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

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Existence and uniqueness of solutions of some nonlinear

equations in Orlicz spaces and weighted Sobolev spaces

L. Aharouch

Facult´e polydisciplinaire Ouarzazate

BP 638 Ouarzazate, Morocco

E-mail: laharouch@gmail.com

A. Benkirane

∗

, J. Bennouna

†

, and A. Touzani

‡

D´epartement de Math´ematiques et Informatique

Facult´e des Sciences Dhar-Mahraz

B.P 1796 Atlas F`es, Morocco

E-mails:

∗

abd.benkirane@gmail.com,

†

jbennouna@hotmail.com,

‡

atouzani@menara.ma

In this paper, we show, in the framework of Orlicz Sobolev spaces, the existence

of renormalized and entropy solutions of some nonlinear equations and the

existence and uniqueness of a unilateral problem. The existence of weak and

renormalized solution of a nonlinear equation is also presented in the framework

of weighted Sobolev spaces.

Keywords: Orlicz Sobolev spaces; Weighted Sobolev spaces; Renormalized

solution; Entropy solution; Weak solution; Boundary value problems.

1. Deﬁnition and existence of renormalized solutions in

Orlicz space

Let Ω be a bounded open subset of IR

with the segment property.

Let M be an N-function satisfying the ∆

-condition and let P be an

N-function such that P << M.

Let A : D(A) ⊂ W

(Ω) → W

−1

(Ω) be a mapping (not deﬁned

everywhere) given by A(u) = −div a(x, u, ∇u), where a : Ω × IR × IR

→

is a caratheodory function satisfying for a.e. x ∈ Ω and all t ∈ IR, ξ, ξ

with ξ 6= ξ

(1 −1) | a(x, t, ξ) |≤ d(x) + k

−1

M(k

| t |) + k

−1

M(k

| ξ |),

(1 −2) [a(x, t, ξ) − a(x, t, ξ)] [ξ − ξ] > 0,

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(1 −3) a(x, t, ξ) ξ ≥ αM(

| ξ |

where d(x) ∈ E

(Ω), d ≥ 0, α, λ ∈ IR

∗

, k

∈ IR

Consider the nonlinear elliptic problem

(1 −4) −div a(x, u, ∇u) − divφ(u) + g(x, u) = f in Ω,

(1 −5) u = 0 on ∂Ω,

where

(1 −6) f ∈ W

−1

(Ω),

and φ = (φ

, . . . , φ

) satisfy

(1 −7) φ ∈ (C

(IR))

Let g(x, t) be a caratheodory function such that for a.e.x ∈ Ω and all t ∈ IR

(1 −8) g(x, t) t ≥ 0,

(1 −9) sup

|t|≤n

| g(., t) |= h

(.) ∈ L

(Ω) ∀n.

Note that no growth hypothesis is assumed on the function φ, which implies

that for a solution u ∈ W

(Ω) the term divφ(u) may be meaningless,

even as a distribution. As in

we deﬁne the following notion of renormalized

solution, which gives a meaning to a possible solution of (1-4)-(1-5).

The notion of renormalized solutions in the usual sens was introduced

by R.J. Diperna and P.L. Lions

for the study of the Boltzmann equations.

This notion was then adapted to the study of the problem (1-1)-(1-2) by

L. Boccardo, D. Giachetti, J.I. Diaz and F. Murat

when the right hand

side is in W

−1,p

(Ω), by J.M. Rakotoson

when the right hand side is in

(Ω), and ﬁnally by G. Dal Maso, F. Murat, L. Orsina and A. Prignet

for the case of right hand side is general measure data.

Deﬁnition 1.1. Assume that (1 −1) −(1 −3), (1 −6) −(1 −9) hold true.

A function u is a renormalized solution of the problem (1 − 4) − (1 − 5) if

(1 −10) u ∈ W

(Ω), g(x, u) ∈ L

(Ω), u g(x, u) ∈ L

(Ω)

(1 −11)

(

−div a(x, u, ∇u) h(u) − div(φ(u)h(u)) + φ(u)h

(u)∇u

+g(x, u)h(u) = fh(u) in D

(Ω), ∀h ∈ C

(IR).

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Remark 1.1. Let us note that in (1 −11) every term is meaningful in the

distributional sense (in contrast with (1 − 4)).

Lemma 1.1. Let Ω be a bounded open subset of IR

with the segment

property. If u ∈ (W

(Ω))

then

Ω

div u dx = 0.

Theorem 1.1. Let M be an N-function satisfying the ∆

-condition. Under

assumptions (1 −1) −(1 −3), (1 −6) − (1 −9), there exists a renormalized

solution u (in the sense of Deﬁnition 1.1) of problem (1 −4) −(1 −5).

Proof of Theorem 1.1 see

In the last theorem, we have supposed the N-function M satisfying the

∆

-condition. In the next theorem we prove the same result without any

restriction on the N-function M (i.e. without the ∆

-condition).

Theorem 1.2. Under assumptions (1 −1) −(1 −3), (1 −6) −(1 −9), there

exists a renormalized solution u (in the sense of Deﬁnition 1.1) of problem

(1 −4) − (1 −5).

Proof of Theorem 1.2 see

2. Deﬁnition and existence of entropy solutions in Orlicz

space

Let Ω be a bounded open subset of IR

with the segment property. Let

M, P be two N-functions such that P << M.

Let A : D(A) ⊂ W

(Ω) → W

−1

(Ω) be a mapping (not deﬁned

everywhere) given by: A(u) = −diva(x, u, ∇u) where a : Ω × IR × IR

→

is a caratheodory function satisfying for a.e. x ∈ Ω and all ξ, ξ ∈ IR

with ξ 6=

ξ:

(2 −1) | a(x, t, ξ) |≤ d(x) + k

−1

M(k

| t |) + k

−1

M(k

| ξ |)

(2 −2) [a(x, t, ξ) − a(x, t, ξ)] [ξ − ξ] > 0

(2 −3) a(x, t, ξ) ξ ≥ αM (

| ξ |

)

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where d(x) ∈ E

(Ω), d ≥ 0, α, λ ∈ IR

∗

, k

∈ IR

. Consider the

nonlinear elliptic problem

(2 −4) −diva(x, u, ∇u) = f − divφ(u)

(2 −5) u = 0 on ∂Ω

where

(2 −6) f ∈ L

(Ω)

and φ = (φ

, . . . , φ

) satisfy

(2 −7) φ ∈ (C

(IR))

Note that no growth hypothesis is assumed on the function φ, which implies

that the term divφ(u) may be meaningless, even as a distribution. The

notion of entropy solution, used in,

allows us to give a meaning to a possible

solution of (2-4)-(2-5).

We introduce the following notation, see,

1311

Deﬁnition 2.1. Let M be an N-function, we deﬁne the following set:

= {Q : Q is an N-function such that

≤

and

QoH

−1

(

1−

)dr < ∞ where H(r) =

M(r)

}

Remark 2.1. Let M(t) = t

and Q(t) = t

, then the condition Q ∈ A

is equivalent to the following conditions:

i) 2 −

< p < N,

ii) q < ˜p =

(p−1)N

N−1

Deﬁnition 2.2. Assume that (2 −1) −(2 −3), (2 −6) −(2 −7) hold true,

and suppose that A

6= ∅. A function u is an entropy solution of problem

(2 −4) − (2 −5) if











u ∈ W

(Ω) ∀Q ∈ A

(u) ∈ W

(Ω) ∀k > 0,

Ω

a(x, u, ∇u) ∇T

[u −v]dx ≤

Ω

[u −v]dx +

Ω

φ(u) ∇T

[u −v]dx

∀v ∈ W

(Ω) ∩ L

∞

(Ω).

Theorem 2.1. Assume that (2 − 1) − (2 − 3), (2 − 6) − (2 − 7) hold true,

and suppose that A

6= ∅, there exists an entropy solution u of problem

(2 −4) − (2 −5) (in the sense of Deﬁnition 2.1).

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Proof of Theorem 2.1 see

Remark 2.2. In the case M(t) = t

, our theorem gives a reﬁnement of the

regularity result (i.e. u ∈ W

1,q

(Ω), q < ˜p =

(p−1)N

N−1

In fact by Theorem 2.1 we have u ∈ W

(Ω) ∀Q ∈ A

(for example

for Q(t) =

˜p

log

(e+t)

, α > 1).

3. Existence and uniqueness of solution of unilateral

problems with L

-data in Orlicz spaces

Let Ω be a bounded open subset of IR

with the segment property.

Let M be an N-function, satisfying the ∆

-condition.

Let A : D(A) ⊂ W

(Ω) → W

−1

(Ω) be a mapping (not deﬁned

everywhere) given by: A(u) = −div a(x, ∇u) where a : Ω × IR

→ IR

a caratheodory function satisfying for a.e. x ∈ Ω and all ξ, ξ ∈ IR

with

ξ 6= ξ:

(3 −1) | a(x, ξ) |≤ d(x) + k

−1

M(k

| ξ |)

(3 −2) [a(x, ξ) − a(x, ξ)] [ξ − ξ] > 0

(3 −3) a(x, ξ) ξ ≥ αM(| ξ |)

where d(x) ∈ E

(Ω), d ≥ 0, α, k

, k

∈ IR

Let

= {v ∈ W

(Ω) ∩ L

∞

(Ω) : v ≥ ψ a.e. in Ω}

where ψ : Ω → IR

is a measurable function on Ω such that:

ψ ∈ W

(Ω) ∩ L

∞

(Ω).

Assume that

(3 −4) f ∈ L

(Ω).

We suppose the regularity assumption on the obstacle function ψ:

(3 −5) there exists ψ ∈ K such that ψ − ψ is continuous on Ω.

Consider the nonlinear elliptic unilateral problem:

(3 −6)











u ∈ W

(Ω) ∀Q ∈ A

u ≥ ψ a.e. in Ω

(u) ∈ W

(Ω) ∀k > 0

Ω

a(x, ∇u) ∇T

[u −v]dx ≤

Ω

[u −v]dx ∀v ∈ K

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It is our purpose, in this paper, to show the existence and uniqueness of

solutions for the problem (3-6) in the setting of the Orlicz Sobolev space

(Ω). Our result, theorem 3.1, generalizes that of Boccardo

and gives

in particular a reﬁnement of his result.

Theorem 3.1. Assume that (3 −1) − (3 − 5) hold true, and suppose that

6= ∅, then there exists an unique solution of problem (3 − 6).

Proof of Theorem 3.1 see

4. Existence of solutions for nonlinear elliptic degenerated

equations

Let Ω be a bounded open subset of IR

. A is a nonlinear operator of the

Leray-lions type from a weighted Sobolev space W

1,p

(Ω, ν) (where ν = ν(x)

is weight function deﬁned on Ω). A is deﬁned by A(u) = −div a(x, u, ∇u)

where a is caratheodory function from Ω × IR × IR

→ IR

satisfying for

a.e. x ∈ Ω, and for all ξ

, ξ,

(4 −1) | a(x, ξ

, ξ) |≤ K(| ξ

|)ν

(x){d(x)+ | ξ

p−1

+ν

1−

| ξ |

p−1

where K verify that sup ess

|s|≤k

K(| s |) < ∞, ∀k > 0, and d(x) ∈ L

(Ω).

(4 −2) [a(x, ξ

, ξ) − a(x, ξ

, η)] [ξ − η] > 0 where ξ 6= η.

Let λ be a continuous function such that λ > 0 and deﬁned on IR

, the

degeneracy of the operator A is expressed by the assumption

(4 −3) a(x, ξ

, ξ) ξ ≥ ν(x) λ(| ξ

|) | ξ |

holds for all ξ

, ξ and let λ

such that

(4 −4) λ

(s) =

[λ(t)]

dt, s ≥ 0 with λ

(∞) = ∞.

Furthermore, we shall assume that for some m

> 0,

(4 −5) inf

m≥m

p.(1−

)

[ inf

m≤t≤m+1

λ(t)]) > 0.

For instance λ(t) = t

−γp

for t ≥ 1, 0 < γ < 1 (small).

We will suppose that

(4 −6) ν ∈ L

loc

(Ω), ν

−1

p−1

∈ L

(Ω),

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(4 −7) ν

−s

∈ L

(Ω) with s ∈ (

, ∞) ∩ [

p −1

, ∞).

We consider the nonlinear elliptic problem

(4 −8) −div a(x, u, ∇u) −divφ(u) + g(x, u) = f,

(4 −9) u = 0 on ∂Ω,

where φ = (φ

, ..., φ

) satisfy

(4 −10) φ ∈ (C

(IR))

Note that no growth hypothesis is assumed on the function φ. Let g(x, t)

be a caratheodory function such that for a.e.x ∈ Ω and all t ∈ IR,

(4 −11) g(x, t) t ≥ 0

(4 −12) sup

|t|≤n

| g(., t) |= h

(.) ∈ L

(Ω) ∀n.

The right-hand side in (4-8) is of the form

(4 −13) f =

i=1

∂f

∂x

where the family {f

i = 1, ..., N} satisﬁes the conditions

∈ L

(Ω, ν

−r

) ,→ L

(Ω, ν

−p

) = (L

(Ω, ν))

∗

for i = 1, . . . , N, with r ≥ p

We distinguish two cases.

The ﬁrst case if r > r

with r

q−N

p−1

and q > N . In this case we

prove in Theorem 4.1 that the solution u of problem (4-8)-(4-9) is bounded,

and then the term divφ(u) has meaningful as a distribution. In Theorem

4.2, we prove that the last problem admits a weak solution (see Deﬁnition

4.1).

The second case if p

≤ r ≤ r

the solution u is not bounded and

the term divφ(u) may be meaningless, even as a distribution. As in

deﬁne the notion of renormalized solution (see Deﬁnition 4.2), which gives

a meaning to a possible solution of (4-8)-(4-9). In Theorem 4.3 we prove

the existence of renormalized solution.

We show in the paper L

∞

estimates for the solutions (see Theorem 4.1)

and the existence of a weak and renormalized solution, and is also showed

the fact that the main operator is degenerate in the space variable.

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4.1. Regularity and existence of weak solutions

Deﬁnition 4.1. Assume that (4 − 1) − (4 − 7), (4 − 10) − (4 − 13) hold

true.A function u is a weak solution of the problem (4 − 8) − (4 − 9) if

(4 −14) u ∈ W

1,p

(Ω, ν) ∩L

∞

(Ω), g(x, u) ∈ L

(Ω), u g(x, u) ∈ L

(Ω)

(4−15)

Ω

a(x, u, ∇u)∇ϕdx+

Ω

φ(u)∇ϕdx+

Ω

g(x, u)ϕdx =

i=1

Ω

∂f

∂x

ϕdx

∀ϕ ∈ D(Ω).

In Theorem 4.1 we prove the boundness of the solution of the problem (4-

8)-(4-9), in Theorem 4.2 we prove the existence of weak solution if r > r

Lemma 4.1. Let Ω be a bounded open subset of IR

. If u ∈ (W

1,p

(Ω, ν))

then

Ω

div u dx = 0.

Lemma 4.2. Let F : IR → IR be uniformly lipschitzian with F (0) = 0. Let

u ∈ W

1,p

(Ω, ν), then F (u) ∈ W

1,p

(Ω, ν).

Let u ∈ W

1,p

(Ω, ν) and let S

θ,h

a real lipschitzian function deﬁned for

θ > 0, h > 0 by

θ,h

(τ) =











1 if τ ≥ θ + h

τ−θ

if θ ≤ τ ≤ θ + h

0 if | τ |≤ θ

τ+θ

if −θ − h ≤ τ ≤ −θ

−1 if τ ≤ −θ − h

then S

θ,h

(u) ∈ W

1,p

(Ω, ν) by Lemma 4.2 and we suppose that u satisﬁes

∀θ ∈]0, sup ess | u | [, ∀h ∈]0, sup ess | u | −θ[,

(4−16)

Ω

a(x, u, ∇u)∇S

θ,h

(u)dx +

Ω

φ(u)∇S

θ,h

(u) +

Ω

g(x, u)S

θ,h

(u)dx

i=1

Ω

∂f

∂x

θ,h

(u)dx

Remark 4.1. It is easy to see that each possible solution of problem (4-

8)-(4-9) veriﬁes the relation (4-16).

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Theorem 4.1. Under assumptions (4 −1) − (4 − 7), (4 − 10) − (4 − 13).

Let u ∈ W

1,p

(Ω, ν) which satisﬁes (4-16), and we suppose furthermore f

∈

(Ω, ν

−r

), with r > r

where r

q−N

p−1

and q > N. Then u is bounded

and we have the following estimate

(4 −17) λ

(k u k

∞

) ≤ C

⇔ k u k

∞

≤ λ

−1

) = M,

where

=k ν

−1

(Ω)

= (

i=1

k f

(Ω,ν

−r

)

p−1

= (

|Ω|

m(1−

)

dσ)

, with

r(p−1)

= 1,

Nα

, α

is the measure of the unit ball of IR

−1

is the converse of the function λ

(see (4-4)).

Proof of Theorem 4.1. The proof is based on the method of relative

rearrangement see

Theorem 4.2. Under assumptions (4 −1) − (4 − 7), (4 − 10) − (4 − 13).

There exist a bounded weak solution u ∈ W

1,p

(Ω, ν) ∩ L

∞

(Ω) (in the sense

of Deﬁnition 4.1) of problem (4 − 8) − (4 − 9).

Proof of Theorem 4.2. See

4.2. Existence of renormalized solutions

If p

≤ r ≤ r

, the solution of the problem (4-8)-(4-9) may not be bounded

and the term div(φ(u)) may have no meaning even a distribution. Then we

consider the nonlinear elliptic problem (4-8)-(4-9) in Ω, and we consider

the following notion of renormalized solution which gives a meaning to a

possible solution of (4-8)-(4-9).

Deﬁnition 4.2. Assume that (4 −1) −(4 −7), (4 −10) −(4 −13) hold true.

A function u is a renormalized solution of the problem (4 − 8) − (4 − 9) if

u ∈ W

1,p

(Ω, ν), g(x, u) ∈ L

(Ω), u g(x, u) ∈ L

(Ω)

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









Ω

a(x, u, ∇u) [h

(u)∇uϕ + h(u)∇ϕ]dx +

Ω

Φ(u)[h

(u)∇uϕ

+ h(u)∇ϕ]dx +

Ω

g(x, u)h(u)ϕdx =

i=1

Ω

∂f

∂x

h(u)ϕdx

∀h ∈ C

(IR), ϕ ∈ W

1,p

(Ω, ν) ∩L

∞

(Ω)

In Theorem 4.3 we state the existence of renormalized solution of (4-8)-(4-

9).

Theorem 4.3. Under assumptions (4 − 1) − (4 − 7), (4 − 10) − (4 − 13),

and furthermore for

p−1

< r ≤ r

with q > N, there exists a renormalized

solution u (in the sense of Deﬁnition 4.2) of problem (4 −8) − (4 −9)

Proof of Theorem 4.3 is given in

References

1. L. Aharouch, J. Bennouna and A. Touzani, Existence of renormalized so-

lution of some elliptic problems in Orlicz spaces, Revista Matematica Com-

plutense, 22 (2009), n1, 91-110.

2. A. Benkirane, J. Bennouna, Existence of Renormalized Solutions for some

elliptic problems involving derivatives of nonlinear terms in Orlicz Spaces,

Lectures notes in pure and applied mathematics , 229, Dekker, New York,

2002.

3. A. Benkirane, J. Bennouna, Existence of Entropy Solutions for some non-

linear problems in L

in Orlicz Spaces, Abstract and Applied Analysis 7

(2002) n 2, 85-102 USA.

4. A. Benkirane, J. Bennouna, Existence and uniqueness of solution of unilat-

eral problems with L

-data in Orlicz spaces, Ital. J. of Pure and App. Math.,

16,(2004) 87-102.

5. A. Benkirane, J. Bennouna, Existence of Solutions for nonlinear elliptic

degenerate equations, Nonlinear Analysis 54 (2003), n9-37, USA.

6. L. Boccardo, D. Giachetti, J.I. Diaz, F. Murat, Existence and Regularity

of Renormalized Solutions of some Elliptic Problems involving derivatives of

nonlinear terms, Journal of diﬀerential equations 106,215-237 (1993)

7. L. Boccardo, Some nonlinear Dirichlet problems in L

involving lower order

terms in divergence form, progress in Elliptic and Parabolic Partiel Diﬀer-

entiel Equations (capri, 1994), Pitman Res. Notes Math. Ser., vol. 350,

Longman, Harlow, 1996, pp. 43-57.

8. L. Boccardo and G. R. Cirmi, Existence and uniqueness of solution of

unilateral problems with L

data, Journal of Convex Analysis, 6, N1 (1999)

195-206.

9. G. Dalmaso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions

of elliptic equations with general measure data, Ann. Scuola Norm. Sup Pisa

Cl. Sci 12 4 (1999) 741-808.