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

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In view of (51) there exists η(ε) > 0 such that

b(h(ε) + kv

∞

)(c(x) +

i=1

∂T

h(ε)+kv

∞

)

∂x

) dx < ε/2 (63)

for all E such that meas E < η(ε).

Finally, combining (61), (62) and (63), one easily has

(x, u

, ∇u

)| dx < ε for all E such that meas E < η(ε),

which implies (57)

Step 5: Passing to the limit.

We take v ∈ K

∩ L

∞

(Ω) as test function in (25), we can write

Ω

a(x, u

, ∇u

)∇(u

− v) dx +

Ω

(x, u

, ∇u

)(u

− v) dx

≤ hf, u

− vi.

(64)

This implies

Ω

a(x, u

, ∇u

)∇(u

− v

) dx +

Ω

a(x, u

, ∇u

)∇(v

− v) dx

Ω

(x, u

, ∇u

)(u

− v) dx

≤ hf, u

− vi.

(65)

By Fatou’s Lemma and the fact that

a(x, u

, ∇u

) * a(x, u, ∇u)

weakly in

i=1

(Ω, w

1−p

) one easily sees that

Ω

a(x, u, ∇u)∇(u −v

) dx +

Ω

a(x, u, ∇u)∇(v

− v) dx

Ω

g(x, u, ∇u)(u −v) dx

≤ hf, u − vi.

(66)

Hence

Ω

a(x, u, ∇u)∇(u −v) dx +

Ω

g(x, u, ∇u)(u −v) dx

≤ hf, u − vi.

(67)

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On the other hand, for h large enough we have

Ω

(x, u

, ∇u

)| dx ≤

Ω

b(h + kv

∞

)(c(x) +

i=1

∂T

h+kv

∞

)

∂x

) dx

{|u

−v

|>h}

|g(x, u

, ∇u

)| dx.

(68)

combining (61), (68) and the fact that u

bounded in W

1,p

(Ω, w), we get

Ω

(x, u

, ∇u

)| dx ≤ c

(69)

taking also v

as a test function in (25), we have

Ω

(x, u

, ∇u

dx ≤ c

(70)

by (69), (70) and Fatou’s Lemma we deduce

g(x, u, ∇u) ∈ L

(Ω), g(x, u, ∇u)u ∈ L

(Ω).

This proves Theorem 3.1.

5. Appendix

Appendix 1: Proof of Lemma 4.2

From H¨older’s inequality, the growth condition (20) we can show that A is

bounded, and by using (26), we have B

bounded. The coercivity follows

from (7), (22) and (26). it remain to show that B

is pseudo-monotone.

Let a sequence (u

) ∈ W

1,p

(Ω, w) such that

* u weakly in W

1,p

(Ω, w),

lim sup

k→+∞

, u

− ui ≤ 0. (71)

Let v ∈ W

1,p

(Ω, w), we will prove that

lim inf

k→+∞

, u

− vi ≥ hB

u, u − vi.

Since (u

)

is bounded in W

1,p

(Ω, w), by (20) we deduce that

(a(x, u

, ∇u

))

is bounded in

i=1

(Ω, w

1−p

), then there exists a func-

tion h ∈

i=1

(Ω, w

1−p

) such that

a(x, u

, ∇u

) * h weakly in

i=1

(Ω, w

1−p

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similarly, it is easy to see that (g

(x, u

, ∇u

))

is bounded in L

(Ω, σ

1−q

then there exists a function ρ

∈ L

(Ω, σ

1−q

) such that

(x, u

, ∇u

) * ρ

weakly in L

(Ω, σ

1−q

It is clear that, (15)

lim inf

k→+∞

, u

− vi = lim inf

k→+∞

Ω

a(x, u

, ∇u

)∇u

dx −

Ω

h∇v dx

Ω

(u − v) dx.

(72)

On the other hand, by condition (21), we have

Ω

(a(x, u

, ∇u

) − a(x, u

, ∇u))(∇u

− ∇u) dx ≥ 0

which implies that

Ω

a(x, u

, ∇u

)∇u

dx ≥ −

Ω

a(x, u

, ∇u)∇u dx +

Ω

a(x, u

, ∇u

)∇u dx

Ω

a(x, u

, ∇u)∇u

dx,

hence

lim inf

k→∞

Ω

a(x, u

, ∇u

)∇u

dx ≥

Ω

h∇u dx. (73)

Combining (72) and (73), we get

lim inf

k→+∞

, u

− vi ≥

Ω

h∇(u −v) dx +

Ω

(u −v) dx. (74)

Now, since v is arbitrary and lim

k→+∞

Ω

(x, u

, ∇u

)(u

− u) dx = 0, we

have by using (71) and (74)

lim

k→+∞

Ω

a(x, u

, ∇u

)∇(u

− u) dx = 0

we deduce that

lim

k→+∞

Ω

(a(x, u

, ∇u

) − a(x, u

, ∇u))∇(u

− u) dx = 0.

In view of Lemma 4.1, we have ∇u

→ ∇u a.e. in Ω, which with (74)

yields

lim inf

k→+∞

, u

− vi ≥ hB

u, u −vi.

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Thus complete the proof of Lemma 4.2.

Appendix 2: We claim that v

∈ K

deﬁned in (35))

We have e

γw

n,k

≤ c

, let η =

Since v

= u

− ηϕ

n,k

), we remark

that

≥



if w

n,k

≤ 0

− w

n,k

if w

n,k

≥ 0

then it suﬃces to prove that u

− w

n,k

≥ ψ we have

− w

n,k







(u) if |u

| ≤ k

− k + T

(u) if u

≥ k

+ k + T

(u) if u

≤ −k

which implies that

− w

n,k

≥







(u) if |u

| ≤ k

(u) if u

≥ k

if u

≤ −k

since u ∈ K

, k ≥ kv

∞

then T

(u) ≥ ψ, which implies that u

−w

n,k

≥ ψ.

Finally since v

∈ W

1,p

(Ω, w) then v

∈ K

References

1. L. Aharouch and Y. Akdim, Existence of solution of degenerated unilateral

Problems with L

data, Annale Math´ematique Blaise Pascal Vol. 11 (2004)

pp 47-66.

2. L. Aharouch, Y. Akdim and E. Azroul, Quasilinear degenerate elliptic

unilateral problems, AAA 2005:1 (2005) 11-31. DOI: 10.1155/AAA.2005.11

3. Y. Akdim, E. Azroul and A. Benkirane, Existence Results for Quasilin-

ear Degenerated Equations via Strong Convergence of Truncations, Revista

Matem´atica Complutence, Madrid, (2004), 17; N 2, 359-379.

4. Y. Akdim, E. Azroul and A. Benkirane, Existence of solution for Quasi-

linear Degenerated Unilateral Problems, Annale Math´ematique Blaise Pascal

Vol. 10 (2003) pp 1-20.

5. P. B

enilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre and J.

L. V

azquez, An L

-theory of existence and uniqueness of nonlinear elliptic

equations, Ann. Scuola Norm. Sup. Pisa 22 (1995), 240-273.

6. L. Boccardo, F. Murat, and J. P. Puel , Existence of bounded solutions for

nonlinear elliptic unilateral problems, Ann. Mat. Pura e Appl., 152 (1988)

183-196.

7. L. Boccardo, T. Gallou

et and F. Murat, A uniﬁed presentation of two

existence results for problems with natural growth, in Progress in PDE, the

Metz surveys 2, M. Chipot editor, Research in Mathematics, Longman, 296

(1993), 127-137.

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8. 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.

9. P. Drabek, F. Nicolosi, Existence of bounded solutions for some degenerated

quasilinear elliptic equations, Ann. Math. Pura Appl. (4) 165 (1993), 217-

238.

10. P. Drabek, A. Kufner and V. Mustonen, Pseudo-monotonicity and de-

generated or singular elliptic operators, Bull. Austral. Math. Soc. 58 (1998),

213-221.

11. P. Drabek, A. Kufner and F. Nicolosi, Non linear elliptic equations, sin-

gular and degenerate cases, University of West Bohemia, (1996).

12. D. Gilbarg and N.S. Trudinger, Elliptic partial diﬀerential equations of

second order, Springer-Verlag, Berlin, (1977).

13. A. Porretta, Existence for elliptic equations in L

having lower order terms

with natural growth,Portugal. Math. 57 (2000), 179-190.

14. A. Elmahi and D. Meskine , Unilateral elleptic problems in L

with natural

growth terms, Journal of Nonlinear and Convex Analysis, 5 N. 1, (2004),

97-112

15. J. Lions, Quelques m´ethodes de r´esolution des probl`emes aux limites non

lin´eaires,Dunod, Paris (1969).

16. B. Opic and A. Kufner, Hardy-type inequalities, Pitman Research Notes in

Mathematics Series vol. 219, Longman Scientiﬁc & Technical, Harlow, 1990,

ISBN 0-582-05198-3.

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205

Existence and multiplicity results for some p(x)-Laplacian

Neumann problems

M. Bendahmane

Al-Imam University, Faculty of Sciences, Department of Mathematics

P. O. Box 90950, Riyadh 11623, Saudi Arabia

E-mail: mostafab@ing-mat.udec.cl

M. Chrif

Department of Mathematics, Faculty of Sciences, Atlas-Fes

Fes, 3000, Morocco

E-mail: moussachrif@yahoo.fr

S. El Manouni

Department of Mathematics, Faculty of Sciences, Al-Imam University

P. O. Box 90950, Riyadh, 11623, Saudi Arabia

E-mail: samanouni@imamu.edu.sa

In this paper, using an equivalent variational approach to a recent Ricceri’s

three critical points theorem,

we obtain the existence of at least three non-

trivial solutions of a Neumann problem for elliptic equations with variable

exponents.

Keywords: p(x)-Laplacian; Elliptic equations; Neumann conditions; Three crit-

ical points theorem.

1. Introduction

Let Ω be a bounded domain of IR

, N ≥ 2 with a smooth boundary ∂Ω.

We consider the following Neumann problem for the corresponding elliptic

problem

)



−div(|∇u|

p(x)−2

∇u) + |u|

p(x)−2

u = λf (x, u) + µg(x, u) in Ω

∂u

∂ν

= 0 on ∂Ω

where p ∈ C(

Ω), p(x) > 1 for every x ∈ Ω. f is a C

-function on Ω ×[0, ∞)

satisfying

(1.1) |f (x, t)| ≤ c

α(x)−1

+ c

∀(x, t)

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

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for some α ∈ C

(

Ω) with

(1.2) α

< p

−

where

= sup

x∈Ω

h(x) and h

−

= inf

x∈

Ω

h(x) ∀h ∈ C

(

Ω).

(1.3) f (x, t) < 0, for all t ∈]0, 1[,

(1.4) f(x, t) > M, for all t > t

where M is a positive constant and t

> 1. The function g is assumed to

be a measurable function with respect to x in Ω for every t in IR, and is a

-function with respect to t in IR for almost every x in Ω and satisﬁes

(1.5) sup

|t|≤s

|g(x, t)| ∈ L

(Ω)

for all s > 0. Here Ω is a bounded domain in IR

with smooth boundary

∂Ω, div(|∇u|

p(x)−2

∇u) is the p(x)−Laplacian, the generalization of the

classical p−Laplacian operator, and ν is the outward unit normal to ∂Ω.

Recently, elliptic equations with variable exponents have been exten-

sively investigated and have received much attention. They have been the

subject of recent developments in nonlinear elasticity theory and electrorhe-

ological ﬂuids dynamics Ru˜zicka.

In that context, let us mention that

there appeared a series of papers on problems which lead to spaces with

variable exponent. We refer the reader to Fan et al.,

8,9

Ru˜zicka

and the

references therein.

Let us point out that when p(x) = p = constant, there is a large lit-

erature which deal with problems involving the p-Laplacian with Dirichlet

boundary conditions for elliptic equations in bounded or unbounded do-

mains and we do not need here to cite them since the reader may easily

reach such papers.

Note also that many papers deal with problems related to the p-

Laplacian with Neumann conditions. We can cite, among others, the articles

Anello et al.

and Bonanno et al.

and the references therein for details.

The case of p(x)−Laplacian with Neumann conditions has been studied by

Dai,

Mihˇailescu

and Shi and Ding,

all the authors used a past result

of Ricceri.

Finally, it would be interesting at this stage to refer the reader to the

recent work Manouni et al.

where the authors established the existence

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

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of at least three nontrivial solutions for elliptic systems involving the p-

Laplacian with Neumann boundary conditions.

Our objective is to study the Neumann problem for such an equation of

the type (P

). Precisely, based on a recent result due to Ricceri,

we are

interested in the existence and multiplicity of weak nontrivial solutions for

the problem (P

) in the Sobolev space W

1,p(x)

(Ω).

Along this paper we ﬁx p

−

> N. Recall that a weak solution of system

) is any u ∈ W

1,p(x)

(Ω) such that

Ω

(|∇u|

p(x)−2

∇u∇ϕ + |u|

p(x)−2

uϕ) dx =

Ω

(λf(x, u) + µg(x, u))ϕ dx

for all ϕ ∈ W

1,p(x)

(Ω).

Remark 1.1. Let us remark that (1.2) and (1.5) guarantees that integrals

given in the right side are well deﬁned.

In particular, we will consider the following Neumann problem

)



−∆

p(x)

u + |u|

p(x)−2

u = λ(|u|

α(x)−2

u −1) + µ|u|

γ(x)−2

u in Ω

∂u

∂ν

= 0 on ∂Ω,

where

(1.6) γ ∈ C

(

Ω), γ > 1

and α satisﬁes

(1.7) α

< p

−

To prove the existence of at least three weak solutions for each of the given

problems (P

)

and (P

), we will use the following result proved in Ricceri

that, on the basis of Bonanno,

can be equivalently stated as follows

Theorem 1.1. Let X be a reﬂexive real Banach space; Φ : X −→ IR is

bounded on each bounded subset of X, continuously Gˆateaux diﬀerentiable

and sequentially weakly lower semi-continuous functional whose Gˆateaux

derivative admits a continuous inverse on X

∗

; J : X −→ IR a continu-

ously Gˆateaux diﬀerentiable functional whose Gˆateaux derivative is com-

pact. Moreover, assume that

(i) lim

kuk−→∞

(Φ(u) + λJ(u)) = +∞ for all λ ∈ ]0, +∞[

and that there are r ∈ IR, u

, u

∈ X such that

(ii) Φ(u

) < r < Φ(u

)

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208

(iii) inf

u∈Φ

−1

(]−∞,r])

J(u) >

(Φ(u

) − r)J(u

) + (r − Φ(u

))J(u

)

Φ(u

) − Φ(u

)

Then, there exist an open interval A of ]0, +∞[ and a positive real number

t such that for every λ ∈ A, and every continuously Gˆateaux diﬀerentiable

functional Ψ : X −→ IR with compact derivative, there exists δ > 0 such

that for each µ ∈ [0, δ], the equation

(u) + λJ

(u) + µΨ

(u) = 0

has at least three solutions in X whose norms are less than t.

2. Preliminaries

We list some well known deﬁnitions and basic properties and recall some

background facts concerning generalized Lebesgue-Sobolev spaces L

p(x)

(Ω),

1,p(x)

(Ω) and W

1,p(x)

(Ω), where Ω is an open subset of IR

and introduce

some notations used below. For more details about these spaces, we refer

the reader to the book of Musielak

and the papers of Kov´aˇcik et al.

and

Fan et al.

8,9

Set

∞

(Ω) = {h; h ∈ L

∞

(Ω), ess inf

x∈Ω

h(x) > 1 for all x ∈

Ω}.

For any h ∈ L

∞

(Ω) we deﬁne

= ess sup

x∈Ω

h(x) and h

−

= ess inf

x∈Ω

h(x).

For any p(x) ∈ L

∞

(Ω), we deﬁne the variable exponent Lebesgue space

p(x)

(Ω) = {u : uis a measurable real-valued function such that

Ω

|u(x)|

p(x)

dx < ∞}.

We deﬁne a norm, the so-called Luxemburg norm, on this space by the

formula

|u|

p(x)

= inf



µ > 0;

Ω



u(x)



p(x)

dx ≤ 1



Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in

many respects: they are Banach spaces (Theorem 2.5, Kov´aˇcik

), the H¨older

inequality holds (Theorem 2.1, Kov´aˇcik

), they are reﬂexive if and only if

1 < p

−

≤ p

< ∞ (Corollary 2.7, Kov´aˇcik

) and continuous functions

are dense if p

< ∞ (Theorem 2.11, Kov´aˇcik

). The inclusion between

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Lebesgue spaces also generalizes naturally (Theorem 2.8, Kov´aˇcik

): if 0 <

|Ω| < ∞ and r

, r

are variable exponents so that r

(x) ≤ r

(x) almost

everywhere in Ω then there exists the continuous embedding L

(x)

(Ω) ,→

(x)

(Ω), whose norm does not exceed |Ω| + 1. We denote by L

(x)

(Ω)

the conjugate space of L

p(x)

(Ω), where 1/p(x) + 1/p

(x) = 1. For any u ∈

p(x)

(Ω) and v ∈ L

(x)

(Ω) the H¨older type inequality



Ω

uv dx



≤



−



|u|

p(x)

|v|

(x)

holds true. An important role in manipulating the generalized Lebesgue-

Sobolev spaces is played by the modular of the L

p(x)

(Ω) space, which is the

mapping

p(x)

: L

p(x)

(Ω) → IR

deﬁned by

p(x)

(u) =

Ω

|u|

p(x)

dx.

If u ∈ L

p(x)

(Ω) and p

< ∞ then the following relations hold

|u|

p(x)

> 1 ⇒ |u|

−

p(x)

≤ ρ

p(x)

(u) ≤ |u|

p(x)

;

|u|

p(x)

< 1 ⇒ |u|

p(x)

≤ ρ

p(x)

(u) ≤ |u|

−

p(x)

;

− u|

p(x)

→ 0 ⇔ ρ

p(x)

− u) → 0.

We also consider the weighted variable exponent Lebesgue spaces. We deﬁne

also the variable Sobolev space

1,p(x)

(Ω) = {u ∈ L

p(x)

(Ω) : |∇u| ∈ L

p(x)

(Ω)}.

On W

1,p(x)

(Ω) we may consider one of the following equivalent norms

kuk

p(x)

= |u|

p(x)

+ |∇u|

p(x)

kuk = inf



µ > 0;

Ω





∇u(x)



p(x)



u(x)



p(x)



dx ≤ 1



Set

p(x)

(u) =

Ω



|∇u|

p(x)

+ |u|

p(x)



dx.