EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 1, No. 4, 2008, (56-71)
ISSN 1307-5543 – www.ejpam.com

Entropy solutions of nonlinear elliptic equations with mea-
surable boundary conditions and without strict monotonocity
conditions

Y. Akdim1,∗, E. Azroul1, Mohamed Rhoudaf2

1 Faculté Poly-disciplinaire de Taza, B.P 638 Taza, Maroc
2 Département de Mathématiques et Informatique Faculté des Sciences Dhar-Mahraz, B.P 1796
Atlas Fès, Morocco

Abstract. We prove some existence results for nonlinear degenerate elliptic problems of the form

Au+ g(x , u) = f − divF,

where A(u) = −diva(x , u,∇u) is a Leray-Lions, operator defined form the weighted Sobolev space

W 1,p
0 (Ω, w) into its dual. The right hand side, f ∈ L1(Ω) and F ∈

N
∏

i=1

Lp′(Ω, w∗i ). Note that the

Carathéodory function a(x , s,ξ) satisfies only the large monotonicity instead of the monotonicity strict
condition. We overcome this difficulty by using the L1-version of Minty’s lemma.

AMS subject classifications: 35J60.

Key words: Entropy solution, boundary value problems,truncations, Weighted Sobolev Space

1. Introduction

On a bounded open domain Ω of IRN N ≥ 2 we consider the Dirichlet problem for the
quasilinear degenerated elliptic equation,

¨

Au+ g(x , u) = µ in Ω
u= 0 on ∂Ω,

(1.1)

where Au = −div(a(x , u,∇u)) is a Leray-Lions operators defined from the weighted Sobolev
space W 1,p

0 (Ω, w) into its dual W−1,p′(Ω, w∗) where w = {wi , 0 ≤ i ≤ N} is collection of

weight functions on Ω, 1< p <∞ and w∗ = {w1−p′

i , 0≤ i ≤ N}.
Here a(x , s,ξ) is a Carathéodory function defined on Ω× IR× IRN and g(x , u) is a nonlinear
term which satisfy some suitable conditions (H1)− (H2) below. The second member µ is a

∗Corresponding author. Email addresses: azroul−elhoussine@yahoo.fr (E. Azroul),
rhoudaf−mohamed@yahoo.fr (M. Rhoudaf)

http://www.ejpam.com 56 c© 2008 EJPAM All rights reserved.



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 57

measure which belongs in L1(Ω)+W−1,p′(Ω, w∗).
The feature of this paper, is to treat a class of problems for which the classical monotone
operator methods (developed by Visik [12], Minty [11], Browder [6], Brézis [5] and Lions
[10] in non weighted case and by Akdim-Azroul [2] in weighted case and others) do not apply.
The reason for this, is that a(.) does not need to satisfy the strict monotonicity condition that
is,

〈a(x , s,ξ)− a(x , s,η),ξ−η〉> 0 for all ξ 6= η ∈ IRN , (1.2)

of a typical Leray-Lions operator but only a large monotonicity that is

〈a(x , s,ξ)− a(x , s,η),ξ−η〉 ≥ 0 for all (ξ,η) ∈ IRN × IRN , (1.3)

where 〈, 〉 denotes the usual inner product in IRN .
The tool we use to overcome the difficulty of the not strict monotonicity (which can not guar-
antees the almost every where convergence of the gradient of approximation solution) is to
investigate some techniques induced by Minty’s lemma. The approach of pseudo-monotonicity
can not be used due to the fact that f ∈ L1(Ω). In order to prove the a.e. convergence of the
gradient of the approximate solution un, the authors in [4] have show that un is bounded in
the Marcinkiewicz space. While in our present work we prove the locally converge in measure
of un ( see step 2 ).
Thus our aim of this paper, is then to prove an existence of solution for the following problem,

(P )
¨

−diva(x , u,∇u) + g(x , u) = µ in Ω
u= 0 on ∂Ω

where µ = f − divF with f ∈ L1(Ω) and F ∈ ΠN
i=1 Lp′(Ω, w∗i ).In the sense of entropy solution

(see definition 2.1 below)
Note that, the existence of such entropy solution is proved by using only the large monotonic-
ity (1.3).
This paper is organized as follows, section 2 contains some preliminaries and basic assump-
tions. In section 3 we give our main general result which is proved in section 4. Section 5 is
devoted to an example which illustrated our abstract hypotheses.

2. Basic assumptions

Let Ω be a bounded open set of IRN , p be a real number such that 1 < p < ∞ and
w = {wi(x), 0 ≤ i ≤ N} be a vector of weight functions, i.e. every component wi(x) is a
measurable function which is positive a.e. in Ω. Further, we suppose in all our considerations
that

wi ∈ L1
loc(Ω), (2.1)

and

w
−1
p−1

i ∈ L1
loc(Ω), (2.2)

for any 0≤ i ≤ N .



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 58

We denote by W 1,p(Ω, w) the space of all real-valued functions u ∈ Lp(Ω, w0) such that
the derivatives in the sense of distributions fulfil

∂ u

∂ x i
∈ Lp(Ω, wi) for all i = 1, ..., N ,

which is a Banach space under the norm

‖u‖1,p,w =





∫

Ω
|u(x)|pw0(x) d x +

N
∑

i=1

∫

Ω
|
∂ u(x)
∂ x i

|pwi(x) d x





1
p

. (2.3)

The condition (2.1) implies that C∞0 (Ω) is a subspace of W 1,p(Ω, w) and consequently, we can
introduce the subspace W 1,p

0 (Ω, w) of W 1,p(Ω, w) as the closure of C∞0 (Ω) with respect to the
norm (2.3). Moreover, the condition (2.2) implies that W 1,p(Ω, w) as well as W 1,p

0 (Ω, w) are
reflexive Banach spaces.

We recall that the dual space of weighted Sobolev spaces W 1,p
0 (Ω, w) is equivalent to

W−1,p′(Ω, w∗), where w∗ = {w∗i = w1−p′

i , i = 1, ..., N} and p′ is the conjugate of p i.e.
p′ = p

p−1
(for more details we refer to [8]).

Assumption(A1)
We assume that the norm :

‖|u‖|=

 

N
∑

i=1

∫

Ω
|
∂ u

∂ x i
|pwi(x) d x

!
1
p

, (2.4)

is equivalent to the usual norm (2.3), and there exists a weight function σ(x) on Ω and a
parameter q, 1< q <∞ such that the Hardy inequality,

�
∫

Ω
|u(x)|qσ(x) d x

�
1
q

≤ c

 

N
∑

i=1

∫

Ω
|
∂ u

∂ x i
|pwi(x) d x

!
1
p

holds for every u ∈W 1,p
0 (Ω, w) with a constant c > 0 independent of u. Moreover, the imbed-

ding,
W 1,p

0 (Ω, w) ,→,→ Lq(Ω,σ), (2.5)

is compact. Let A be a nonlinear operator from W 1,p
0 (Ω, w) into its dual W−1,p′(Ω, w∗) defined

as
A(u) =−div(a(x , u,∇u))

where a(x , s,ξ) : Ω× IR× IRN → IRN is a Caradhéodory vector-valued function satisfies the
following assumption.



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 59

Assumption(A2)
For i = 1, ..., N

|ai(x , s,ξ)| ≤ βw
1
p

i (x) [k(x) +σ
1
p′ |s|

q
p′ +

N
∑

j=1

w
1
p′

j (x)|ξ j|p−1], (2.6)

for a.e., x ∈ Ω, all (s,ξ) ∈ IR× IRN , some function k(x) ∈ Lp′(Ω)( 1
p
+ 1

p′
= 1) and β > 0. Here

σ and q are as in (A1).

〈a(x , s,ξ)− a(x , s,η),ξ−η〉 ≥ 0 for all (ξ,η) ∈ IRN × IRN , (2.7)

〈a(x , s,ξ),ξ〉 ≥ α
N
∑

i=1

wi|ξi|p, (2.8)

where α is strictly positive constant.
Moreover, the function g(x , s) is a Carathéodory function satisfying

g(x , s)s ≥ 0. (2.9)

sup
|s|≤n
|g(x , s)|= hn(x) ∈ L1(Ω) (2.10)

We recall that, for k > 1 and s in IR, the truncation is defined as

Tk(s) =

¨

s if |s| ≤ k
k s
|s| if |s|> k.

Lemma 2.1. ( cf. [1] ) Assume that (A1) holds. Let (un) be a sequence of W 1,p
0 (Ω, w) such that

un * u weakly in W 1,p
0 (Ω, w). Then Tk(un)* Tk(u) weakly in W 1,p

0 (Ω, w).

3. Main Existence Theorem

Consider the following problem:

(P )
¨

−diva(x , u,∇u) + g(x , u) = f − div(F) in Ω
u= 0 on ∂Ω

where f ∈ L1(Ω) and F ∈
N
∏

i=1

Lp′(Ω, w∗i ).

Definition 3.1. .
An entropy solution of (P ) is a measurable function u such that Tk(u) belongs to W 1,p

0 (Ω, w)
for every k > 0 and such that
∫

Ω
〈a(x , u,∇u),∇Tk[u−ϕ]〉 d x+

∫

Ω
g(x , u)Tk[u−ϕ] d x =

∫

Ω
f Tk[u−ϕ] d x+

∫

Ω
〈F,∇Tk[u−ϕ]〉 d x

for every ϕ ∈W 1,p
0 (Ω, w)∩ L∞(Ω).



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 60

Theorem 3.1. Under the assumptions (A1) and (A2) there exist an entropy solution u of the
problem (P ). i.e. u is a solution of (P ) in the following sense.
∫

Ω
〈a(x , u,∇u),∇Tk[u−ϕ]〉 d x+

∫

Ω
g(x , u)Tk[u−ϕ] d x =

∫

Ω
f Tk[u−ϕ] d x+

∫

Ω
〈F,∇Tk[u−ϕ]〉 d x

for every ϕ ∈W 1,p
0 (Ω, w)∩ L∞(Ω), for every k > 0.

Remark 3.1. The statement of Theorem 3.1 generalizes in weighted case the analogous in [4]
and [3](with g ≡ 0).

4. Proof of Existence Theorem

4.1. Main Lemma

Lemma 4.1. Let u be a measurable function such that Tk(u) belongs to W 1,p
0 (Ω, w) for every

k > 0. Then
∫

Ω
〈a(x , u,∇ϕ),∇Tk[u−ϕ]〉 d x ≤

∫

Ω
f Tk[u−ϕ] d x +

∫

Ω
〈F,∇Tk[u−ϕ]〉 d x . (4.1)

is equivalent to
∫

Ω
〈a(x , u,∇u),∇Tk[u−ϕ]〉 d x+

∫

Ω
g(x , u)Tk[u−ϕ] d x =

∫

Ω
f Tk[u−ϕ] d x+

∫

Ω
〈F,∇Tk[u−ϕ]〉 d x .

(4.2)
for every ϕ in W 1,p

0 (Ω, w)∩ L∞(Ω), and for every k > 0.

Proof
In fact (4.2) implies (4.1) is easily proved adding and subtracting

∫

Ω
〈a(x , u,∇ϕ),∇Tk[u−ϕ]〉 d x

and then using assumption (2.7). Thus, it remains to prove that (4.1) implies (4.2). Let h and
k be positive real numbers, let λ ∈ ]− 1, 1[ and ψ ∈W 1,p

0 (Ω, w)∩ L∞(Ω).
Choose, ϕ = Th(u−λTk(u−ψ)) ∈W 1,p

0 (Ω, w)∩ L∞(Ω) as test function in (4.1), we have:

Ihk ≤ Jhk (4.3)

with

Ihk =

∫

Ω
〈a(x , u,∇Th(u−λTk(u−ψ))),∇Tk(u− Th(u−λTk(u−ψ)))〉 d x

+

∫

Ω
g(x , u)Tk(u− Th(u−λTk(u−ψ))) d x = I ′hk + I ′′hk



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 61

and

Jhk =

∫

Ω
f Tk(u− Th(u−λTk(u−ψ))) d x +

∫

Ω
〈F,∇Tk(u− Th(u−λTk(u−ψ)))〉 d x .

Put
Ahk = {x ∈ Ω, |u− Th(u−λTk(u−ψ))| ≤ k}
and
Bhk = {x ∈ Ω, |u−λTk(u−ψ)| ≤ h}.
Then, we obtain

I ′hk =

∫

Akh∩Bhk

〈a(x , u,∇Th(u−λTk(u−ψ))),∇Tk(u− Th(u−λTk(u−ψ)))〉 d x

+

∫

Akh∩BC
hk

〈a(x , u,∇Th(u−λTk(u−ψ))),∇Tk(u− Th(u−λTk(u−ψ)))〉 d x

+

∫

AC
kh

〈a(x , u,∇Th(u−λTk(u−ψ))),∇Tk(u− Th(u−λTk(u−ψ)))〉 d x .

Since ∇Tk(u− Th(u−λTk(u−ψ))) is different to zero only on Akh, we have
∫

AC
kh

〈a(x , u,∇Th(u−λTk(u−ψ))),∇Tk(u− Th(u−λTk(u−ψ)))〉 d x = 0. (4.4)

Moreover, if x ∈ BC
hk, we have ∇Th(u−λTk(u−ψ)) = 0 and using (2.8), we deduce that,

∫

Akh∩BC
hk

〈a(x , u,∇Th(u−λTk(u−ψ))),∇Tk(u− Th(u−λTk(u−ψ)))〉 d x

=

∫

Akh∩BC
hk

〈a(x , u, 0),∇Tk(u− Th(u−λTk(u−ψ)))〉 d x = 0. (4.5)

From (4.4) and (4.5), we obtain

I ′hk =

∫

Akh∩Bhk

〈a(x , u,∇Th(u−λTk(u−ψ))),∇Tk(u− Th(u−λTk(u−ψ)))〉 d x .

Letting h→+∞, and |λ| ≤ 1, we have

Akh→ {x , |λ||Tk(u−ψ)| ≤ k}= Ω, (4.6)

Bhk→ Ω which implies Akh ∩ Bhk→ Ω. (4.7)



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 62

Which and using Lebesgue theorem, we conclude that

lim
h→+∞

∫

Akh∩Bhk

〈a(x , u,∇Th(u−λTk(u−ψ))),∇Tk(u− Th(u−λTk(u−ψ)))〉 d x

= λ

∫

Ω
〈a(x , u,∇(u−λTk(u−ψ)),∇Tk(u−ψ)〉 d x .

(4.8)

i.e.,

lim
h→+∞

I ′hk = λ

∫

Ω
〈a(x , u,∇(u−λTk(u−ψ)),∇Tk(u−ψ)〉 d x . (4.9)

moreover it is easy to see that,

lim
h→+∞

∫

Ω
g(x , u)Tk(u− Th(u−λTk(u−ψ))) d x = λ

∫

Ω
g(x , u)Tk[u−ψ] d x

thus implies that,

lim
h→+∞

Ihk = λ

∫

Ω
〈a(x , u,∇(u−λTk(u−ψ)),∇Tk(u−ψ)〉 d x+λ

∫

Ω
g(x , u)Tk[u−ψ] d x (4.10)

On the other hand, we have,

Jhk =

∫

Ω
f Tk(u− Th(u−λTk(u−ψ))) d x +

∫

Ω
〈F,∇Tk(u− Th(u−λTk(u−ψ)))〉 d x .

Then

lim
h→+∞

∫

Ω
f Tk(u− Th(u−λTk(u−ψ))) d x +

∫

Ω
〈F,∇Tk(u− Th(u−λTk(u−ψ)))〉 d x

= λ

∫

Ω
f Tk[u−ψ] d x +λ

∫

Ω
〈F,∇Tk[u−ψ]〉 d x

i.e.,

lim
h→+∞

Jhk = λ

∫

Ω
f Tk[u−ψ] d x +λ

∫

Ω
〈F,∇Tk[u−ψ]〉 d x . (4.11)

Together (4.10), (4.11) and passing to the limit in (4.3), we obtain,

λ

�
∫

Ω
〈a(x , u,∇(u−λTk(u−ψ),∇Tk(u−ψ)〉 d x +

∫

Ω
g(x , u)Tk[u−ψ] d x

�

≤ λ
�
∫

Ω
f Tk[u−ψ] d x +

∫

Ω
〈F,∇Tk[u−ψ]〉 d x

�



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 63

for every ψ ∈ W 1,p
0 (Ω, w) ∩ L∞(Ω), and for k > 0. Choosing λ > 0 dividing by λ, and then

letting λ tend to zero , we obtain
∫

Ω
〈a(x , u,∇u),∇Tk[u−ϕ]〉 d x+

∫

Ω
g(x , u)Tk[u−ψ] d x ≤

∫

Ω
f Tk[u−ϕ] d x+

∫

Ω
〈F,∇Tk[u−ϕ]〉 d x .

(4.12)
For λ < 0 , dividing by λ, and then letting λ tend to zero , we obtain
∫

Ω
〈a(x , u,∇u),∇Tk[u−ϕ]〉 d x+

∫

Ω
g(x , u)Tk[u−ψ] d x ≥

∫

Ω
f Tk[u−ϕ] d x+

∫

Ω
〈F,∇Tk[u−ϕ]〉 d x .

(4.13)
Combining (4.12) and (4.13), we conclude the following equality :
∫

Ω
〈a(x , u,∇u),∇Tk[u−ϕ]〉 d x+

∫

Ω
g(x , u)Tk[u−ψ] d x =

∫

Ω
f Tk[u−ϕ] d x+

∫

Ω
〈F,∇Tk[u−ϕ]〉 d x .

(4.14)
This completes the proof of Lemma 4.1.

4.2. Proof of Theorem 3.1

1. Approximate problem and a priori estimate
Let fn be a sequence function of L∞(Ω) which is strongly convergent to f in L1(Ω) such

that ‖ fn‖L1 ≤ ‖ f ‖L1 , and let un be a solution in W 1,p
0 (Ω, w) of the problem

¨

−diva(x , un,∇un) + gn(x , un) = fn− div(F) in Ω
un = 0 on ∂Ω (4.15)

where

gn(x , s) =
g(x , s)

1+ 1
n
|g(x , s)|

θn(x) and θn(x) = T 1
n
(σ

1
q (x))

which exists thanks to [7].
Choosing Tk(un) as test function in (4.15), we have

∫

Ω
〈a(x , un,∇un),∇Tk(un)〉 d x+

∫

Ω
gn(x , un)Tk(un) d x =

∫

Ω
fnTk(un) d x+

∫

Ω
〈F,∇Tk(un)〉 d x

using ∇Tk(un) =∇unχ{|un|≤k} and thanks to assumption (2.8), we obtain

∫

Ω
〈a(x , un,∇un),∇Tk(un)〉 d x ≥ α

N
∑

i=1

∫

Ω
wi|
∂ Tk(un)
∂ x i

|p d x

then since gn(x , un)Tk(un)≥ 0 we have,

α

N
∑

i=1

∫

Ω
wi|
∂ Tk(un)
∂ x i

|p d x ≤ k‖ f ‖L1 +
N
∑

i=1

∫

Ω
Fi|
∂ Tk(un)
∂ x i

| d x



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 64

≤ k‖ f ‖L1 +
N
∑

i=1

∫

Ω
Fiw

−1
p

i (
α

2
)
−1
p |
∂ Tk(un)
∂ x i

|w
1
p

i (
α

2
)

1
p d x

by Young’s inequality, we obtain

α

N
∑

i=1

∫

Ω
wi|
∂ Tk(un)
∂ x i

|p d x ≤ k‖ f ‖L1 +
c(α)
p′
|F‖∏ Lp′ (Ω,w∗i )

+
α

2

N
∑

i=1

∫

Ω
wi|
∂ Tk(un)
∂ x i

|p d x .

Then,
α

2

N
∑

i=1

∫

Ω
wi|
∂ Tk(un)
∂ x i

|p d x ≤ k(‖ f ‖L1 +
c(α)
p′
‖F‖∏ Lp′ (Ω,w∗i )

for k > 1, which implies that

 

N
∑

i=1

∫

Ω
|
∂ Tk(un)
∂ x i

|pwi(x)d x

!
1
p

≤ ck
1
p ∀k > 1. (4.16)

2: Locally convergence of un in measure
We prove that un converges to some function u locally in measure (and therefore, we can

always assume that the convergence is a.e. after passing to a suitable subsequence), we shall
show that un is a Cauchy sequence in measure in any ball BR.
Let k > 0 large enough, by using (2.5), we have

k meas({|un|> k} ∩ BR) =

∫

{|un|>k}∩BR

|Tk(un)| d x ≤
∫

BR

|Tk(un)| d x

≤
�
∫

Ω
|Tk(un)|pw0 d x

�
1
p

.

 

∫

BR

w1−p′

0 d x

!
1
q′

≤ cR

 

∫

Ω

N
∑

i=1

|
∂ Tk(un)
∂ x i

|pwi(x) d x

!
1
p

≤ c1k
1
p .

which implies

meas({|un|> k} ∩ BR)≤
c1

k1− 1
p

∀k > 1. (4.17)

We have, for every δ > 0,

meas({|un− um|> δ} ∩ BR) ≤ meas({|un|> k} ∩ BR) +meas({|um|> k} ∩ BR)
+meas{|Tk(un)− Tk(um)|> δ}.

(4.18)

Since Tk(un) is bounded in W 1,p
0 (Ω, w), there exists some vk ∈W 1,p

0 (Ω, w), such that

Tk(un)* vk weakly in W 1,p
0 (Ω, w)

Tk(un)→ vk strongly in Lq(Ω,σ) and a.e. in Ω.



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 65

Consequently, we can assume that Tk(un) is a Cauchy sequence in measure in Ω.
Let ε > 0, then by (4.17) and (4.18), there exists some k(ε) > 0 such that meas({|un − um| >
δ} ∩ BR) < ε for all n, m ≥ n0(k(ε),δ, R). This proves that (un) is a Cauchy sequence in
measure in BR, thus converges almost everywhere to some measurable function u. Then

Tk(un)* Tk(u) weakly in W 1,p
0 (Ω, w),

Tk(un)→ Tk(u) strongly in Lq(Ω,σ) and a.e in Ω.
(4.19)

3. Equi-integrability of nonlinearities
we need to prove that

gn(x , un)→ g(x , u) strongly in L1(Ω) (4.20)

in particular it is enough to prove the equi-integrable of gn(x , un) to this purpose. We take
Tl+1(un)− Tl(un) as test function in (4.15), we obtain

∫

Ω
〈a(x , un,∇un),∇(Tl+1(un)− Tl(un))〉 d x +

∫

Ω
gn(x , un)(Tl+1(un)− Tl(un)) d x

=

∫

Ω
f (Tl+1(un)− Tl(un)) d x

+
N
∑

i=1

∫

Ω
Fi∇(Tl+1(un)− Tl(un)) d x

which implies that,
∫

{l≤|un|≤l+1}
〈a(x , un,∇un),∇un〉 d x +

∫

{|un|≥l+1}
|gn(x , un)| d x

≤ c

∫

{|un|≥l}
| f | d x +

N
∑

i=1

∫

{l≤|un|≤l+1}
Fiw

−1
p

i (
α

2
)
−1
p |∇un|(

α

2
)

1
p d x

by Young’s inequality, we obtain
∫

{l≤|un|≤l+1}
〈a(x , un,∇un),∇un〉 d x +

∫

{|un|≥l+1}
|gn(x , un)| d x

≤ c

∫

{|un|≥l}
| f | d x +

c(α)
p′

N
∑

i=1

∫

{|un|≥l}
|Fi|p

′
w1−p′

i d x

+α
2

N
∑

i=1

∫

{l≤|un|≤l+1}
|∇un|pwi d x

thus by (2.8), we have

∫

{|un|≥l+1}
|gn(x , un)| d x ≤ c

∫

{|un|≥l}
| fn| d x +

c(α)
p′

N
∑

i=1

∫

{|un|≥l}
|Fi|p

′
w1−p′

i d x .



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 66

Let ε > 0, then there exist l(ε)≥ 1 such that
∫

{|un|>l(ε)}
|gn(x , un)| d x ≤

ε

2
. (4.21)

For any measurable subset E ⊂ Ω, we have
∫

E

|gn(x , un)| d x ≤
∫

E∩{|un|≤l(ε)}
|gn(x , un)| d x +

∫

E∩{|un|>l(ε)}
|gn(x , un)| d x

≤
∫

E

|hl(ε)(x)| d x +

∫

E∩{|un|>l(ε)}
|gn(x , un)| d x .

In view to (2.10) there exist η(ε)> 0 such that
∫

E

|hl(ε)(x)| d x ≤
ε

2
(4.22)

for all E such that meas(E)< η(ε).

Finally, by combining (4.21) and (4.22) one easily has

∫

E

|gn(x , un)| d x ≤ ε, for all E such

that meas(E)< η(ε).
4. An intermediate Inequality

In this step, we shall prove that for ϕ ∈W 1,p
0 (Ω, w)∩ L∞(Ω), we have

∫

Ω
〈a(x , un,∇ϕ),∇Tk[un−ϕ]〉 d x +

∫

Ω
gn(x , un)Tk[un−ϕ] d x

≤
∫

Ω
fnTk[un−ϕ] d x +

∫

Ω
〈F,∇Tk[un−ϕ]〉 d x . (4.23)

We choose now Tk(un−ϕ) as test function in (4.15), with
ϕ in W 1,p

0 (Ω, w)∩ L∞(Ω), we obtain
∫

Ω
〈a(x , un,∇un),∇Tk[un−ϕ]〉 d x +

∫

Ω
gn(x , un)Tk[un−ϕ] d x

=

∫

Ω
fnTk[un−ϕ] d x +

∫

Ω
〈F,∇Tk[un−ϕ]〉 d x .

Adding and subtracting the term

∫

Ω
〈a(x , un,∇ϕ),∇Tk[un−ϕ]〉 d x i.e.,

∫

Ω
〈a(x , un,∇un),∇Tk[un−ϕ]〉 d x +

∫

Ω
〈a(x , un,∇ϕ),∇Tk[un−ϕ]〉 d x

−
∫

Ω
〈a(x , un,∇ϕ),∇Tk[un−ϕ]〉 d x +

∫

Ω
gn(x , un)Tk[un−ϕ] d x

=

∫

Ω
fnTk[un−ϕ] d x +

∫

Ω
〈F,∇Tk[un−ϕ]〉 d x

(4.24)



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 67

Thanks to assumption (2.7) and the definition of truncation function, we have
∫

Ω




[a(x , un,∇un)− a(x , un,∇ϕ)
�

,∇Tk[un−ϕ]〉 d x ≥ 0 (4.25)

Combining (4.24) and (4.25), we obtain (4.23).
5. Passing to the limit

We shall prove that for ϕ ∈W 1,p
0 (Ω, w)∩ L∞(Ω), we have

∫

Ω
〈a(x , u,∇ϕ),∇Tk[u−ϕ]〉 d x+

∫

Ω
g(x , u)Tk[u−ϕ] d x ≤

∫

Ω
f Tk[u−ϕ] d x+

∫

Ω
〈F,∇Tk[u−ϕ]〉 d x .

Firstly, we claim that
∫

Ω
〈a(x , un,∇ϕ),∇Tk[un−ϕ]〉 d x →

∫

Ω
〈a(x , u,∇ϕ),∇Tk[u−ϕ]〉 d x as n→+∞.

Since TM (un) * TM (u) weakly in W 1,p
0 (Ω, w),with M = k + ‖ϕ‖∞, then by Lemma 2.1, we

have
Tk(un−ϕ)* Tk(u−ϕ) in W 1.p

0 (Ω, w), (4.26)

which gives

∂ Tk

∂ x i
(un−ϕ)*

∂ Tk

∂ x i
(u−ϕ) weakly in Lp(Ω, wi) ∀i = 1, .., N . (4.27)

Show that
ai(x , TM (un),∇ϕ)→ ai(x , TM (u),∇ϕ) strongly in Lp′(Ω, w∗i )

Thanks to assumption (2.6), we obtain

|ai(x , TM (un),∇ϕ)|p
′
w
−p′

p

i ≤ β[k(x) + |TM (un)|
q
p′σ

1
p′ +

N
∑

j=1

|
∂ ϕ

∂ x i
|p−1w

1
p′

i ]
p′

≤ γ[k(x)p
′
+ |TM (un)|qσ+

N
∑

j=1

|
∂ ϕ

∂ x i
|pwi], (4.28)

with β and γ are positive constants. Since TM (un)* TM (u) weakly in W 1,p
0 (Ω, w)

and W 1,p
0 (Ω, w) ,→,→ Lq(Ω,σ), then TM (un) → TM (u) strongly in Lq(Ω,σ) and a.e. in Ω,

hence
|ai(x , TM (un),∇ϕ)|p

′
w∗i → |ai(x , TM (u),∇ϕ)|p

′
w∗i a.e.in Ω.

and

γ






k(x)p

′
+ |TM (un)|qσ+

N
∑

j=1

|
∂ ϕ

∂ x i
|pwi






→ γ






k(x)p

′
+ |TM (u)|qσ+

N
∑

j=1

|
∂ ϕ

∂ x i
|pwi






a.e. in Ω.



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 68

Then, By Vitali’s theorem, we deduce that

ai(x , TM (un),∇ϕ)→ ai(x , TM (u),∇ϕ) strongly in Lp′(Ω, w∗i ), as n→+∞. (4.29)

Combining (4.27) and (4.29), we obtain
∫

Ω
〈a(x , un,∇ϕ),∇Tk[un−ϕ]〉 d x →

∫

Ω
〈a(x , u,∇ϕ),∇Tk[u−ϕ]〉 d x , as n→+∞. (4.30)

Secondly, we show that
∫

Ω
fnTk[un−ϕ] d x →

∫

Ω
f Tk[u−ϕ] d x . (4.31)

We have fnTk[un −ϕ]→ f Tk[u−ϕ] a.e. in Ω and | fnTk[un −ϕ]| ≤ k| fn| and k| fn| → k| f |
in L1(Ω), then by using Vitali’s theorem, we obtain (4.31).
Similarly thanks to (4.20) we can show that

∫

Ω
gn(x , un)Tk[un−ϕ] d x →

∫

Ω
g(x , u)Tk[u−ϕ] d x as n→∞. (4.32)

Show that:
∫

Ω
〈F,∇Tk[un−ϕ]〉 d x →

∫

Ω
〈F,∇Tk[u−ϕ]〉 d x . (4.33)

In view of (4.27) and since F ∈
N
∏

i=1

Lp′(Ω, w∗i ), we obtain (4.33).

Thanks to (4.30) , (4.31) and (4.33) allow to pass to the limit in the inequality (4.23), so
that ∀ϕ ∈W 1,p

0 (Ω, w)∩ L∞(Ω), we deduce
∫

Ω
〈a(x , u,∇ϕ),∇Tk[u−ϕ]〉 d x ≤

∫

Ω
f Tk[u−ϕ] d x +

∫

Ω
〈F,∇Tk[u−ϕ]〉 d x .

In view of Main Lemma, we can deduce that u is an entropy solution of the problem (P ).
This completes the proof of Theorem 3.1.

Remark 4.1. In the case where F ≡ 0, if we suppose that the second member are nonnegative,
then we obtain a nonnegative solution.

Indeed, If we take v = Th(u+) in (P), we have
∫

Ω
〈a(x , u,∇u),∇Tk(u− Th(u

+))〉 d x

+

∫

Ω
g(x , u)Tk(u− Th(u

+)) d x

≤
∫

Ω
f Tk(u− Th(u

+)) d x .



Y. Akdim, E. Azroul, and M. Rhoudaf / Eur. J. Pure Appl. Math, 1 (2008), (56-71) 69

Since g(x , u)Tk(u− Th(u+))≥ 0, we deduce
∫

Ω
〈a(x , u,∇u),∇Tk(u− Th(u

+))〉 d x ≤
∫

Ω
f Tk(u− Th(u

+)) d x ,

we remark also, by using f ≥ 0
∫

Ω
f Tk(u− Th(u

+)) d x ≤
∫

{u≥h}
f Tk(u− Th(u)) d x .

On the other hand, thanks to (2.8), we conclude

α

∫

Ω

N
∑

i=1

|
∂ Tk(u−)
∂ x i

|pwi d x ≤
∫

{u≥h}
f Tk(u− Th(u)) d x .

Letting h tend to infinity, we can easily deduce

Tk(u
−) = 0, ∀k > 0,

which implies that
u≥ 0.

5. Example

Let us consider the following special case:

ai(x ,η,ξ) = wi(x)|ξi|p−1sgn(ξi) i = 1, ..., N ,

g(x , s) = ρs|s|r ρ > 0 and r > 0

with wi(x) is a weight function (i = 1, ..., N).
For simplicity, we shall suppose that:

wi(x) = w(x) for i = 1, ..., N − 1, wN (x)≡ 0

it is easy to show that ai(x , s,ξ) are Caracthéodory function satisfying the growth condition
(2.6) and the coercivity (2.8). On the other hand, the monotonicity condition is verified. In
fact,

N
∑

i=1

(ai(x , s,ξ)− ai(x , s, ξ̂))(ξi − ξ̂i) = w(x)
N−1
∑

i=1

(|ξi|p−1sgn(ξi)− |ξ̂i|p−1sgn(ξ̂i))(ξi − ξ̂i)≥ 0

for almost all x ∈ Ω and for all ξ, ξ̂ ∈ IRN . This last inequality can not be strict, since for ξ 6= ξ̂
with ξN 6= ξ̂N and ξi = ξ̂i , i = 1, ..., N − 1. The corresponding expression is zero.



REFERENCES 70

In particular, let us use special weight functions w and σ expressed in terms of the distance
to the bounded ∂Ω. Denote d(x) = dist(x ,∂Ω) and set

w(x) = dλ(x), σ(x) = dµ(x).

In this case, the Hardy inequality reads

�
∫

Ω
|u(x)|qdµ(x) d x

�
1
q

≤ c

 

N−1
∑

i=1

∫

Ω
|
∂ u

∂ x i
|pdλ(x) d x

!
1
p

.

The corresponding imbedding is compact if:
(i) For, 1< p ≤ q <∞,

λ < p− 1,
N

q
−

N

p
+ 1≥ 0,

µ

q
−
λ

p
+

N

q
−

N

p
+ 1> 0. (5.1)

(ii) For 1≤ q < p <∞,

λ < p− 1,
µ

q
−
λ

p
+

1

q
−

1

p
+ 1> 0. (5.2)

Remark 5.1. 1.Condition (5.1) or (5.2) are sufficient for the compact imbedding (2.5) to hold;
for example [ [7], Example 1, [8] Example 1.5], and [9], Theorems 19.17, 19.22].

Finally, the hypotheses of Theorem 3.1 are satisfied. Therefor the following problem






































Tk(u) ∈W 1,p
0 (Ω, w)

∫

Ω

N
∑

i=1

wi(x)|
∂ u

∂ x i
|p−1sgn(

∂ u

∂ x i
)
∂ Tk(u−ϕ)

∂ x i
d x

+

∫

Ω
uexp(u)Tk(u−ϕ) d x =

∫

Ω
f Tk(u−ϕ) d x +

∫

Ω
F∇Tk(u−ϕ) d x

f ∈ L1(Ω), F ∈
N
∏

i=1

Lp′(Ω, w∗i ) and ∀ϕ ∈W 1,p
0 (Ω, w)∩ L∞(Ω)

has at last one solution.

References

[1] Y. Akdim, E. Azroul and A. Benkirane, Existence of Solution for Quasilinear Degenerated Elliptic
Equations, Electronic J. Diff. Equ., Vol. 2001, N 71, (2001) pp 1-19.

[2] Y. Akdim, E. Azroul and A. Benkirane, Psudo-monotonicity and Degenerated Elliptic operator of
second order, Electronic J. Diff. Equ., conference 09, 2003, N 71, (2001) pp 9-24.

[3] L. Boccardo,A remark on some nonlinear elliptic problems, Electron. J. Diff. Eqns. Conf. 08, 2002,
pp. 47-52.

[4] L. Boccardo, L. Orsina, Existence Results for Dirichlet Problem in L1 via Minty’s lemma, Applicable
Ana (1999) pp 309-313.



REFERENCES 71

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Hilbert, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). North-Holland
Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973,
MR 50 6= 1060 Zbl 252.47055.

[6] F. E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis
(Berkeley, 1968), Proc. Sympos. Pure Math., no. XVI, AMS, Providence, 1970, pp. 1-60, MR
42 6= 4855.

[7] P. Drabek, A. KUFNER and V. Mustonen, Pseudo-monotonicity and degenerated or singular elliptic
operators, Bull. Austral. Math. Soc. Vol. 58 (1998), 213-221.

[8] P. Drabek, A. Kufner and F. Nicolosi, Non linear elliptic equations, singular and degenerate cases,
University of West Bohemia, (1996).

[9] A. Kufner, Weighted Sobolev Spaces, John Wiley and Sons, (1985).
[10] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris

(1969).
[11] G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962),
[12] M.L. Visik, Solvability of the first boundary value problem for quasilinear equations with rapidly

increasing coefficients in Orlicz classes, Dok1. Akad. Nauk SSSR 151, 1963, pp. 758-761=Sovier
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