Electronic Journal of Qualitative Theory of Differential Equations 2006, No. 19, 1-18;http://www.math.u-szeged.hu/ejqtde/
Quasilinear degenerated equations with
L
1
datum and without coercivity in
perturbation terms
∗
L. AHAROUCH, E. AZROUL and A. BENKIRANE
D´epartement de Math´ematiques et Informatique Facult´e des Sciences Dhar-Mahraz
B.P 1796 Atlas F`es, Morocco
Abstract
In this paper we study the existence of solutions for the generated boundary value prob-lem, with initial datum being an element of L1(Ω) +W−1,p′(Ω, w∗)
−diva(x, u,∇u) +g(x, u,∇u) =f−divF
where a(.) is a Carath´eodory function satisfying the classical condition of type Leray-Lions hypothesis, whileg(x, s, ξ) is a non-linear term which has a growth condition with respect to
ξ and no growth with respect tos, but it satisfies a sign condition on s.
1
.
Introduction
Let Ω be a bounded subset ofIRN (N ≥2), 1< p <∞, andw={wi(x); i= 0, ..., N},be a
collection of weight functions on Ω i.e., each wi is a measurable and strictly positive function
everywhere on Ω and satisfying some integrability conditions (see section 2). Let us consider the non-linear elliptic partial differential operator of order 2 given in divergence form
Au=−div(a(x, u,∇u)) (1.1)
It is well known that equation Au=h is solvable by Drabek, Kufner and Mustonen in [7] in the case whereh∈W−1,p′
(Ω, w∗).
In this paper we investigate the problem of existence solutions of the following Dirichlet problem
Au+g(x, u,∇u) =µ in Ω. (1.2)
∗AMS Subject Classification: 35J15, 35J20, 35J70.
whereµ∈L1(Ω) +
N
Y
i=1
Lp′(Ω, w1i−p′).
In this context of nonlinear operators, if µ belongs to W−1,p′(Ω, w∗) existence results for problem (1.2) have been proved in [2], where the authors have used the approach based on the strong convergence of the positive part u+ε (resp. ngative partu−ε ).
The case where µ∈L1(Ω) is investigated in [3] under the following coercivity condition,
|g(x, s, ξ)| ≥β
N
X
i=1
wi|ξi|p for |s| ≥γ, (1.3)
Let us recall that the results given in [2, 3] have been proved under some additional conditions on the weight function σ and the parameterq introduced in Hardy inequality.
The main point in our study to prove an existence result for some class of problem of the kind (1.2), without assuming the coercivity condition (1.3). Moreover, we didn’t supose any restriction for weight functionσ and parameterq.
It would be interesting at this stage to refer the reader to our previous work [1]. For different appproach used in the setting of Orlicz Sobolev space the reader can refer to [4], and for same results in theLp case, to [10].
The plan of this is as follows : in the next section we will give some preliminaries and some technical lemmas, section 3 is concerned with main results and basic assumptions, in section 4 we prove main results and we study the stability and the positivity of solution.
2
.
Preliminaries
Let Ω be a bounded open subset ofIRN(N ≥2). Let 1< p <∞, and letw={wi(x); 0≤i≤N},
be a vector of weight functions i.e. every componentwi(x) is a measurable function which is
strictly positive a.e. in Ω. Further, we suppose in all our considerations that for 0≤i≤N
wi ∈L1loc(Ω) and w
−p−11
i ∈L1loc(Ω). (2.1)
We define the weighted space with weight γ in Ω as
Lp(Ω, γ) ={u(x) :uγ1p ∈L1(Ω)}, which is endowed with, we define the norm
kukp,γ =
Z
Ω|u(x)|
pγ(x)dx
1
p
.
We denote byW1,p(Ω, w) the weighted Sobolev space of all real-valued functionsu∈Lp(Ω, w
0) such that the derivatives in the sense of distributions satisfy
∂u ∂xi
∈Lp(Ω, wi) for all i= 1, ..., N.
This set of functions forms a Banach space under the norm
kuk1,p,w =
Z
Ω
|u(x)|pw0dx+
N
X
i=1
Z
Ω
|∂u
∂xi
|pwi(x)dx
!1p
To deal with the Dirichlet problem, we use the space
X=W01,p(Ω, w)
defined as the closure of C0∞(Ω) with respect to the norm (2.2). Note that, C0∞(Ω) is dense inW01,p(Ω, w) and (X,k.k1,p,w) is a reflexive Banach space.
We recall that the dual of the weighted Sobolev spacesW01,p(Ω, w) is equivalent toW−1,p′(Ω, w∗), where w∗ ={w∗i =w1i−p′}, i= 1, ..., N and p′ is the conjugate of p i.e. p′ = p−p1. For more details we refer the reader to [8].
We introduce the functional spaces, we will need later.
For p ∈(1,∞), T01,p(Ω, w) is defined as the set of measurable functions u : Ω −→ IR such that fork >0 the truncated functionsTk(u)∈W01,p(Ω, w).
We give the following lemma which is a generalization of Lemma 2.1 [5] in weighted Sobolev spaces.
Lemma 2.1. For everyu∈ T01,p(Ω, w), there exists a unique measurable function v: Ω−→
IRN such that
∇Tk(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 by v=∇u.
Lemma 2.2. Let λ∈IR and letuandvbe two functions which are finite almost everywhere, and which belongs to T01,p(Ω, w). 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 2.1.
The proof of this lemma is similar to the proof of Lemma 2.12 [6] for the non weighted case.
Definition 2.1. LetY be a reflexive Banach space, a bounded operator B fromY to its dual
Y∗ is called pseudo-monotone if for any sequenceun∈Y withun⇀ u weakly inY. Bun⇀ χ
weakly inY∗ andlim sup
n→∞ hBun, uni ≤ hχ, ui, we have
Bun=Bu and hBun, uni → hχ, ui as n→ ∞.
Now, we state the following assumptions. (H1)-The expression
kukX = N
X
i=1
Z
Ω
|∂u
∂xi
|pwi(x)dx
!1p
, (2.3)
is a norm defined on X and is equivalent to the norm (2.2). (Note that (X,kukX) is a
uniformly convex (and reflexive) Banach space.
-There exist a weight function σ on Ω and a parameter q, 1 < q <∞,such that the Hardy inequality
Z
Ω
|u|qσ(x)dx
1q
≤C
N
X
i=1
Z
Ω
|∂u
∂xi
|pwi(x)dx
!1p
holds for everyu∈X with a constant C >0 independent ofu. Moreover, the imbeding
X ֒→Lq(Ω, σ) (2.5)
determined by the inequality (2.4) is compact.
We state the following technical lemmas which are needed later.
Lemma 2.3 [2]. Let g ∈ Lr(Ω, γ) and let gn ∈ Lr(Ω, γ), with kgnkΩ,γ ≤ c,1 < r < ∞. If
gn(x)→g(x) a.e. in Ω, then gn⇀ g weakly inLr(Ω, γ).
Lemma 2.4 [2]. Assume that (H1) holds. Let F : IR → IR be unifomly Lipschitzian,
with F(0) = 0. Let u ∈ W01,p(Ω, w). Then F(u) ∈ W01,p(Ω, w). Moreover, if the set D of discontinuity points of F′ is finite, then
∂F(u)
∂xi
=
(
F′(u)∂x∂u
i a.e. in {x∈Ω :u(x)∈/ D} 0 a.e. in {x∈Ω :u(x)∈D}.
From the previous lemma, we deduce the following.
Lemma 2.5 [2]. Assume that (H1) holds. Let u ∈ W01,p(Ω, w), and let Tk(u), k ∈ IR+, be
the usual truncation, thenTk(u)∈W01,p(Ω, w). Moreover, we have Tk(u)→u strongly in W01,p(Ω, w).
3
.
Main results
Let Ω be a bounded open subset of IRN (N ≥ 2). Consider the second order operator
A:W01,p(Ω, w)−→W−1,p′
(Ω, w∗) in divergence form
Au=−div(a(x, u,∇u)),
wherea: Ω×IR×IRN →IRNis a Carath´eodory function Satisfying the following assumptions:
(H2) Fori= 1, ..., N
|ai(x, s, ξ)| ≤βw
1
p
i (x)[k(x) +σ
1
p′
|s|pq′
+
N
X
j=1 w
1
p′
j (x)|ξj|p−1], (3.1)
[a(x, s, ξ)−a(x, s, η)](ξ−η)>0 for all ξ6=η∈IRN, (3.2)
a(x, s, ξ)ξ ≥α
N
X
i=1
wi(x)|ξi|p. (3.3)
wherek(x) is a positive function in Lp′(Ω) and α, β are positive constants. Assume thatg: Ω×IR×IRN −→IR is a Carath´eodory function satisfying : (H3) g(x, s, ξ) is a Carath´eodory function satisfying
|g(x, s, ξ)| ≤b(|s|)(
N
X
i=1
wi(x)|ξi|p+c(x)), (3.5)
where b :IR+ → IR+ is a positive increasing function and c(x) is a positive function which belong toL1(Ω).
Furthermore we suppose that
µ=f−divF, f ∈L1(Ω), F ∈
N
Y
i=1
Lp′(Ω, w1i−p′), (3.6)
Consider the nonlinear problem with Dirichlet boundary condition
(P)
u∈ T01,p(Ω, w), g(x, u,∇u)∈L1(Ω)
Z
Ω
a(x, u,∇u)∇Tk(u−v)dx+
Z
Ω
g(x, u,∇u)Tk(u−v)dx
≤ Z
Ω
f Tk(u−v)dx+
Z
Ω
F∇Tk(u−v)dx
∀v∈W01,p(Ω, w)∩L∞(Ω)∀k >0.
We shall prove the following existence theorem
Theorem 3.1. Assume that (H1)−(H3) hold true. Then there exists at least one solution
of the probl`em (P).
Remark 3.1. If wi =σ =q = 1, the result of the preceding theorem coincides with those of
Porretta (see [10]).
4
.
Proof of main results
In order to prove the existence theorem we need the following
Lemma 4.1 [2]. Assume that (H1) and (H2) are satisfied, and let (un)n be a sequence in
W01,p(Ω, w) such that
1) un⇀ u weakly in W01,p(Ω, w),
2) Z
Ω[a(x, un,∇un)−a(x, un,∇u)]∇(un−u)dx→0
then,
un→u in W01,p(Ω, w). We give now the proof of theorem 3.1.
STEP 1. The approximate problem.
Letfn be a sequence of smooth functions which strongly converges tof inL1(Ω).
We Consider the sequence of approximate problems:
un ∈W01,p(Ω, w),
Z
Ωa(x, un,∇un)∇v dx+
Z
Ωgn(x, un,∇un)v dx =
Z
Ωfnv dx+
Z
ΩF∇v dx
∀v∈W01,p(Ω, w).
wheregn(x, s, ξ) = 1+g1(x,s,ξ)
n|g(x,s,ξ)|
θn(x) withθn(x) =nT1/n(σ1/q(x)).
Note thatgn(x, s, ξ) satisfises the following conditions
gn(x, s, ξ)s≥0, |gn(x, s, ξ)| ≤ |g(x, s, ξ)| and |gn(x, s, ξ)| ≤n.
We define the operator Gn :X −→X∗ by,
hGnu, vi=
Z
Ω
gn(x, u,∇u)v dx
and
hAu, vi =
Z
Ω
a(x, u,∇u)∇v dx
Thanks to H¨older’s inequality, we have for all u∈X and v∈X,
Z
Ωgn(x, u,∇u)v dx
≤
Z
Ω|gn(x, u,∇u)|
q′
σ−q
′
q dx
q1′ Z
Ω|v|
qσ dx
1
q
≤ n
Z
Ω
σq′/qσ−q′/q dx
1
q′
kvkq,σ
≤ CnkvkX,
(4.2)
the last inequality is due to (2.3) and (2.4).
Lemma 4.2. The operator Bn = A+Gn from X into its dual X∗ is pseudomonotone.
Moreover, Bn is coercive, in the following sense:
< Bnv, v >
kvkX
−→+∞ if kvkX −→+∞, v ∈W01,p(Ω, w).
This Lemma will be proved below.
In view of Lemma 4.2, there exists at least one solution un of (4.1) (cf. Theorem 2.1 and
Remark 2.1 in Chapter 2 of [11] ).
STEP 2. A priori estimates.
Taking v=Tk(un) as test function in (4.1), gives
Z
Ωa(x, un,∇un)∇Tk(un)dx+
Z
Ωgn(x, un,∇un)Tk(un)dx =
Z
ΩfnTk(un)dx+
Z
ΩFn∇Tk(un)dx and by using in fact thatgn(x, un,∇un)Tk(un)≥0, we obtain
Z
{|un|≤k}
a(x, un,∇un)∇undx≤ck+
Z
Ω
Fn∇Tk(un)dx.
Thank’s to Young’s inequality and (3.3), one easily has
α
2
Z
Ω
N
X
i=1
|∂Tk(un)
∂xi
|pwi(x)dx≤c1k. (4.3)
STEP 3. Almost everywhere convergence of un.
aloways assume that the convergence is a.e. after passing to a suitable subsequence). To prove this, we show that un is a Cauchy sequence in measure in any ballBR.
Letk >0 large enough, we have
kmeas({|un|> k} ∩BR) =
Z
{|un|>k}∩BR
|Tk(un)|dx≤
Z
BR
|Tk(un)|dx
≤ Z
Ω|Tk(un)|
pw
0dx
1pZ
BR
w10−p′dx
1
q′
≤ c0
Z
Ω
N
X
i=1
|∂Tk(un)
∂xi
|pwi(x)dx
! 1
p
≤ c1k
1
p.
Which implies that
meas({|un|> k} ∩BR)≤
c1
k1−1p
∀k >1. (4.4)
Moreover, we have, for everyδ >0,
meas({|un−um|> δ} ∩BR) ≤ meas({|un|> k} ∩BR) + meas({|um|> k} ∩BR)
+meas{|Tk(un)−Tk(um)|> δ}.
(4.5) Since Tk(un) is bounded inW01,p(Ω, w),there exists some vk∈W01,p(Ω, w), such that
Tk(un)⇀ vk weakly in W01,p(Ω, w)
Tk(un)→vk strongly in Lq(Ω, σ) and a.e. in Ω.
Consequently, we can assume that Tk(un) is a Cauchy sequence in measure in Ω.
Letε >0, then, by (4.4) and (4.5), there exists somek(ε)>0 such that meas({|un−um|>
δ} ∩BR)< ε for all n, m ≥n0(k(ε), δ, R). This proves that (un)n is a Cauchy sequence in
measure in BR, thus converges almost everywhere to some measurable function u. Then
Tk(un)⇀ Tk(u) weakly in W01,p(Ω, w),
Tk(un)→Tk(u) strongly in Lq(Ω, σ) and a.e. in Ω.
STEP 4. Strong convergence of truncations.
We fixk >0, and let h > k >0. We shall use in (4.1) the test function
(
vn = φ(wn)
wn = T2k(un−Th(un) +Tk(un)−Tk(u)), (4.6)
withφ(s) =seγs2, γ= (b(αk))2.
It is well known that
φ′(s)−b(k)
α |φ(s)| ≥
1
2 ∀s∈IR, (4.7)
It follows that
Z
Ωa(x, un,∇un)∇wnφ ′(w
n)dx+
Z
Ωgn(x, un,∇un)φ(wn)dx =
Z
Ω
fnφ(wn)dx+
Z
Ω
F∇φ(wn)dx.
Sinceφ(wn)gn(x, un,∇un)>0 on the subset{x∈Ω,|un(x)|> k}, we deduce from (4.8) that
Z
Ω
a(x, un,∇un)∇wnφ′(wn)dx+
Z
{|un|≤k}
gn(x, un,∇un)φ(wn)dx
≤ Z
Ω
fnφ(wn)dx+
Z
Ω
F∇φ(wn)dx.
(4.9)
Denote by ε1h(n), ε2h(n), ... various sequences of real numbers which converge to zero as n
tends to infinity for any fixed value ofh.
We will deal with each term of (4.9). First of all, observe that
Z
Ω
fnφ(wn)dx=
Z
Ω
f φ(T2k(u−Th(u)))dx+ε1h(n) (4.10)
and
Z
ΩF∇φ(wn)dx=
Z
ΩF∇T2k(u−Th(u))φ ′(T
2k(u−Th(u)))dx+ε2h(n). (4.11)
Splitting the first integral on the left hand side of (4.9) where|un| ≤k and |un|> k, we can
write,
Z
Ωa(x, un,∇un)∇wnφ ′(w
n)dx
=
Z
{|un|≤k}
a(x, Tk(un),∇Tk(un))[∇Tk(un)− ∇Tk(u)]φ′(wn)dx
+
Z
{|un|>k}
a(x, un,∇un)∇wnφ′(wn)dx.
(4.12)
Setting m = 4k +h, using a(x, s, ξ)ξ ≥ 0 and the fact that ∇wn = 0 on the set where
|un|> m,we have
Z
{|un|>k}
a(x, un,∇un)∇wnφ′(wn)dx
≥ −φ′(2k)Z
{|un|>k}
|a(x, Tm(un),∇Tm(un))||∇Tk(u)|dx,
(4.13) and sincea(x, s,0) = 0 ∀s∈IR, we have
Z
{|un|≤k}
a(x, Tk(un),∇Tk(un))[∇Tk(un)− ∇Tk(u)]φ′(wn)dx
=
Z
Ω
a(x, Tk(un),∇Tk(un))[∇Tk(un)− ∇Tk(u)]φ′(wn)dx.
(4.14)
Combining (4.13) and (4.14), we get
Z
Ωa(x, un,∇un)∇wnφ ′(w
n)dx
≥ Z
Ωa(x, Tk(un),∇Tk(un))[∇Tk(un)− ∇Tk(u)]φ ′(w
n)dx
−φ′(2k)
Z
{|un|>k}
|a(x, Tm(un),∇Tm(un))||∇Tk(u)|dx.
The second term of the right hand side of the last inequality tends to 0 asntends to infinity.
On the other hand, the term of the right hand side of (4.16) reads as
Z by using the continuity of the Nymetskii operator, while ∂(Tk(un))
∂xi ⇀
In the same way, we have
−
Combining (4.16)-(4.19), we get
Z
The second term of the left hand side of (4.9), can be estimated as
Since c(x) belongs to L1(Ω) it is easy to see that
On the other side, we have
Z
As above, by letting ngo to infinity, we can easily see that each one of last two integrals of the right-hand side of the last equality is of the form ε8h(n) and then
(4.9)-(4.11), (4.20) and (4.24), we get
Z
which and (4.7) implies that
It remains to show, for our purposes, that the all terms on the right hand side of (4.25) converge to zero as h goes to infinity. The only difficulty that exists is in the last term. For the other terms it suffices to apply Lebesgue’s theorem.
We deal with this term. Let us observe that, if we takeφ(T2k(un−Th(un))) as test function and thanks to the sign condition (3.4), we get
α
Using the Young inequality we have
α
again because the norm is lower semi-continuity, we get
Consequently, in view of (4.26) and (4.27), we obtain Finally, the strong convergence inL1(Ω) off
n, we have, as first nand thenhtend to infinity,
Therefore by (4.25), letting hgo to infinity, we conclude,
lim
n→∞
Z
Ω[a(x, Tk(un),∇Tk(un))−a(x, Tk(un),∇Tk(u))][∇Tk(un)− ∇Tk(u)]dx= 0, which and using Lemma 4.1 implies that
Tk(un)→Tk(u) strongly in W01,p(Ω, w) ∀k >0. (4.28)
STEP 5. Passing to the limit.
By using Tk(un−v) as test function in (4.1), with v∈W01,p(Ω, w)∩L∞(Ω), we get By Fatou’s lemma and the fact that
a(x, Tk+kvk∞(un),∇Tk+kvk∞(un))⇀ a(x, Tk+kvk∞(u),∇Tk+kvk∞(u))
For the second term of the right hand side of (4.29), we have
since∇Tk(un−v)⇀∇Tk(u−v) weakly in N
Y
i=1
Lp(Ω, wi),while F ∈ N
Y
i=1
Lp′(Ω, wi1−p′). On the other hand, we have
Z
Ω
fnTk(un−v)dx−→
Z
Ω
f Tk(u−v)dx as n→ ∞. (4.32)
To conclude the proof of theorem, it only remains to prove
gn(x, un,∇un)→g(x, u,∇u) strongly in L1(Ω), (4.33)
in particular it is enough to prove the equiintegrable of gn(x, un,∇un). To this purpose, we
takeTl+1(un)−Tl(un) as test function in (4.1), we obtain
Z
{|un|>l+1}
|gn(x, un,∇un)|dx≤
Z
{|un|>l}
|fn|dx.
Letε >0. Then there exists l(ε)≥1 such that
Z
{|un|>l(ε)}
|gn(x, un,∇un)|dx < ε/2. (4.34)
For any measurable subsetE ⊂Ω, we have
Z
E
|gn(x, un,∇un)|dx ≤
Z
E
b(l(ε)) c(x) +
N
X
i=1 wi|
∂(Tl(ε)(un))
∂xi
|p !
dx
+
Z
{|un|>l(ε)}
|gn(x, un,∇un)|dx.
In view of (4.28) there existsη(ε)>0 such that
Z
Eb(l(ε)) c(x) + N
X
i=1 wi|
∂(Tl(ε)(un))
∂xi |
p
!
dx < ε/2
for all E such that measE < η(ε).
(4.35)
Finally, by combining (4.34) and (4.35) one easily has
Z
E|gn(x, un,∇un)|dx < ε for all E such that measE < η(ε),
which shows thatgn(x, un,∇un) are uniformly equintegrable in Ω as required.
Thanks to (4.30)-(4.33) we can pass to the limit in (4.29) and we obtain thatu is a solution of the problem (P).
This completes the proof of Theorem 3.1.
Remark 4.1. Note that, we obtain the existence result withowt assuming the coercivity condition. However one can overcome this difficulty by introduced the functionwn=T2k(un−
Th(un) +Tk(un)−Tk(u)) in the test function(4.6).
Proof of Lemma 4.2.
show that Bn is pseudo-monotone.
Let a sequence (uk)k ∈W01,p(Ω, w) such that
uk⇀ u weakly in W01,p(Ω, w), Bnuk⇀ χ weakly in W−1,p
′
(Ω, w∗),
and lim sup
k→∞
hBnuk, uki ≤ hχ, ui.
We will prove that
χ=Bnu and hBnuk, uki → hχ, ui as k→+∞.
Since (uk)kis a bounded sequence inW01,p(Ω, w), we deduce that (a(x, uk,∇uk))kis bounded
in
N
Y
i=1
Lp′(Ω, w1i−p′), then there exists a functionh∈
N
Y
i=1
Lp′(Ω, wi1−p′) such that
a(x, uk,∇uk)⇀ h weakly in N
Y
i=1
Lp′(Ω, w1i−p′) as k→ ∞,
similarly, it is easy to see that (gn(x, uk,∇uk))kis bounded in Lq
′
(Ω, σ1−q′), then there exists a function kn∈Lq
′
(Ω, σ1−q′) such that
gn(x, uk,∇uk)⇀ kn weakly in Lq
′
(Ω, σ1−q′) as k→ ∞.
It is clear that, for allw∈W01,p(Ω, w), we have
hχ, wi = lim
k→+∞hBnuk, wi
= lim
k→+∞
Z
Ω
a(x, uk,∇uk)∇w dx
+ lim
k→+∞
Z
Ω
gn(x, uk,∇uk).w dx.
Consequently, we get
hχ, wi=
Z
Ω
h∇w dx+
Z
Ω
kn.w dx ∀w∈W01,p(Ω, w). (4.36)
On the one hand, we have
Z
Ωgn(x, uk,∇uk).uk dx−→
Z
Ωkn.u dx as k→ ∞, (4.37) and, by hypotheses, we have
lim sup
k→∞
Z
Ωa(x, uk,∇uk)∇ukdx+
Z
Ωgn(x, uk,∇uk).ukdx
≤ Z
Ωh∇u dx+
Z
Ωkn.u dx, therefore
lim sup
k→∞
Z
Ω
a(x, uk,∇uk)∇ukdx≤
Z
Ω
By virtue of (3.2), we have
Z
Ω
(a(x, uk,∇uk)−a(x, uk,∇u))(∇uk− ∇u)dx >0. (4.39)
Consequently
Z
Ωa(x, uk,∇uk)∇uk dx ≥ −
Z
Ωa(x, uk,∇u)∇u dx+
Z
Ωa(x, uk,∇uk)∇u dx +
Z
Ωa(x, uk,∇u)∇ukdx, hence
lim inf
k→∞
Z
Ωa(x, uk,∇uk)∇ukdx≥
Z
Ωh∇u dx. This implies by using (4.38)
lim
k→∞
Z
Ωa(x, uk,∇uk)∇uk dx=
Z
Ωh∇u dx. (4.40) By means of (4.36), (4.37) and (4.40), we obtain
hBnuk, uki → hχ, ui as k−→+∞.
On the other hand, by (4.40) and the fact that a(x, uk,∇u) → a(x, u,∇u) strongly in N
Y
i=1
Lp′(Ω, w1i−p′) it can be easily seen that
lim
k→+∞
Z
Ω
(a(x, uk,∇uk)−a(x, uk,∇u))(∇uk− ∇u)dx= 0,
and so, thanks to Lemma 4.1
∇un→ ∇u a.e. in Ω.
We deduce then that
a(x, uk,∇uk)→a(x, u,∇u) weakly in N
Y
i=1
Lp′(Ω, w1i−p′),
and gn(x, uk,∇uk)→g(x, u,∇u) weakly in Lq
′
(Ω, wi1−q′).
Thus implies thatχ=Bnu.
Corollary 4.1. Let1< p <∞. Assume that the hypothesis(H1)−(H3) holds, letfn be any
sequence of functions in L1(Ω) which converge tof weakly in L1(Ω)and let un the solution
of the following problem
(Pn′)
un∈ T01,p(Ω, w), g(x, un,∇un)∈L1(Ω)
Z
Ωa(x, un,∇un)∇Tk(un−v)dx+
Z
Ωg(x, un,∇un)Tk(un−v)dx
≤ Z
ΩfnTk(un−v)dx,
∀ v∈W01,p(Ω, w)∩L∞(Ω), ∀k >0.
Then there exists a subsequence of un still denoted un such that un converges to u almost
Proof. We give a brief proof.
Step 1. A priori estimates.
As before we take v= 0 as test function in (Pn′), we get
Z
Ω
N
X
i=1 wi|
∂Tk(un)
∂xi
|pdx≤C1k. (4.41)
Hence, by the same method used in the first step in the proof of Theorem 3.1 there exists a functionu∈ T01,p(Ω, w) and a subsequence still denoted byun such that
un→u a.e. in Ω, Tk(un)⇀ Tk(u) weakly in W01,p(Ω, w), ∀k >0.
Step 2. Strong convergence of truncation.
The choice of v=Th(un−φ(wn)) as test function in (Pn′), we get, for alll >0
Z
Ωa(x, un,∇un)∇Tl(un−Th(un−φ(wn)))dx+
Z
Ωg(x, un,∇un)Tl(un−Th(un−φ(wn))dx
≤ Z
ΩfnTl(un−Th(un−φ(wn)))dx. Which implies that
Z
{|un−φ(wn)|≤h}
a(x, un,∇un)∇Tl(φ(wn))
+
Z
Ω
g(x, un,∇un)Tl(un−Th(un−φ(wn)))dx
≤ Z
ΩfnTl(un−Th(un−φ(wn)))dx. Lettingh tend to infinity and choosing l large enough, we deduce
Z
Ω
a(x, un,∇un)∇φ(wn)dx+
Z
Ω
g(x, un,∇un)φ(wn)dx
≤ Z
Ω
fnφ(wn)dx,
the rest of the proof of this step is the same as in step 4 of the proof of Theorem 3.1.
Step 3. Passing to the limit.
This step is similar to the step 5 of the proof of Theorem 3.1, by using the Egorov’s theorem in the last term of (P′
n).
Remark 4.2. In the case whereF = 0, if we suppose that the second member is nonnegative, then we obtain a nonnegative solution.
Indeed. If we take v=Th(u+) in (P), we have
Z
Ωa(x, u,∇u)∇Tk(u−Th(u +))dx
+
Z
Ωg(x, u,∇u)Tk(u−Th(u +))dx
≤ Z
Ω
Since g(x, u,∇u)Tk(u−Th(u+))≥0, we deduce
Z
Ω
a(x, u,∇u)∇Tk(u−Th(u+))dx≤
Z
Ω
f Tk(u−Th(u+))dx,
and remark also that by using f ≥0 we have
Z
Ωf Tk(u−Th(u
+))dx≤Z
{u≥h}f Tk(u−Th(u))dx.
On the other hand, thanks to (3.3), we conclude
α
Z
Ω
N
X
i=1 wi|
∂Tk(u−)
∂xi
|pdx≤ Z
{u≥h}
f Tk(u−Th(u))dx.
Lettingh tend to infinity, we can easily deduce
Tk(u−) = 0, ∀k >0,
which implies that
u≥0.
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L. Ahrouch (E-mail: l−aharouch@yahoo.fr)
E. Aazroul(E-mail: azroul−elhoussine@yahoo.fr )
D´epartement de Math´ematiques et Informatique, Facult´e des Sciences Dhar-Mahraz,
B.P. 1796 Atlas, F`es, Maroc.
