behaviour of Some
Equations in Orlicz Spaces
M
|∇un |
µ
Z
dx +
M
E∩{|un |>k}
|∇un |
µ
dx.
Let > 0. By virtue of the modular convergence of the truncates, there
exists some η(, k) such that for any E measurable
Z
|∇un |
(4.4)
|E| < η(, k) ⇒
M
dx < , ∀n.
µ
2
E∩{|un |≤k}
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Choosing T1 (un − Tk (un )), with k > 0 a test function in (Pn ) we obtain:
hAun , T1 (un − Tk (un ))i
Z
|∇un |
n−1
+ |g(x, un )|
T1 (un − Tk (un ))dx
g(x, un )M
µ
Ω
Z
=
f T1 (un − Tk (un ))dx,
Ω
which implies
Z
Asymptotic Behaviour of Some
Equations in Orlicz Spaces
n
|g(x, un )| M
{|un |>k+1}
|∇un |
µ
Z
dx ≤
|f |dx.
D. Meskine and A. Elmahi
{|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
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Z
{|un |>k+1}
|g(x, un )|n M
|∇un |
µ
dx < , ∀n.
2
By setting t() = max {k + 1, γ} we obtain
Z
|∇un |
(4.5)
M
dx < , ∀n.
µ
2
{|un |>t()}
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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
|∇u|
|∇un |
M
→M
strongly in L1 (Ω).
µ
µ
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|
M
dx → 0 as n → +∞,
2µ
Ω
which completes the proof.
Asymptotic Behaviour of Some
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D. Meskine and A. Elmahi
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Remark 4.1. 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
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we obtain
Z
Z
|∇un |
n−1
|g(x, un )|
g(x, un )M
θh (un )dx ≤
f θh (un )dx.
µ
Ω
Ω
and then, by letting h → +∞,
Z
|∇un |
n
|g(x, un )| M
dx ≤ C.
µ
Ω
Asymptotic Behaviour of Some
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References
[1] R. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
[2] A. BENKIRANE AND A. ELMAHI, Almost everywhere convergence of
the gradients of solutions to elliptic equations in Orlicz spaces and application, Nonlinear Anal. T.M.A., 28 (11) (1997), 1769–1784.
[3] A. BENKIRANE AND A. ELMAHI, An existence theorem for a strongly
nonlinear elliptic problem in Orlicz spaces, Nonlinear Anal. T.M.A., 36
(1999), 11–24.
Asymptotic Behaviour of Some
Equations in Orlicz Spaces
[4] A. BENKIRANE AND A. ELMAHI, A strongly nonlinear elliptic equation having natural growth terms and L1 data, Nonlinear Anal. T.M.A., 39
(2000), 403–411.
D. Meskine and A. Elmahi
[5] A. BENKIRANE, A. ELMAHI AND D. MESKINE, On the limit of some
nonlinear elliptic problems, Archives of Inequalities and Applications, 1
(2003), 207–220.
Contents
[6] L. BOCCARDO AND F. MURAT, Increase of power leads to bilateral
problems, in Composite Media and Homogenization Theory, G. Dal Maso
and G. F. Dell’Antonio (Eds.), World Scientific, Singapore, 1995, pp. 113–
123.
[7] A. DALL’AGLIO AND L. ORSINA, On the limit of some nonlinear elliptic
equations involving increasing powers, Asympt. Anal., 14 (1997), 49–71.
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[8] J.-P. GOSSEZ, Nonlinear elliptic boundary value problems for equations
with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc.,
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[9] J.-P. GOSSEZ, Some approximation properties in Orlicz-Sobolev spaces,
Studia Math., 74 (1982), 17–24.
[10] J.-P. GOSSEZ, A strongly nonlinear elliptic problem in Orlicz-Sobolev
spaces, Proc. A.M.S. Symp. Pure Math., 45 (1986), 455–462.
[11] J.-P. GOSSEZ AND V. MUSTONEN, Variational inequalities in OrliczSobolev spaces, Nonlinear Anal. T.M.A., 11 (1987), 379–392.
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