Ammari H., Benkirane A., Touzani A. (editors) Recent Developments in Nonlinear Analysis

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10. R.J. Diperna, P.L. Lions, on the cauchy problem for Boltzmann equations:

Global existence and weak stability, ann of Math, 130 (1989), 321-366.

11. M. Kbiri Alaoui, Sur certains problems elliptiques avec second membre

mesure ou L

loc

, These soutenu `a l’Universit´e Sidi Moh. Ben Abd. Facult´e

des Sciences Dhar Mehraz Fes Maroc (1999)

12. J.M. Rakotoson, Uniqueness of renormalized solutions in a T -set for L

data

problems and the link between various formulations, Indiana University Math.

Jour., vol. 43, 2(1994).

13. G. Talenti, Nonlinear Elliptic Equations, Rearrangements of functions and

Orlicz spaces, Ann. Mat. Pura Appl.(4) 120 (1979), 159-184.

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181

Existence of solutions for variational degenerated unilateral

problems

L. Aharouch, E. Azroul, and M. Rhoudaf

D´epartement de Math´ematiques et Informatique

Facult´e des Sciences Dhar-Mahraz

B.P 1796 Atlas F`es, Morocco

An existence result is proved for a variational degenerated unilateral problems

associated to the following equations

Au + g(x, u, ∇u) = f,

where A is a Leray-Lions operator acting from the weighted Sobolev space

1,p

(Ω, w) into its dual W

−1,p

(Ω, w

∗

), while g(x, s, ξ) is a nonlinear term

which has a growth condition with respect to ξ and a sign condition on s,

i.e. g(x, s, ξ).s ≥ 0 for every s ∈ IR and for every x and ξ in their respective

domains. The source term f is supposed to belong to W

−1,p

(Ω, w

∗

Keywords: Degenerate unilateral problem; Existence result.

1. Introduction

Let Ω be a bounded open subset of IR

(N ≥ 2), p be a real number such

that 1 < p < ∞ and w = {w

(x); 0 ≤ i ≤ N}, be a collections of weight

functions on Ω, i.e. each w

(x) is a measurable a.e. strictly positive function

on Ω satisfying some intergrability conditions (see section 2).

In this paper we are interested in the study of the degenerated obstacle

problem associated to the following Dirichlet problem



Au + g(x, u, ∇u) = f in Ω

u ≡ 0 on ∂Ω,

(1)

where Au = −div(a(x, u, ∇u)) is a Leray-Lions operator act-

ing from W

1,p

(Ω, w) into its dual W

−1,p

(Ω, w

∗

) with w

∗

1−p

; 0 ≤ i ≤ N

, p

p−1

is the conjugate exponent of p and where

g(x, u, ∇u) is a nonlinearity term satisfying some p-growth condition with

respect to ∇u, and satisﬁes the sign condition g(x, u, ∇u)u ≥ 0, but has

unrestricted growth with respect to u.

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The source term f is assumed to belong to W

−1,p

(Ω, w

∗

Let us start by the case of equation and recall that in the particular

case where g(x, u, ∇u) = −C

|u|

p−2

u the following degenerated equation

−div(a(x, u, ∇u)) − C

|u|

p−2

u = h(x, u, ∇u),

has been studied by Drabek-Nicolosi

under some more degeneracy and

some additional assumptions on h and a. While the existence solution for

the variational Dirichlet problem (1) is treated in the work

but under the

following integrability condition

−1/q−1

∈ L

loc

(Ω) for some q such that 1 < q < ∞, (2)

where q is the so-called Hardy exponent and σ is the so-called Hardy weight

(see (14)) below).

Now we turn our attention to the degenerated unilateral case and we

will give some known about the following problem











Find u ∈ K

, g(x, u, ∇u) ∈ L

(Ω), g(x, u, ∇u)u ∈ L

(Ω)

Ω

a(x, u, ∇u)∇(u − v) dx +

Ω

g(x, u, ∇u)(u −v) dx

≤ hf, u − vi,

∀ v ∈ K

∩ L

∞

(Ω),

(3)

where the convex set K

is deﬁned as

= {v ∈ W

1,p

(Ω, w); v ≥ ψ a.e. in Ω},

with an obstacle ψ which is a measurable function on Ω.

Akdim et al. have proved in

the existence of a solution for the problem

(3) under the following conditions

−

q−1

∈ L

loc

(Ω) with 1 < q < p + p

, (4)

and

∈ L

∞

(Ω) ∩ W

1,p

(Ω, w). (5)

For that, the authors in

have approximated the nonlinear term g by some

function involving χ

Ω

where Ω

is a sequence of compacts covering the

bounded open set Ω and χ

Ω

is a characteristic function, i.e.

(x, s, ξ) =

g(x, s, ξ)

1 + ε|g(x, s, ξ)|

Ω

(x).

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So there are consider the following approximate unilateral problem











∈ K

such that

Ω

a(x, u

, ∇u

)∇(u

− v) dx +

Ω

(x, u

, ∇u

)(u

− v) dx

≤ hf, u

− vi,

∀ v ∈ K

(6)

Note that, the hypotheses (4) and (5) used in

have played an important role

for to assure the boundedness, coercivity and pseudo-monotonicity of the

operator associated to the approximate problem (6) and also for to prove

the boundedness of the approximate solution u

in the space W

1,p

(Ω, w)

(see

for more details).

The aim of this paper is then to study the existence solution for the same

degenerated unilateral problem (3) but without assuming the condition (4)

nor (5).

To overcome the diﬃculties mentioned above, we have changed in the

present paper the classical coercivity,

a(x, s, ξ)ξ ≥ α

i=1

(x)|ξ

by the following one,

a(x, s, ξ)(ξ −∇v

) ≥ α

i=1

(x)|ξ

− δ(x), (7)

where v

is some element of K

∩ L

∞

(Ω) and δ(x) is an element of L

(Ω)

and also we have replaced the approximation term g

(of the nonlinearity

g) by the following one,

(x, s, ξ) =

g(x, s, ξ)

1 +

|g(x, s, ξ)|

(x), (8)

where the function θ

(x) is deﬁned according to the Hardy weight σ and

hardy exponent q, i.e.,

(x) = nT

1/n

(σ

1/q

(x)).

1/n

is the truncation operator at height 1/n see (18)).

It would be interesting at this stage to refer the reader to our previous

work

in which the same unilateral problem with L

-data is studied under

the integrability condition (4) and under the regularity condition (5). An-

other work in this direction can be found in

where the existence solution

of the same unilateral problem (with L

-data) is proved by assuming the

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

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previous hypotheses (4) and (5), but the sign condition (22) is violated and

the classical growth condition of the nonlinearity g (see (23)) is replaced by

|g(x, s, ξ)| ≤ c(x) + ρ(s)

i=1

|ξ

, (9)

(with ρ ∈ L

(Ω), ρ ≥ 0).

We refer also to the work,

where the authors have solved an analogous

problem in the classical Sobolev space but where the obstacle function is

supposed to satisfy the condition ψ

∈ W

1,p

(Ω) ∩ L

∞

(Ω).

The outline of this paper is as follows. After giving some preliminary

results about the weighted Sobolev space in section 2, we formulate in

section 3 our problem and we give the main existence result, which its

proof is giving in section 4. And we achieve the paper by an appendix in

section 5 where some intermediate results are proved.

2. Preliminaries

Let Ω be a bounded open subset of IR

(N ≥ 2). Let 1 < p < ∞ and

let w = {w

(x); i = 0, . . . , 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,

∈ L

loc

(Ω) (10)

and

−

p−1

∈ L

loc

(Ω), for 0 ≤ i ≤ N. (11)

We deﬁne the weighted space with weight γ in Ω as,

(Ω, γ) = {u(x), uγ

∈ L

(Ω)},

which is normed by,

kuk

p,γ



Ω

|u(x)|

γ(x) dx



We denote by W

1,p

(Ω, w) the weighted Sobolev space of all real-valued func-

tions u ∈ L

(Ω, w

) such that the derivatives in the sense of distributions

satisfy,

∂u

∂x

∈ L

(Ω, w

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

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This set of functions forms a Banach space under the norm

kuk

1,p,w

Ω

|u(x)|

dx +

i=1

Ω

∂u

∂x

(x) dx

. (12)

To deal with the Dirichlet problem, we use the space W

1,p

(Ω, w) deﬁned as

the closure of C

∞

(Ω) with respect to the norm (12). Note that C

∞

(Ω) is

dense in W

1,p

(Ω, w) and (W

1,p

(Ω, w), k.k

1,p,w

) is a reﬂexive Banach space.

We recall that the dual 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 = 1, . . . , 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 assumptions.

) The expression,

kuk =

i=1

Ω

∂u

∂x

(x) dx

(13)

is a norm on W

1,p

(Ω, w) equivalent to the norm (12).

There exists a weight function σ strictly positive a.e. in Ω and a parameter

q, 1 < q < ∞, such that the Hardy inequality



Ω

|u|

σ(x) dx



≤ C

i=1

Ω

∂u

∂x

(x) dx

, (14)

holds for every u ∈ W

1,p

(Ω, w) with a constant C > 0 independent of u.

Moreover, the imbedding

1,p

(Ω, w) ,→ L

(Ω, σ) (15)

determined by the inequality (14) is compact.

Note that, (W

1,p

(Ω, w), k k) is a uniformly convex (and thus reﬂexive)

Banach space.

Remark 2.1. Assume that w

(x) = 1 and in addition the integrability

condition:

There exists ν ∈]

, ∞[∩[

p−1

, ∞[ such that w

−ν

∈ L

(Ω) holds for all

i = 1, . . . , N (which is stronger than (11)). Then

kuk =

i=1

Ω

∂u

∂x

(x) dx

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is a norm deﬁned on W

1,p

(Ω, w) and it is equivalent to (12). Moreover

1,p

(Ω, w) ,→,→ L

(Ω)

for all 1 ≤ q < p

∗

if pν < N(ν + 1) and for all q ≥ 1 if pν ≥ N(ν + 1),

where p

pν

ν+1

and p

∗

N−p

Npν

N(ν+1)−pν

is the Sobolev conjugate of

(see

). Thus the hypotheses (H

) is satisﬁed for σ ≡ 1.

Remark 2.2. If we use the special weight functions w and σ expressed in

terms of the distance to the boundary ∂Ω. Denote d(x) = dist(x, ∂Ω) and

set

w(x) = d

(x), σ(x) = d

(x).

In this case, the Hardy inequality reads



Ω

|u|

(x) dx



≤ C



Ω

|∇u|

(x) dx



(i) For, 1 < p ≤ q < ∞,

λ < p − 1,

−

+ 1 ≥ 0,

−

+ 1 > 0. (16)

(ii) For, 1 ≤ q < p < ∞,

λ < p −1,

−

+ 1 > 0. (17)

The conditions (16) or (17) are suﬃcient for the compact imbedding (15)

to hold (see for example [,

Example 1], [,

Example 1.5, p.34], and [,

theorems 19.17 and 19.22]).

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

Lemma 2.1. cf.

3,15

Let g ∈ L

(Ω, γ) and let g

∈ L

(Ω, γ), with kg

Ω,γ

≤

c, 1 < r < ∞. If g

(x) → g(x) a.e. in Ω, then g

* g weakly in L

(Ω, γ).

Lemma 2.2. cf.

3,12

Assume that (H

) 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) /∈ D}

0 a.e. in {x ∈ Ω : u(x) ∈ D}.

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We introduce the truncation operator. For a given constant k > 0 we deﬁne

the cut function T

: IR → IR as

(s) =



s if |s| ≤ k

k sign(s) if |s| > k.

(18)

For a function u = u(x), x ∈ Ω, we deﬁne the truncated function T

u =

(u) pointwise: for every x ∈ Ω the value of (T

u) at x is just T

(u(x)).

From Lemma 2.2, we deduce the following.

Lemma 2.3. cf.

Assume that (H

) holds. Let u ∈ W

1,p

(Ω, w) and let

(u) be the usual truncation (k ∈ IR

). Then T

(u) ∈ W

1,p

(Ω, w). More-

over, we have

(u) → u strongly in W

1,p

(Ω, w).

We states that every element of W

−1,p

(Ω, w

∗

) can be decomposed as f

−

divF where f

∈ L

(Ω, w

1−p

) and F ∈

i=1

(Ω, w

1−p

3. Main results

Let Ω be an open bounded subset of IR

, N ≥ 2. Given a measurable

function ψ : Ω → IR (so-called obstacle) and consider its associated convex

set,

= {u ∈ W

1,p

(Ω, w); u ≥ ψ a.e. in Ω}. (19)

Let A be the nonlinear operator from W

1,p

(Ω, w) into its dual

−1,p

(Ω, w

∗

) deﬁned by,

Au = −div(a(x, u, ∇u)),

where a : Ω × IR × IR

→ IR

is a Carath´eodory function satisfying the

following assumptions:

) For i = 1, . . . , N

(x, s, ξ)| ≤ w

(x)[k(x) + σ

|s|

j=1

(x)|ξ

p−1

], (20)

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

, (21)

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

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for a.e. x in Ω, s ∈ IR, where k(x) is a positive function in L

(Ω).

) g(x, s, ξ) is a Carath´eodory function which satisﬁes the following clas-

sical conditions,

g(x, s, ξ).s ≥ 0 (22)

and

|g(x, s, ξ)| ≤ b(|s|)(

i=1

(x)|ξ

+ c(x)), (23)

where b : IR

→ IR

is a positive increasing function and c(x) is a positive

function in L

(Ω).

Our main result in this note is the following

Theorem 3.1. Assume that (H

)−(H

) and (7) hold and f ∈ W

−1,p

(Ω).

Then there exists at least one solution of the following unilateral problem,











u ∈ K

g(x, u, ∇u) ∈ L

(Ω), g(x, u, ∇u)u ∈ L

(Ω)

Ω

a(x, u, ∇u)∇(u −v) dx +

Ω

g(x, u, ∇u)(u − v) dx

≤ hf, u − vi,

∀ v ∈ K

∩ L

∞

(Ω).

(24)

Remark 3.1. Note that if we take w ≡ 1 and σ ≡ 1 in the statement of

the previous theorem, then we get a new result in the nondegenerated case.

4. Proof of Theorem 3.1

For the reason of simplicity we take f = −divF where F ∈

i=1

(Ω, w

1−p

The following lemma play an important rˆole in the proof of our main result,

Lemma 4.1.

4,15

Assume that (H

), (H

) and (7) are satisﬁed, and let

)

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

then, u

→ u in W

1,p

(Ω, w).

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We shall give the proof of Theorem in several steps.

STEP 1: Construction of the approximate unilateral problem and

existence of a solution.

Let us approximate the nonlinear function g by g

deﬁned through (8). We

consider the following approximate unilateral problem











∈ K

hAu

, u

− vi+

Ω

(x, u

, ∇u

)(u

− v) dx ≤ hf, u

− vi

∀v ∈ K

(25)

It is easy prove that g

satisﬁes the following conditions:

(x, s, ξ)s ≥ 0, |g

(x, s, ξ)| ≤ |g(x, s, ξ)| and |g

(x, s, ξ)| ≤ n

for a.e. x in Ω, s ∈ IR and all ξ ∈ IR

Thus, we can deﬁne the operator G

: W

1,p

(Ω, w) −→ W

−1,p

(Ω, w

∗

)

by,

u, vi =

Ω

(x, u, ∇u)v dx

and

hAu, vi =

Ω

a(x, u, ∇u)∇v dx.

By virtue of H¨older’s inequality and due to (13) and (14), we have for all

u ∈ W

1,p

(Ω, w) and all v ∈ W

1,p

(Ω, w),



Ω

(x, u, ∇u)v dx



≤



Ω

(x, u, ∇u)|

−





Ω

|v|

σ dx



≤ n



Ω

−q



kvk

q,σ

≤ C

kvk.

(26)

Deﬁnition 4.1. Let Y be a reﬂexive Banach space. A bounded operator B

from Y to its dual Y

∗

is called pseudo-monotone if for any sequence u

∈ Y

with u

* u weakly in Y , and lim sup

n→+∞

hBu

, u

− ui ≤ 0, we have

lim inf

n→+∞

hBu

, u

− vi ≥ hBu, u − vi ∀ v ∈ Y.