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Moreover, u ∈ C([0, T ], L

(Ω)) and u

converges to u in

C([0, T ], L

(Ω)).

The proof is similar of Proposition 7.3.1 in.

Step 5: In this step we prove that u satisﬁes (15). Using in (18) the

admissible test function T

k+c

) − T

) leads to

Ω

[ϕ

k+c

)(T ) − ϕ

)(T )]dx +

a(Du

)[DT

k+c

) − DT

)]dxdt

k+c

) − T

)]dxdt +

Ω

[ϕ

k+c

) − ϕ

)]dx.

(45)

In (45) and in what follows the (t, x) dependence on a(x, t, d) is dropped

for convenience. Since the function ϕ

k+c

(r) − ϕ

(r) is positive, using the

convergence of f

to f in L

(Q), the convergence of u

to u

in L

(Ω),

the fact that |T

k+c

(r) −T

(r)| ≤ C, the linear growth of ϕ

at inﬁnity, and

(3.1), we deduce from (45)that:

lim sup

n→∞

a(Du

)[DT

k+c

) − DT

)]dxdt

≤

f[T

k+c

(u) − T

(u)]dxdt +

Ω

[ϕ

k+c

) − ϕ

)]dx.

(46)

Since DT

k+c

) − DT

) = 0 when (t, x) does not belong to the set

{(t, x); k ≤ |u

(t, x)| ≤ k + c}, we have almost pointwise in Q

a(Du

)[DT

k+c

) − DT

)]

= a(DT

k+c

) − DT

))[DT

k+c

) −DT

)]

(47)

Since, by (20), T

k+c

) − T

) weakly converge to T

k+c

(u) − T

(u)

in X as n tends to ∞, we obtain

α lim sup

n→∞

i=1



∂[T

k+c

(u) − T

(u)]

∂x



(x)dxdt

≤

f[T

k+c

(u) − T

(u)]dxdt +

Ω

[ϕ

k+c

) − ϕ

)]dx.

(48)

When k tends to inﬁnity, the right-hand side (48) tends to zero: indeed

k+c

(u) − T

(u) is dominated by C and tends almost everywhere to zero,

while



Ω

[ϕ

k+c

) − ϕ

)]



≤ C

{|u

(x)|>k}

(x)|dx.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

341

This proves that u satisﬁes (15).

Step 6: Passing to the limits. We now return to the proof of the

existence result of theorem (3.1). We notice that the function u deﬁned in

Lemma 20 meets the regularity requirements (15), (13) and (16) of Deﬁ-

nition 3.1. The initial condition is easily recovered by passing to the limit

(t = 0) = u

and using (3.1). It thus only remains to prove that u satisﬁes equa-

tion (16). Since u

belongs to L

(0, T ; W

1, p

(Ω, w)) with

∂u

∂t

(0, T ; W

−1,p

(Ω, w

∗

)) , we have for any S ∈ C

∞

(R) with S

∈ C

∞

(R)

∂S(u

)

∂t

= S

)

∂u

∂t

in D

(Q).

Using this fact in equation (18) leads to:

∂S(u

)

∂t

− div[S

)a(Du

)] + S

)a(Du

)Du

= fS

) in D

(Q).

(49)

To pass to the limit in (49) as n tends to +∞, let M be such that supp S

⊂

[−M, M], so that

)a(Du

) = S

)a(DT

)).

The pointwise convergence of u

to u as n tends to +∞ and the bounded

character of S

permit us to conclude that

)a(Du

) * S

(u)a(DT

(u)) in

i=1

(Q, w

∗

). (50)

as n tends to +∞. S

(u)a(DT

(u)) has been denoted by S

(u)a(Du) in

equation (16).

As far as the ’energy’ term

)a(Du

)Du

= S

)a(DT

))DT

)

is concerned, remark that

lim

n→+∞

[a(DT

)) − a(DT

(u))][DT

) − DT

(u)]dxdt = 0.

Due to the monotonicity (9), the above equality shows that

[a(DT

))−a(DT

(u))][DT

)−DT

(u)] → 0 strongly in L

(Q)

(51)

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

342

as n tends to +∞. Again using (51), (20) and (44) then implies that

a(DT

))DT

) * a(DT

(u))DT

(u) weakly in L

(Q) (52)

as n tends to +∞. Appealing now to the pointwise convergence of S

)

to S

(u) and since S

is bounded, a straightforward application of Egoroﬀ’s

theorem leads to

)a(DT

))DT

) * S

(u)a(DT

(u))DT

(u)

weakly in L

(Q), (53)

as n tends to +∞.

Finally, the pointwise convergence (3.1) and the L

(Q)−strong convergence

of f

to f yield

→ S

(u)f strongly in L

(Q), (54)

as n tends to +∞. The L

∞

−weak* convergence of S(u

) to S(u) together

with (49), (50), (53) and (54) proves that (16) is satisﬁed in the sense of

distributions. This completes the proof of the existence result of Theorem

3.1.

Proof of uniqueness result

We now turn to the proof of the uniqueness result of Theorem 3.1. In fact,

we will prove a comparison result for renormalised solutions.

Lemma 3.4. Assume that u

and u

lie in L

(Ω), that f

and f

lie in

(Q) and that they satisfy:

≤ u

, (55)

≤ f

. (56)

Then if u

and u

are two renormalised solutions of problem (18) respec-

tively for the data (f

, u

) and (f

, u

), we have

≤ u

almost everywhere in Q. (57)

Remark 2. Uniqueness of a solution of (13)-(16) is a direct application of

Lemma 3.4.

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References

1. L.Aharouch, E.Azroul and M.Rhoudaf, Strongly nonlinear variational

parabolic problems in weighted sobolev spaces. The Australian journal of

Mathematical Analysis and Applications, Vol.5, Issue 2, Article 13, pp. 1-

25,2008 .

2. L. Boccardo and T.Gallouet. On some nonlinear elliptic and parabolic equa-

tions involving measure data.J.Funct.Anal.87(1989),149-69.

3. D. Blanchard. Truncations and monotonocity methods for parabolic equa-

tions. Nonlinear Anal.21 (1993),725-43.

4. D. Blanchard, F.Murat. Renormalized solutions of nonlinear parabolic prob-

lems with L

data: existence and uniqueness. Proceedings of the Royal Soci-

ety of Edinburgh, 127A, 1137-1152, 1997.

5. J.Berkovits and V.Mustonen. Topological degree for perturbations of linear

maximal monotone mappings and applications to a class of parabolic problem

. Rendiconti di Matematica, Serie VII, volume 12, Roma (1992),597-621.

6. P. Drabek, A. Kufner and F. Nicolosi, Non linear elliptic equations, singular

and degenerated cases, University of West Bohemia, (1996).

7. J. L. Lions quelques m´ethodes de r´esolution des probl`eme aux limites non

lineaires, Dundo, Paris (1969).

8. P. L. Lions and R.J.Diperna, On the cauchy problem for Boltzman equations:

global existence and weak stability. Ann.of Math.(2)130(1989),321-366.

9. M.Rhoudaf, Etude de certains problemes elliptiques et paraboliques non lin-

eaires avec second membre L

ou dans le dual , Th`ese de doctorat d’ etat,

Facult´e des Sciences Dhar-Mhraz, F`es, Maroc, (2006) .

10. E.Zeidler , Nonlinear Functional Analysis and its Applications, Springer-

Verlag(New York- Heidlberg, (1990)).

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

344

Solvability of quasilinear degenerated elliptic equations with L

data

Youssef Akdim

Laboratoire d’Informatique Math´ematique Automatique et Optoel´ectronique,

Facult´e Poly-disciplinaire, Taza, Morocco

E-mail: akdimyoussef@yahoo.fr

In this paper we prove the existence of solutions for quasilinear degenerated

elliptic operators A(u) + g(x, u, ∇u) = f , where A is a Leray-Lions operator

from W

1,p

(Ω, w) 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 on s. The right hand side f is assumed to belong to

(Ω).

Keywords: Quasilinear degenerate elliptic operator; Existence result.

1. Introduction

In this paper, we shall be concerned with the existence of solutions for

quasilinear degenerated elliptic equations of the type

(P)



Au + g(x, u, ∇u) = f in D

(Ω),

u ∈ W

1,p

(Ω, w), g(x, u, ∇u) ∈ L

(Ω),

where

Au = −div(a(x, u, ∇u))

is a weighted Leray-Lions operator from the weighted Sobolev space X =

1,p

(Ω, w) into X

∗

= W

−1,p

(Ω, w

∗

) and where g is a nonlinear lower order

term having natural growth (|g(x, s, ξ)| ≤ b(|s|)(c(x) +

i=1

|ξ

)) which

satisﬁes the sign condition g(x, s, ξ)s ≥ 0. We also assume that g(x, s, ξ)

having an ”exact natural growth” i.e., |g(x, s, ξ)| ≥ ρ

i=1

|ξ

. The

right hand side f is assumed to belong to L

(Ω). It will turn out that for

any solution u, g(x, u, ∇u) ∈ L

(Ω), but for a general v ∈ X, g(x, v, ∇v)

can be more singular.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

345

Drabek and Nicolosi in

proved the existence of bounded solution for the

degenerated problem (P) where g(x, u, ∇u) = −c

|u|

p−2

u, more precisely

for the problem,

Au −c

|u|

p−2

u = f(x, u, ∇u)

with some more general degeneracy, but under some other assumptions on

f and a(x, s, ξ). The existence result for the problem (P) (respectively, uni-

lateral problem) where f lies in the dual space W

−1,p

(Ω, w

∗

) is studied in

(respectively,

). Our aim in this paper is to prove the existence results of

the problem (P) by using the approach based on the strong convergence of

the positive part u

(resp. u

−

) of u

, where u

is a solution of the approx-

imate problem. Note that, the same result is proved in

by using another

approach based on the strong convergence of the truncations. On the other

hand, in the non weighted case Boccardo Gallou¨et

have proved the exis-

tence of at least one solution for the problem (P). Let us point out that

another work in this direction can be found in.

4,6

Our results theorem gen-

eralizes those obtained in

and,

in the weighted case.

The present paper is organized as follows: In section 2, we give some prelim-

inaries and some technical lemmas. In section 3, we give and prove our main

general result. The fourth part is devoted to an example which illustrates

our abstract hypothesis.

2. Preliminaries

Let Ω be a bounded open subset of IR

(N ≥ 1), let 1 < p < ∞, and

let w = {w

(x)}, 0 ≤ i ≤ N be a vector of weight functions; i.e. every

component w

(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,

∈ L

loc

(Ω) (1)

−

p−1

∈ L

loc

(Ω) . (2)

We deﬁne the weighted space with weight γ on Ω as

(Ω, γ) = {u = u(x), uγ

1/p

∈ L

(Ω)}.

In this space, we deﬁne the norm

kuk

p,γ



Ω

|u(x)|

γ(x) dx



1/p

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346

We denote by W

1,p

(Ω, w) the space of all real-valued functions u ∈

(Ω, w

) such that the derivatives in the sense of distributions satisfy

∂u

∂x

∈ L

(Ω, w

) for all i = 1, . . . , N .

This set of functions forms a Banach space under the norm

kuk

1,p,w



Ω

|u(x)|

(x) dx +

i=1

Ω

∂u(x)

∂x

(x) dx



1/p

. (3)

To deal with the Dirichlet problem, we use the space

X = W

1,p

(Ω, w)

deﬁned as the closure of C

∞

(Ω) with respect to the norm (3). Note that,

∞

(Ω) is dense in W

1,p

(Ω, w) and (X, k.k

1,p,w

) is a reﬂexive Banach space.

We recall that the dual space of the weighted Sobolev spaces W

1,p

(Ω, w) is

equivalent to W

−1,p

(Ω, w

∗

), where w

∗

= {w

∗

= w

1−p

} i = 0, . . . , N, and

is the conjugate of p i.e. p

p−1

. For more details, we refer the reader

to.

Now, we state the following assumption.

(H1) The expression

k|u|k

= (

i=1

Ω

∂u(x)

∂x

(x) dx)

1/p

(4)

is a norm deﬁned on X and is equivalent to the norm (3).

Note that (X, k|.|k

) is a uniformly convex (and thus reﬂexive) Banach

space. There exist a weight function σ on Ω and a parameter q, 1 < q < ∞,

such that

1−q

∈ L

loc

(Ω), (5)

with q

q−1

and such that the Hardy inequality,



Ω

|u(x)|

σ dx



1/q

≤ c



i=1

Ω

∂u(x)

∂x

(x) dx



1/p

, (6)

holds for every u ∈ X with a constant c > 0 independent of u. Moreover,

the imbedding

X ,→ L

(Ω, σ), (7)

determined by the inequality (6) is compact.

December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent

347

Remark 2.1. If we assume that w

(x) ≡ 1 and that there exists ν ∈

]

, ∞[∩[

p−1

, ∞[ such that w

−ν

∈ L

(Ω) for all i = 1, . . . , N, (which is an

integrability condition, stronger than (2)), then

k|u|k



i=1

Ω

∂u(x)

∂x

(x) dx



1/p

is a norm deﬁned on W

1,p

(Ω, w) and equivalent to (3). Also we have that

1,p

(Ω, w) ,→ L

(Ω)

for 1 ≤ q < p

∗

, pν < N (ν + 1), and q ≥ 1 is arbitrary for pν ≥ N(ν + 1)

where p

pν

ν+1

. Where p

∗

N−p

Npν

N(ν+1)−pν

is the Sobolev conjugate

of p

(see

). Thus the hypotheses (H1) is veriﬁed (for σ ≡ 1).

Now, we recall the following technical lemmas which are needed later. Note

that this lemmas are proved in.

Lemma 2.1.

Let g ∈ L

(Ω, γ) and let g

∈ L

(Ω, γ), with kg

r,γ

≤ c,

1 < r < ∞. If g

(x) → g(x) a.e. in Ω, then g

* g in L

(Ω, γ), where *

denotes weak convergence and γ is a weight function on Ω.

Lemma 2.2.

Assume that (H1) holds. Let F : IR → IR be uniformly

Lipschitzian, with F (0) = 0. Let u ∈ W

1,p

(Ω, w). Then F (u) ∈ W

1,p

(Ω, w).

Moreover, if the set D of discontinuity points of F

is ﬁnite, then

∂(F ◦u)

∂x

(

(u)

∂u

∂x

a.e. in {x ∈ Ω : u(x) 6∈ D}

0 a.e. in {x ∈ Ω : u(x) ∈ D}.

Lemma 2.3.

Assume that (H1) holds. Let u ∈ W

1,p

(Ω, w), and let T

(u),

k ∈ IR

, be the usual truncation then T

(u) ∈ W

1,p

(Ω, w). Moreover, we

have

(u) → u strongly in W

1,p

(Ω, w).

Lemma 2.4.

Assume that (H1) holds. Let u ∈ W

1,p

(Ω, w), then u

max(u, 0) and u

−

= max(−u, 0) lie in W

1,p

(Ω, w). Moreover, we have

∂(u

)

∂x



∂u

∂x

, if u > 0

0, if u ≤ 0

∂(u

−

)

∂x



0, if u ≥ 0

−

∂u

∂x

, if u < 0.

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348

Lemma 2.5.

Assume that (H1) holds. Let (u

) be a sequence of

1,p

(Ω, w) such that u

* u weakly in W

1,p

(Ω, w). Then, u

* u

weakly

in W

1,p

(Ω, w) and u

−

* u

−

weakly in W

1,p

(Ω, w).

3. Main result

Let A be the nonlinear operator from W

1,p

(Ω, w) into the dual

−1,p

(Ω, w

∗

) deﬁned as

Au = −div(a(x, u, ∇u)) ,

where a : Ω ×IR ×IR

→ IR

is a Carath´eodory vector-function satisfying

the following assumptions:

(H2) For i = 1, . . . , N,

(x, s, ξ)| ≤ βw

1/p

(x)[k(x) + σ

1/p

|s|

j=1

1/p

(x)|ξ

p−1

] (8)

[a(x, s, ξ) −a(x, s, η)](ξ − η) > 0 for all ξ 6= η ∈ IR

(9)

a(x, s, ξ).ξ ≥ α

i=1

|ξ

, (10)

where k(x) is a positive function in L

(Ω) and α, β are positive con-

stants.

(H3) g(x, s, ξ) is a Carath´eodory function satisfying

g(x, s, ξ)s ≥ 0 , (11)

|g(x, s, ξ)| ≤ b(|s|)(

i=1

|ξ

+ c(x)), (12)

There exists ρ

> 0 and ρ

> 0 such that, for |s| ≥ ρ

|g(x, s, ξ)| ≥ ρ

i=1

|ξ

(13)

where b : IR

→ IR

is a continuous increasing function and c(x) is

positive function which lies in L

(Ω).

Finally, we assume that

f ∈ L

(Ω). (14)

Consider the nonlinear problem with Dirichlet boundary conditions

(P)



Au + g(x, u, ∇u) = f in D

(Ω),

u ∈ W

1,p

(Ω, w), g(x, u, ∇u) ∈ L

(Ω).

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349

We state our main result as follows.

Theorem 3.1. Under assumptions (H1)-(H3) and (14), there exists at

least one solution of the problem (P).

Remark 3.1. The statement of Theorem 3.1 generalizes in the weighted

case the one in.

5,6

Remark 3.2. In the case f ∈ W

−1,p

(Ω, w), it is true that g(x, u, ∇u)u ∈

(Ω), which in contrast is in general false if we assume that f ∈ L

(Ω)

(cf. Remark 3

Remark 3.3. In the case f ≥ 0, the solution of the problem (P) is positive

it suﬃcient to take v = −T

−

) k > 0.

In order to prove Theorem 3.1, we needed the following lemma. Which is

proved in

Lemma 3.1.

Assume that (H1) and (H2) are satisﬁed, and let (u

) be

a sequence in W

1,p

(Ω, w) such that u

* u weakly in W

1,p

(Ω, w) and

Ω

[a(x, u

, ∇u

) − a(x, u

, ∇u)]∇(u

− u) dx → 0. (15)

Then, u

→ u in W

1,p

(Ω, w).

Proof of Theorem 3.1

Step (1) The approximate problem and a priori estimate

Let f

be a sequence of smooth functions which strongly converges to f in

(Ω) and kf

(Ω)

≤ c

for some constants c

Now, consider the following approximate problem

)



A(u

) + g(x, u

, ∇u

) = f

∈ W

1,p

(Ω, w).

Which has a solution by the Theorem 3.1 in.

Multiplying (P

) by T

) ∈

X and since

Ω

g(x, u

, ∇u

) ≥ 0, we obtain

Ω

a(x, u

, ∇T

))∇T

) ≤

Ω

) ≤ kc

In view of (10), we have

i=1

Ω

∂T

)

∂x

dx ≤