Document de travail 2013-11
Mai 2013
“Existence and nonexistence of
nontrivial solutions for some
nonlinear elliptic systems involving
the (p(x), q(x))-Laplacian”
Warda SAIFIA & Jean VELIN
Université des Antilles et de la Guyane. Faculté de Droit et d’Economie de la Martinique. Campus de Schoelcher - Martinique FWI
B.P. 7209 - 97275 Schoelcher Cedex - Tél. : 0596. 72.74.00 - Fax. : 0596. 72.74.03
www.ceregmia.eu
Centre d’Etude et de Recherche en Economie,
Gestion, Modélisation et Informatique Appliquée
Existence and Nonexistence of Nontrivial
Solutions 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 nontrivial solutions.
Generalization of the well-known Pohozaev and Pucci-Serrin idendities and some nonexistence results for Dirichlet problem involving the (p(x), q(x))-Laplacian system are obtained.
Key Words: Fibering method, non-existence theorem, p(x)-Laplacian, Generalized
Pohozeav identity, Pucci Serrin identity.
1
Introduction
After the pioneer article of Kovacik and Rokosnik [26] concerning the
Lp(x) (Ω) and W 1,p(x) (Ω) spaces, many researches led in these kinds of
variable exponent spaces. We refer to [15] for the properties of such
spaces and [8, 19] for the applications of variable exponent on partial
differential equations
In the recent years, the theory of problem with p(x)-Laplacian has a large
application 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, University of Antilles-Guyane, Campus de Fouillole, 97159 Pointe-à-Pitre Guadeloupe
(FWI). E-mails: [email protected], 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 of
the 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
(1.1)
on Ω.
Here Ω designates a bounded, regular open set in RN with a smooth
boundary ∂Ω, p, q : Ω −→ R two functions of class C(Ω), p(x), q(x) > 1
for every x ∈ Ω and c a function which may changes sign and for every
real function p ∈ L∞ (Ω), we denote:
p− = inf p(x) and p+ = sup p(x), (1 ≤ p− ≤ p+ < +∞),
Ω̄
Ω̄
∞
L∞
+ (Ω) = {p ∈ L (Ω), p− ≥ 1}.
Concerning in existence and nonexistence results of the type of this system, we cite a study presented in [6]. The authors use the fibering method
introduced by S. Pohozeav. They proved the existence of multiple solutions for a Dirichlet problem associated with a quasilinear system involving a pair of (p, q)-Laplacian operators.
More recently, employing the fibering method, the second author of our
paper has proved the existence of multiple positive solutions for a class
of (p, q)-gradient elliptic systems including systems like (1.1). For more
details, 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 and
nonexistence of a non-trivial solution (u, v) ∈ W01,p (Ω) × W01,q (Ω) of the
following system















−∆p u = u|u|α−1 |v|β+1 in Ω
−∆q v = v|v|β−1 |u|α+1 in Ω
u=v=0
on Ω.
2
(1.2)
When
(α + 1)
N −q
N −p
+ (β + 1)
>1
Np
Nq
(1.3)
and Ω is a strictly starshaped open domain in RN ,
they have proved nonexistence results.
On other side, when
(α + 1)
and
N −p
N −q
+ (β + 1)
<1
Np
Nq
(1.4)
α+1 β+1
+
6= 1, existence results have discussed.
p
q
In [12], the authors obtain nonexistence for Dirichlet problem governed 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 needed
for the framework of the paper. We also recall some tools defined
by 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 establish a Pohozaev-type identity for the system (1.1). By using
this adapted identity, we deal with the non-existence results of non
trivial 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 nontrivial solution.
6. To acheive the study, in connexion with the non-existence, we establish that the solution is bounded below in Ω.
3
2
Notations and Preliminaries
In this section, and throughout the study, we recall some definitions and
properties on the generalized Lebesgue space Lp(x) (Ω) and generalized
Sobolev spaces W 1,p(x) (Ω). Ω ⊂ RN is an open set. For more details, the
reader can consult for instance [11, 17, 18, 19, 21, 22, 26, 27, 28].
The generalized Lebesgue space Lp(x) (Ω) consists in all measurable functions u defined on Ω for which the p(x)-modular
ρp(.) (u) =
Z
|u(x)|p(x) dx
Ω
is finite. The Luxemberg norm on this space is defined as:


kukp = inf λ > 0; ρp(.) (u) =

p(x)
Z u(x) dx
Ω λ 

≤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.
0
For given p ∈ L∞
+ (Ω), we define the conjugate function p (x) as
p0 (x) =
p(x)
.
p(x) − 1
The following results show the close relation between the convex modular
ρp(.) and the norm k.kLp(.) (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 kukLp(.) (Ω) = a ⇔ %
u
=1
a
2. kukLp(.) (Ω) < 1(= 1, > 1) ⇔ %p(.) (u) < 1(= 1, > 1)
−
+
3. If kukLp(.) > 1 then kukpLp(.) ≤ %p(.) (u) ≤ kukpLp(.)
+
4. If kukLp(.) < 1 then kukpLp(.) ≤ %p(.) (u) ≤ kukp−
Lp(.)
4
p(.)
(Ω) and u ∈
Proposition 2.2 [26, 17] Let p ∈ L∞
+ (Ω), (un ) ⊂ L
p(.)
L (Ω).
The following assertions are equivalent:
(i) lim ku − un kLp(.) = 0
n→+∞
(ii) lim %p(.) (u − un ) = 0.
n→+∞
p(.)
(Ω)
Theorem 2.1 (see[17, 15, 28]). Consider p, q, r ∈ L∞
+ (Ω), u ∈ L
q(.)
et v ∈ L (Ω) such that:
1
1
1
+
=
p(x) q(x)
r(x)
then
"
kuvkLr(.) (Ω)
e. a in Ω
#
1
1
≤
+
kukLp(.) (Ω) kvkLq(.) (Ω)
−
(p/r)
(q/r)−
for all u ∈ Lp(.) (Ω), v ∈ Lq(.) (Ω). It is immediate to make this remark
0
Remark 2.1 Let p ∈ L∞
+ (Ω) and let p : Ω → [1, +∞[ be the conjugate
function of p.
There is a constant Cp > 0 such that;
Z
Ω
|uv| ≤ Cp kukLp(.) kvkLp0 (.)
0
for all u ∈ Lp(.) (Ω), v ∈ Lp (.) (Ω).
We also have the following imbedding theorem and we refer the reader
to 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 any
p ∈ L∞
+ (Ω) and m ∈ N∗, we define
W m,p(.) (Ω) = {u ∈ Lp(.) (Ω) : Dα u ∈ Lp(.) (Ω)pour tout|α| ≤ m},
kukm,p(.) =
X
|α|≤m
5
kDα ukLp(.) (Ω)
The pair (W m,p(.) (Ω), k · km,p(.) ) is a separable Banach space (reflexive
if p− > 1) which is called generalized Sobolev space (also known as
1,p(.)
Sobolev space with variable exponent). We will denote by W0 (Ω) the
closure of C0∞ (Ω) in W m,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 continuous 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| ≤ 21 .
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 Ω̄, then C (Ω̄) is dense
in W 1,p(.) (Ω).
Theorem 2.3 Let Ω ⊂ RN be a bounded open set with Lipschitz boundary and let p ∈ C(Ω̄) be a function which satisfies p− > 1.
Define the Sobolev conjugate exponent p∗ : Ω̄ → R̄ of p
∗
p (x) =





N p(x)
N − p(x)
si
p(x) < N




∞
si
p(x) ≥ N.
then the imbedding W m,p(.) (Ω) ,→ Lq(.) (Ω) is continuous and holds for
every function q ∈ C(Ω̄) which satisfies; 1 < q(x) < p∗ (x) for all x ∈ Ω̄.
3
Main results
We define
6
• p, q : Ω −→ IR+ \ {0} as two functions belonging in CB1 (Ω) ∩ C(Ω̄),
• c : Ω −→ IR; c+ (x) 6= 0, c− (x) 6= 0,
• p− = min p(x), q − = min q(x), p+ = max p(x), q + = max q(x).
x∈Ω
x∈Ω
x∈Ω
x∈Ω
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 C 1 . Let
– p, q : Ω −→ IR functions of class CB1 (Ω) ∩ C(Ω), p− , q − > 1,
– c(.) ∈ CB1 (Ω \ C), with meas(C) = 0.
Assume that
– Ω be a bounded domain of class C 1 , starshaped with respect to
the origin,
– (p, q) ∈ CB1 (Ω) ∩ C(Ω̄); p− , q − > 1 and (x.∇p) ≥ 0, (x · ∇q) ≥
0,
–
hx, ∇c(x)i ≤ 0 for any x in Ω.
–
(α + 1)
(3.1)
N − p+
N − q+
+
(β
+
1)
≥ 1.
N p+
N q+
(3.2)
2
Then (1.1) has no nontrivial classical solution (u, v) ∈ C 2 (Ω) ∩ C 1 (Ω̄)
which satisfies:
|∇u(x)| ≥ e1/p(x) and |∇v(x)| ≥ e1/q(x) a.e x ∈ Ω,
and
Z
(3.3)
c(x)|u|α+1 |v|β+1 dx > 0.
Ω
• 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 C 1 . Let p, q : Ω → IR functions of class CB1 (Ω) ∩ C(Ω);
p− , q− > 1. Assume that:
p+ ≤ α + 1
(α + 1)
or
q + ≤ β + 1,
N − p−
N − q−
+
(β
+
1)
< 1.
N p−
N q−
(3.4)
1,p(x)
Then (1.1) admits at least a nontrivial solution (u∗ , v ∗ ) ∈ W0
1,q(x)
W0
(Ω). Moreover, one have
(Ω)×
1.
|∇u∗ (x)| ≥ e1/p(x) and |∇v ∗ (x)| ≥ e1/q(x) a.e x ∈ Ω. (3.5)
2.
Z
c(x)|u∗ |α+1 |v ∗ |β+1 dx > 0.
Ω
Remark 3.1 Let us remark that conditions (3.2) and (3.4) seem to
generalize to (p(x), q(x))− gradient elliptic systems conditions (1.3) and
(1.4) well known when (p, q)− gradient elliptic systems are considered.
N − p−
+
Obviously, conditions (3.2) and (3.4) imply respectively 1 ≤ (α+1)
N p−
N − p+
N − q+
N − q−
and
(α
+
1)
+
(β
+
1)
< 1.
(β + 1)
N q−
N p+
N q+
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+1 v|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.
!
∂u
∂
|∇u|p(x)−2
.
• ∆p(x) u =
∂xi
∂xi
Proposition 4.1 Let Ω be a bounded open set of RN , with boundary ∂Ω
of class C 1 . Assume that
• p, q : Ω −→ IR of class CB1 (Ω) ∩ C(Ω), p− , q − > 1,
• c(.) ∈ CB1 (Ω \ C), with meas(C) = 0 and
hx, ∇c(x)i ≤ 0 for any x in Ω.
Then, for every classical solution (u, v) ∈ C 2 (Ω) ∩ C 1 (Ω) (1.1), the following identity holds:
α + 1 Z 1 − p(x)
β + 1 Z 1 − q(x)
p(x)
|∇u| hx, νidσ +
|∇v|q(x) hx, νidσ
N
p(x)
N
q(x)
∂Ω
∂Ω
!
!
α + 1 Z N − p(x)
β + 1 Z N − q(x)
− a1 |∇u|p(x) dx +
− a2 |∇v|q(x) dx
=
N
p(x)
N
q(x)
Ω
Ω
+
+
β + 1 Z hx, ∇qi
α + 1 Z hx, ∇pi
p(x)
p(x)
(ln
|∇u|
−
1)|∇u|
dx
+
(ln |∇v|q(x) − 1)|∇v|q(x) dx
N
N
Ω p2 (x)
Ω q 2 (x)
Z (
Ω
)
(α + 1)a1 + (β + 1)a2 − N c(x)|u|α+1 |v|β+1 dx −
Z
hx, ∇ci|u|α+1 |v|β+1 dx.
Ω
for all a1 and a2 ∈ RN .
In order to prove Proposition 4.1, we need the following result, which
generalizes the variational identity of Pucci-Serrin[33] .
Proposition 4.2 Let Ω be a bounded open set of RN with boundary ∂Ω
of class C 1 . Assume that
• p, q : Ω −→ IR of class CB1 (Ω) ∩ C(Ω), p− , q − > 1,
• c(.) ∈ CB1 (Ω \ C), C ⊂ Ω with meas(C) = 0.
9
2
Then, for every classical solution (u, v) ∈ C 2 (Ω) ∩ C 1 (Ω)
lem (1.1), the following identity holds:

of the prob-
!
∂ 
α+1
β+1
xi
|∇u|p(x) +
|∇v|q(x) − c(x)|u|α+1 |v|β+1 −
∂xi
p(x)
q(x)
!
!

∂u
∂v
∂v 
∂u
+ a1 u |∇u|p(x)−2
− (β + 1) xj
+ a2 v |∇v|q(x)−2
(α + 1) xj
∂xj
∂xi
∂xj
∂xi
"
#
"
#
N − p(x)
N − q(x)
= (α + 1)
− a1 |∇u|p(x) + (β + 1)
− a2 |∇v|q(x)
p(x)
q(x)
+
hx, ∇pi
hx, ∇qi
(ln |∇u|p(x) − 1)|∇u|p(x) + 2
(ln |∇v|q(x) − 1)|∇v|q(x)
2
p (x)
q (x)
+ {(α + 1)a1 + (β + 1)a2 − N } c(x)|u|α+1 |v|β+1 − hx, ∇ci|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 computation.
Proof of Proposition 4.1.
Throughout the proof, for x = (xi )i=1;··· ,N , y = (yi )i=1;··· ,N two vectors in
IRN , the classical inner product is simply denoted xi yi and the notation
N
X
2
is omitted. Let (u, v) ∈ CB2 ∩ C 1 (Ω̄)
be a classical solution of the
i=1
problem (1.1).
According to Proposition 4.2, (u, v) satisfies the identity (4.1). Integrating by part over Ω, we get:
10
"
Z
∂Ω
α+1
β+1
|∇u|p(x) +
|∇v|q(x) − c(x)|u|α+1 |v|β+1
p(x)
q(x)
!
!
#
∂u
∂u
∂v
∂v
−(α + 1) xj
+ a1 u |∇u|p(x)−2
− (β + 1) xj
+ a2 v |∇v|q(x)−2
νi dσ
∂xj
∂xi
∂xj
∂xi
= (α + 1)
Z
Ω
Z "
Ω
+
!
!
Z
N − p(x)
N − q(x)
− a1 |∇u|p(x) dx + (β + 1)
− a2 |∇v|q(x) dx
p(x)
q(x)
Ω
#
1
1
p(x)
p(x)
hx.∇pi(ln
|∇u|
−
1)|∇u|
+
hx, ∇qi(ln |∇v|q(x) − 1)|∇v|q(x) dx
p2 (x)
q 2 (x)
Z (
Ω
)
α+1
(α + 1)a1 + (β + 1)a2 − N c(x)|u|
β+1
|v|
dx −
Z
hx, ∇ci|u|α+1 |v|β+1 dx.
Ω
(4.2)
ν is the unit outer normal to the boundary ∂Ω, since u = 0 on ∂Ω, it
follows clearly that
∂u
= (∇u.ν)νi , ∀i = 1, · · · , N, x on ∂Ω,
∂xi
then we can write:
xj
∂u
∂u ∂u
|∇u|p(x)−2 νi = xj [(∇u.ν)νj ]
|∇u|p(x)−2 νi
∂xj ∂xi
∂xi
=
∂u ∂u
|∇u|p(x)−2 (x.ν)
∂xi ∂xi
= |∇u|p(x) (x.ν) on ∂Ω
Using the relation (4.2) and the fact that u|∂Ω = 0 in the left hand side
of this relation, the statement of Proposition 4.1 occurs.
Remark 4.1 Before proving Proposition 4.2, let us note that the set of
functions c satisfying to hypothesis (3.1), is non-empty. Indeed, let x0
be 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 define
R0
the set Ω1 as follow Ω1 = x ∈ Ω; 0 ≤ kxk ≤
. For instance, we
2
11
define the function c as follow:
(
c(x) =
2
−ekxk
2
e−kxk
if
if
x ∈ Ω1
x ∈ Ω \ Ω1 .
c changes sign into Ω and we also have for any x ∈ Ω, hx, ∇c(x)i ≤ 0.
Moreover, c ∈ L∞ (Ω).
Proof of Theorem 3.1.
Suppose that there exists a nontrivial classical solution (u, v) ∈ C 2 (Ω) ∩
C 1 (Ω̄) of the problem (1.1). So that, (u, v) satisfies the statement of
Proposition 4.1.
Since Ω ⊂ RN is strictly starshaped with respect to the origin, we have
x.ν > 0 on ∂Ω thus:
β+1Z
1
1
α+1Z
p(x)
|∇u|
< x, ν > dσ−
|∇v|q(x) < x, ν > dσ < 0
−
N
N
∂Ω p̃(x)
∂Ω q̃(x)
where
1
p(x) − 1 1
q(x) − 1
=
,
=
.
p̃(x)
p(x)
q̃(x)
q(x)
In other hand, choosing a1 ∈ IR and a2 ∈ IR such that
a1
a2
(α + 1) + (β + 1) = 1
N
N
and using the relations (3.2), (3.3), we get
α + 1 Z 1 − p(x)
β + 1 Z 1 − q(x)
p(x)
|∇u| hx, νidσ +
|∇v|q(x) hx, νidσ
N
p(x)
N
q(x
∂Ω
∂Ω
!
!
α + 1 Z N − p(x)
β + 1 Z N − q(x)
p(x)
=
− a1 |∇u| dx +
− a2 |∇v|q(x) dx
N
p(x)
N
q(x)
Ω
Ω
(
)
+
Z
Ω
(α + 1)a1 + (β + 1)a2 − N c(x)|u|α+1 |v|β+1 dx −
Z
hx, ∇ci|u|α+1 |v|β+1 dx
Ω
N − p+ Z
N − q+ Z
a1 Z
p(x)
q(x)
≥ (α + 1)
|∇u| dx + (β + 1)
|∇v| dx − (α + 1)
|∇u|p(x) dx−
+
N p+ Ω
N
q
N
Ω
Ω
)
Z (
a2 Z
q(x)
(β + 1)
|∇v| dx +
(α + 1)a1 + (β + 1)a2 − N c(x)|u|α+1 |v|β+1 dx
N
Ω
Ω
Z
− hx, ∇ci|u|α+1 |v|β+1 dx
Ω
(
)Z
Z
N − p+
N − q+
α+1
β+1
≥ (α + 1)
+ (β + 1)
−1
c(x)|u| |v| dx − hx, ∇ci|u|α+1 |v|β+1 dx.
+
Np
N q+
Ω
Ω
12
Now, because (u, v) is a solution, let us notice that
c(x)|u|α+1 |v|β+1 dx =
Ω
Z
Z
Z
|∇u|p(x) dx =
|∇v|q(x) dx > 0. Moreover, from hypothesis (3.1), we
Ω
get that the right hand is positive. So on, a contradiction occurs and the
proof is complete.
Ω
5
Existence Results via the Fibering Method
Throughout this section, Ω denotes a bounded regular open set in RN
1,p(x)
1,q(x)
and X0 (x) = W0
(Ω) × W0
(Ω). The generalized Sobolev spaces
1,p(x)
1,q(x)
W0
(Ω) and W0
(Ω) are equipped with the Luxembourg norm kukW 1,p(x) (Ω)
0
and kukW 1,q(x) (Ω) respectively. For a best reading, we denote as kukW 1,p(x) (Ω) =
0
0
ku|k1,p(x) and kukW 1,q(x) (Ω) = kuk1,q(x) .
0
Before starting this section, we make a fundamental remark
N − p−
N − q−
+ (β + 1)
≤ 1, we can
N p−
N q−
Z
also establish that c(x)|z|α+1 |w|β+1 dx possesses a sens. The functional
Remark 5.1 Assuming (α + 1)
Ω
c belongs in L∞ (Ω), it suffices to verify that |z|α+1 |w|β+1 belongs in L1 (Ω).
α+1 β+1
α+1 β+1
Indeed, since we have
+ + > 1 and also
+ − > 1.
+
p
q
p−
q
So, there exists a pair (p̂, q̂) such that
1.
p− < p̂ <
N p−
N − p−
(5.1)
q − < q̂ <
N q−
N − q−
(5.2)
and
2.
α+1 β+1
+
= 1.
p̂
q̂
N p−
N p(x)
N q−
N q(x)
<
and
<
,
N − p−
N − p(x)
N − q−
N − q(x)
N − p−
N − q−
the assumption (α+1)
+(β+1)
≤ 1 implies for any x ∈ Ω,
N p−
N q−
(5.1) and (5.2) become
Remark 5.2 So that, since
p− < p̂ <
N p(x)
N − p(x)
13
and
N q(x)
.
N − q(x)
q− < q̂ <
1,p(x)
From compactness results, we can conclude that the imbedding W0
(Ω) ,→
1,q(x)
p̂
q̂
L (Ω) and W0
(Ω) ,→ L (Ω) are continuous and compact. So on, employing the Hölder inequality, we get the following estimation
Z
α+1
β+1
c(x)|u|
|v|
dx
Ω
5.1
β+1
β+1
α+1
≤ kckL∞ (Ω) kukα+1
Lp̂ (Ω) kvkLq̂ (Ω) ≤ Cstkuk1,p(x) kvk1,q(x) .
Definition of a weak solution for (1.1)
Let us consider again the system (1.1). First, we recall the definition of
the weak solution.
Definition 5.1 We say that (u, v) ∈ X0 (x) is a weak solution of (1.1) if
Z
|∇u|p(x)−2 ∇u∇φ dx =
c(x)u|u|α−1 |v|β+1 u φ dx
Ω
Ω
Z
Z
|∇v|q(x)−2 ∇v∇ ψ dx =
Z
c(x)|u|α+1 v|v|β−1 u ψ dx
Ω
Ω
for any (φ, ψ) ∈ X0 (x).
5.2
Fibering Method for quasilinear systems
Fibering method has been introduced by S. Pohozaev in [29] (see also
[31, 32]). One can consult various applications of this method (see for
instance [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)
Z
Ω
−
Z
Z
1
1
p(x)
|∇u| dx + (β + 1)
|∇v|q(x) dx
p(x)
Ω q(x)
c(x)|u|α+1 |v|β+1 dx.
Ω
Clearly, critical points of the functional J are weak solutions of the problem (1.1). The fibering method applied to this problem consists in the
following scheme.
14
We express (u, v) ∈ X in the form
u = rz
v = ρw
1,p(x)
where the functions z and w belong respectively in W0
(Ω) \ {0} and
1,q(x)
W0
(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
Existence of a fibering parameter (r∗ (z, w), ρ∗ (z, w))
5.3
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 the
following Proposition:
Proposition 5.1 Let (z, w) be fixed in X0 (x) such that
0. Assume
q + ≤ β + 1 or p+ ≤ α + 1.
Z
c(x)|z|α+1 |w|β+1 dx >
Ω
(5.3)
Then there exists a pair (r∗ , ρ∗ ) ∈ R∗+ × R∗+ depending on (z, w) such
that
 Z


r∗p(x) |∇z|p(x) dx


 Ω
Z





ρ
∗q(x)
q(x)
|∇w|
= ρ∗β+1 r∗α+1
dx = ρ
Z
c(x)|z|α+1 |w|β+1 dx,
Ω
(5.4)
∗ β+1 ∗α+1
r
Ω
Z
α+1
c(x)|z|
β+1
|w|
dx.
Ω
Proof
Assume that (u, v) ∈ X0 (x) is a critical point of J, then any fibering
parameter (r, ρ) is characterized as follow
∂J
∂J
(rz, ρw) = 0 and
(rz, ρw) = 0.
∂r
∂ρ
(5.5)
That means






Z
|r|p(x)−2 r|∇z|p(x) dx = |ρ|β+1 |r|α−1 r
Ω
Z





Z
c(x)|z|α+1 |w|β+1 dx,
Ω
|ρ|q(x)−2 |ρ||∇w|q(x) dx = |ρ|
β−1
Ω
|ρ||r|α+1
Z
c(x)|z|α+1 |w|β+1 dx.
Ω
(5.6)
15
It follows
Z
Z
|r|p(x) |∇z|p(x) dx =
Ω
|ρ|q(x) |∇w|q(x) dx.
(5.7)
Ω
We fix ρ in (5.6). Else, we derive
Z
q(x)−(β+1)
|ρ|
ΩZ
|r| = 

q(x)
|∇w|

α+1
c(x)|z|
β+1
|w|

dx 
1
α+1


dx
.
Ω
Let us consider the functional f˜ defined as follow
Z
f˜ : ρ 7−→
Z
Ω
q(x)−(β+1)
 Ω |ρ|
 Z

|∇w|
α+1
c(x)|z|
q(x)
β+1
|w|
 p(t)
dx 
α+1
|∇z|p(t) dt−


dx
Z
|ρ|q(t) |∇w|q(t) dt.
Ω
Ω
Z


Z

We put a(z, w) =
q(x)
|∇w|
 p(t)
α+1
dx
Ω
α+1
c(x)|z|
guish the following situations:
β+1
|w|
dx



|∇z|p(t) . Let us distin-
Ω
• ρ > 1,
So that,

Z
Ω
p(x) q − −(β+1)
[
ρ
]
α+1


q(x)
a(z, w) − ρ
q(x) 
|∇w|
dx ≤ f˜(ρ) ≤
Z
Ω
[
]
p(x) q + −(β+1)
ρ
α+1
a(z, w) − ρq(x) |∇
(5.8)
• 0 < ρ < 1. In this way, we obtain

Z
Ω
ρ
[
]

p(x) q + −(β+1)
α+1
q(x)
a(z, w) − ρ
q(x) 
|∇w|

dx ≤ f˜(ρ) ≤
Z
Ω
ρ
p(x) q − −(β+1)
[
]
α+1
(5.9)
Assuming (5.3), it follows that
p(x) [q + − (β + 1)]
< q(x),
α+1
∀x ∈ Ω.
Consequently, from (5.8) and (5.9), we observe that
lim f˜(ρ) = −∞ and lim f˜(ρ) > 0.
ρ→+∞
ρ→0
16
(5.10)
a(z, w) − ρq(x) |∇
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+1 β+1
+ + −1>0
p+
q
(5.11)
α+1 β+1
+ − − 1 > 0.
p−
q
(5.12)
and consequently
2. When q(x) and p(x) are constant, (5.11) and (5.12) are reduced to
the well-know condition
1<
5.4
α+1 β+1
+
.
p
q
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∗ ) defined
as in (5.4) is such that
A(z)
q̄−(β+1)
d¯
B(w)
β+1
d¯
q̄
C(z, w) d¯
A(z)
α+1
d¯
B(w)
p̄−(α+1)
d¯
p̄
A(z) q̃−(β+1)
B(w)
d˜
≤r ≤
(
C z, w) dq̃˜
∗
∗
≤ρ ≤
A(z)
α+1
d˜
B(w)
p̃−(α+1)
d˜
p̃
C(z, w) d¯
β+1
d˜
(5.13)
(5.14)
,
C(z, w) d˜
where
A(z) =
Z
p(x)
|∇z|
B(w) =
dx,
Ω
Z
q(x)
|∇w|
dx,
Ω
C(z, w) =
Z
c(x)|z|α+1 |w|β+1 dx.
Ω
(5.15)
Proof Let us introduce the following function defined on X0 (x):
˜ w) = (α + 1)
J(z,
Z
Ω
−r
Z
1 ∗q(x)
1 ∗p(x)
p(x)
r
|∇z| dx + (β + 1)
ρ
|∇w|q(x) dx
p(x)
Ω q(x)
∗α+1 ∗β+1
ρ
Z
c(x)|z|α+1 |w|β+1 dx.
Ω
17
So that, employing (5.4) and (5.7), we get
Z
α+1 β+1
˜ w)
+ + −1
r∗p(x) |∇z|p(x) dx ≤ J(z,
+
p
q
Ω
(5.16)
Z
α+1 β+1
+ − −1
r∗p(x) |∇z|p(x) dx.
−
p
q
Ω
(5.17)
and
˜ w) ≤
J(z,
We consider the functions:
G : (z, w, r, ρ) 7−→
Z
r
p(x)
p(x)
|∇z|
dx − ρ
β+1 α+1
r
Z
Ω
c(x)|z|α+1 |w|β+1 dx
Ω
(5.18)
and
G̃ : (z, w, r, ρ) 7−→
Z
ρ
q(x)
p(x)
|∇w|
dx − ρ
β+1 α+1
r
Ω
Z
c(x)|z|α+1 |w|β+1 dx.
Ω
(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, G̃(z, w, r∗ , ρ∗ ) = 0.
(5.20)
Let us put
1.
(
pe =
(
p+ if 0 < r < 1
p− if 1 ≤ r,
pb =
p− if 0 < r < 1
p+ if 1 ≤ r.
(5.21)
2.
b + 1) − pbq,
b
db = p̄(β + 1) + q(α
e + 1) + q(α
e + 1) − pq̃.
e
de = p(β
Consequently, for any r > 0, the estimates follow
1.
b pe ≤ p+ ,
p− ≤ p,
b qe ≤ q + ,
q − ≤ q,
(5.22)
2.
b de ≤ d
d2 ≤ d,
1
(5.23)
where
d1 = p+ (β+1)+q + (α+1)−p− q − , d2 = p− (β+1)+q − (α+1)−p+ q + ,
18
3.
qbpe qepb q + p+
q − p−
≤ b, e ≤
.
d1
d2
d
d
(5.24)
So that, from (5.20) and (5.21), we deduce:






r





ρ∗eq
∗p
e
Z
p(x)
|∇z|
∗β+1 ∗α+1
dx − ρ
r
Z
Ω
Z
c(x)|z|α+1 |w|β+1 dx
≤ 0
Ω
Z
|∇w|q(x) dx − ρ∗β+1 r∗α+1
Ω
c(x)|z|α+1 |w|β+1 dx ≤ 0
Ω
(5.25)
and






0 ≤ r∗pb





∗b
q
Z
|∇z|p(x) − ρ∗β+1 r∗α+1
Z
Ω
c(x)|z|α+1 |w|β+1 dx
Ω
(5.26)
0 ≤ ρ
Z
q(x)
|∇w|
−ρ
∗β+1 ∗α+1
r
Z
Ω
α+1
c(x)|z|
β+1
|w|
dx.
Ω
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 J˜
Into (5.16) and (5.32), we insert the estimations Zfor r∗ determined by
c(x)z|α+1 |w|β+1 dx.
(5.13). For any (z, w) ∈ X0 (x), we set C(z, w) =
Ω
Before continuing, let us denote
γ+ =
α+1 β+1
α+1 β+1
+ + and γ − = − + − .
+
p
q
p
q
In this way, we obtain:
(q −(β+1)p)
γ+ − 1
A
b
d
b
b
β+1
B db
b
qe
p
C(z, w) db
˜ w)
≤ J(z,
(5.27)
and
(q −(β+1)p)
˜ w) ≤ γ − − 1
J(z,
19
A
b
d
b
b
β+1
B db
e
qb
p
C(z, w) de
.
(5.28)
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
b
qe
p
C(z, w) db
˜ w) ≤ (γ − − 1)
≤ J(z,
1
e
q p̄
C(z, w) de
.
Let us consider the subsets
E 1 = {(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
˜ w) ≤ (γ − − 1)
≤ J(z,
1
C(z, w)
q + p+
d2
.
2. ∀(z, w) ∈ E 1 ,
(γ + − 1)
1
C(z, w)
q + p+
d2
˜ w) ≤ (γ − − 1)
≤ J(z,
1
C(z, w)
q − p−
d1
.
Let us consider the optimal problem
1
.
(z,w)∈E; C(z,w)>0 C(z, w)
inf
We claim that the infimun value is attainted in E. To assert this, we need
the following lemma:
Lemma 5.1 The infinimum problem
1
(z,w)∈E; c(z,w)>0 C(z, w)
inf
admits a solution.
20
(5.30)
Proof The infimum problem is equivalent to the maximizing problem
Z
sup
α+1
c(x)|z|
β+1
|w|
dx; (z, w) ∈ E, C(z, w) > 0 .
(5.31)
Ω
We set
Z
α+1
M = sup
c(x)|z|
β+1
|w|
dx; (z, w) ∈ E, C(z, w) > 0 .
Ω
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
β+1
0 < C(z, w) ≤ kck∞ Kkzkα+1
1,p(.) kwk1,q(.) ≤ Constant.
We follow the ideas of [6] and we show that there exists (zM , wM ) ∈ E
such 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 such
that
A(zn ) = 1, B(wn ) = 1
and
C(zn , wn ) → M > 0).
It is easy to see that (zn , wn ) is bounded in X0 (x). It follows that
1.p(x)
zn * z̄ weakly in W0
(Ω) and zn → z̄ strongly in Lp̂ (Ω).
1.q(x)
Similarly, wn * w̄ weakly in W0
(Ω) and zn → z̄ strongly in Lq̂ (Ω).
Consequently
C(zn , wn ) → C(z̄, w̄).
Moreover, since z 7→
Z
|∇z|p(x) dx is a semimodular in the sens of Defi-
Ω
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 * z̄ weakly in
1.p(x)
W0
(Ω), we deduce
Z
|∇z̄|p(x) ≤ limninf
Z
|∇w̄|q(x) ≤ limninf
Z
Ω
and also
Z
Ω
21
Ω
Ω
|∇z¯n |p(x) = 1
|∇w¯n |q(x) = 1.
Assume now by contradiction that
Z
p(x)
|∇z̄|
< 1 and
Ω
From Proposition 2.1, we also have kz̄k1,p(x) < 1 and
Let us set
Z
|∇w̄|p(x) < 1.
Ω
kw̄k1,p(x)
< 1.
a = kz̄k1,p(x) = k∇z̄kLp(x) and b = kw̄k1,q(x) = k∇w̄kLq(x) .
1
Z p(x)
1 1 dx
Using again Proposition 2.1, it derives % ∇ z̄ = ∇ z̄ a
a
Ω
q(x)
Z 1 1 ∇
and also % ∇ w̄ =
w̄ dx = 1. Obviously, we see
b
that
b
Ω
1 1
z̄, w̄ ∈ E.
a b
On other hand
C
1
1
zn , wn
a
b
goes to C
1 1
z̄, w̄
a b
for n → +∞.
However, we notice that
C
α+1 β+1
1 1
1
z̄, w̄ =
a b
a
1
b
α+1 β+1
C(z̄, w̄) =
1
a
1
b
M.
Since a < 1, b < 1, we get
1 1
z̄, w̄ > M.
C
a b
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 arguments
remain valid if we assume 1 ≤ C(z ∗ , w∗ )).
˜ w) exists and denoting C ∗ = C(z ∗ , w∗ ), we get
Consequently inf J(z,
(z,w)∈E
1
(γ + − 1)
C
q − p−
∗ d1
≤
inf
(z,w)∈E1
˜ w) ≤ (γ − − 1)
J(z,
1
+ +
C ∗ q d2p
We are going to show that the infimum of the functional J˜ is attained
on E.
22
=
5.6
˜ w)
Existence for the minimizing problem inf J(z,
(z,w)∈E
We are looking for (z, w) ∈ E satisfying
˜ w) :
inf J(z,
Z
|∇z|p(x) dx = 1,
Z
Ω
|∇w|q(x) dx = 1.
(5.32)
Ω
Lemma 5.2 Let (zn , wn ) ∈ E be a minimizing sequence of (5.32) then
the sequence (un , vn ) in the form
un = r(zn , wn )zn and vn = ρ(zn , wn )wn
is a Palais-Smale sequence for the functionnal J.
That means:
J(un , vn ) ≤ m,
J (un , vn ) → 0, in the sense of the norm k.kX0∗ (x).
0
(5.33a)
(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) ∈
2
CB1 (Ω) ∩ C(Ω̄) .
Then for any (u, v) ∈ X0 (x), there exit t(u) > 0, θ(v) > 0such that
!
1
1
u,
v ∈ E.
t(u) θ(v)
1,p(x)
Proof For any fixed u in W0
function as follow:
f (u, ·) : t 7−→
(Ω) \ {0}, we define on ]0, +∞[ a
Z p(x)
Ω
1
t
|∇u|p(x) dx − 1.
For any t > 1, we have:
p− Z
p+ Z
1
1
|∇u|p(x) dx − 1 ≤ f (u, t) ≤
t
t
Ω
Now, taking t < 1, we get
p− Z
1
p(x)
|∇u| dx − 1 ≤ f (u, t) ≤
t
Ω
It follows from the above inequality that
23
Ω
p+ Z
1
t
|∇u|p(x) dx − 1.
Ω
|∇u|p(x) dx − 1.
• 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 exists
tu ∈]0, +∞[ such that
Z
1
p(x)
Ω tu
|∇u|p(x) dx = 1.
Similarly, we can prove that, there exists θv > 0 such that:
Z
1
q(x)
Ω θv
|∇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) and
v 7−→ θ(v) defined as in Lemma 5.3 possess C 1 -regularity respectivily
from Uu,tu to IR and Vv,θv to IR.
Here, Uu,tu is a neighbourhood of (u, tu ) lying on the open set, U =
1,p(x)
W0
(Ω) \ {0}×]0, +∞[ and Vv,θv is a neighbourhood of (v, θv ) lying
1,q(x)
on the open set V = W0
(Ω) \ {0}×]0, +∞[.
Proof After a simple computation, it is easily seen that
∂f
1Z
1
(u, t) = −
p(x) p(x) |∇u|p(x) dx
∂t
t Ω
t
Replace t by tu , we have
∂f
(u, tu )
∂t
>
p−
> 0.
tu
Hence by the implicit function theorem, there exists Uu,tu , a neigh1,p(x)
bourhood of (u, tu ) lying on the open set U = W0
(Ω) \ {0}×]0, +∞[,
1
and a function of class C : u 7−→ t(u) from Uu,tu to IR.
1,q(x)
Particularly, for all u in W0
(Ω), we get
∂f
(u, tu ) · φ
t0 (u) · φ = − ∂u
.
∂f
(u, tu )
∂t
24
(5.34)
Since, we have
Z
∂f
1
(u, tu ) · φ = p(x) p(x) |∇u|p(x)−2 ∇u · ∇φdx,
∂u
Ω
tu
then more precisely, (5.34) becomes
Z
t0 (u) · φ = −
p(x)
Ω
1
p(x)
tu
|∇u|p(x)−2 ∇u · ∇φdx
1 Z
1
p(x) p(x) |∇u|p(x) dx
tu Ω
tu
(5.35)
.
In the same way, we also have
Z
θ0 (v) · ψ = −
q(x)
Ω
1
q(x)
θv
|∇v|q(x)−2 ∇v · ∇ψdx
1
1 Z
q(x) q(x) |∇v|q(x) dx.
θv Ω
θv
(5.36)
.
1,p(x)
×
Remark 5.1 Let us introduce the functional J¯ defined on IR×W0
1,q(x)
IR × W0
as follow:
¯ u, ρ, v) = J(ru, ρv).
J(r,
(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):
¯
˜ w),
J(r(z,
w), z, ρ(z, w), w) = J(z,
(5.38)
the functional J˜ is defined as in section 3.
!
u
v
Moreover, from definition (5.37), remarking (u, v) = t(u)
, θ(v)
,
t(u)
θ(v)
we also deduce that the functional J becomes (u, v) ∈ X0 (x)
u
v
J(u, v) = J t(u)
, θ(v)
t(u)
θ(v)
!
!
u
v
= J¯ t(u),
, θ(v),
. (5.39)
t(u)
θ(v)
Proof of Lemma 5.2
We inspire us by the work of [3]. For a best understanding, some of
notation used remain the same than in [3].
25
We set
u
π(u) = (π1 (u), π2 (u)) = t(u),
t(u)
and
!
!
v
τ (v) = (τ1 (v), τ2 (v)) = θ(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
˜ n , wn ) ≤ m + 1 .
m ≤ J(z
n
On other hand, after applying the Ekeland variational principle, we have
˜0
J (φn , ψn )
≤
X0∗ (x)
1
k(φn , ψn )kX0 (x) ∀(φn , ψn ) ∈ T(zn ,wn ) ∈ E,
n
where T(zn ,wn ) is the tangent space to E at the point (zn , wn ).
J 0 (un , vn ) · (φ, ψ) = J˜0 (zn , wn ) · (π20 (un ) · φ, τ20 (vn ) · ψ)) .
(5.40)
Therefore, it follows
|J 0 (un , vn ) · (φ, ψ)| ≤
1
||(π20 (un ) · φ, τ20 (vn ) · ψ)| |X0 (x) .
n
Remenbering that X0 (x) is equipped with the cartesian norm
k · kX0 (x) = k · k1,p(x) + k · k1,q(x) ,
the following estimate occurs
1
||(π20 (un ) · φ| |1,p(x) + ||τ20 (vn ) · ψ)| |1,q(x) .
n
(5.41)
Setting t̃n = t(un ), by definition of π2 , we check that
|J 0 (un , vn ) · (φ, ψ)| ≤
φ
π20 (un , vn ) · φ =
−
t̃n
un
Z
p(x)
Ω
1
p(x)
t̃n
|∇un |p(x)−2 ∇un · ∇φdx
1 Z
1
p(x) p(x) |∇un |p(x) dx
t̃ n Ω
t̃n
26
.
Thus,
||φ||1,p(x)
+
t̃n
||π20 (un , vn ) · φ| |1,p(x) ≤
||φ||1,p(x)
+
t̃n
≤
Z
1
||un ||1,p(x) p(x) p(x) |∇un |p(x)−2 ∇un
Ω
t̃n
Z
1
t̃ n
p(x)
Ω
1
p(x)
t̃n
|∇un |p(x) dx
Z
1
p(x) p(x) |∇un |p(x)−2 ∇un
Ω
t̃
Z n
p(x)
Ω
· ∇φdx
1
|∇un |p(x) dx
p(x)
t̃n
Particularly, by appling the Hölder inequality for p(x)-Lebesgue space
[25, 26, 17] successively, we find
Z
|∇un |p(x)−2 ∇un
p(x)
p(x)−2
Ω
t̃n
u
|∇u |p(x)−1 ||φ||1,p(x)
∇φ n
·
·
dx ≤ p + p(x)−1 p(x)
p(x)−1
t̃
t̃n
t̃n
n
n
L
(Ω)
||φ||
1,p(x)
= p+ ·
.
t̃n
(5.42a)
Z
Z
1
1
|∇un |p(x) dx ≥ p− .
(5.42b)
p(x) p(x) |∇un |p(x) dx ≥ p−
p(x)
Ω t̃n
Ω
t̃n
The above remarks allow us to estimate
||π20 (un , vn )
· φ| |1,p(x)
p+
≤ 1+ −
p
!
||φ||1,p(x)
.
t̃n
Finally, from properties on Lp(x) (Ω) and W 1,p(x) (Ω) spaces (see for instance [17]),
||π20 (un , vn ) · φ| |1,p(x)
p+
≤ 1+ −
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 ) <
and also
m<
!Z
α+1 β+1
+ − −1
|∇un |p(x) dx
p−
q
Ω
!Z
α+1 β+1
+ − −1
|∇vn |q(x) dx.
−
p
q
Ω
27
.
· ∇φdx
(
p− if kun k1,p(x) ≤ 1,
Thus, we set pn =
and qn =
p+ if kun k1,p(x) > 1
In vertue to Theorem 1.3 [17], we get more precisely,
(
q−
q+
if kvn k1,q(x) ≤ 1,
if kvn k1,q(x) > 1.
!
m < J(un , vn ) <
α+1 β+1
n
+ − − 1 kun kp1,p(x)
p−
q
and also
!
α+1 β+1
n
+ − − 1 kvn kq1,q(x)
.
p−
q
m<
!1/pn
||φ||1,p(x)
.
m1/pn
!1/qn
||ψ||1,q(x)
.
m1/qn
!
· φ| |1,p(x)
p+
≤ 1+ −
p
α+1 β+1
·
+ − −1
p−
q
!
||τ20 (vn ) · ψ| |1,q(x)
q+
≤ 1+ −
q
α+1 β+1
+ − −1
·
p−
q
||π20 (un )
Similarly,
We conclude that
lim ||J 0 (un , vn )||X0∗ (x) = 0.
n→+∞
Lemma 5.5 Assume
α+1 β+1
+ + − 1 > 0,
p+
q
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,
Z
Ω
p(x)
|∇un |
Z
dx+
Ω
q(x)
|∇vn |
Z
dx−
Ω
α+1
c(x)|un |
β+1
|vn |
dx−
Z
Ω
c(x)|un |α+1 |vn |β+1 dx = 0.
(5.43)
28
So, on the other hand, because (zn , wn ) is a minimizing sequence for
˜ w), we have
inf J(z,
(z,w)∈E
Z
Z
1
1
1
p(x)
q(x)
|∇un | dx+(β+1)
|∇vn | dx− c(x)|un |α+1 |vn |β+1 dx < m+ .
m ≤ (α+1)
n
Ω q(x)
Ω
Ω p(x)
(5.44)
Combining (5.43) and (5.44), one conclude that
Z
m≤
!
!
Z
Z
β+1
α+1
− 1 |∇un |p(x) dx+
− 1 |∇vn |q(x) dx+ c(x)|un |α+1 |vn |β+1 dx < m+
p(x)
q(x)
Ω
Ω
Z
Ω
Recall that
Z
p(x)
|∇un |
Ω
dx =
Z
Ω
q(x)
|∇vn |
dx =
Z
Ω
c(x)|un |α+1 |vn |β+1 dx,
We obtain
m≤
Z
Ω
!
Z
α+1
β+1
1
p(x)
|∇un | dx +
− 1 |∇vn |q(x) dx < m + .
p(x)
q(x)
n
Ω
More precisely, after some easy calculations, we obtain
!Z
α+1 β+1
|∇un |p(x) dx ≤
+ + −1
p+
q
Ω
!Z
α+1 β+1
|∇vn |p(x) dx < m+1.
+
−1
p(x)
q(x)
Ω
Arguing similarly, we find
!Z
α+1 β+1
+ + −1
|∇vn |q(x) dx < m + 1.
+
p
q
Ω
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 previous lemmas that lim J(un , vn ) = m and lim ||J 0 (un , vn )||X0∗ (x) =
n→+∞
n→+∞
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
1.p(x)
un * u∗ in W0
1.q(x)
vn * v ∗ in W0
29
(Ω),
(Ω).
1,p(x)
• Step 2: Stong convergence of un and vn in W0
1,q(x)
(Ω) (resp. W0
(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
h
i
J 0 (um , vm ) − J 0 (ul , vl ) (um − ul , 0)
= (α + 1)
− (α + 1)
Z
ZΩ
Ω
(|∇um |p(x)−2 ∇um − |∇ul |p(x)−2 ∇ul )(∇um − ul )dx
h
i
c(x) |vm |(β+1 |um |α−1 um − |vl |(β+1 |ul |α−1 ul (um − ul )dx.
Thus, after a suitable rearrangement, we get
Z
Ω
(|∇um |p(x)−2 ∇um − |∇ul |p(x)−2 ∇ul )(∇um − ul )dx
i
1 h 0
J (um , vm ) − J 0 (ul , vl ) (um − ul , 0)dx
Zα + 1
=
h
+
Ω
i
c(x) |vm |β+1 |um |α−1 um − |vl |β+1 |ul |α−1 ul (um − ul )dx.
We claim that
Z
Ω
i
h
c(x) |vm |(β+1 |um |α−1 um −|vl |(β+1 |ul |α−1 ul (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
≤
Z
h
c(x)
|vm |β+1 |um |α−1 vm
Ω
Z
Ω
β+1
− |vl |
c(x)|vm |β+1 |um |α |um − ul |dx +
α−1
|ul |
Z
Ω
ul (um − ul )dx
i
c(x)|vl |β+1 |ul |α |um − ul |dx
α
≤ kck∞ kvm kβ+1
Lq̂ (Ω) kum kLp̂ (Ω) kum − ul kLp̂ (Ω)
α
+ kck∞ kvl kβ+1
Lq̂ (Ω) kul kLp̂ (Ω) kum − ul kLp̂ (Ω)
≤ Ckum − ul kLp̂ (Ω) .
(5.46)
Before continuing, let us recall a fundamental convergence property.
Indeed, it is well known (see [17] for instance) that the imbedding
1.p(x)
1.q(x)
W0
(Ω) ,→ Lδ(x) (Ω) (resp W0
(Ω) ,→ Lγ(x) (Ω)) with δ(x) <
N p(x)
N q(x)
(resp. γ(x) <
) is compact.
N − p(x)
N − q(x)
30
Choose γ(x) = p̂, it follows that un converges strongly to u∗ in
Lp̂ (Ω) and so on, un is a Cauchy sequence in sense of Lp̂ (Ω) norm.
Consequently, (5.46) occurs.
Futhermore, following [25], there exist constants C1 , C2 , C3 , C4 such
that
hF (∇um )−F (∇ul ), um −ul i ≥





































if 1 < p(x) < 2,
C1 kum − ul k21,p(x) ,
2p
/p1,1
0,1
C2 kum − ul k1,p(x)
,
if 2 ≥ p(x),
C3 kum −
p0,2
ul k1,p(x)
,
p
1,2
C4 kum − ul k1,p(x)
,
(5.47)
where
1. F (ξ) = |ξ|p(x)−2 ξ,
∀ξ ∈ IRN ,
2. p0,j = inf p(x), p1,j = sup p(x),
x∈Ωj
3. Ω1 = {x ∈ Ω;
j = 1; 2,
x∈Ωj
1 < p(x) < 2} and Ω2 = {x ∈ Ω;
2 ≥ p(x)} .
Then, combining (5.33b), (5.45) and (5.47), we conclude that un
1,p(x)
converges strongly to u∗ in W0
(Ω). Similar argues allow to prove
1,q(x)
that the sequence vn converges to v ∗ strongly in W0
(Ω).
• Step 3: (u∗ , v ∗ ) is a solution of (1.1) involving a fibering decomposition
We show u∗ = r̄z̄ and v ∗ = ρ̄w̄ involve a solution of (1.1) via the
fibering method. Let us recall that z̄ and w̄ are respectively the
weak limit of zn and wn is the weak limit of wn . The sequences rn
and ρn are defined as in (5.4). Moreover, using (5.13), extracting
if necessary subsequences, we can assume that rn and ρn , converge
in IR. Let r̄ and ρ̄ be such that rn → r̄ and ρn → ρ̄ as n tends to
+∞.
31
1
1
1
un − u∗ =
[(r̄ − rn ) u∗ + r̄ (un − u∗ )] ,
rn
r̄
rn r̄
and the convergences results announced above, it is clear that
Because the formulation
u∗ − → 0,
rn
r̄ 1,p(x)
un
In other words, since
as n tends to + ∞.
un
= zn , we deduce that zn converges strongly
rn
u∗
1,p(x)
(Ω).
in W0
r̄
1,p(x)
Thus, since zn converges weakly to z̄ in W0
(Ω), we deduce from
above and also from uniqueness
to
z̄ =
u∗
.
r̄
On the other hand
ku∗ k1,p(x) ≤ lim inf kun k1,p(x) ≤ lim sup kun k1,p(x) .
n
n
So
kz̄k1,p(x) r̄ ≤ lim inf kun k1,p(x) ≤ lim sup kun k1,p(x)
n
n
thus
kz̄k1,p(x) r ≤ lim inf rn kzn k ≤ lim sup kun k1,p(x) .
n
n
Since kzn k1,p(x) = 1 and kun k1,p(x) ≤ kun − u∗ k1,p(x) + ku∗ k1,p(x) ,
we obtain
kz̄k1,p(x) r̄ ≤ r̄ ≤ kz̄k1,p(x) r̄
thus after dividing by r̄ > 0, it occurs kz̄k1,p(x) = 1.
In the same way
kw̄k1,q(x) = 1.
We can conclude that (z̄, w̄) is solution of the conditional problem
(5.32).
Now, the material needed to prove Theorem 3.2 is complete. In other
words, we establish that the boundary value problem (1.1) admits at
least a solution.
32
5.7
Proof of Theorem 3.2
Proof
The previous lemmas imply that (z̄, w̄) is a contitional critical point for
˜
J.
From the Euler-Lagrange characterization, we deduce that there exists a
pair (m1 , m2 ) in IR2 such that for any (h, k) ∈ X0 (x),
˜ w̄) · (h, k) = m1 ∇A(z̄, w̄) · (h, k) + m2 ∇B(z̄, w̄) · (h, k). (5.48)
∇J(z̄,
In (5.48), we choose h = z̄, k = w̄, we obtain
J˜0 (z̄, w̄)(z̄, w̄) = 0.
(5.49)
Combining (5.48) and (5.49), we obtain
(
m1 A(1) · (z̄, w̄) + m2 B (1) · (z̄, w̄) = 0
m1 A(2) · (z̄, w̄) + m2 B (2) · (z̄, w̄) = 0.
Here, A(1) , B (1) (resp. A(2) and B (2) ) denote the first derivatives respect
with z (resp. w). But

det 

A(1) · (z̄, w̄) B (1) · (z̄, w̄)
A(2) · (z̄, w̄) B (2) · (z̄, w̄)



> p− q − A(z̄)B(w̄) = p− q − > 0.
It follows that
m1 = m2 = 0.
Consequently,
J˜0 (z̄, w̄) = 0,
or again,
J 0 (r̄z̄, ρ̄w̄) = 0
Finaly, we can conclude that (u∗ , v ∗ ) = (r̄z̄, ρ̄w̄) 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) then
necessarily we shoud have
Z
|∇u∗ |p(x) dx ≥ e
and
Ω
Z
Ω
33
|∇v ∗ |q(x) dx ≥ e.
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 there
exist two functions gp and gq satisfies: gp ≤ u∗ and gq ≤ v ∗ for all x on
the set Br0 \ B r20 such that
Br0 = {x ∈ Ω; |x| ≤ r0 }.
Proof .
We define the function
2
2
gp = kp e−αp |x| − e−αp r0 .
Consequently, we have
−div |∇gp |p(x)−2
∂gp 2
= −2p(x)−1 αpp(x)−1 rp(x)−2 e−αp r (p(x)−1)
∂xi
h
−
i
2
< x, ∇p > ln(2αp re−αp r + (p(x) − 1)(2αp r2 − 1) − (n − 1) .
On the set Br0 \ B r20 , we consider the function
2
f (r) = (2αp re−αp r − 1)
such that r > 1 and αp > 0.
Clearly,
1
2
∀αp > , 2αp re−αp r − 1 < 0.
2
So,
2
ln(2αp re−αp r ) < 0.
Finally, if we suppose that

1
n
hx, ∇pi > 0 and αp > sup  , 1 + −
,
2
p +1
−a
2r0p−−2
!
1
p− −2
we conclude then
(1)p
p(x)−2 ∂gp
∀a < 0; −div |∇gp |
34
∂xi
!
− agp < 0.



,
Multiplying (−∆p(x) u = c(x)u|u|α−1 |v|β+1 ) in (1.1) and (1)p by the test
function
ϕp = (gp − u∗ )+
and integrating over the set
Bp+ = {x ∈ Br0 /B r20
ϕp > 0},
we obtain
Z
0 ≤
Bp+
(|∇gp |p(x)−2 ∇gp − |∇u∗ p |p(x)−2 ∇u∗ p )∇ϕp dx
≤a
Z
g p(x)−2 ϕp −
+ p
Z
Bp+
Bp
u∗ |u∗ |(α−1) |v ∗ |β+1 ϕp dx ≤ 0.
Which implies that
mes(Bp+ ) = 0.
Therefore
gp ≤ u∗
in Br0 /B r20 .
Using the similar arguments we find
gq ≤ v ∗
in Br0 /B r20 .
Proof of Theorem 6.1.
Let (u∗ , v ∗ ) be a solution of the system (1.1), then
Z
|∇u∗ |p(x) dx =
Ω
Z
|∇v ∗ |q(x) dx =
Z
|u∗ |α+1 |v ∗ |β+1 dx.
Ω
Ω
Using the previous lemma then we have
Z
|u∗ |α+1 |v ∗ |β+1 dx ≥
Z
Br0
Br0
|gp |α+1 |gq |β+1 dx.
Now we set
Kpq = min(kp , kq )
and
K=
Z
h
2
2
e−αp kxk − e−αp r0
iα+1 h
2
Br0
Finally we choose k1 and k2 such that
α+β+2
Kpq
K ≥ meas(Br0 )e.
35
2
e−αq kxk − e−αq r0
iβ+1
dx.
Therefore
Z
|∇u∗ |p(x) dx =
Br0
Z
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 .
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