Annales Mathematiques Blaise Pascal 10 , 1 - 2 0 ( 20 3 )
Existence
Degenerated
of Solution for Quasilinear
Ellipt ic Unilateral Problems
Youssef Akdim
Elhoussine Azroul
Abdelmoujib Benkirane
Abstract
An existence theorem is proved , for a quasilinear degenerated ellip of
+ g(x, u ,
∇u),
Au
ticinequalitywhereA is a Leray involving Lionsnonlinear
operatoroperators from W0 1,p the
−
(Ω, f ormw)into its dual, while
g(x, s, ξ) is a nonlinear term which has a growth condition with respect
to ξ and no growth with respect to s, but it satisfies
a sign condition
0
on s, the second term belongs to W −1,p (Ω, w∗ ).
1
Introduction
Let Ω be a bounded open set of RN , p be a real number such that 1 < p < ∞
and w = {wi (x), 0 ≤ i ≤ N } be a vector of weight functions on Ω, i . e . each
wi (x) is a me a−s urable a.e. strictly positive on Ω, satisfying some integrability
conditions ( see section 2 ) . This paper is concerned with the existence of
solution of unilateral degenerate problems associated to a nonlinear operator
of the form
Au + g(x, u, ∇u).
The principal part A is a differential operator of second order in divergence
0
form of Leray - Lions type acting from W01,p (Ω, w) into it ’ s dual W −1,p (Ω, w∗ ),
i. e.
Au = −div(a(x, u, ∇u))
(1.1)
and g is a nonlinear lower order term having natural growth with respect to
| ∇u |, with respect to | u | we do not assume any growth restrictions , but
we assume the sign - condition .
Bensoussan , Boccardo and Murat have
proved in the first part of [ 3 ] , the existence of a solution for the problem
Au + g(x, u, ∇u) = f,
1
Y0 . Akdim , E . Azroul , A . Benkirane
where f ∈ W −1,p (Ω). In the second part of [ 3 ] , the authors have extended
the last result to variational inequalities , more precisely , they have proved
the existence of at l east one solution of the following unilateral problem :









,p
Z
hAu, v − ui +
g(x, u, ∇u)(v − u)dx ≥ hf, v − ui
Ω
Kψ
f oru∈ allW0v1,p ∈
(Ω ) u
1
g(x, u, ∇u) ∈ L (Ω)
≥ ψa.e.inΩ
g(x, u, ∇u)u ∈ L1 (Ω),
∞( Ω)
≥ a.e. in
W0 , 1, p(vΩ) ψ∩L∞ (Ω ) .ΩT he }, withsame ψameasurableresultisalso
(Ω)
L
f unctionwhereKψ = on{vΩ ∈ W01 suchthat
∩ ψ+
∈
proved in [ 2 ] where f ∈ L1 (Ω). It is our purpose in this paper , to study
the variational degenerated inequal - ities .
More precisely , we prove the
existence of solution to the problem (P) ( see section 4 ) , in the framework of
weighted Sobolev space .
We obtain the
existenceu−
ε )
ε+
(resp .
the positive
u
of uε resultsbystrongly converges provingthattou+ (resp .u− ) partinW01 ,p(Ω , w), negativewhereuε partisa
solution of the approximate problem (Pε ) ( see section 4 ) .
Let us point
out ,
that another work in this direction can be found in [ 6 ] and [ 1 ] in the case
of equation . Note that , this paper can be seen as a generalization of [ 3 ] in
weighted case
and as a continuation of [ 1 ] where the case of equation is treated .
This
paper is organized as follows :
Section 2 contains some preliminaries ,
section 3 is concerned with the basic assumptions and some technical lemmas
, in section 4 we state and prove main results .
2
Preliminaries
Let Ω be a bounded open subset of RN (N ≥ 1), let 1 < p < ∞,
and
l et w = {wi (x), 0 ≤ i ≤ N } be a vector of weight functions , i . e .
every
component wi (x) is a measurable function which is strictly positive a.e. in Ω.
Further , we suppose in all our considerations that
wi ∈ L1loc (Ω)
(2.1)
and
1
−p−1
wi
∈ L1loc (Ω)
2
(2.2)
Existence of Solution for Quasilinear Degenerated Elliptic . .
.
f orany0 ≤ i ≤ N.
We define the weighted space Lp (Ω, γ), where γ is a weight function on Ω by ,
Lp (Ω, γ) = {u = u(x),
uγ 1p ∈ Lp (Ω)}
with the norm
Z
k u kp,γ = ( | u(x) |p γ(x)dx)1p .
Ω
Now ,
1,p
we denote by W (Ω, w) the space of all real - valued functions u ∈
such that the derivatives in the se line − n se of distributions fulfil
L − p(Ω, w0 )
∂ ∂ xui ∈ Lp (Ω, wi )f oralli = 1, ..., N,
which is a Banach space under the norm
1
!
k u k 1, p, w =
(
p
R
Ω
| u(x) | w0 (x)dx +
Pi=1 R
N
Ω
|
u(x)
∂ ∂ xi |p
wi (x)dx
p
. (2.3)
Since we shall deal with the Dirichlet problem , we shall use the space
the
W edef inedC∞0 recall(Ω)asis thatdense
1of p
∞0(Ω) with
closure
indual W0 , space (CΩ,
the
w)of andweighted (respect,X
0
0
alent to W −1,p (Ω, w∗ ), where w∗ = {wi∗ = wi1−p , ∀i = 0, ..., N }, where p0 is
the conjugate of p i . e .p0 = pp− 1 ( for more details we refer to [ 5 ] ) .
Definition :
Let Y be a separable reflexive Banach space , the operator B
from Y to its dual Y ∗ is called of the calculus of variations type ,
if B is
bounded and is of the form ,
B(u) = B(u, u),
(2.5)
where (u, v) → B(u, v) is an operator from Y × Y into Y ∗ satisfying the
following properties :
(
∀u ∈ Y,
v → B(u, v) isboundedhemicontinuousf romY intoY ∗
and(B(u, u) − B(u, v), u − v) ≥ 0,
(2. 6) 3
k . k1,p,w
Sobolev )isaref le
∀v ∈ Y,
(
Y . Akdim , E . Azroul , A . Benkirane
is bounded hemicontinuous from Y into Y ∗ , (2.7)
u → B(u, v)
if un * uweaklyinY andif (B(un , un ) − B(un , u), un − u) → 0
then,
B(un , v) * B(u, v)weaklyinY ∗ ,
∀v ∈ Y,
(2. 8)
{
un *
thenif
(uB(weakly un , v), inun ) Y→ and(ψ, ifu) B.(un , v) * ψweaklyinY ∗ ,
,
(2.9)
Definition :
Let Y be a reflexive Banach space , a bounded mapping B
from Y to Y ∗ is called pseudo - monotone if for any sequence un ∈ Y with
un * u
weakly in Y and lim sup hBun , un − ui ≤ 0, one has
n→∞
lim inf hBun , un − vi ≥ hBu, u − vi
for all v ∈ Y.
n→∞
3 Basic assumption and
lem mas We start by the following assumptions .
some technical
Assumption(H1 )
The expression
k| u |k X =
i=1 Z
X
u(x)
Ω
(N
| ∂∂
xi
|p wi (x)dx)1p
is a norm defined on X and it ’ s equivalent to the norm ( 2 . 3 ) . And there
exist a weight function σ on Ω and a parameter q, such that
1 < q < p + p0 ,
(3.1)
and
0
σ 1−q ∈ L1loc (Ω),
(3.2)
with q0 = qq− 1 and that the Hardy inequality ,
Z
i=one−line
X Z
u(x)
( | u(x) |q σ(x)line − dx )1q ≤ c
| ∂∂ xi |p wi parenleft − linex)dx)1p , (3.3)
Ω
Ω
(N
4
Existence of Solution for Quasilinear Degenerated Elliptic . .
.
holds for every u ∈ X with a constant c > 0 independent of u, and moreover ,
the imbedding
X →→ Lq (Ω, σ),
(3.4)
expressed by the inequality ( 3 . 3 ) is compact .
Notice that (X, k| . |k X) is a uniformly convex ( and thus reflexive ) Banach
space .
Remark : If we assume that w0 (x) ≡ 1 and in addition the integrability
condition :
there exists ν ∈]Np , ∞[∩[p 1− 1, ∞[ such that wi−ν ∈ L1 (Ω) ∀i = 1, ..., N, which is
stronger than ( 2 . 2 ) .
Then
1
!
i=1 Z
X
k| u |k X =
(N
|
Ω
u(x)
∂∂ xi |p
wi (x)dx
p
,
is a norm defined on W01,p (Ω, w) and it ’ s equivalent to ( 2 . 3 ) , moreover
W01,p (Ω, w) →→ Lq (Ω),
for all 1
where p1
conjugate
for σ ≡
1
1
<
<
q
q
≤ q < p∗1 if pν < N (ν + 1) and for all q ≥ 1 if pν ≥ N (ν + 1),
= pνν+ 1 and p∗1 = N p1N −p1 = (N N pνν+1) −pν is the Sobolev
of p1 ( see [ 5 ] ) .
Thus the hypotheses (H1 ) are verified
1 and for all
< min {p∗1, p + p0 braceright − line if pν line − less N (ν + 1) and for all
< p + p0 if
pν ≥ N (ν + 1).
0
Let A be a nonlinear operator from W01,p (Ω, w) into its dual W −1,p (Ω, w∗ )
defined by ( 1 . 1 ) , i . e . ,
Au = −div(a(x, u, ∇u)),
N
N
where a : Ω × R × R → R is a Carathéodory vector - function satisfying the
following a − s sumptions :
Assumption(H2 )
| ai (x, s, ξ) |≤
βwpi1 (x)[k(x)
+σ
1p0
q0
|s|p +
N vlinew0p j1 (x)
| ξj |
p−1
] f oralli = 1, ..., N,
j=1
(3. 5) 5
Y . Akdim , E . Azroul , A . Benkirane
[a(x, s, ξ) − a(x, s, η)](ξ − η) > 0,
f orallξ 6= η ∈ RN ,
(3.6)
N
a(x, s, ξ).ξ ≥ α
X
wi | ξi |
p
(3.7)
,
i=1
where k(x)
constants .
0
is a positive function in Lp (Ω)
and α, β are strictly positive
Assumption(H3 )
Let g(x, s, ξ)
ions :
be a Carathéodory function satisfying the following assump - t
g(x, s, ξ)s ≥ 0

N
P
wi | ξi |p +c(x)  ,
| g(x, s, ξ) |≤ b(| s |) 
i=1
(3.8)

(3.9)
where b : R+ → R+ is a continuous increasing function and c(x) is a positive
function which lies in L1 (Ω). We consider ,
0
f ∈ W −1,p (Ω, w∗ ).
(3.10)
Now we recall some lemmas introduced in [ 1 ] which will be used later .
Lemma 3.1 : (cf.[1]) Let g ∈ Lr (Ω, γ) and let gn ∈ Lr (Ω, γ), with k gn kr,γ ≤
c (1 < r < ∞). If
gn(x) → g(x)a.e. in Ω, then gn * g weakly in Lr (Ω, γ),
where γ is a weight function on Ω.
1]) Assume
(0)
Lemma3.2Lipschitzian:, (cf.
with [F
Moreover , if the s et
∂(F ∂x◦ i u)
LemmaTk (u),
k 3∈ .3
={
D
Let
F :R
Letthat u(∈ H1 )W0holds.
→ F (u)R ∈ beunif ormlyW01,p (Ω
1,p (Ω, w ). T hen
= 0.
of dis continuity points of
0F
0
(u) ∂ u
∂ xi
1])
e.
aa.
. e.
F0
inin {{xx ∈∈ ΩΩ ::
that(H
)
is finite , then
}
(x)
uu (x) element − negationslash∈ DD }.
p(Ω,
(Ω,
),
1
holds.
Let u
: R+, (cfbe .[the Assumeusualtruncation
∈ 0W 10 W 1,p w) .w M oreover andlet
,
Tk (u )
then
∈,
we have
Tk (u) → ustronglyinW01,p (Ω, w).
6
Existence of Solution for Quasilinear Degenerated Elliptic . .
.
(cf .
[ 1])
that
(H 1)
.
Let
u
and u− Assume=max(− u, 0)
lie holdsinW 1,p (Ω , w ).
Lemma3.4:
u+ =max (u, )0
0
∈ W01,p (Ω
(u+ )
∂∂
−
(u )
∂∂
sucht − linehatun * uweaklyinW01,p (Ω, w).
])
1,p
−
) andu−
(Ω,w).
n *u weaklyinW0
1,p
inW0 (Ω,
1,p
:
W0 (Ω,w)Lemma3 .5
(cf.
[1])
w
xi
= line − bracelef tbigg
(H )
and
(H2 )
Assumethat(H1 )
are s atisfied , and let
un * uweaklyinW01,p (Ω, w)
and
Z
[a(x, un , ∇un ) − a(x, un , ∇u)]∇(un − u)dx → 0.
Ω
Then ,
un → ustronglyinW01,p (Ω, w).
Main
general
result
Let ψ be a measurable function
with values in R such that ,
ψ + ∈ W01,p (Ω, w) ∩ L∞ (Ω).
(4.1)
Kψ = {v ∈ W01,p (Ω, w) ∩ L∞ (Ω) v ≥ ψa.e.inΩ}.
(4.2)
Set
Remark that ( 4 . 1 ) implies that Kψ 6= ∅. Consider the nonlinear problem
with Dirichlet boundary condition ,
(P)





Z
hAu, v − ui +
g(x, u, ∇u)(v − u)dx ≥ hf, v − ui
Ω
v




∈
all
f oru∈
W0 1,p (Kψ Ω, w) u ≥ ψa.e.inΩ
g(x, u, ∇u) ∈ L1 (Ω) g(x, u, ∇u)u ∈ L1 (Ω),
7
={
T hen,
1
. [
Assume that
Lemmabeasequence 3.6 : (cf
, w)such that
of W0 11,p (Ω
(un )
4
xi
holds.
Let(un )
Y . Akdim , E . Azroul , A . Benkirane
here u is the solution of the problem (P).
Our main result is the following . Theorem 4 . 1 :
Assume that the
assumption (H1 ) − (H3 ), (3.10) and ( 4 . 1 )
hold , then , there exists at least one s o luti on of (P).
Remarks :
1 ) The statement of Theorem 4 . 1 , generalizes in weighted case the
analo gous one in [ 3 ] . 2 ) If we take ψ = −∞, we obtain the existence result
for the equation
case(see[1]).
Proof of Theorem 4 . 1 Step ( 1 ) The approximate problem and a
priori estimate .
Let Ωε be a sequence of compact subsets of Ω such that Ωε increase to Ω as
ε → 0.
We consider the sequence of approximate problems :
hAu ∈ ε
∈ , vWW01 (Ω, w)
(Pε )bracelef tmid−bracelef tbt
uεv
,p(Ω,w)
v ≥ ψa.e.inΩ01,− p u
uε ≥ψa.e.inΩ,
R
u )dx
gε(x,uε ,∇uε )(v− ε
ε i+ Ω
where
(x,
ξ)
gε(x, s, ξ) = 1 +g ε|g( s,
x, s, ξ) | χΩε (x),
and where χΩε is the charac t − line eristic function of Ωε . Note that gε(x, s, ξ)
satisfies the following conditions ,
gε(x, s, ξ)s ≥ 0,
| gε(x, s, ξ) |≤| g(x, s, ξ) |
and | gε(x, s, ξ) |≤ 1ε .
X → X ∗ by ,
Z
hGε u, vi =
gε(x, u, ∇u)vdx.
We define the operator Gε :
Ω
8
≥ hf, v−uε i
Existence of Solution for Quasilinear Degenerated Elliptic . .
.
Thanks to Hölder ’ s inequality we have for all u ∈ X and v ∈ X,
R
Z
|
gε(x, u, ∇u)vdx |
≤≤ Ω
Ω
0
0q kv qq
(x, u,
1− 0 ∇dx) u)1 q 0 σ− k q,σ dx)1q 0
|
|gε σ q
(Ωε
R
(1ε
Z
( | v |q σdx)1q
Ω
≤
cε k| v |k .
(4. 3)
The last inequality is due to ( 3 . 2 ) and parenlef t − line3.4).
Lemma 4 . 2 :
The operator Bε = A + Gε from X into its dual X ∗ is
pseudo - monotone .
Moreover , Bε is coercive , in th e fo llowing s ense
:
{
that
thereexists
v i → v0 + ∈∞ Kψ if k| suchv|k
B vk,v− 0
hε
|v| k
→ ∞,
v ∈ Kψ .
This lemma wi line − l l be proved below . In view of lemma 4.2, (Pε ) has a
solution by the classical result ( cf . theorem 8 . 2 chapter 2 [ 7 ] ) . Let
v = ψ + as test function in (Pε ), we easily deduce that
Z
gε(x, uε , ∇uε )(uε − ψ + ) ≥ 0, then, hAuε , uε i ≤ hf, uε − ψ + i + hAuε , ψ + i,
Ω
i. e.
Z
a(x, uε , ∇uε )∇uε dx ≤ hf, uε − ψ + i +
i=1 Z
X
Ω
N
Ω
ai (x, uε , ∇uε )∂∂ψ
+
xi dx,
then ,
α
i=1 Z
X
N
Ω
wi | ∂∂uε xi |p dx
+
=
α k| uε |kp ≤
k f k X ∗ (k| uε |k + k| ψ + |k)+
i=1 R
R
X
0
+
0
(Ω | ai (x, uε , ∇uε ) |p wi1−p dx)1p0 (Ω | ∂∂ψ xi |p wi dx)1p
N
≤
+ c
i=1
X
R
0
(Ω (k p + | uε |q σ +
N
k f k X ∗ (k| uε |k + k| ψ + |k)+
j=1
X
N
9
| ∂ ∂ xujε |p wj )dx)1p0 k| ψ + |k .
Y . Akdim , E . Azroul , A . Benkirane
Using ( 3 . 4 ) the last inequality becomes
α k| uε |kp ≤ c1 k| uε |k
+c2 k| uε |k pq0 + c3 k| uε |k p − 1 + c4
thatwhere ucεi arevariousremainspositiveboundedin constantsW0 1, p(Ω, w. )T hen,i.e.
thanks to
( 3 . 1 ) , we can deduce
k| uε |k vline ≤ β0,
(4.4)
where β0 is a positive constant . Extracting a subsequence ( still denoted by
we get
uε * u weakly in X and a.e. in Ω.
Note that u ≥ ψa.e. in Ω. Step ( 2 ) We study the convergence of the
positive part of uε .
Let k > 0. Define ku+ = min {u+ , k}. We shall fix k, and use the notation ,
uε )
+
zε = ε+
u − ku .
(4.5)
Assertion ( i ) We claim that ,
Z
lim sup
ε→0
Ω
+ +
+
+
[a(x, uε , ∇u+
ε ) − a(x, uε , ∇ku )]∇(εu − uk ) dx ≤ Qk,
(4.6)
whereQk → 0, if k → +∞.
test
f unction
Indeed3.4,wehave , Considerzthe
vε∈
1
(Ω, w
) andε+
ε ∈W0 ,p
z
=
ε+
3
z . By lemma 3.
uε W0 1, −p(Ω
, w) . And since k and→ lemma∞and
∈ L∞ (Ω) we can assume that k ≥ ψa.e. in Ω, by the choice of k, the above
test function is admissible for (Pε ). Multiplying (Pε ) by vε we obtain ,
ψ+
hAuε , zε+ i
Z
+
+
gε(x, uε , ∇uε )ε+
z dx ≤ hf, zε i.
Ω
If ε+
z > 0, we have uε > 0 and from (3.8), gε(x, uε , ∇uε ) ≥ 0, then ,
Z
+
a(x, uε , ∇uε )∇ε+
z dx ≤ hf, εz i.
Ω
+
Since uε = ε+
u in {x ∈ Ω/ εz > 0}, then ,
Z
+
+
a(x, uε , ∇u+
ε )∇εz dx ≤ hf, εz i,
Ω
10
(4.7)
Existence of Solution for Quasilinear Degenerated Elliptic . .
.
which implies that ,
Z
Ω
,
+
+ Ω
+
[a(x, uε , ∇ε+
u ) − a(x, uε ≤− a(x, uε , ∇uk )∇(εu − uk )∇+
uk
R
+ +
+
)]∇(ε+
u −uk ) dx≤
dx + hf, zε+ i.
+
have +ε
, we
zwe → have(u, −ku+ )+ a.e. in Ω, moreover ε+
( 4 . 8 ) AsεW 1, →
z is
p( Ω, 0w ) , hence,
0
bounded in
1,p
+
+ +
ε+
z * (u − ku ) weaklyinW0 (Ω, w).
QN
+
p0
∗
a−s
Since a(x, uε , ∇u+
sing
i=1 L (Ω, wi ), we obtain by p
k ) − a(x, u, ∇ku ) in
the limit in ε in ( 4 . 8 ) the inequality ( 4 . 6 ) with Qk defined by ,
Z
Qk = −
Ω
+ +
+
+ +
+
a(x, u, ∇u+
k )∇(u − ku ) dx + hf, (u − uk ) i.
to
(4.9)
Because , (u+ − ku+ )+ → 0 in W01,p (Ω, w) as k → ∞, we have Qk → 0 as
k → ∞.
Assertion ( ii ) Let us show that ,
Z
+
−[a(x, uε , ∇u+
ε ) − a(x, uε , ∇ku
lim sup
ε→0
zε−
≤
zε−
k, i.e.,
∈
L∞ (Ω)
andsincezε−
Ω
.2wehavevε ∈W01,p (Ω,w),thenclearly,vε
∈
ϕλ(s)
=
seλs2 .
W ehave,
0
W
1,p
(Ω,w),henceusinglemma
W econsiderf orthat, 3 thet
0
≤Indeed.
is an admissible test function . Multiplying (Pε ) by vε we obtain ,
Z
a(x, uε , ∇uε )∇zε− ϕ0λ (zε− )dx +
Ω
Z
gε(x, uε , ∇uε )ϕλ(zε− )dx ≥ hf, ϕλ(zε− )i.
Ω
( 4 . 1 1 ) Define ,
Eε = {x ∈ Ω,
+
ε+
u (x) ≤ ku (x)}andFε = {x ∈ Ω,
0 ≤ uε (x) ≤ ku+ (x)}.
Since ϕλ(zε− ) = 0 in Eεc, we have ,
Z
gε(x, uε , ∇uε )ϕλ(zε− )dx
Z
=
Ω
Eε
11
gε(x, uε , ∇uε )ϕλ(zε− )dx.
(4.12)
Y . Akdim , E . Azroul , A . Benkirane
When uε ≤ 0, we have gε(x, uε , ∇uε ) ≤ 0 and since ϕλ(zε− ) ≥ 0, we obtain
Z
Z
gε(x, uε , ∇uε )ϕλ(zε− )dx
≤
Eε
gε(x, uε , ∇uε )ϕλ(zε− )dx
Fε
Z
≤
Fε
i=1
X
b(| uε |)[
wi | ∂∂uε xi |p +c(x)]ϕλ(zε− )dx
N
Z
≤
b(k)
Fε
Z
i=1
X
[
wi | ∂∂uε xi |p +c(x)]ϕλ(zε− )dx
N
a(x, uε , ∇uε )∇uε ϕλ(zε− )dx + b(k)
≤ b(α k)
Fε
Z
c(x)ϕλ(zε− )dx.
(4.13)
Fε
As in the proof of theorem 1 . 1 in [ 3 ] , we can show that ,
− 0
−
+u )∇zε ϕλ (zε )
uε ,
x,
dx+
R u,∇
R
ku ,
+ Ω
ε
(
,
ε
b
b(α k)
∇+u )∇(εu −
+
Ω a (x
dx
a(x,
k
+u )ϕ
−)
λ(zε
k
+ one−line R R
≤ 2
+
+
+ −
( Ω [a(x,
+ − Ωb
uε ,∇u+
ε )−a(x,uε ,∇ku )]∇(εu −uk ) dx≤
k) (
αaΩ
+ 0
+
−
[a(x, uε , ∇uε )−a(x, uε , ∇ε+
u )]∇ku ϕλ (ku )dx+h−f, ϕλ(zε )i
( 4 . 14 )
(k)2
f o − linerλ = b4
α2 .
For short notation , we rewrite the above inequality as ,
1
2
3
4
5
Iεk ≤ Iεk
+ Iεk
+ Iεk
+ Iεk
+ Iεk
.
(4.15)
Extracting a subsequence such that ,
bracelef tmid − bracelef tbt
a(x,u
u )
ε
anda(x,uεε ,, ∇∇ u+
** γγ1 2 weaklyweakly inin
)
N
Y
0
L p ( Ω, w ∗i)
Lp0 (Ω, wi∗ ).
ε
QN
i=1
i=1
(4.16)
Lemma 4 . 3 :
that ,
1
1)Iεk
→
Ik1
( cf .
Z
=
Ω
[ 1 ] ) For k fixed and letting
ε → 0,
we claim
+
0
+
+ −
[γ1 − γ2]∇u+
k ϕλ (uk )dx + h−f, ϕλ((u − ku ) )i.
+ −
−
0
+
Ωa(x, u, ∇ku+ )∇((u+ − u+
k ) )ϕλ ((u − uk ) )dx.
12
2
2)Iεk
→ Ik2 =
Existence of Solution for Quasilinear Degenerated Elliptic . .
.
3
3)Iεk
→
Ik3
Z
= b(α k)
Ω
4
4)Iεk
→ Ik4 = b(α k)
Z
Ω
+ −
+
γ2∇u+
k ϕλ((u − uk ) )dx.
+ −
+
a(x, u, ∇ku+ )∇(u+ − u+
k )ϕλ((u − uk ) )dx.
Ωc(x)ϕλ((u+ − ku+ )− )dx.
5
5)Iεk
→ Ik5 = b(k)
In view of lemma 4 . 3 and (u+ − ku+ )− = 0 and ϕλ(0) = 0 we have ,
lim sup Iεk ≤
ε→0
Ik1
+
Ik2
+
Ik3
+
Ik4
+
Ik5
Z
=
[γ1(x) − γ2(x)]∇ku+ ϕ0λ (ku+ )dx.
Ω
Moreover , if uε < 0 we have (uε )+
k = 0, hence ,
+
(a(x, uε , ∇uε ) − a(x, uε , ∇ε+
u ))(uε )k = 0a.e.inΩ
which implies that (γ1(x) − γ2(x))ku+ = 0, and so that ,
lim sup Iεk ≤ 0,
ε→0
thus , ( 4 . 1 0 ) follows . Assertion ( iii ) Let us show that ,
1,p
+
ε+
u → u stronglyinW0 (Ω, w). (4.17)
From ( 4 . 6 ) and ( 4 . 1 0 ) we have ( as in the proof of theorem 1 . 1 in [ 3
]),
Z
lim sup
ε→0
Ω
, uε ,
Ω
u
R
[a(x, uε , ∇ε+
u ) − a(x≤Q k + [γ2(x) − a∇u+ )]∇(ε+ − (
u
+
)
x, u,
+kdx∇u ≤ )]∇(+uk − u+ )dx.
Now letting k → ∞ and using lemma 3 . 6 we obtain ( 4 . 1 7 ) .
Step ( 3 ) We study the convergence of the negative part of
Similarly to the preceding step , we shall prove that
−
u−
ε →u
stronglyinW01,p (Ω, w). (4.18)
Assertion ( j ) Let us show that ,
Z
lim sup
ε→0
(4. 19) 13
Ω
−
− +
−
−[a(x, uε , −∇u−
ε ) − a(x, uε , −∇uk )]∇(uε − uk ) dx ≤ Q̃k,
uε
Y . Akdim , E . Azroul , A . Benkirane
whereQ̃k → 0, if k → +∞.
Indeed .
We define
−
u−
k = min{u , k}
−
yε = u−
ε − uk .
and
Consider the test function vε = uε + yε+ in (Pε ), it is clearly admissible . Multiplying (Pε ) by vε , we have ,
Z
a(x, uε , ∇uε )∇yε+ dx +
Z
Ω
gε(x, uε , ∇uε )yε+ dx ≥ hf, yε+ i.
Ω
Since yε+ > 0 implies uε < 0, then from ( 3 . 8 ) , we have gε(x, uε , ∇uε )
hence gε(x, uε , ∇uε )yε+ ≤ 0a.e. in Ω, then ,
Z
≤ 0,
a(x, uε , ∇uε )∇yε+ dx ≥ hf, yε+ i.
Ω
Since uε = −u−
yε+ > 0} we can also write
ε on the set {x ∈ Ω,
+
hf, yε i, which implies that ,
R
Ω
+
a(x, uε , −∇u−
ε )∇yε dx ≥
Z
−
have y+ +ε
AsεW 1, →
yε
p( Ω, 0w we), then
0
−
− +
−
[a(x, uε , −∇u−
ε ) − a(x, uε , −∇uk )]∇(uε − uk ) dx ≤
Ω
Z
− +
−
+
a(x, uε , −∇u−
k )∇(uε − uk ) dx − hf, yε i.
Ω
+
)
a.e.
(u
− −− u+
in
Ω
since
−
→*(u− − uk ) k weakly in ,W 1 and
w)(yε+ f oriskf ixedbounded ).
,p(Ω ,
0
Passing to the limit in ε we obtain ( 4 . 1 9 ) , with Q̃k defined by ,
Z
Q̃k =
Ω
− +
− +
−
−
a(x, u, −∇u−
k )∇(u − uk ) dx − hf, (u − uk ) i.
1,p
+
Because (u− − u−
k ) → 0 strongly in W0 (Ω, w) as k → ∞ we obtain that
Q̃k − 0ask − ∞.
Assertion ( jj ) Let us show that ,
Z
lim sup
ε→0
Ω
−
− −
−
[a(x, uε , −∇u−
ε ) − a(x, uε , −∇uk )]∇(uε − uk ) dx ≤ 0.
(4.20)
Indeed .
Considering the following test function vε = uε − δε ϕλ(yε− ) where
2
δε > 0 such that δε eλ(y−ε) ≤ 1 this function is admissible ( cf .
[ 3 ] ) , then ,
hAuε , −δε ϕλ(yε− )i − δε
Z
gε(x, uε , ∇uε )ϕλ(yε− )dx ≥ −hf, δε ϕλ(yε− )i,
Ω
14
in
Existence of Solution for Quasilinear Degenerated Elliptic . .
.
i. e. ,
hAuε , ϕλ(yε− )i +
Z
gε(x, uε , ∇uε )ϕλ(yε− )dx ≤ hf, ϕλ(yε− )i,
Ω
with this choice ( 4 . 20 ) follows as in ( 4 . 1 0 ) . Finally combining ( 4 .
1 9 ) and ( 4 . 20 ) , we deduce as in ( 4 . 1 7 ) the assertion ( 4
. 1 8 ) . Step ( 4 ) Convergence of uε . From ( 4 . 1 7 ) and ( 4 . 1 8 )
we deduce that for a subsequence
uε → u strongly in W01,p (Ω, w) and a.e. in Ω (4.21)
∇uε → ∇ua.e.inΩ,
(4.22)
which implies that ,
{
u )
uε , ∇ ε → (x , u, u) a.e
Ω
g g εε ((x,
x, uε, ∇ uε )u ε g→ g(x , ∇u, ∇u )u ina.e . inΩ. (4.23)
On the other hand , multiplying (Pε ) by uε and using ( 3 . 7 ) , ( 3 . 8 ) , ( 4
. 3 ) , ( 4 . 4 ) , we obtain
Z
0≤
gε(x, uε , ∇uε )uε dx ≤ β̃,
(4.24)
Ω
where β̃ is some positive constant . For any measurable subset E of Ω and
any m > 0 we have ,
Z
Z
Z
| gε(x, uε , ∇uε ) | dx =
| gε(x, uε , ∇uε ) | dx +
ε
E∩Xm
E
| gε(x, uε , ∇uε ) | dx
ε
E∩Ym
where
{
= {x ∈
Y εm
{ x ∈ ΩΩ ::
ε =
Xm
||uuε (x)
}
ε(x)
|| ≤> mm }.
(4.25)
From ( 3 . 9 ) , ( 4 . 24 ) , ( 4 . 25 ) we have ,
Z
| gε(x, uε , ∇uε ) | dx
E
R
ε
E∩Xm
Z
≤
b(m)
(
E
( 4 . 26 ) 1 5
i=line−one
X
N
| gε(x, uε , ∇uε ) | dx + 1m
≤
R
Ω
1
wi | ∂∂uε xi |p +c(x))dx + β̃m
.
gε(x, uε , ∇uε )uε dx
Y . Akdim , E . Azroul , A . Benkirane
Q
p
Since the sequence (∇uε ) strongly converges in N
i=1 L (Ω, wi ), then ( 4 . 26 )
implies the equi - integrability of gε(x, uε , ∇uε ). Thanks to ( 4 . 23 ) and Vitali
’ s theorem one easily has
gε(x, uε , ∇uε ) → g(x, u, ∇u) stronglyinL1 (Ω).
(4.27)
Moreover , since gε(x, uε , ∇uε )uε ≥ 0 a.e. in Ω, then by using ( 4 . 23 ) , ( 4 .
24 ) and Fatou ’ s lemma we have
g(x, u, ∇u)u ∈ L1 (Ω).
(4.28)
From ( 4 . 2 1 ) and ( 4 . 27 ) we can pass to the limit in
Z
hAuε , v − uε i +
gε(x, uε , ∇uε )(v − uε )dx ≥ hf, v − uε i
Ω
and we obtain ,
{
i
R
,
v− u
hAu
v ∈ +W0
f or any
g(x , u,
u)(v −
≥ f,
u)
1, Ωp ( Ω, w )∩ ∇L∞ (Ω ) v dx≥ψ hp.p . vin
− Ωui
.
(4.29)
Proof of lemma 4 . 2 By the proposition 2 . 6 chapter 2 [ 7 ] , it is sufficient
to show that Bε is of the
calculus of variations type .
Indeed put ,
b1 (u, v, w̃) =
i=1 Z
X
ai (x, u, ∇v)∇w̃dx
Ω
N
Z
gε(x, u, ∇u)w̃dx.
b2 (u, w̃) =
Ω
The form w̃ − b1 (u, v, w̃) + b2 (u, w̃) is continuous in X.
0
Bε (u, v) ∈ W −1,p (Ω, w∗ )
Then ,
b1 (u, v, w̃) + b2 (u, w̃) = b(u, v, w̃) = hBε (u, v), w̃i,
and we have
Bε (u, u) = Bε u.
Using ( 3 . 5 ) and Hölder ’ s inequality we can show that A is bounded [ 4 ] ,
and
thanks to (4.3), Bε is bounded .
Then , it is sufficient to check (2.6) − (2.9).
16
Existence of Solution for Quasilinear Degenerated Elliptic . .
.
Show that ( 2 . 6 ) and ( 2 . 7 ) are true .
By ( 3 . 6 ) we have ,
(Bε (u, u) − Bε (u, v), u − v) = b1 (u, u, u − v) − b1 (u, v, u − v) ≥ 0.
The operator v → Bε (u, v) is bounded hemicontinuous 0.
Indeed , we have
ai (x, u, ∇(v1 + λv2 )) → ai (x, u, ∇v1 ) strongly in Lp (Ω, wi∗ ) as λ → 0.
( 4 . 30 )
0
0
On the other hand , (gε(x, u1 + λu2 , ∇(u1 + λu2 )))λ is bounded in Lq (Ω, σ1−q ) and
gε(x, u1 + λu2 , ∇(u1 + λu2 )) → gε(x, u1 , ∇u1 ) a.e. in Ω hence lemma
3 . 1 gives
0
0
gε(x, u1 + λu2 , ∇(u1 + λu2 )) * gε(x, u1 , ∇u1 ) weakly in Lq (Ω, σ 1−q ) as λ → 0. ( 4 . 3
1 ) Using ( 4 . 30 ) and ( 4 . 3 1 ) , we can write
b(u, v1 + λv2 , w̃) − b(u, v1 , w̃) asλ → 0
∀u, vi , w̃ ∈ X.
Similarly we can prove ( 2 . 7 ) . Proof of assertion ( 2 . 8 ) . Assume that
un * u weakly in X and (B(un , un )−
W ehave :
=
i=1 Z
X
N
(B(un , un ) − B(un , u), un − u)
(ai (x, un , ∇un ) − ai (x, un , ∇u))∇(un − u)dx → 0,
Ω
then , by lemma 3 . 6 we have ,
un → ustronglyinX,
which gives
b(un , v, w̃) − b(u, v, w̃) ∀w̃ ∈ X,
i. e. ,
Bε (un , v) * Bε (u, v)weaklyinX ∗ .
It remains to prove ( 2 . 9 ) .
un * u
Assume that
weaklyinX
(4.32)
and that
B(un , v) * ψweaklyinX ∗ . (4.33)
17
B(un , u), un − u) → 0.
Y . Akdim , E . Azroul , A . Benkirane
Thanks to ( 3 . 4 ) , ( 3 . 5 ) and ( 4 . 32 ) , we obtain
0
ai (x, un , ∇v) → ai (x, u, ∇v) strongly in Lp (Ω, wi∗ ) as n → ∞, then ,
b1 (un , v, un ) → b1 (u, v, u).
(4.34)
On the other hand , by Hölder ’ s inequality ,
Z
| b2 (un , un − u) |
≤
≤
Z
q
0
| gε(x, un , ∇un ) |q σ −q0 dx)1q0 ( | un − u |q σdx)1q
Ω
Ω
Z
0
1ε (
σ −q q dx)1q0 k un − u k Lq(Ω, σ) → 0asn → ∞,
(
Ωε
i. e. ,
b2 (u − linen, un − u) − 0asn → ∞,
(4.35)
but in view of ( 4 . 33 ) and ( 4 . 34 ) , we obtain
b2 (un , u) = (Bε (un , v), u) − b1 (un , v, u) → (ψ, u) − b1 (u, v, u)
and from ( 4 . 35 ) we have ,
b2 (un , un ) → (ψ, u) − b1 (u, v, u).
Then ,
(Bε (un , v), un ) = b1 (un , v, un ) + b2 (un , un ) → (ψ, u).
Now , let us show that Bε is co ercive .
Let v0 ∈ Kψ . From Hölder ’ s
inequality , the growth condition ( 3 . 5 ) and the compact imbed - ding ( 3 .
4 ) we have ,
hAv, v0 i
=
i=1 Z
X
N
≤
i=1
X
N
Ω
ai (x, v, ∇v)∂∂v0 xi dx
Z
Z
0
v0
p
( | ai (x, v, ∇v) |p wpi−p0 dx)1p0 ( | ∂∂x
i | wi dx)1p
Ω
Ω
1
≤
Z
line−j=1
X
0
c1 k| v0 |k ( k(x)p + | v |q σ +
| ∂ ∂ xv j |line−p wj dx)p0
Ω
N
≤
c2 (c3 +
18
k| v |k 0qp +
k| v |k p − 1),
Existence of Solution for Quasilinear Degenerated Elliptic . .
.
where ci are various constants ,
thanks to ( 3 . 7 ) we obtain ,
,
,
hAvk|v| vki − hAvk|v| v0 ki
≥ α k| v |kp−1 − k| v |kp−2 − k| v |k pq0 − 1−
In view of ( 3 . 1 ) we have p − 1 > pq0 − 1.
k| v c |k .
Then ,
line − angbracketlef tAv, kv|v| − kv0 i
− ∞as k| v |k −∞,
since hGε v, vi ≥ 0 and hGε v, v0 i is bounded we have
v
hBεvk, |v| − kv0 line − angbracketright
boundednessRemark:
T he
≥
hAv, kv|v − |k
0
εv,
|v| vk i
v0 i − hGk
− ∞as
k| v |k
appears necessary ,
in order
of assumption(uε )ε inW0 (3.1parenright−line
(Ω, w
) andthe
p,
coercivity of the toproveoperato
line−one
assumption
boundedness
(3.2).
W hileW0the
1,p (Ω , w)
.T hus,when (3g .2≡ )is0, necessarywedon0 t toproveneed to thesuppose
of Gε in
References
[ 1 ] Y . Akdim , E . Azroul , and A . Benkirane . Existence of solutions
for
quasilinear degenerated elliptic equations .
Electronic
J.
Diff .
Eqns . ,
200 1 ( 7 1 ) : 1 – 1 9 , 200 1 .
[ 2 ] A . Benkirane and A . Elmahi . Strongly nonlinear elliptic unilateral
prob lem having natural growth terms and L1 data . Rendiconti di Matematica
, 1 8 : 289 – 303 , 1 998 .
[ 3 ] A . Bensoussan , L . Boccardo , and F . Murat . On a non linear partial
dif ferential equation having natural growth terms and unbounded solution .
Ann . Inst . Henri Poincaré , 5 ( 4 ) : 347 – 364 , 1 988 .
[ 4 ] P . Drabek , A . Kufner , and V . Mustonen . Pseudo - monotonicity
and de generated or singular elliptic operators . Bull . Austral . Math . Soc . ,
58 : 2 1 3 – 22 1 , 1 998 .
19
Y . Akdim , E . Azroul , A . Benkirane
[5]
P . Drabek , A . Kufner , and F . Nicolosi .
Quasilinear e l liptic
equations with
degenerations and singularities .
De Gruyter Series in Nonlinear
Analysis
and Applications , New York , 1 997 .
[ 6 ] P . Drabek and F . Nicolosi . Existence of bounded solutions for
some
degenerated quasilinear elliptic equations .
Annali di Mathematica
pura
ed applicata , CLXV : 2 1 7 – 238 , 1 993 .
[ 7 ] J . L . Lions . Quelques méthodes de résolution des probl è mes aux
limites
non linéaires .
Dunod , Paris , 1 969 .
Youssef Akdim
Elhoussine Azroul
Faculté des Sciences
Faculté des Sciences
Dhar - Mahraz
Dhar - Mahraz
Département de Mathématiques
Département de Mathématiques
et Informatique
et Informatique
B . P 1 796 Atlas F è s .
B . P 1 796 Atlas F è s .
F è s
F è s
MAROC
MAROC
y - akdim @ car a − m ai l . c om
e lazroul @ car m − a ai l . c om
Abdelmoujib Benkirane
Faculté des Sciences Dhar - Mahraz
Département de Mathématiques
et Informatique
B . P 1 796 Atlas F è s .
F è s
MAROC
abdelmouj ib @ i a − m. net . ma
20