Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 131, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR ELLIPTIC SYSTEMS IN RN
INVOLVING THE p(x)-LAPLACIAN
ALI DJELLIT, ZAHRA YOUBI, SAADIA TAS
Abstract. This article presents sufficient conditions for the existence of nontrivial solutions for a nonlinear elliptic system. To establish this result, we use
a classical existence theorem in reflexive Banach spaces, under some growth
conditions on the non-linearities.
1. Introduction
In this article we establish the existence of nontrivial weak solution for nonlinear
elliptic system
∂F
−∆p(x) u =
(x, u, v) in RN
∂u
(1.1)
∂F
N
−∆q(x) v =
(x, u, v) in R
∂v
Here p(x) and q(x) are continuous real-valued functions such that 1 < p(x), q(x) <
N (N ≥ 2) for all x ∈ RN . The real-valued function F belongs to C 1 (RN × R2 ),
and ∆p(x) is the so-called p(x)-Laplacian operator; i.e., ∆p(x) u = div(|∇u|p(x) ∇u).
This decade bears witness to a considerable sum of results on non standard
growth conditions problems. This abundance is due to the recent research developments in elasticity problems, electrorheological fluids, image processing, flow in
porous media, etc.; see for example [2, 12].
In a natural way, the introduction of the generalized Lebesgue-Sobolev spaces
turned out to be crucial [3, 5, 8]. In this way, many authors could successfully
deal with p(x)-Laplacian problems [7, 8]. Many additional works concern elliptic
systems in relationship to standard and nonstandard growth conditions. We refer
the readers to [1, 10, 15] and the references therein. In [4, 14], the authors show
the existence of nontrivial solutions for the (p, q)-Laplacian system
∂F
−∆p u =
(x, u, v) in RN
∂u
∂F
−∆q v =
(x, u, v) in RN
∂v
where the potential function F satisfies mixed and subcritical growth conditions
and, in addition, to be intimately connected with the first eigenvalue of p-Laplacian
2000 Mathematics Subject Classification. 35J50, 35J92.
Key words and phrases. p(x)-Laplacian operator; critical point, variational system.
c
2012
Texas State University - San Marcos.
Submitted May 21, 2012. Published August 15, 2012.
1
2
A. DJELLIT, Z. YOUBI, S. TAS
EJDE-2012/131
operator. They apply the Mountain Pass theorem to obtain the nontrivial solutions
of the system.
In [6], the authors obtained the existence and multiplicity of solutions for the
vector valued elliptic system
∂F
(x, u, v) in Ω
∂u
∂F
(x, u, v) in Ω
−∆p(x) v =
∂v
u = v = 0 on ∂Ω,
−∆p(x) u =
where Ω ⊂ RN is a bounded domain with a smooth boundary ∂Ω, N ≥ 2, (p, q) ∈
[C(Ω)]2 , F ∈ C(RN × R2 , R). Existence and multiplicity results are subjected to
some natural growth conditions which guarantee the Mountain Pass geometry and
Palais-Smale condition.
In [16], the authors studied the system
∂F
(x, u, v) in RN
∂u
∂F
(x, u, v) in RN
−∆p(x) v + |v|q(x)−2 v =
∂v
The potential function F needs to satisfy Caratheodory conditions. Using critical
point theory, they establish existence results in sub-linear and super-linear cases.
In [12], by the Mountain Pass theorem, the authors show the existence of nontrivial solutions for the following (p(x), q(x))-Laplacian system
−∆p(x) u + |u|p(x)−2 u =
∂F
(x, u, v) in RN
∂u
∂F
−∆q(x) v =
(x, u, v) in RN
∂v
−∆p(x) u =
where F ∈ C 1 (RN × R2 , R) verifies some mixed growth conditions.
With regard to existence results, we use critical point theory. Our main goal is
to establish that the energy functional of the system is lower semi- continuous and
coercive in reflexive Banach space.
2. Notation and hypotheses
To discuss system (1.1), we recall some results on generalized Lebesgue-Sobolev
spaces.
Let E(Ω) be a space of functions defined on Ω. We set
E+ (Ω) = {h ∈ E(Ω) : inf h(x) > 1}.
x∈Ω
N
So, for all h ∈ C+ (R ), we set
h− := inf h(x),
x∈RN
h+ := sup h(x).
x∈RN
Let M (RN ) be the set of all measurable real-valued functions defined on RN . For
p ∈ C+ (RN ), we designates the variable exponent Lebesgue space by
Z
p(x)
N
N
L
(R ) = {u ∈ M (R ) :
|u(x)|p(x) dx < ∞},
RN
EJDE-2012/131
EXISTENCE OF SOLUTIONS
3
equipped with the so called Luxemburg norm
Z
|u|p(x) := |u|Lp(x) (RN ) = inf{λ > 0 :
|
RN
u(x) p(x)
|
dx ≤ 1}.
λ
This is a Banach space. Define the Lebesgue-Sobolev space W 1,p(x) (RN ) by
W 1,p(x) (RN ) = {u ∈ Lp(x) (RN ) : |∇u| ∈ Lp(x) (RN )},
equipped with the norm
kuk1,p(x) = kukW 1,p(x) (RN ) = |u|p(x) + |∇u|p(x) .
1,p(x)
The space W0
(Ω) is defined as the closure of C0∞ (Ω) in W 1,p(x) (Ω) with respect
1,p(x)
to the norm kuk1,p(x) . For u ∈ W0
(Ω), we can define an equivalent norm
kuk = |∇u|p(x) ; since the well known Poincaré inequality holds.
Next, we recall some previous results. This way, we want to make the proofs of
the main results as transparent as possible.
Proposition 2.1 ([5, 9]). If p ∈ C+ (RN ), then the spaces Lp(x) (RN ), W 1,p(x) (RN )
1,p(x)
and W0
(RN ) are separable and reflexive Banach spaces.
0
Proposition 2.2 ([5, 9]). The topological dual space of Lp(x) (RN ) is Lp (x) (RN ),
where
1
1
+ 0
= 1.
p(x) p (x)
0
Moreover for any (u, v) ∈ Lp(x) (RN ) × Lp (x) (RN ), we have
Z
1
1
uvdx| ≤ ( − + 0 − )|u|p(x) |v|p0 (x) ≤ 2|u|p(x) |v|p0 (x) .
|
p
(p
)
N
R
R
Set ρ(u) = RN |u|p(x) dx.
Proposition 2.3 ([5, 9]). For all u ∈ Lp(x) (RN ), we have
−
−
+
+
min{|u|pp(x) , |u|pp(x) } ≤ ρ(u) ≤ max{|u|pp(x) , |u|pp(x) }.
In addition, we have
(i) |u|p(x) < 1 (resp. = 1, > 1) ⇔ ρ(u) < 1 (resp. = 1, > 1);
−
+
+
−
(ii) |u|p(x) > 1 ⇒ |u|pp(x) ≤ ρ(u) ≤ |u|pp(x) ;
(iii) |u|p(x) > 1 ⇒ |u|pp(x) ≤ ρ(u) ≤ |u|pp(x) ;
(iv) ρ( |u|up(x) ) = 1.
Proposition 2.4 ([5]). Let p(x) and q(x) be measurable functions such that p(x) ∈
L∞ (RN ) and 1 ≤ p(x)q(x) ≤ ∞ almost every where in RN . If u ∈ Lq(x) (RN ),
u 6= 0. Then
−
+
|u|p(x)q(x) ≤ 1 ⇒ |u|pp(x)q(x) ≤ |u|p(x) q(x) ≤ |u|pp(x)q(x) ,
+
−
|u|p(x)q(x) ≥ 1 ⇒ |u|pp(x)q(x) ≤ |u|p(x) q(x) ≤ |u|pp(x)q(x) .
In particular, if p(x) = p is a constant, then ||u|p |q(x) = |u|ppq(x) .
Proposition 2.5 ([9]). If u, un ∈ Lp(x) (RN ), n = 1, 2, . . . , then the following
statements are mutually equivalent:
(1) limn→∞ |un − u|p(x) = 0,
4
A. DJELLIT, Z. YOUBI, S. TAS
EJDE-2012/131
(2) limn→∞ ρ(un − u) = 0,
(3) un → u in measure in RN and limn→∞ ρ(un ) = ρ(u).
Let p∗ (x) be the critical Sobolev exponent of p(x) defined by
(
N p(x)
for p(x) < N
∗
p (x) = N −p(x)
+∞
for p(x) ≥ N
and let C 0,1 (RN ) be the Lipschitz-continuous functions space.
0,1
Proposition 2.6 ([3, 5]). If p(x) ∈ C+
(RN ), then there exists a positive constant
c such that
1,p(x)
|u|p∗ (x) ≤ ckukp(x) , ∀u ∈ W0
(RN ).
N
+
Let p ∈ L∞
+ (R ) be an uniformly continuous function such that p < N and let
N
Ω ⊂ R be a bounded domain.
N
∗
Proposition 2.7 ([3, 5]). (1) If q ∈ L∞
+ (R ) and p(x) ≤ q(x) p (x), for
N
1,p(x)
N
q(x)
N
all x ∈ R , then the embedding W
(R ) ,→ L
(R ) is continuous but not
compact.
(2) If p is continuous on Ω and q is a measurable function on Ω, with p(x) q(x) p∗ (x), for all x ∈ Ω, then the embedding W 1,p(x) (Ω) ,→ Lq(x) (Ω) is compact.
Observe that the solution of (1.1) will belong to the product space
1,p(x)
Wp(x),q(x) := W0
1,q(x)
(RN ) × W0
(RN ),
equipped with the norm
k(u, v)kp(x) = |∇u|p(x) + |∇v|p(x) .
0
Wp(x),q(x)
The space
is the topological dual of Wp(x),q(x) equipped with the usual
dual norm. For (u, v) in Wp(x),q(x) , let us define the functionals I, J, K
Z
F (u, v) =
F (x, u(x), v(x))dx,
RN
Z
Z
1
1
p(x)
J(u, v) =
|∇u|
dx +
|∇v|q(x) dx,
RN p(x)
RN q(x)
I(u, v) = J(u, v) − F (u, v).
Hypotheses. We assume some growth conditions:
(H1) F ∈ C 1 (RN × R2 , R) and F (x, 0, 0) = 0.
(H2) There exist positive functions ai , bi such that
−
+
∂F
(x, u, v)| ≤ a1 (x)|u|p1 −1 + a2 (x)|v|p1 −1 ,
∂u
−
+
∂F
|
(x, u, v)| ≤ b1 (x)|u|q1 −1 + b2 (x)|v|q1 −1 ,
∂v
where 1 < p1 (x), q1 (x) < inf(p(x), q(x)), and p(x), q(x) > N2 , for all x ∈
RN . The weight-functions ai and bi , i = 1, 2, belong respectively to the
generalized Lebesgue spaces Lαi (RN ) and Lβ (RN ), where
|
α1 (x) =
p(x)
p∗ (x)q ∗ (x)
,
, β(x) = ∗
p(x) − 1
p (x)q ∗ (x) − p∗ (x) − q ∗ (x)
α2 (x) =
q(x)
.
q(x) − 1
EJDE-2012/131
EXISTENCE OF SOLUTIONS
5
(H3) There exist constants R > 0, θ > 0, and a positive function H : RN × R2 →
R such that for x ∈ RN , |u|, |v| ≤ R and t > 0 sufficiently small, we have
F (x, t1/p(x) u, t1/q(x) v) ≥ tθ H(x, u, v).
Assumption (H3) implies that the potential function F is sufficiently positive in
a neighborhood of zero.
Lemma 2.8. Under assumptions (H1)–(H2), the functional F is well defined and
Frechet differentiable. Its derivative is
Z
∂F
∂F
F 0 (u, v)(ω, z) =
(x, u, v)ω +
(x, u, v)zdx, ∀(u, v), (ω, z) ∈ Wp(x),q(x) .
∂u
∂v
N
R
Proof. The functional F is well defined on Wp(x),q(x) . Indeed, for all pair of realvalued functions (u, v) ∈ Wp(x),q(x) , we have in virtue of (H1) and (H2),
Z u
∂F
F (x, u, v) =
(x, s, v)ds + F (x, 0, v)
∂s
Z v
Z0 u
∂F
∂F
(x, s, v)ds +
(x, 0, s)ds + F (x, 0, 0).
=
∂s
0 ∂s
0
Then
−
+
+
F (x, u, v) ≤ c1 [a1 (x)|u|p1 + a2 (x)|v|p1 −1 |u| + b2 (x)|v|q1 ]
Since W
1,p(x)
N
s(x)p(x)
(R ) ,→ L
(2.1)
N
(R ) for s(x) > 1, we have
p− p−
p−
1
|u| 1 = |u|p1− p(x) ≤ ckukp(x)
.
p(x)
1
So, taking into account Hölder inequality, Propositions 2.2, 2.4, 2.6, 2.7 and (H2),
we obtain
Z
F (u, v) =
F (x, u, v) dx
RN
p−
p+ −1
q+
≤ c2 |a1 |α1 (x) |u|p1− p(x) + |a2 |β(x) |v|(p1+ −1)q∗ (x) |u|p∗ (x) + |b2 |α2 (x) |v|q1+ q(x)
1
≤
p−
1
c3 (|a1 |α1 (x) kukp(x)
1
+
p+
1 −1
|a2 |β(x) kvkq(x)
kukp(x)
1
+
q1+
|b2 |α2 (x) kvkq(x)
)
<∞
The proof is complete.
Similarly, we show that F 0 is also well defined. Indeed, for all (u, v), (ω, z) ∈
Wp(x),q(x) , we can write
Z
Z
∂F
∂F
F 0 (u, v)(ω, z) =
(x, u, v)ωdx +
(x, u, v)z dx
∂u
∂v
N
N
R
R
Z
−
+
≤
(a1 (x)|u|p1 −1 + a2 (x)|v|p1 −1 )ω dx
RN
Z
−
+
+
(b1 (x)|u|q1 −1 + b2 (x)|v|q1 −1 )z dx
RN
Following Hölder inequality, we obtain
−
+
F 0 (u, v)(ω, z) ≤ c4 (|a1 |α1 (x) ||u|p1 −1 |p∗ (x) |ω|p(x) + |a2 |β(x) ||v|p1 −1 |q∗ (x) |ω|p∗ (x)
−
+
+ |b1 |β(x) ||u|q1 −1 |p∗ (x) |z|q∗ (x) + |b2 |α2 (x) ||v|q1 −1 |q∗ (x) |z|q(x) )
6
A. DJELLIT, Z. YOUBI, S. TAS
EJDE-2012/131
The above propositions yield
p− −1
p+ −1
1
1
F 0 (u, v)(ω, z) ≤ c5 (|a1 |α1 (x) kukp(x)
kωkp(x) + |a2 |β(x) kvkq(x)
kωkp(x)
q − −1
q + −1
1
1
+ |b1 |β(x) kukp(x)
kzkq(x) + |b2 |α2 (x) kvkq(x)
kzkq(x) ) < ∞.
Moreover F is Frechet differentiable; namely, for any fixed point (u, v) ∈ Wp(x),q(x) ,
and for any ε > 0, there exist δ = δε,u,v > 0 such that for all (ω, z) ∈ Wp(x),q(x) ,
satisfying k(ω, z)kp(x),q(x) < δ we have
|F (u + ω, v + z) − F (u, v) − F 0 (u, v)(ω, z)| ≤ εk(ω, z)kp(x),q(x) .
0
First, let BR be the ball in RN centered at the origin and of radius R. Set BR
=
N
R − BR .
1,p(x)
1,q(x)
It is well-known that the functional FR defined on W0
(BR ) × W0
(BR )
by
Z
FR (u, v) =
F (x, u, v)dx
BR
1,p(x)
1,q(x)
belongs to C 1 (W0
(BR ) × W0
(BR )), by in virtue of (H1) and (H2). In addi1,p(x)
1,q(x)
1,p(x)
tion, the operator FR0 defined from W0
(BR ) × W0
(BR ) to (W0
(BR ) ×
1,p(x)
0
W0
(BR )) by
Z
∂F
∂F
FR0 (u, v)(ω, z) =
(x, u, v)ω +
(x, u, v)zdx,
∂u
∂v
BR
is compact (see [9]). Clearly, for all (u, v), (ω, z) ∈ Wp(x),q(x) , we can write
|F (u + ω, v + z) − F (u, v) − F 0 (u, v)(ω, z)|
≤ |FR (u + ω, v + z) − FR (u, v) − FR0 (u, v)(ω, z)|
Z
∂F
∂F
(F (x, u + ω, v + z) − F (x, u, v)) −
+|
(x, u, v)ω −
(x, u, v)zdx|
0
∂u
∂v
BR
According to a classical theorem, there exist ζ1 , ζ2 ∈]0, 1[, such that
Z
∂F
∂F
(F (x, u + ω, v + z) − F (x, u, v)) −
(x, u, v)ω −
(x, u, v)zdx
0
∂u
∂v
BR
Z
∂F
∂F
(x, u + ζ1 ω, v)ω +
(x, u, v + ζ2 z)z
=
0
∂u
∂v
BR
∂F
∂F
−
(x, u, v)ω −
(x, u, v)z dx.
∂u
∂v
Consequently, by growth conditions (H2), we obtain
Z
∂F
∂F
|
(F (x, u + ω, v + z) − F (x, u, v)) −
(x, u, v)ω −
(x, u, v)z dx|
0
∂u
∂v
BR
Z
−
−
≤
a1 (x)(|u + ζ1 ω|p1 −1 + |u|p1 −1 )ωdx
0
BR
Z
+
+
+
−
−
a2 (x)(|v + ζ2 z|p1 −1 + |v|p1 −1 )ω dx
0
BR
Z
+
0
BR
b1 (x)(|u + ζ1 ω|q1 −1 + |u|q1 −1 )zdx
EJDE-2012/131
EXISTENCE OF SOLUTIONS
Z
+
7
+
b2 (x)(|v + ζ2 z|q1 −1 + |v|q1 −1 )z dx.
+
0
BR
By an elementary inequality, Propositions 2.4, 2.6 and the fact that
|ai |Lp0 (x) (B 0 ) → 0,
|ai |Lβ(x) (BR0 ) → 0
|bi |Lq0 (x) (B 0 ) → 0,
|bi |Lβ(x) (BR0 ) → 0,
R
R
(2.2)
for R sufficiently large and i = 1, 2, we obtain the estimate
Z
∂F
∂F
(F (x, u + ω, v + z) − F (x, u, v) −
(x, u, v)ω −
(x, u, v)z) dx
0
∂u
∂v
BR
≤ ε(kωkp(x) + kzkq(x) ).
We prove now that F 0 is continuous on Wp(x),q(x) . To this end, we let (un , vn ) →
(u, v) in Wp(x),q(x) as n → ∞. Then for any (ω, z) ∈ Wp(x),q(x) , we have
|F 0 (un , vn )(ω, z) − F 0 (u, v)(ω, z)|
≤ |FR0 (un , vn )(ω, z) − FR0 (u, v)(ω, z)| + |
Z
+|
0
BR
Z
0
BR
∂F
∂F
(x, un , vn ) −
(x, u, v) ωdx|
∂u
∂u
∂F
∂F
(x, un , vn ) −
(x, u, v) z dx|
∂v
∂v
Note that
|FR0 (un , vn )(ω, z) − FR0 (u, v)(ω, z)| → 0
1,p(x)
as n → ∞,
1,q(x)
(BR ) (see [9]). Using (H2) once
(BR ) × W0
since FR0 is continuous on W0
again and (2.1), the other terms on the wrigth-hand side of the above inequality
tend to zero.
Lemma 2.9. Under assumptions (H1)–(H2), F is lower weakly semicontinuous in
Wp(x),q(x) .
Proof. Let (un , vn ) be a weakly convergent sequence to (u, v) in Wp(x),q(x) . In the
same way, we write
Z
(F (x, un , vn ) − F (x, u, v))dx|
|F (un , vn ) − F (u, v)| ≤ |FR (un , vn ) − FR (u, v)| + |
0
BR
Since the restriction operator is continuous, the sequence (un , vn ) is weakly con1,p(x)
1,q(x)
vergent to (u, v) in W0
(BR ) × W0
(BR ). However FR is weakly lower semicontinuous. This result comes from growth conditions (H1) and (H2), and Sobolev
compact inclusion
1,p(x)
W0
1,q(x)
(BR ) × W0
(BR ) ,→ Ls(x) (BR ) × Lt(x) (BR ),
for all (s, t) ∈ [p(x), p∗ (x)[×[q(x), q ∗ (x)[. Using (2.1) and (2.2), both the terms on
the right-hand side of the last inequality tend to zero.
We remark that the C 1 −functional J is weakly lower semi-continuous, and its
derivative is given by
Z
Z
0
p(x)−2
J (u, v)(ω, z) =
|∇u|
∇u∇ω dx +
|∇v|q(x)−2 ∇v∇z dx
RN
RN
8
A. DJELLIT, Z. YOUBI, S. TAS
EJDE-2012/131
The Euler-Lagrange functional associated to the system (1.1) takes the form
Z
Z
1
1
p(x)
q(x)
I(u, v) =
|∇u|
+
|∇v|
dx −
F (x, u, v) dx.
q(x)
RN p(x)
RN
In other words I(u, v) = J(u, v) − F (u, v). Observe that the weak solutions of the
system (1.1) are precisely the critical points of the functional I.
Lemma 2.10. Under assumptions (H1)–(H2), the functional I is coercive.
Proof. We have
Z
1
1
|∇u|p(x) +
|∇v|q(x) − F (x, u, v) dx
I(u, v) =
p(x)
q(x)
N
R
Z
1
1
≥
|∇u|p(x) + + |∇v|q(x) dx
+
p
q
N
R
Z
−
+
+
−
(a1 (x)|u|p1 + a2 (x)|v|p1 −1 |u| + b2 (x)|v|q1 )dx
RN
1
1
≥ + ρ(∇u) + + ρ(∇v)
p
q
−
+
+
− (|a1 |α1 (x) ||u|p1 |p(x) + |a2 |β(x) ||v|p1 −1 |q∗ (x) |u|p∗ (x) + |b2 |α2 (x) ||v|q1 |q(x) ).
By Propositions 2.3, 2.4, 2.6 and the Young inequality, we obtain
−
−
1
1
p−
1
I(u, v) ≥ + kukpp(x) + + kvkqq(x) − |a1 |α1 (x) kukp(x)
p
q
p+ − 1
1
p+
p+
q1+
1
1
+ |a2 |β(x) ( 1 + kvkq(x)
+ + kukp(x)
) + |b2 |α2 (x) kvkq(x)
p1
p1
−
−
1
1
p−
1
≥ + kukpp(x) + + kvkqq(x) − c6 |a1 |α1 (x) kukp(x)
p
q
p+
q1+
1 −1
+ |a2 |β(x) kvkq(x)
+ |a2 |β(x) kukp(x) + |b2 |α2 (x) kvkq(x)
Clearly, I(u, v) tends to infinity as k(u, v)kp(x),q(x) → ∞, since 1 < p1 (x), q1 (x) <
inf(p(x), q(x)).
Theorem 2.11. Under assumptions (H1)–(H3), the system (1.1) has a non-trivial
weak solution.
Proof. By lemmas 2.8, 2.9 and 2.10, the functional I is weakly lower semi-continuous
and coercive in Wp(x),q(x) . Consequently, the functional I has a global minimum
(see [13, Theorem 12]. On the other hand I is C 1 . Hence this minimum is necessarily characterized by a critical point of I, which is a weak solution of (1.1).
This solution is nontrivial. Indeed, as I(0, 0) = 0, it is sufficient to show that
there exists (u1 , v1 ) ∈ Wp(x),q(x) such that I(u1 , v1 ) < 0. Let R > 0, θ < 1 and
(0, 0) 6= (ϕ, ψ) ∈ C0∞ (RN ) × C0∞ (RN ) with |ϕ|, |ψ| ≤ R. According to (H3), one
has
I(t1/p(x) ϕ, t1/q(x) ψ)
= J(t1/p(x) ϕ, t1/q(x) ψ) − F (t1/p(x) ϕ, t1/q(x) ψ)
Z
Z
1
1
p(x)
q(x)
+ − |∇ψ|
]dx −
F (x, t1/p(x) ϕ, t1/q(x) ψ)dx
≤t
[ − |∇ϕ|
q
RN
RN p
EJDE-2012/131
EXISTENCE OF SOLUTIONS
9
Z
1
1
θ
ρ(∇ϕ)
+
ρ(∇ψ)]
−
t
H(x, ϕ, ψ)dx
p−
q−
RN
−
−
+
+
1
1
≤ t[ − max{|∇ϕ|pp(x) , |∇ϕ|pp(x) } + − max{|∇ψ|qq(x) , |∇ψ|qq(x) }]
p
q
Z
− tθ
H(x, ϕ, ψ)dx
≤ t[
RN
−
−
+
+
1
1
≤ t[ − max{k∇ϕkpp(x) , k∇ϕkpp(x) } + − max{k∇ψkqq(x) , k∇ψkqq(x) }]
p
q
Z
− tθ
H(x, ϕ, ψ) dx < 0,
RN
for t > 0 sufficiently small.
Acknowledgements. This work was partially supported by PNR (ANDRU) contracts.
References
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[11] T. C. Halsey; Electrorheological fluids, Science 258 (1992) 761-766.
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systems in RN involving the (p(x), q(x))-Laplacian, J. Inequal. Appl. Art. Id 612938 (2008)
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[13] M. Struwe; Variational methods: Application to nonlinear Partial Differential Equations and
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Ali Djellit
Mathematics, Dynamics and Modelization Laboratory, Badji-Mokhtar Annaba University, Annaba 23000, Algeria
E-mail address: a djellit@hotmail.com
10
A. DJELLIT, Z. YOUBI, S. TAS
EJDE-2012/131
Zahra Youbi
Mathematics, Dynamics and Modelization Laboratory, Badji-Mokhtar Annaba University, Annaba 23000, Algeria
E-mail address: zahra.youbi@yahoo.fr
Saadia Tas
Applied Mathematics Laboratory, Abderrahmane Mira Bejaia University, Bejaia, Algeria
E-mail address: tas saadia@yahoo.fr