Document de travail 2013-11

“Existence and nonexistence of nontrivial solutions for some

nonlinear elliptic systems involving the (p(x), q(x))-Laplacian”

Warda SAIFIA & Jean VELIN

Mai 2013

Centre d’Etude et de Recherche en Economie, Gestion, Modélisation et Informatique Appliquée

Université des Antilles et de la Guyane. Faculté de Droit et d’Economie de la Martinique. Campus de Schoelcher - Martinique FWIB.P. 7209 - 97275 Schoelcher Cedex - Tél. : 0596. 72.74.00 - Fax. : 0596. 72.74.03

www.ceregmia.eu

Existence and Nonexistence of NontrivialSolutions for Some Nonlinear Elliptic

Systems Involving the (p(x), q(x))-Laplacian

Warda SAIFIA and ∗ Jean VELIN †

Abstract

In view of the fibring method, we prove the existence of non-trivial solutions.Generalization of the well-known Pohozaev and Pucci-Serrin iden-dities and some nonexistence results for Dirichlet problem involv-ing the (p(x), q(x))-Laplacian system are obtained.

Key Words: Fibering method, non-existence theorem, p(x)-Laplacian, GeneralizedPohozeav identity, Pucci Serrin identity.

1 IntroductionAfter the pioneer article of Kovacik and Rokosnik [26] concerning theLp(x)(Ω) and W 1,p(x)(Ω) spaces, many researches led in these kinds ofvariable exponent spaces. We refer to [15] for the properties of suchspaces and [8, 19] for the applications of variable exponent on partialdifferential equationsIn the recent years, the theory of problem with p(x)-Laplacian has a largeapplication in nonlinear electrorheological fluids, and elastic mechanics,∗Department of mathematics; University of Annaba; PO12; El Hadjar, 23000,

Annaba. Algeria†Department of Mathematic and Computer, Laboratory CEREGMIA, Univer-

sity of Antilles-Guyane, Campus de Fouillole, 97159 Pointe-à-Pitre Guadeloupe(FWI). E-mails: jean-velin@univ-ag.fr, AMS Classification: 35J20, 35J35, 35J45,35J50, 35J60, 35J70

1

image processing and flow in porous media see [1, 5, 9, 10, 20, 27, 34, 43].

The aim of this paper is to study the existence and non-existence ofthe weak solutions for the following (p, q)-gradient elliptic system:

−∆p(x)u = c(x)u|u|α−1|v|β+1 in Ω

−∆q(x)v = c(x)v|v|β−1|u|α+1 in Ω

u = v = 0 on Ω.

(1.1)

Here Ω designates a bounded, regular open set in RN with a smoothboundary ∂Ω, p, q : Ω −→ R two functions of class C(Ω), p(x), q(x) > 1for every x ∈ Ω and c a function which may changes sign and for everyreal function p ∈ L∞(Ω), we denote:

p− = infΩp(x) and p+ = sup

Ωp(x), (1 ≤ p− ≤ p+ < +∞),

L∞+ (Ω) = p ∈ L∞(Ω), p− ≥ 1.

Concerning in existence and nonexistence results of the type of this sys-tem, we cite a study presented in [6]. The authors use the fibering methodintroduced by S. Pohozeav. They proved the existence of multiple solu-tions for a Dirichlet problem associated with a quasilinear system involv-ing a pair of (p, q)-Laplacian operators.

More recently, employing the fibering method, the second author of ourpaper has proved the existence of multiple positive solutions for a classof (p, q)-gradient elliptic systems including systems like (1.1). For moredetails, the reader can consult [39, 40].Systems structured as (1.1) have been investigated for instance in [38].The authors have presented some results dealing with existence andnonexistence of a non-trivial solution (u, v) ∈ W 1,p

0 (Ω)×W 1,q0 (Ω) of the

following system

−∆pu = u|u|α−1|v|β+1 in Ω

−∆qv = v|v|β−1|u|α+1 in Ω

u = v = 0 on Ω.

(1.2)

2

When(α + 1)N − p

Np+ (β + 1)N − q

Nq> 1 (1.3)

and Ω is a strictly starshaped open domain in RN ,they have proved nonexistence results.On other side, when

(α + 1)N − pNp

+ (β + 1)N − qNq

< 1 (1.4)

and α + 1p

+ β + 1q6= 1, existence results have discussed.

In [12], the authors obtain nonexistence for Dirichlet problem gov-erned by the p(x)-Laplacian operator.

Our paper follows this organization:

1. In the first part of our paper, we recall the position of this work.

2. A second section presents some notation and preliminaries neededfor the framework of the paper. We also recall some tools definedby the theory of variable exponents Lebesgue and Sobolev spaces.

3. The third section announces the main results.

4. In the fourth section, following the ideas explained in [12], we es-tablish a Pohozaev-type identity for the system (1.1). By usingthis adapted identity, we deal with the non-existence results of nontrivial solutions.

5. In the next section, after recalling the spirit of the fibering method,we show via this method that (1.1) admits at least one weak non-trivial solution.

6. To acheive the study, in connexion with the non-existence, we es-tablish that the solution is bounded below in Ω.

3

2 Notations and PreliminariesIn this section, and throughout the study, we recall some definitions andproperties on the generalized Lebesgue space Lp(x)(Ω) and generalizedSobolev spaces W 1,p(x)(Ω). Ω ⊂ RN is an open set. For more details, thereader can consult for instance [11, 17, 18, 19, 21, 22, 26, 27, 28].The generalized Lebesgue space Lp(x)(Ω) consists in all measurable func-tions u defined on Ω for which the p(x)-modular

ρp(.)(u) =∫

Ω|u(x)|p(x)dx

is finite. The Luxemberg norm on this space is defined as:

‖u‖p = inf

λ > 0; ρp(.)(u) =∫

Ω

∣∣∣∣∣u(x)λ

∣∣∣∣∣p(x)

dx ≤ 1

.Equipped with this norm, Lp(x)(Ω) is a Banach space.If p(x) is constant, Lp(x)(Ω) is reduced to the standard Lebesgue space.For given p ∈ L∞+ (Ω), we define the conjugate function p′(x) as

p′(x) = p(x)p(x)− 1 .

The following results show the close relation between the convex modularρp(.) and the norm ‖.‖Lp(.)(Ω).Let us recall main results on generalized Lebesgue spaces. We start by

Proposition 2.1 Let p ∈ L∞+ (Ω).

1. If u ∈ Lp(.)(Ω) then ‖u‖Lp(.)(Ω) = a⇔ %(u

a

)= 1

2. ‖u‖Lp(.)(Ω) < 1(= 1, > 1)⇔ %p(.)(u) < 1(= 1, > 1)

3. If ‖u‖Lp(.) > 1 then ‖u‖p−

Lp(.)≤ %p(.)(u) ≤ ‖u‖p

+

Lp(.)

4. If ‖u‖Lp(.) < 1 then ‖u‖p+

Lp(.)≤ %p(.)(u) ≤ ‖u‖p−

Lp(.)

4

Proposition 2.2 [26, 17] Let p ∈ L∞+ (Ω), (un) ⊂ Lp(.)(Ω) and u ∈Lp(.)(Ω).

The following assertions are equivalent:(i) lim

n→+∞‖u− un‖Lp(.) = 0

(ii) limn→+∞

%p(.)(u− un) = 0.

Theorem 2.1 (see[17, 15, 28]). Consider p, q, r ∈ L∞+ (Ω), u ∈ Lp(.)(Ω)et v ∈ Lq(.)(Ω) such that:

1p(x) + 1

q(x) = 1r(x) e. a in Ω

then‖uv‖Lr(.)(Ω) ≤

[1

(p/r)− + 1(q/r)−

]‖u‖Lp(.)(Ω)‖v‖Lq(.)(Ω)

for all u ∈ Lp(.)(Ω), v ∈ Lq(.)(Ω). It is immediate to make this remark

Remark 2.1 Let p ∈ L∞+ (Ω) and let p′ : Ω→ [1,+∞[ be the conjugatefunction of p.There is a constant Cp > 0 such that;∫

Ω|uv| ≤ Cp‖u‖Lp(.)‖v‖Lp′(.)

for all u ∈ Lp(.)(Ω), v ∈ Lp′(.)(Ω).

We also have the following imbedding theorem and we refer the readerto Kovacik and Rokosnik[26], Fan and Zhao[17]

Proposition 2.3 Let Ω ⊂ RN be a bounded open set and let p, q ∈L∞+ (Ω).If p(x) ≤ q(x) a.e in Ω, then Lq(.)(Ω) → Lp(.)(Ω).

Now, we recall main results about generalized Sobolev space. For anyp ∈ L∞+ (Ω) and m ∈ N∗, we define

Wm,p(.)(Ω) = u ∈ Lp(.)(Ω) : Dαu ∈ Lp(.)(Ω)pour tout|α| ≤ m,

‖u‖m,p(.) =∑|α|≤m

‖Dαu‖Lp(.)(Ω)

5

The pair (Wm,p(.)(Ω), ‖ · ‖m,p(.)) is a separable Banach space (reflexiveif p− > 1) which is called generalized Sobolev space (also known asSobolev space with variable exponent). We will denote by W 1,p(.)

0 (Ω) theclosure of C∞0 (Ω) in Wm,p(.)(Ω).

Proposition 2.4 Let Ω ⊂ RN be a bounded open set and let p, q ∈L∞+ (Ω).If

p(x) ≤ q(x) a.e in Ω,

then W 1,q(.)(Ω) → W 1,p(.)(Ω).

Definition 2.1 We say that a function P : A→ R is ln-Hölder contin-uous on A provided that there exists a constant C > 0 such that

|P (x)− P (y)| ≤ C

− ln |x− y|

for all x, y ∈ A, |x− y| ≤ 12 .

The following density result holds.

Theorem 2.2 Let Ω ⊂ RN be a bounded open set with Lipschitz bound-ary and p ∈ L∞+ . If p is ln- Hölder continuous on Ω, then C∞(Ω) is densein W 1,p(.)(Ω).

Theorem 2.3 Let Ω ⊂ RN be a bounded open set with Lipschitz bound-ary and let p ∈ C(Ω) be a function which satisfies p− > 1.Define the Sobolev conjugate exponent p∗ : Ω→ R of p

p∗(x) =

Np(x)N − p(x) si p(x) < N

∞ si p(x) ≥ N.

then the imbedding Wm,p(.)(Ω) → Lq(.)(Ω) is continuous and holds forevery function q ∈ C(Ω) which satisfies; 1 < q(x) < p∗(x) for all x ∈ Ω.

3 Main resultsWe define

6

• p, q : Ω −→ IR+ \0 as two functions belonging in C1B(Ω)∩ C(Ω),

• c : Ω −→ IR; c+(x) 6= 0, c−(x) 6= 0,

• p− = minx∈Ω

p(x), q− = minx∈Ω

q(x), p+ = maxx∈Ω

p(x), q+ = maxx∈Ω

q(x).

Let us now announce the main results of this paper:

• Non-existence result for the (p(x), q(x))-Laplacian system(1.1):

Theorem 3.1 Let Ω be a bounded open set of RN , with boundary∂Ω of class C1. Let

– p, q : Ω −→ IR functions of class C1B(Ω) ∩ C(Ω), p−, q− > 1,

– c(.) ∈ C1B(Ω \ C), with meas(C) = 0.

Assume that

– Ω be a bounded domain of class C1, starshaped with respect tothe origin,

– (p, q) ∈ C1B(Ω)∩C(Ω); p−, q− > 1 and (x.∇p) ≥ 0, (x ·∇q) ≥

0,–

〈x,∇c(x)〉 ≤ 0 for any x in Ω. (3.1)

–(α + 1)N − p

+

Np+ + (β + 1)N − q+

Nq+ ≥ 1. (3.2)

Then (1.1) has no nontrivial classical solution (u, v) ∈(C2(Ω) ∩ C1(Ω)

)2

which satisfies:

|∇u(x)| ≥ e1/p(x) and |∇v(x)| ≥ e1/q(x) a.e x ∈ Ω, (3.3)

and ∫Ωc(x)|u|α+1|v|β+1dx > 0.

• Existence result for the (p(x), q(x))-Laplacian system (1.1)

7

Theorem 3.2 Let Ω be a bounded open set of RN , with boundary∂Ω of class C1. Let p, q : Ω→ IR functions of class C1

B(Ω)∩ C(Ω);p−, q− > 1. Assume that:

p+ ≤ α + 1 or q+ ≤ β + 1,

(α + 1)N − p−

Np−+ (β + 1)N − q

−

Nq−< 1. (3.4)

Then (1.1) admits at least a nontrivial solution (u∗, v∗) ∈ W 1,p(x)0 (Ω)×

W1,q(x)0 (Ω). Moreover, one have

1.

|∇u∗(x)| ≥ e1/p(x) and |∇v∗(x)| ≥ e1/q(x) a.e x ∈ Ω. (3.5)

2.∫

Ωc(x)|u∗|α+1|v∗|β+1dx > 0.

Remark 3.1 Let us remark that conditions (3.2) and (3.4) seem togeneralize to (p(x), q(x))− gradient elliptic systems conditions (1.3) and(1.4) well known when (p, q)− gradient elliptic systems are considered.

Obviously, conditions (3.2) and (3.4) imply respectively 1 ≤ (α+1)N − p−

Np−+

(β + 1)N − q−

Nq−and (α + 1)N − p

+

Np+ + (β + 1)N − q+

Nq+ < 1.

4 A Pohozaev-type identity for (p(x),q(x))-Laplacian and Nonexistence Results

Consider the elliptic system with Dirichlet boundary condition:

−∆p(x)u = c(x)u|u|α−1|v|β+1 in Ω

−∆q(x)v = c(x)|u|α+1v|v|β−1 in Ω

u = v = 0 on Ω

where

8

• Ω ⊂ IRN is a bounded open set with a regular boundary ∂Ω,

• p, q, c are defined as in the previous section.

• ∆p(x)u = ∂

∂xi

(|∇u|p(x)−2 ∂u

∂xi

).

Proposition 4.1 Let Ω be a bounded open set of RN , with boundary ∂Ωof class C1. Assume that

• p, q : Ω −→ IR of class C1B(Ω) ∩ C(Ω), p−, q− > 1,

• c(.) ∈ C1B(Ω \ C), with meas(C) = 0 and

〈x,∇c(x)〉 ≤ 0 for any x in Ω.

Then, for every classical solution (u, v) ∈ C2(Ω) ∩ C1(Ω) (1.1), the fol-lowing identity holds:

α + 1N

∫∂Ω

1− p(x)p(x) |∇u|p(x)〈x, ν〉dσ + β + 1

N

∫∂Ω

1− q(x)q(x) |∇v|q(x)〈x, ν〉dσ

= α + 1N

∫Ω

(N − p(x)p(x) − a1

)|∇u|p(x)dx+ β + 1

N

∫Ω

(N − q(x)q(x) − a2

)|∇v|q(x)dx

+α + 1N

∫Ω

〈x,∇p〉p2(x) (ln |∇u|p(x) − 1)|∇u|p(x)dx+ β + 1

N

∫Ω

〈x,∇q〉q2(x) (ln |∇v|q(x) − 1)|∇v|q(x)dx

+∫

Ω

(α + 1)a1 + (β + 1)a2 −N

c(x)|u|α+1|v|β+1dx−

∫Ω〈x,∇c〉|u|α+1|v|β+1dx.

for all a1 and a2 ∈ RN .

In order to prove Proposition 4.1, we need the following result, whichgeneralizes the variational identity of Pucci-Serrin[33] .

Proposition 4.2 Let Ω be a bounded open set of RN with boundary ∂Ωof class C1. Assume that

• p, q : Ω −→ IR of class C1B(Ω) ∩ C(Ω), p−, q− > 1,

• c(.) ∈ C1B(Ω \ C), C ⊂ Ω with meas(C) = 0.

9

Then, for every classical solution (u, v) ∈(C2(Ω) ∩ C1(Ω)

)2of the prob-

lem (1.1), the following identity holds:

∂

∂xi

xi(α + 1p(x) |∇u|

p(x) + β + 1q(x) |∇v|

q(x) − c(x)|u|α+1|v|β+1)−

(α + 1)(xj∂u

∂xj+ a1u

)|∇u|p(x)−2 ∂u

∂xi− (β + 1)

(xj∂v

∂xj+ a2v

)|∇v|q(x)−2 ∂v

∂xi

= (α + 1)[N − p(x)p(x) − a1

]|∇u|p(x) + (β + 1)

[N − q(x)q(x) − a2

]|∇v|q(x)

+〈x,∇p〉p2(x) (ln |∇u|p(x) − 1)|∇u|p(x) + 〈x,∇q〉

q2(x) (ln |∇v|q(x) − 1)|∇v|q(x)

+ (α + 1)a1 + (β + 1)a2 −N c(x)|u|α+1|v|β+1 − 〈x,∇c〉|u|α+1|v|β+1,(4.1)

for all a1 and a2 ∈ R.

The proof of Proposition 4.2 can be established by a simple compu-tation.

Proof of Proposition 4.1.Throughout the proof, for x = (xi)i=1;··· ,N , y = (yi)i=1;··· ,N two vectors inIRN , the classical inner product is simply denoted xiyi and the notationN∑i=1

is omitted. Let (u, v) ∈(C2B ∩ C1(Ω)

)2be a classical solution of the

problem (1.1).According to Proposition 4.2, (u, v) satisfies the identity (4.1). Integrat-ing by part over Ω, we get:

10

∫∂Ω

[(α + 1p(x) |∇u|

p(x) + β + 1q(x) |∇v|

q(x) − c(x)|u|α+1|v|β+1)

−(α + 1)(xj∂u

∂xj+ a1u

)|∇u|p(x)−2 ∂u

∂xi− (β + 1)

(xj∂v

∂xj+ a2v

)|∇v|q(x)−2 ∂v

∂xi

]νidσ

= (α + 1)∫

Ω

(N − p(x)p(x) − a1

)|∇u|p(x)dx+ (β + 1)

∫Ω

(N − q(x)q(x) − a2

)|∇v|q(x)dx

∫Ω

[1

p2(x)〈x.∇p〉(ln |∇u|p(x) − 1)|∇u|p(x) + 1

q2(x)〈x,∇q〉(ln |∇v|q(x) − 1)|∇v|q(x)

]dx

+∫

Ω

(α + 1)a1 + (β + 1)a2 −N

c(x)|u|α+1|v|β+1dx−

∫Ω〈x,∇c〉|u|α+1|v|β+1dx.

(4.2)ν is the unit outer normal to the boundary ∂Ω, since u = 0 on ∂Ω, it

follows clearly that

∂u

∂xi= (∇u.ν)νi, ∀i = 1, · · · , N, x on ∂Ω,

then we can write:

xj∂u

∂xj

∂u

∂xi|∇u|p(x)−2νi = xj [(∇u.ν)νj]

∂u

∂xi|∇u|p(x)−2νi

= ∂u

∂xi

∂u

∂xi|∇u|p(x)−2(x.ν)

= |∇u|p(x)(x.ν) on ∂Ω

Using the relation (4.2) and the fact that u|∂Ω = 0 in the left hand sideof this relation, the statement of Proposition 4.1 occurs.

Remark 4.1 Before proving Proposition 4.2, let us note that the set offunctions c satisfying to hypothesis (3.1), is non-empty. Indeed, let x0be in ∂Ω such that dist(0, ∂Ω) = dist(0, x0). We set R0 = dist(0, ∂Ω).Obviously, we notice that the ball B(0, R0) is contained in Ω. We definethe set Ω1 as follow Ω1 =

x ∈ Ω; 0 ≤ ‖x‖ ≤ R0

2

. For instance, we

11

define the function c as follow:

c(x) =−e‖x‖2 if x ∈ Ω1e−‖x‖

2 if x ∈ Ω \ Ω1.

c changes sign into Ω and we also have for any x ∈ Ω, 〈x,∇c(x)〉 ≤ 0.Moreover, c ∈ L∞(Ω).

Proof of Theorem 3.1.Suppose that there exists a nontrivial classical solution (u, v) ∈ C2(Ω) ∩C1(Ω) of the problem (1.1). So that, (u, v) satisfies the statement ofProposition 4.1.Since Ω ⊂ RN is strictly starshaped with respect to the origin, we havex.ν > 0 on ∂Ω thus:

−α + 1N

∫∂Ω

1p(x) |∇u|

p(x) < x, ν > dσ−β + 1N

∫∂Ω

1q(x) |∇v|

q(x) < x, ν > dσ < 0

where 1p(x) = p(x)− 1

p(x) ,1

q(x) = q(x)− 1q(x) .

In other hand, choosing a1 ∈ IR and a2 ∈ IR such that

(α + 1)a1

N+ (β + 1)a2

N= 1

and using the relations (3.2), (3.3), we getα + 1N

∫∂Ω

1− p(x)p(x) |∇u|p(x)〈x, ν〉dσ + β + 1

N

∫∂Ω

1− q(x)q(x |∇v|q(x)〈x, ν〉dσ

= α + 1N

∫Ω

(N − p(x)p(x) − a1

)|∇u|p(x)dx+ β + 1

N

∫Ω

(N − q(x)q(x) − a2

)|∇v|q(x)dx

+∫

Ω

(α + 1)a1 + (β + 1)a2 −N

c(x)|u|α+1|v|β+1dx−

∫Ω〈x,∇c〉|u|α+1|v|β+1dx

≥ (α + 1)N − p+

Np+

∫Ω|∇u|p(x)dx+ (β + 1)N − q

+

Nq+

∫Ω|∇v|q(x)dx− (α + 1)a1

N

∫Ω|∇u|p(x)dx−

(β + 1)a2

N

∫Ω|∇v|q(x)dx+

∫Ω

(α + 1)a1 + (β + 1)a2 −N

c(x)|u|α+1|v|β+1dx

−∫

Ω〈x,∇c〉|u|α+1|v|β+1dx

≥

(α + 1)N − p+

Np+ + (β + 1)N − q+

Nq+ − (α + 1)a1

N− (β + 1)a2

N

∫Ωc(x)|u|α+1|v|β+1dx+

≥

(α + 1)N − p+

Np+ + (β + 1)N − q+

Nq+ − 1∫

Ωc(x)|u|α+1|v|β+1dx−

∫Ω〈x,∇c〉|u|α+1|v|β+1dx.

12

Now, because (u, v) is a solution, let us notice that∫

Ωc(x)|u|α+1|v|β+1dx =∫

Ω|∇u|p(x)dx =

∫Ω|∇v|q(x)dx > 0. Moreover, from hypothesis (3.1), we

get that the right hand is positive. So on, a contradiction occurs and theproof is complete.

5 Existence Results via the Fibering MethodThroughout this section, Ω denotes a bounded regular open set in RN

and X0(x) = W1,p(x)0 (Ω) × W

1,q(x)0 (Ω). The generalized Sobolev spaces

W1,p(x)0 (Ω) andW 1,q(x)

0 (Ω) are equipped with the Luxembourg norm ‖u‖W

1,p(x)0 (Ω)

and ‖u‖W

1,q(x)0 (Ω) respectively. For a best reading, we denote as ‖u‖W 1,p(x)

0 (Ω) =‖u|‖1,p(x) and ‖u‖W 1,q(x)

0 (Ω) = ‖u‖1,q(x).

Before starting this section, we make a fundamental remark

Remark 5.1 Assuming (α + 1)N − p−

Np−+ (β + 1)N − q

−

Nq−≤ 1, we can

also establish that∫

Ωc(x)|z|α+1|w|β+1dx possesses a sens. The functional

c belongs in L∞(Ω), it suffices to verify that |z|α+1|w|β+1 belongs in L1(Ω).Indeed, since we have α + 1

p+ + β + 1q+ > 1 and also α + 1

p−+ β + 1

q−> 1.

So, there exists a pair (p, q) such that

1.p− < p <

Np−

N − p−(5.1)

andq− < q <

Nq−

N − q−(5.2)

2. α + 1p

+ β + 1q

= 1.

Remark 5.2 So that, since Np−

N − p−<

Np(x)N − p(x) and Nq−

N − q−<

Nq(x)N − q(x) ,

the assumption (α+1)N − p−

Np−+(β+1)N − q

−

Nq−≤ 1 implies for any x ∈ Ω,

(5.1) and (5.2) become

p− < p <Np(x)N − p(x)

13

andq− < q <

Nq(x)N − q(x) .

From compactness results, we can conclude that the imbeddingW 1,p(x)0 (Ω) →

Lp(Ω) and W 1,q(x)0 (Ω) → Lq(Ω) are continuous and compact. So on, em-

ploying the Hölder inequality, we get the following estimation∣∣∣∣∣∫

Ωc(x)|u|α+1|v|β+1dx

∣∣∣∣∣ ≤ ‖c‖L∞(Ω)‖u‖α+1Lp(Ω)‖v‖

β+1Lq(Ω) ≤ Cst‖u‖α+1

1,p(x)‖v‖β+11,q(x).

5.1 Definition of a weak solution for (1.1)Let us consider again the system (1.1). First, we recall the definition ofthe weak solution.

Definition 5.1 We say that (u, v) ∈ X0(x) is a weak solution of (1.1) if∫Ω|∇u|p(x)−2∇u∇φ dx =

∫Ωc(x)u|u|α−1|v|β+1u φ dx

∫Ω|∇v|q(x)−2∇v∇ ψ dx =

∫Ωc(x)|u|α+1v|v|β−1u ψ dx

for any (φ, ψ) ∈ X0(x).

5.2 Fibering Method for quasilinear systemsFibering method has been introduced by S. Pohozaev in [29] (see also[31, 32]). One can consult various applications of this method (see forinstance [2, 3, 4, 6, 7, 13, 23, 24, 35, 36, 41, 42]).We define the functional J : X0(x) −→ R by

J(u, v) = (α + 1)∫

Ω

1p(x) |∇u|

p(x)dx+ (β + 1)∫

Ω

1q(x) |∇v|

q(x)dx

−∫

Ωc(x)|u|α+1|v|β+1dx.

Clearly, critical points of the functional J are weak solutions of the prob-lem (1.1). The fibering method applied to this problem consists in thefollowing scheme.

14

We express (u, v) ∈ X in the form

u = rz v = ρw

where the functions z and w belong respectively in W 1,p(x)0 (Ω) \ 0 and

W1,q(x)0 (Ω) \ 0. r and ρ are real numbers.

Since we look for nontrivial solutions, we must assume that r 6= 0 and ρ 6=0

5.3 Existence of a fibering parameter (r∗(z, w), ρ∗(z, w))Now if (u, v) ∈ X0(x) is a critical point of J then a fibering parameter(r(z, w), ρ(z, w)) associated to (z, w) ∈ X0(x) \ (0, 0) is defined by thefollowing Proposition:

Proposition 5.1 Let (z, w) be fixed in X0(x) such that∫

Ωc(x)|z|α+1|w|β+1dx >

0. Assumeq+ ≤ β + 1 or p+ ≤ α + 1. (5.3)

Then there exists a pair (r∗, ρ∗) ∈ R∗+ ×R∗+ depending on (z, w) suchthat

∫Ωr∗p(x)|∇z|p(x)dx = ρ∗β+1r∗α+1

∫Ωc(x)|z|α+1|w|β+1dx,

∫Ωρ∗q(x)|∇w|q(x)dx = ρ∗ β+1r∗α+1

∫Ωc(x)|z|α+1|w|β+1dx.

(5.4)

ProofAssume that (u, v) ∈ X0(x) is a critical point of J, then any fibering

parameter (r, ρ) is characterized as follow

∂J

∂r(rz, ρw) = 0 and ∂J

∂ρ(rz, ρw) = 0. (5.5)

That means

∫Ω|r|p(x)−2r|∇z|p(x)dx = |ρ|β+1|r|α−1r

∫Ωc(x)|z|α+1|w|β+1dx,

∫Ω|ρ|q(x)−2|ρ||∇w|q(x)dx = |ρ| β−1|ρ||r|α+1

∫Ωc(x)|z|α+1|w|β+1dx.

(5.6)

15

It follows ∫Ω|r|p(x)|∇z|p(x)dx =

∫Ω|ρ|q(x)|∇w|q(x)dx. (5.7)

We fix ρ in (5.6). Else, we derive

|r| =

∫

Ω|ρ|q(x)−(β+1)|∇w|q(x)dx∫Ωc(x)|z|α+1|w|β+1dx

1

α+1

.

Let us consider the functional f defined as follow

f : ρ 7−→∫

Ω

∫

Ω|ρ|q(x)−(β+1)|∇w|q(x)dx∫Ωc(x)|z|α+1|w|β+1dx

p(t)α+1

|∇z|p(t)dt−∫

Ω|ρ|q(t)|∇w|q(t)dt.

We put a(z, w) =

∫

Ω|∇w|q(x)dx∫

Ωc(x)|z|α+1|w|β+1dx

p(t)α+1

|∇z|p(t). Let us distin-

guish the following situations:

• ρ > 1,So that,∫

Ω

ρ p(x)[q−−(β+1)]α+1 a(z, w)− ρq(x)|∇w|q(x)

dx ≤ f(ρ) ≤∫

Ω

ρ p(x)[q+−(β+1)]α+1 a(z, w)− ρq(x)|∇w|q(x)

dx.(5.8)

• 0 < ρ < 1. In this way, we obtain

∫Ω

ρ p(x)[q+−(β+1)]α+1 a(z, w)− ρq(x)|∇w|q(x)

dx ≤ f(ρ) ≤∫

Ω

ρ p(x)[q−−(β+1)]α+1 a(z, w)− ρq(x)|∇w|q(x)

dx.(5.9)

Assuming (5.3), it follows that

p(x) [q+ − (β + 1)]α + 1 < q(x), ∀x ∈ Ω. (5.10)

Consequently, from (5.8) and (5.9), we observe that

limρ→+∞

f(ρ) = −∞ and limρ→0

f(ρ) > 0.

16

Using the Mean Value Theorem, there exists a pair (r∗, ρ∗) ∈ R∗+×R∗+depending on z, w and satisfying (5.4) and so on (5.5).

Remark 5.3 1. Under the assumption (5.3), we also have

α + 1p+ + β + 1

q+ − 1 > 0 (5.11)

and consequentlyα + 1p−

+ β + 1q−

− 1 > 0. (5.12)

2. When q(x) and p(x) are constant, (5.11) and (5.12) are reduced tothe well-know condition

1 < α + 1p

+ β + 1q

.

5.4 Estimation on the fibering parameters r∗ and ρ∗

More precisely, we have

Proposition 5.2 Let (z, w) be fixed in X0(x). The pair (r∗, ρ∗) definedas in (5.4) is such that

A(z)q−(β+1)

d B(w)β+1d

C(z, w)q

d

≤ r∗ ≤A(z) q−(β+1)

dB(w)

β+1d

C(z, w) qd

(5.13)

A(z)α+1d B(w)

p−(α+1)d

C(z, w)p

d

≤ ρ∗ ≤ A(z)α+1d B(w)

p−(α+1)d

C(z, w)p

d

, (5.14)

where

A(z) =∫

Ω|∇z|p(x)dx, B(w) =

∫Ω|∇w|q(x)dx, C(z, w) =

∫Ωc(x)|z|α+1|w|β+1dx.

(5.15)

Proof Let us introduce the following function defined on X0(x):

J(z, w) = (α + 1)∫

Ω

1p(x)r

∗p(x)|∇z|p(x)dx+ (β + 1)∫

Ω

1q(x)ρ

∗q(x)|∇w|q(x)dx

−r∗α+1ρ∗β+1∫

Ωc(x)|z|α+1|w|β+1dx.

17

So that, employing (5.4) and (5.7), we get

[α + 1p+ + β + 1

q+ − 1] ∫

Ωr∗p(x)|∇z|p(x)dx ≤ J(z, w) (5.16)

and

J(z, w) ≤[α + 1p−

+ β + 1q−

− 1] ∫

Ωr∗p(x)|∇z|p(x)dx. (5.17)

We consider the functions:

G : (z, w, r, ρ) 7−→∫

Ωrp(x)|∇z|p(x)dx− ρβ+1rα+1

∫Ωc(x)|z|α+1|w|β+1dx

(5.18)and

G : (z, w, r, ρ) 7−→∫

Ωρq(x)|∇w|p(x)dx− ρβ+1rα+1

∫Ωc(x)|z|α+1|w|β+1dx.

(5.19)It is obvious that for any (z, w) fixed in X0(x), r∗ and ρ∗ defined by (5.4)in Proposition 5.1, we have

G(z, w, r∗, ρ∗) = 0, G(z, w, r∗, ρ∗) = 0. (5.20)

Let us put

1.

p =p+ if 0 < r < 1p− if 1 ≤ r,

p =p− if 0 < r < 1p+ if 1 ≤ r.

(5.21)

2.

d = p(β + 1) + q(α + 1)− pq, d = p(β + 1) + q(α + 1)− pq.

Consequently, for any r > 0, the estimates follow

1.p− ≤ p, p ≤ p+, q− ≤ q, q ≤ q+, (5.22)

2.d2 ≤ d, d ≤ d1 (5.23)

whered1 = p+(β+1)+q+(α+1)−p−q−, d2 = p−(β+1)+q−(α+1)−p+q+,

18

3.q−p−

d1≤ qp

d,qp

d≤ q+p+

d2. (5.24)

So that, from (5.20) and (5.21), we deduce:r∗p

∫Ω|∇z|p(x)dx− ρ∗β+1r∗α+1

∫Ωc(x)|z|α+1|w|β+1dx ≤ 0

ρ∗q∫

Ω|∇w|q(x)dx− ρ∗β+1r∗α+1

∫Ωc(x)|z|α+1|w|β+1dx ≤ 0

(5.25)and

0 ≤ r∗p∫

Ω|∇z|p(x) − ρ∗β+1r∗α+1

∫Ωc(x)|z|α+1|w|β+1dx

0 ≤ ρ∗q∫

Ω|∇w|q(x) − ρ∗β+1r∗α+1

∫Ωc(x)|z|α+1|w|β+1dx.

(5.26)

We can combine the relations (5.25) and (5.26) to obtain estimates(5.13) and (5.14). The proof is complete.

5.5 A conditional critical point of JInto (5.16) and (5.32), we insert the estimations for r∗ determined by(5.13). For any (z, w) ∈ X0(x), we set C(z, w) =

∫Ωc(x)z|α+1|w|β+1dx.

Before continuing, let us denote

γ+ = α + 1p+ + β + 1

q+ and γ− = α + 1p−

+ β + 1q−

.

In this way, we obtain:

(γ+ − 1

)A

(q−(β+1)p)d B

β+1d

C(z, w)q p

d

≤ J(z, w) (5.27)

and

J(z, w) ≤(γ− − 1

)A

(q−(β+1)p)d B

β+1d

C(z, w)q p

d

. (5.28)

19

Before continuing, let us consider the set

E = (u, v) ∈ X; A(z) = 1, B(w) = 1. (5.29)

Consequently, (5.27) and (5.28) become

(γ+ − 1) 1

C(z, w)q p

d

≤ J(z, w) ≤ (γ− − 1) 1

C(z, w)q p

d

.

Let us consider the subsets

E1 = (u, v) ∈ E; C(z, w) > 1,

E1 = (u, v) ∈ E; C(z, w) ≤ 1.

Else, using estimates (5.24), we get

1. ∀(z, w) ∈ E1,

(γ+ − 1) 1

C(z, w)q− p−d1

≤ J(z, w) ≤ (γ− − 1) 1

C(z, w)q+ p+d2

.

2. ∀(z, w) ∈ E1,

(γ+ − 1) 1

C(z, w)q+ p+d2

≤ J(z, w) ≤ (γ− − 1) 1

C(z, w)q− p−d1

.

Let us consider the optimal problem

inf(z,w)∈E; C(z,w)>0

1C(z, w) .

We claim that the infimun value is attainted in E. To assert this, we needthe following lemma:

Lemma 5.1 The infinimum problem

inf(z,w)∈E; c(z,w)>0

1C(z, w) (5.30)

admits a solution.

20

Proof The infimum problem is equivalent to the maximizing problem

sup∫

Ωc(x)|z|α+1|w|β+1dx; (z, w) ∈ E, C(z, w) > 0

. (5.31)

We set

M = sup∫

Ωc(x)|z|α+1|w|β+1dx; (z, w) ∈ E, C(z, w) > 0

.

Firstly, from Remarks 5.1 and 5.2, we observe that M is finite. Indeed,from the end of Remark 5.2, for any (z, w) ∈ E, we get

0 < C(z, w) ≤ ‖c‖∞K‖z‖α+11,p(.)‖w‖

β+11,q(.) ≤ Constant.

We follow the ideas of [6] and we show that there exists (zM , wM) ∈ Esuch that C(z, w) ≤ C(zM , wM) for any (z, w) ∈ E.

Let (zn, wn) be a maximizing sequence of (5.31) (i.e (zn, wn) is suchthat

A(zn) = 1, B(wn) = 1

andC(zn, wn)→M > 0).

It is easy to see that (zn, wn) is bounded in X0(x). It follows thatzn z weakly in W 1.p(x)

0 (Ω) and zn → z strongly in Lp(Ω).Similarly, wn w weakly in W 1.q(x)

0 (Ω) and zn → z strongly in Lq(Ω).Consequently

C(zn, wn)→ C(z, w).

Moreover, since z 7→∫

Ω|∇z|p(x)dx is a semimodular in the sens of Defi-

nition 2.1.1 [11], applying Theorem 2.2.8 (see again [11]), we obtain that%(·) is weakly lower semicontinuous and so on since zn z weakly inW

1.p(x)0 (Ω), we deduce∫

Ω|∇z|p(x) ≤ lim inf

n

∫Ω|∇zn|p(x) = 1

and also ∫Ω|∇w|q(x) ≤ lim inf

n

∫Ω|∇wn|q(x) = 1.

21

Assume now by contradiction that∫

Ω|∇z|p(x) < 1 and

∫Ω|∇w|p(x) < 1.

From Proposition 2.1, we also have ‖z‖1,p(x) < 1 and ‖w‖1,p(x) < 1.Let us set

a = ‖z‖1,p(x) = ‖∇z‖Lp(x) and b = ‖w‖1,q(x) = ‖∇w‖Lq(x) .

Using again Proposition 2.1, it derives %(∣∣∣∣∇(1

az)∣∣∣∣) =

∫Ω

∣∣∣∣∇(1az)∣∣∣∣p(x)

dx =

1 and also %(∣∣∣∣∇(1

bw)∣∣∣∣) =

∫Ω

∣∣∣∣∇(1bw)∣∣∣∣q(x)

dx = 1. Obviously, we seethat (1

az,

1bw)∈ E.

On other hand

C(1azn,

1bwn

)goes to C

(1az,

1bw)

for n→ +∞.

However, we notice that

C(1az,

1bw)

=(1a

)α+1 (1b

)β+1C(z, w) =

(1a

)α+1 (1b

)β+1M.

Since a < 1, b < 1, we get

C(1az,

1bw)> M.

So on, a contradiction occurs.

So, we have either 0 < C(z∗, w∗) ≤ 1, either 1 ≤ C(z∗, w∗).Let us assume for a moment 0 < C(z∗, w∗) ≤ 1 (Similar argumentsremain valid if we assume 1 ≤ C(z∗, w∗)).Consequently inf

(z,w)∈EJ(z, w) exists and denoting C∗ = C(z∗, w∗), we get

(γ+ − 1) 1

C∗q− p−d1

≤ inf(z,w)∈E1

J(z, w) ≤ (γ− − 1) 1C∗ q

+ p+

d2

We are going to show that the infimum of the functional J is attainedon E.

22

5.6 Existence for the minimizing problem inf(z,w)∈E

J(z, w)

We are looking for (z, w) ∈ E satisfying

inf J(z, w) :∫

Ω|∇z|p(x)dx = 1,

∫Ω|∇w|q(x)dx = 1. (5.32)

Lemma 5.2 Let (zn, wn) ∈ E be a minimizing sequence of (5.32) thenthe sequence (un, vn) in the form

un = r(zn, wn)zn and vn = ρ(zn, wn)wnis a Palais-Smale sequence for the functionnal J.That means:

J(un, vn) ≤ m, (5.33a)J ′(un, vn)→ 0, in the sense of the norm ‖.‖X∗0 (x). (5.33b)

The proof of this lemma requires several lemmas and remarks.Lemma 5.3 Let E be the set defined as in (5.29).Assume that the functions p and q are such that

(p, q) ∈(C1B(Ω) ∩ C(Ω)

)2.

Then for any (u, v) ∈ X0(x), there exit t(u) > 0, θ(v) > 0such that(1t(u)u,

1θ(v)v

)∈ E.

Proof For any fixed u in W1,p(x)0 (Ω) \ 0, we define on ]0,+∞[ a

function as follow:

f(u, ·) : t 7−→∫

Ω

(1t

)p(x)|∇u|p(x)dx− 1.

For any t > 1, we have:

(1t

)p+ ∫Ω|∇u|p(x)dx− 1 ≤ f(u, t) ≤

(1t

)p− ∫Ω|∇u|p(x)dx− 1.

Now, taking t < 1, we get(1t

)p− ∫Ω|∇u|p(x)dx− 1 ≤ f(u, t) ≤

(1t

)p+ ∫Ω|∇u|p(x)dx− 1.

It follows from the above inequality that

23

• for t large enough, f(u, t) goes to −1 as t→ +∞,

• for t small enough, f(u, t) goes to +∞ as t→ 0.

By applying the Mean Value Theorem, we conclude that there existstu ∈]0,+∞[ such that ∫

Ω

1tp(x)u

|∇u|p(x)dx = 1.

Similarly, we can prove that, there exists θv > 0 such that:∫Ω

1θq(x)v

|∇v|q(x)dx = 1.

The proof is complete.

Lemma 5.4 Let (u, v) ∈ X0(x) be fixed. The functions u 7−→ t(u) andv 7−→ θ(v) defined as in Lemma 5.3 possess C1-regularity respectivilyfrom Uu,tu to IR and Vv,θv to IR.Here, Uu,tu is a neighbourhood of (u, tu) lying on the open set, U =W

1,p(x)0 (Ω) \ 0×]0,+∞[ and Vv,θv is a neighbourhood of (v, θv) lying

on the open set V = W1,q(x)0 (Ω) \ 0×]0,+∞[.

Proof After a simple computation, it is easily seen that

∂f

∂t(u, t) = −1

t

∫Ωp(x) 1

tp(x) |∇u|p(x)dx

Replace t by tu, we have ∣∣∣∣∣∂f∂t (u, tu)∣∣∣∣∣ > p−

tu> 0.

Hence by the implicit function theorem, there exists Uu,tu , a neigh-bourhood of (u, tu) lying on the open set U = W

1,p(x)0 (Ω) \ 0×]0,+∞[,

and a function of class C1 : u 7−→ t(u) from Uu,tu to IR.Particularly, for all u in W 1,q(x)

0 (Ω), we get

t′(u) · φ = −∂f

∂u(u, tu) · φ∂f

∂t(u, tu)

. (5.34)

24

Since, we have

∂f

∂u(u, tu) · φ =

∫Ωp(x) 1

tp(x)u

|∇u|p(x)−2∇u · ∇φdx,

then more precisely, (5.34) becomes

t′(u) · φ = −

∫Ωp(x) 1

tp(x)u

|∇u|p(x)−2∇u · ∇φdx

1t u

∫Ωp(x) 1

tp(x)u

|∇u|p(x)dx. (5.35)

In the same way, we also have

θ′(v) · ψ = −

∫Ωq(x) 1

θq(x)v

|∇v|q(x)−2∇v · ∇ψdx

1θ v

∫Ωq(x) 1

θq(x)v

|∇v|q(x)dx.. (5.36)

Remark 5.1 Let us introduce the functional J defined on IR×W 1,p(x)0 ×

IR×W 1,q(x)0 as follow:

J(r, u, ρ, v) = J(ru, ρv). (5.37)

Thus, particularly this definition implies, for any (z, w) ∈ X0(x)\(0, 0),r(z, w) and ρ(z, w) by (5.18), (5.19) and (5.20):

J(r(z, w), z, ρ(z, w), w) = J(z, w), (5.38)

the functional J is defined as in section 3.

Moreover, from definition (5.37), remarking (u, v) =(t(u) u

t(u) , θ(v) v

θ(v)

),

we also deduce that the functional J becomes (u, v) ∈ X0(x)

J(u, v) = J

(t(u) u

t(u) , θ(v) v

θ(v)

)= J

(t(u), u

t(u) , θ(v), v

θ(v)

). (5.39)

Proof of Lemma 5.2We inspire us by the work of [3]. For a best understanding, some ofnotation used remain the same than in [3].

25

We setπ(u) = (π1(u), π2(u)) =

(t(u), u

t(u)

)and

τ(v) = (τ1(v), τ2(v)) =(θ(v), v

θ(v)

).

From these notation, (5.39) becomes for any (u, v) ∈ X0(x)

J(u, v) = J (π(u), τ(v)) .

Now, we consider a minimizing sequence (zn, wn) ∈ E, then

m ≤ J(zn, wn) ≤ m+ 1n.

On other hand, after applying the Ekeland variational principle, we have

∣∣∣∣∣∣∣∣∣∣J ′(φn, ψn)

∣∣∣∣∣∣∣∣∣∣X∗0 (x)

≤ 1n‖(φn, ψn)‖X0(x) ∀(φn, ψn) ∈ T(zn,wn) ∈ E,

where T(zn,wn) is the tangent space to E at the point (zn, wn).

J ′(un, vn) · (φ, ψ) = J ′(zn, wn) · (π′2(un) · φ, τ ′2(vn) · ψ)) . (5.40)

Therefore, it follows

|J ′(un, vn) · (φ, ψ)| ≤ 1n||(π′2(un) · φ, τ ′2(vn) · ψ)| |X0(x).

Remenbering that X0(x) is equipped with the cartesian norm

‖ · ‖X0(x) = ‖ · ‖1,p(x) + ‖ · ‖1,q(x),

the following estimate occurs

|J ′(un, vn) · (φ, ψ)| ≤ 1n

(||(π′2(un) · φ| |1,p(x) + ||τ ′2(vn) · ψ)| |1,q(x)

).

(5.41)Setting tn = t(un), by definition of π2, we check that

π′2(un, vn) · φ = φ

tn−un

∫Ωp(x) 1

tp(x)n

|∇un|p(x)−2∇un · ∇φdx

1t n

∫Ωp(x) 1

tp(x)n

|∇un|p(x)dx.

26

Thus,

||π′2(un, vn) · φ| |1,p(x) ≤||φ||1,p(x)

tn+||un||1,p(x)

∣∣∣∣∣∫

Ωp(x) 1

tp(x)n

|∇un|p(x)−2∇un · ∇φdx∣∣∣∣∣

1t n

∫Ωp(x) 1

tp(x)n

|∇un|p(x)dx

≤||φ||1,p(x)

tn+

∣∣∣∣∣∫

Ωp(x) 1

tp(x)n

|∇un|p(x)−2∇un · ∇φdx∣∣∣∣∣∫

Ωp(x) 1

tp(x)n

|∇un|p(x)dx.

Particularly, by appling the Hölder inequality for p(x)-Lebesgue space[25, 26, 17] successively, we find∣∣∣∣∣∫

Ωp(x) |∇un|

p(x)−2

up(x)−2n

∇untn· ∇φtndx

∣∣∣∣∣ ≤ p+∣∣∣∣∣ |∇un|p(x)−1

tp(x)−1n

∣∣∣∣∣L

p(x)p(x)−1 (Ω)

·||φ||1,p(x)

tn

= p+ ·||φ||1,p(x)

tn.

(5.42a)∫Ωp(x) 1

tp(x)n

|∇un|p(x)dx ≥ p−∫

Ω

1tp(x)n

|∇un|p(x)dx ≥ p−. (5.42b)

The above remarks allow us to estimate

||π′2(un, vn) · φ| |1,p(x) ≤(

1 + p+p−

)||φ||1,p(x)

tn.

Finally, from properties on Lp(x)(Ω) and W 1,p(x)(Ω) spaces (see for in-stance [17]),

||π′2(un, vn) · φ| |1,p(x) ≤(

1 + p+

p−

)||φ||1,p(x)

||un||1,p(x).

Since r(zn, wn) and ρ(zn, wn) are given by estimates (5.25) and (5.26),we deduce that

m < J(un, vn) <(α + 1p−

+ β + 1q−

− 1)∫

Ω|∇un|p(x)dx

and alsom <

(α + 1p−

+ β + 1q−

− 1)∫

Ω|∇vn|q(x)dx.

27

Thus, we set pn =p− if ‖un‖1,p(x) ≤ 1,p+ if ‖un‖1,p(x) > 1 and qn =

q− if ‖vn‖1,q(x) ≤ 1,q+ if ‖vn‖1,q(x) > 1.

In vertue to Theorem 1.3 [17], we get more precisely,

m < J(un, vn) <(α + 1p−

+ β + 1q−

− 1)‖un‖pn1,p(x)

and alsom <

(α + 1p−

+ β + 1q−

− 1)‖vn‖qn1,q(x).

||π′2(un) · φ| |1,p(x) ≤(

1 + p+

p−

)·(α + 1p−

+ β + 1q−

− 1)1/pn ||φ||1,p(x)

m1/pn.

Similarly,

||τ ′2(vn) · ψ| |1,q(x) ≤(

1 + q+

q−

)·(α + 1p−

+ β + 1q−

− 1)1/qn ||ψ||1,q(x)

m1/qn.

We conclude thatlim

n→+∞||J ′(un, vn)||X∗0 (x) = 0.

Lemma 5.5 Assume

α + 1p+ + β + 1

q+ − 1 > 0,

then (un, vn) is bounded in X0(x).

Proof of Lemma 5.5

Since rn = r(zn, wn), ρn = ρ(zn, wn), zn and wn satisfy identities (5.18),(5.19) and (5.20), it follows that,∫

Ω|∇un|p(x)dx+

∫Ω|∇vn|q(x)dx−

∫Ωc(x)|un|α+1|vn|β+1dx−

∫Ωc(x)|un|α+1|vn|β+1dx = 0.

(5.43)

28

So, on the other hand, because (zn, wn) is a minimizing sequence forinf

(z,w)∈EJ(z, w), we have

m ≤ (α+1)∫

Ω

1p(x) |∇un|

p(x)dx+(β+1)∫

Ω

1q(x) |∇vn|

q(x)dx−∫

Ωc(x)|un|α+1|vn|β+1dx < m+ 1

n.

(5.44)Combining (5.43) and (5.44), one conclude that

m ≤∫

Ω

(α + 1p(x) − 1

)|∇un|p(x)dx+

∫Ω

(β + 1q(x) − 1

)|∇vn|q(x)dx+

∫Ωc(x)|un|α+1|vn|β+1dx < m+ 1

n.

Recall that∫Ω|∇un|p(x)dx =

∫Ω|∇vn|q(x)dx =

∫Ωc(x)|un|α+1|vn|β+1dx,

We obtain

m ≤∫

Ω

α + 1p(x) |∇un|

p(x)dx+∫

Ω

(β + 1q(x) − 1

)|∇vn|q(x)dx < m+ 1

n.

More precisely, after some easy calculations, we obtain(α + 1p+ + β + 1

q+ − 1)∫

Ω|∇un|p(x)dx ≤

(α + 1p(x) + β + 1

q(x) − 1)∫

Ω|∇vn|p(x)dx < m+1.

Arguing similarly, we find(α + 1p+ + β + 1

q+ − 1)∫

Ω|∇vn|q(x)dx < m+ 1.

It shows that the sequence is bounded in X0(x).Lemma 5.6 The problem (5.32) admits a solution.

Proof . We divide the proof in three steps.• Step 1: Weak convergence of un and vn.

Let (zn, wn) ∈ E a minimizing sequence. It is known from the previ-ous lemmas that lim

n→+∞J(un, vn) = m and lim

n→+∞||J ′(un, vn)||X∗0 (x) =

0 and that (un, vn)is bounded in X0(x).Extracting if necessary to a subsequence, there exists a pair (u∗, v∗)in X0(x) such that

un u∗ in W 1.p(x)0 (Ω),

vn v∗ in W 1.q(x)0 (Ω).

29

• Step 2: Stong convergence of un and vn inW 1,p(x)0 (Ω) (resp. W 1,q(x)

0 (Ω))

To do it, let us establish that un and vn are two Cauchy sequences.

Firstly, easy calculations insure for any m ∈ IN, l ∈ IN[J ′(um, vm)− J ′(ul, vl)

](um − ul, 0)

= (α + 1)∫

Ω(|∇um|p(x)−2∇um − |∇ul|p(x)−2∇ul)(∇um − ul)dx

− (α + 1)∫

Ωc(x)

[|vm|(β+1|um|α−1um − |vl|(β+1|ul|α−1ul

](um − ul)dx.

Thus, after a suitable rearrangement, we get∫Ω

(|∇um|p(x)−2∇um − |∇ul|p(x)−2∇ul)(∇um − ul)dx

= 1α + 1

[J ′(um, vm)− J ′(ul, vl)

](um − ul, 0)dx

+∫

Ωc(x)

[|vm|β+1|um|α−1um − |vl|β+1|ul|α−1ul

](um − ul)dx.

We claim that∫Ωc(x)

[|vm|(β+1|um|α−1um−|vl|(β+1|ul|α−1ul

](um−ul)→ 0, m, l → +∞.

(5.45)Indeed in view of (5.45) and according to Remarks 5.1 and 5.2(notation used remain the same), we observe that∣∣∣∣∣

∫Ωc(x)

[|vm|β+1|um|α−1vm − |vl|β+1|ul|α−1ul

](um − ul)dx

∣∣∣∣∣≤

∫Ωc(x)|vm|β+1|um|α|um − ul|dx+

∫Ωc(x)|vl|β+1|ul|α|um − ul|dx

≤ ‖c‖∞‖vm‖β+1Lq(Ω)‖um‖

αLp(Ω)‖um − ul‖Lp(Ω)

+ ‖c‖∞‖vl‖β+1Lq(Ω)‖ul‖

αLp(Ω)‖um − ul‖Lp(Ω)

≤ C‖um − ul‖Lp(Ω).(5.46)

Before continuing, let us recall a fundamental convergence property.Indeed, it is well known (see [17] for instance) that the imbeddingW

1.p(x)0 (Ω) → Lδ(x)(Ω) (resp W

1.q(x)0 (Ω) → Lγ(x)(Ω)) with δ(x) <

Np(x)N − p(x) (resp. γ(x) < Nq(x)

N − q(x)) is compact.

30

Choose γ(x) = p, it follows that un converges strongly to u∗ inLp(Ω) and so on, un is a Cauchy sequence in sense of Lp(Ω) norm.Consequently, (5.46) occurs.Futhermore, following [25], there exist constants C1, C2, C3, C4 suchthat

〈F (∇um)−F (∇ul), um−ul〉 ≥

if 1 < p(x) < 2,C1‖um − ul‖2

1,p(x),

C2‖um − ul‖2p0,1/p1,11,p(x) ,

if 2 ≥ p(x),C3‖um − ul‖

p0,21,p(x),

C4‖um − ul‖p1,21,p(x),

(5.47)where

1. F (ξ) = |ξ|p(x)−2 ξ, ∀ξ ∈ IRN ,

2. p0,j = infx∈Ωj

p(x), p1,j = supx∈Ωj

p(x), j = 1; 2,

3. Ω1 = x ∈ Ω; 1 < p(x) < 2 and Ω2 = x ∈ Ω; 2 ≥ p(x) .

Then, combining (5.33b), (5.45) and (5.47), we conclude that unconverges strongly to u∗ inW 1,p(x)

0 (Ω). Similar argues allow to provethat the sequence vn converges to v∗ strongly in W 1,q(x)

0 (Ω).

• Step 3: (u∗, v∗) is a solution of (1.1) involving a fibering decompo-sitionWe show u∗ = rz and v∗ = ρw involve a solution of (1.1) via thefibering method. Let us recall that z and w are respectively theweak limit of zn and wn is the weak limit of wn. The sequences rnand ρn are defined as in (5.4). Moreover, using (5.13), extractingif necessary subsequences, we can assume that rn and ρn, convergein IR. Let r and ρ be such that rn → r and ρn → ρ as n tends to+∞.

31

Because the formulation 1rnun−

1ru∗ = 1

rnr[(r − rn)u∗ + r (un − u∗)] ,

and the convergences results announced above, it is clear that∥∥∥∥unrn − u∗

r

∥∥∥∥1,p(x)

→ 0, as n tends to +∞.

In other words, since unrn

= zn, we deduce that zn converges strongly

to u∗

rin W 1,p(x)

0 (Ω).

Thus, since zn converges weakly to z inW 1,p(x)0 (Ω), we deduce from

above and also from uniqueness

z = u∗

r.

On the other hand

‖u∗‖1,p(x) ≤ lim infn‖un‖1,p(x) ≤ lim sup

n‖un‖1,p(x).

So‖z‖1,p(x)r ≤ lim inf

n‖un‖1,p(x) ≤ lim sup

n‖un‖1,p(x)

thus‖z‖1,p(x)r ≤ lim inf

nrn‖zn‖ ≤ lim sup

n‖un‖1,p(x).

Since ‖zn‖1,p(x) = 1 and ‖un‖1,p(x) ≤ ‖un − u∗‖1,p(x) + ‖u∗‖1,p(x),we obtain

‖z‖1,p(x)r ≤ r ≤ ‖z‖1,p(x)r

thus after dividing by r > 0, it occurs ‖z‖1,p(x) = 1.In the same way

‖w‖1,q(x) = 1.

We can conclude that (z, w) is solution of the conditional problem(5.32).

Now, the material needed to prove Theorem 3.2 is complete. In otherwords, we establish that the boundary value problem (1.1) admits atleast a solution.

32

5.7 Proof of Theorem 3.2ProofThe previous lemmas imply that (z, w) is a contitional critical point forJ .From the Euler-Lagrange characterization, we deduce that there exists apair (m1,m2) in IR2 such that for any (h, k) ∈ X0(x),

∇J(z, w) · (h, k) = m1∇A(z, w) · (h, k) +m2∇B(z, w) · (h, k). (5.48)

In (5.48), we choose h = z, k = w, we obtain

J ′(z, w)(z, w) = 0. (5.49)

Combining (5.48) and (5.49), we obtainm1A

(1) · (z, w) +m2B(1) · (z, w) = 0

m1A(2) · (z, w) +m2B

(2) · (z, w) = 0.

Here, A(1), B(1) (resp. A(2) and B(2)) denote the first derivatives respectwith z (resp. w). But

det

A(1) · (z, w) B(1) · (z, w)

A(2) · (z, w) B(2) · (z, w)

> p−q−A(z)B(w) = p−q− > 0.

It follows thatm1 = m2 = 0.

Consequently,J ′(z, w) = 0,

or again,J ′(rz, ρw) = 0

Finaly, we can conclude that (u∗, v∗) = (rz, ρw) is a critical point of J.

6 Low bounds for the solution (u∗, v∗)Theorem 6.1 If (u∗, v∗) ∈ X is a solution of the system (1.1) thennecessarily we shoud have∫

Ω|∇u∗|p(x)dx ≥ e and

∫Ω|∇v∗|q(x)dx ≥ e.

33

For the proof of this theorem we need the following lemma

Lemma 6.1 Let (u∗, v∗) be a solution of the problem (1.1), then thereexist two functions gp and gq satisfies: gp ≤ u∗ and gq ≤ v∗ for all x onthe set Br0 \B r0

2such that

Br0 = x ∈ Ω; |x| ≤ r0.

Proof .We define the function

gp = kp(e−αp|x|

2 − e−αpr20).

Consequently, we have

−div(|∇gp|p(x)−2∂gp

∂xi

)= −2p(x)−1αp(x)−1

p rp(x)−2e−αpr2(p(x)−1)

−[< x,∇p > ln(2αpre−αpr

2 + (p(x)− 1)(2αpr2 − 1)− (n− 1)].

On the set Br0 \B r02, we consider the function

f(r) = (2αpre−αpr2 − 1)

such that r > 1 and αp > 0.Clearly,

∀αp >12 , 2αpre−αpr

2 − 1 < 0.

So,ln(2αpre−αpr

2) < 0.Finally, if we suppose that

〈x,∇p〉 > 0 and αp > sup

12 , 1 + n

p− + 1 ,(−a

2rp−−20

) 1p−−2

,we conclude then

(1)p ∀a < 0;−div(|∇gp|p(x)−2∂gp

∂xi

)− agp < 0.

34

Multiplying (−∆p(x)u = c(x)u|u|α−1|v|β+1) in (1.1) and (1)p by the testfunction

ϕp = (gp − u∗)+

and integrating over the set

B+p = x ∈ Br0/B r0

2ϕp > 0,

we obtain

0 ≤∫B+p

(|∇gp|p(x)−2∇gp − |∇u∗p|p(x)−2∇u∗p)∇ϕpdx

≤ a∫B+p

gp(x)−2p ϕp −

∫B+p

u∗|u∗|(α−1)|v∗|β+1ϕpdx ≤ 0.

Which implies thatmes(B+

p ) = 0.Therefore

gp ≤ u∗ in Br0/B r02.

Using the similar arguments we find

gq ≤ v∗ in Br0/B r02.

Proof of Theorem 6.1.Let (u∗, v∗) be a solution of the system (1.1), then∫

Ω|∇u∗|p(x)dx =

∫Ω|∇v∗|q(x)dx =

∫Ω|u∗|α+1|v∗|β+1dx.

Using the previous lemma then we have∫Br0

|u∗|α+1|v∗|β+1dx ≥∫Br0

|gp|α+1|gq|β+1dx.

Now we setKpq = min(kp, kq)

and

K =∫Br0

[e−αp‖x‖

2 − e−αpr20]α+1[

e−αq‖x‖2 − e−αqr2

0]β+1

dx.

Finally we choose k1 and k2 such that

Kα+β+2pq K ≥ meas(Br0)e.

35

Therefore ∫Br0

|∇u∗|p(x)dx =∫Br0

|∇v∗|q(x)dx ≥ meas(Br0)e.

Thus|∇u∗|p(x) ≥ e a.e in Br0 .

Similary|∇v∗|q(x) ≥ e a.e in Br0 .

References[1] E. Acerbi and G. Mingione, Regularity results for stationary electrhe-

ological fluids, Archive for rational mechanic and analysis, 164 (3)(2002) 213-259.

[2] K. Adriouch, Sur les systèmes elliptiques quasi-linéaires etanisotropiques avec exposants critiques de Sobolev, Thèse de Doc-torat, Université de La Rochelle

[3] K. Adriouch and A. El Hamidi, The Nehari manifold for systems ofnonlinear elliptic equations, Nonlinear Analysis 64 (2006) 2149-2167.

[4] L. Antonio-Ribeiro de Santana, Y. Bozhkov and W. Castro FerreiroJr, Species survival versus eigeinvalues, Abstract and Applied Anal-ysis 2004 (2) (2004) 115-135.

[5] S. N. Antontsev an S. I. Shmarev, A model porous medium equationwith variable exponent of nonlinearity: existence, uniqueness and lo-calization properties of solutions, Nolinear Analysis: theory, Methodsand Applications, 60 (3) (2005) 515-554.

[6] Y. Bozhkova, E. Mitidieri, Existence of multiple solutions for quasi-linear systems via Fibering method, J. Differential Equations 190(2003) 239-267.

[7] K. Brown and T-F. Wu, A fibering map approach to a semilinearelliptic boundary value problem, E.J.D.E 2007 (69) (2007) 1-9.

36

[8] J. Chabrowski and Y. Fu, existence of solutions for p(x)- Laplacianproblem on a bounded domain, Journal of Mathematical Analysisand applications 306 (2) (2005) 604-618.

[9] Y. Chen, S. Levine, and m. Rao, Variable exponent, linear growtyFunctionals in image restoration, SIAM journal on Applied Mathe-matics, 66 (4) (2006) 1383-1406.

[10] L. Diening. (2000) Theoretical and numerical results for eletrorheo-logical fluids, PND. Thesis, university of Frieburg, Germany.

[11] L. Diening, P. Harjulehto, P. Hästö and M. Ružička, (2011),Lebesgue and Sobolev spaces with variable exponents, Lecture Notesin Mathematics, Springer-Verlag Berlin Heidelberg.

[12] G. Dinca. F. Issaia, Generalized Pohozeav and Pucci-Serrin Identi-ties and non-existence results for p(x)-laplacian type equations, Ren-diconti del circolo matimatico di palermo 59 (2010) 1-46.

[13] P. Drabeck and S.I. Pohozaev, Positive solutions for the p-Laplacian:application of the fibering method, Proceedings of the Royal Societyof Edinburgh, 127A (1997) 703-726.

[14] D. Edminds and J. Rakosnik, Sobolev embedding with variable expo-nent, Studia mathematical, 143 (3) (2000) 267-293.

[15] X.L. Fan, J. Shen and D. Zhao, Sobolev Embedding theorems forspaces W k,p(x)(Ω), J.math.Anal.Appl, 262 (2001) 749-760.

[16] X. L.Fan, S. Y. Wang., and D. Zhao, Density of C∞(Ω) in W 1.p(x)

with discontinous exponent p(x), Math. Nachr., 279 (1-2) (2006) 142-149.

[17] X.L. Fan and D. Zhao, On the spaces Lp(x)(Ω) and W 1,p(x)(Ω),J.Math.Anal.Appl 263 (2001) 424-446.

[18] X. Fan, Solutions for p(x)-Laplacian Dirichlet problems with singu-lar coefficients, Journal of Mathematical Analysis and applications,312 (1) (2005)464-477.

[19] M. Galewski, New variation method for p(x)-Laplacian equation,Bulletin of the Australian Mathematical society, 72 (1) (2005) 53-65.

37

[20] T. C. Halsey, Electrorheological fluids, Science 258 (1992) 761-766.

[21] H. Hudzik, On generalized Orlicz-Sobolev space, Funct. Approxima-tio Comment. Math. 4 (1976) 37-51.

[22] H. Hudzik A generalization of Sobolev spaces.I, Funct. Approxima-tio Comment. Math. 2 (1976) 67-73.

[23] Y.S. Il’yasov, The Pokhozhaev identity and the fibering method, Diff.Equations, 38 (10) (2002) 1453-1459.

[24] D.A. Kandilakis and M. Magiropoulos, Existence results for ap-Laplacian problem with competing nonlinear boundary conditions,E.J.D.E, 2011 (95) (2011) 1-6.

[25] Yun-Ho Kim, L. Wang and C. Zhang, Global bifurcation fora class of degenerate elliptic equations with variable exponents,J.Math.Anal.Appl. 371 (2010) 624-637.

[26] O. Kovacik and J. Rakosnik, On the spaces Lp(x)(Ω) and W 1,p(x)(Ω),Czechoslovak Math. Journal 41(4) (1991) 592-618.

[27] M. Minailescu and V. Rodulescu, A mulplicity result for nonlineardegenerate problem arising in the theory of electrorheological fluids,Procedings of the royal society of london A, 426 (2073) (2006) 2625-2641.

[28] J. Musielak, Orliez Spaces and Modular Spaces, Lecture Notes inMathematics 1034, Springer-Verlag, Berlin (1983).

[29] S. I. Pohozaev, On a constructive method in the calculus of varia-tions, Dokl. Akad. NaukSSSR 298 (1988), pp. 1330-1333(in Russian).

[30] S.I.Pohozeav, Eigenfunctions of the equation −∆u + λf(u) = 0,Soviet Math. Dokl. 6 (1965) 1408-1411.

[31] S.I. Pohozeav, Nonlinear variationnal problems Via the FiberingMethod, Steklov mathematical Institute, Russian academy of sciences,Gubkina str.8, (1991) Moscow, Russia.

[32] S. I. Pohozaev, On the global fibrering method in variational prob-lems, Proceedings of the Steklov Institute of Mathematics, 219 (1997)281-328.

38

[33] P. Pucci, J. Serrin, A general variationnal identity, Indiana Univ.Math. 35 (3) (1986) 681-703.

[34] M. Ruzicka, Eletrorheological Fluids, Modeling and mathematicaltheory, vol.1748 of Lecture Notes in Mathemathematics, Springer,Berlin, Germany, 2000

[35] A. Salvatore, Some multiplicity results for a superlinear elliptic prob-lem in RN , Topological Methods in Nonlinear Analysis, 21 (2003)29-39.

[36] A. Salvatore, Multiple solutions for elliptic systems with nonlinear-ities of arbitrary growth J.Diff. Eq., 244 (2008) 2529-2544.

[37] S. Samoko, Convolution type operators in Lp(x)(RN), Integral trans-form. Spec. Funct., 7 (1-2) (1998) 123-144.

[38] F de Thélin and J.Vélin, Existence and non-existence of non-trivialsolutions for quasilinear elliptic systems, Rev. Mat. Univ. Complu-tence Madrid 6 (1993) 153-154.

[39] J.Vélin, On an existence result for a class of (p, q)-gradient ellipticsystems via a fibering method Nonlinear Analysis T.M.A 75 (2012)6009-6033.

[40] J.Vélin, Multiple solutions for a class of (p, q)-gradient elliptic sys-tems via a fibering method To appear in Proceeding A of The RoyalSociety of Edinburgh.

[41] T-F. Wu, Multiple positive solutions of a nonlinear boundaryvalue problem involving a sign-changing weight, Nonlinear Analysis:T.M.A, 74 (12) (2011) 4223-4233.

[42] G. Yang and M. Wang, Existence of multiple positive solutions for ap-Laplacian system with sign-changing weight, Computer and Math-ematics with applications, 55 (4) (2008) 636-653.

[43] V. V. Zhikov, Averaging of functionals of calculus of variationsand elasticity theory, Izvestiya Akademii Nauk SSSR.Seriya Mathe-maticheskaya, 50 (4) (1986) 675-710, (Russian).

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