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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 612938, 16 pages
doi:10.1155/2008/612938
Research Article
Existence of Solutions for a Class of Elliptic
Systems in
R
N
Involving the px,qx-Laplacian
S. Ogras, R. A. Mashiyev, M. Avci, and Z. Yucedag
Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey
Correspondence should be addressed to R. A. Mashiyev,
Received 4 April 2008; Accepted 17 July 2008
Recommended by M. Garcia Huidobro
In view of variational approach, we discuss a nonlinear elliptic system involving the px-
Laplacian. Establishing the suitable conditions on the nonlinearity, we proved the existence of
nontrivial solutions.
Copyright q 2008 S. Ogras et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
The paper concerns the existence of nontrivial solutions for the following nonlinear elliptic
system:
−Δ
px
u
∂F
∂u
x, u, υ in R
N
,
−Δ
qx
υ
∂F
∂υ
x, u, υ in R
N
,
P,Q
where px and qx are two functions such that 1 <px,qx <NN ≥ 2, for every
x ∈ R
N
. However, F ∈ C
1
R
N
× R
2
and Δ
px
is the px-Laplacian operator defined by
Δ
px
u div|∇u|
px−2
∇u. Using a variational approach, the authors prove the existence of
nontrivial solutions.
Over the last decades, the variable exponent Lebesgue space L
px
and Sobolev space
W
1,px
1–5 have been a subject of active research stimulated mainly by the development
of the studies of problems in elasticity, electrorheological fluids, image processing, flow in
porous media, calculus of variations, and differential equations with px-growth conditions
6–13.
Among these problems, the study of px-Laplacian problems via variational methods
is an interesting topic. A lot of researchers have devoted their work to this area 14–22.
2 Journal of Inequalities and Applications
The operator Δ
px
u : div|∇u|
px−2
∇u is called px-Laplacian, where p is a continuous
nonconstant function. In particular, if px ≡ p constant, it is the well-known p-Laplacian
operator. However, the px-Laplace operator possesses more complicated nonlinearity than
p-Laplace operator due to the fact that Δ
px
is not homogeneous. This fact implies some
difficulties, as for example, we cannot use the Lagrange multiplier theorem and Morse
theorem in a lot of problems involving this operator.
In literature, elliptic systems with standard and nonstandard growth conditions have
been studied by many authors 23–28, where the nonlinear function F have different and
mixed growth conditions and assumptions in each paper.
In 29, the authors show the existence of nontrivial solutions for the following p-
Laplacian problem:
−Δ
p
u
∂F
∂u
x, u, υ in R
N
,
−Δ
q
υ
∂F
∂υ
x, u, υ in R
N
,
1.1
where F ∈ C
1
R
N
× R
2
yields some mixed growth conditions and the primitive F being
intimately connected with the first eigenvalue of an appropriate system. Using a weak
version of the Palais-Smale condition, that is, Cerami condition, they apply the mountain
pass theorem to get the nontrivial solutions of the the system.
In 30, the author obtains the existence and multiplicity of solutions for the following
problem:
− div|∇u|
px−2
∇u
∂F
∂u
x, u, υ in Ω,
− div|∇υ|
qx−2
∇υ
∂F
∂υ
x, u, υ in Ω,
u 0, υ 0onΩ,
1.2
where Ω ⊂ R
N
is a bounded domain with a smooth boundary ∂Ω,N ≥ 2, p, q ∈
C
Ω
2
,px > 1,qx > 1, for every x ∈ Ω. The function F is assumed to be continuous
in x ∈
Ω and of class C
1
in u, υ ∈ R. Introducing some natural growth hypotheses on the
right-hand side of the system which will ensure the mountain pass geometry and Palais-
Smale condition for the corresponding Euler-Lagrange functional of the system, the author
limits himself to the subcritical case for function F to obtain the existence and multiplicity
results.
In the paper 31, Xu and An deal with the following problem:
− div|∇u|
px−2
∇u|u|
px−2
u
∂F
∂u
x, u, υ in R
N
,
− div|∇υ|
qx−2
∇υ|υ|
qx−2
υ
∂F
∂υ
x, u, υ in R
N
,
u, υ ∈ W
1,px
R
N
× W
1,qx
R
N
,
1.3
where N ≥ 2,px,qx are functions on R
N
. The function F is assumed to satisfy
Caratheodory conditions and to be L
∞
in x ∈ R
N
and C
1
in u, υ ∈ R. By the critical point
S. Ogras et al. 3
theory, the authors use the two basic results on the existence of solutions of the system; these
results correspond to the sublinear and superlinear cases for p 2, respectively.
Inspired by the above-mentioned papers, we concern the existence of nontrivial
solutions of problem P,Q. We know that in the study of px-Laplace equations in R
N
,the
main difficulty arises from the l ack of compactness. So, establishing some growth conditions
on the right-hand side of the system which will ensure the mountain pass geometry
and Cerami condition for the corresponding Euler-Lagrange functional J and applying a
subcritical case for function F, we will overcome this difficulty.
2. Notations and preliminaries
We will investigate our problem P,Q in the variable exponent Sobolev space W
1,px
0
R
N
,
so we need to recall some theories and basic properties on spaces L
px
R
N
and W
1,px
R
N
.
Set
C
R
N
h ∈ CR
N
:inf
x∈R
N
hx > 1
. 2.1
For every h ∈ C
R
N
, denote
h
−
: inf
x∈R
N
hx,h
: sup
x∈R
N
hx. 2.2
Let us define by UR
N
the set of all measurable real-valued functions defined on R
N
.
For p ∈ C
R
N
, we denote the variable exponent Lebesgue space by
L
px
R
N
u ∈UR
N
:
R
N
|ux|
px
dx < ∞
, 2.3
which is equipped with the norm, so-called Luxemburg norm 1, 3, 4:
|u|
px
: |u|
L
px
R
N
inf
λ>0:
R
N
ux
λ
px
dx ≤ 1
, 2.4
and L
px
R
N
, |·|
L
px
R
N
becomes a Banach space, we call it generalized Lebesgue space.
Define the variable exponent Sobolev space W
1,px
R
N
by
W
1,px
R
N
{u ∈ L
px
R
N
: |∇u|∈L
px
R
N
}, 2.5
and it can be equipped with the norm
u
1,px
: u
W
1,px
|u|
px
|∇u|
px
∀u ∈ W
1,px
R
N
. 2.6
The space W
1,px
0
R
N
is denoted by the closure of C
∞
0
R
N
in W
1,px
R
N
and it is
equipped with the norm for all u ∈ W
1,px
0
R
N
:
u
px
|∇u|
px
∀u ∈ W
1,px
0
R
N
. 2.7
If p
−
> 1, then the spaces L
px
R
N
,W
1,px
R
N
, and W
1,px
0
R
N
are separable and
reflexive Banach spaces.
4 Journal of Inequalities and Applications
Proposition 2.1 see 1, 3, 4. The conjugate space of L
px
R
N
is L
p
x
R
N
, where 1/p
x
1/px1. For any u ∈ L
px
R
N
and v ∈ L
p
x
R
N
, we have
R
N
uv dx
≤
1
p
−
1
p
−
|u|
px
|v|
p
x
≤ 2|u|
px
|v|
p
x
. 2.8
Proposition 2.2 see 1, 3, 4. Denote
px
u
R
N
|ux|
px
dx for all u ∈ L
px
R
N
, one has
min
|u|
p
−
px
, |u|
p
px
≤
px
u ≤ max
|u|
p
−
px
, |u|
p
px
. 2.9
Proposition 2.3 see 1. Let px and qx be measurable functions such that px ∈ L
∞
R
N
and 1 ≤ pxqx ≤∞, for a.e. x ∈ R
N
. Let u ∈ L
qx
R
N
,u
/
0. Then,
|u|
pxqx
≤ 1 ⇒|u|
p
pxqx
≤
|u|
px
qx
≤|u|
p
−
pxqx
,
|u|
pxqx
≥ 1 ⇒|u|
p
−
pxqx
≤
|u|
px
qx
≤|u|
p
pxqx
.
2.10
In particular, if pxp is constant, then
||u|
p
|
qx
|u|
p
pqx
. 2.11
Proposition 2.4 see 3, 4. If u, u
n
∈ L
px
R
N
,n 1, 2, ,then the following statements are
equivalent to each other:
1 lim
n→∞
|u
n
− u|
px
0,
2 lim
n→∞
u
n
− u0,
3 u
n
→ u in measure in R
N
and lim
n→∞
u
n
u.
Definition 2.5. 1 <px <Nand for all x ∈ R
N
,letdefine
p
∗
x
⎧
⎪
⎨
⎪
⎩
Npx
N − px
if px <N,
∞ if px ≥ N,
where p
∗
x is the so-called critical Sobolev exponent of px.
Proposition 2.6 see 1, 32. Let px ∈ C
0,1
R
N
, that is, Lipschitz-continuous function defined
on R
N
, then there exists a positive constant c such that
|u|
p
∗
x
≤ cu
px
, 2.12
for all u ∈ W
1,px
0
R
N
.
In the following discussions, we will use the product space
W
px,qx
: W
1,px
0
R
N
× W
1,qx
0
R
N
, 2.13
S. Ogras et al. 5
which is equipped with the norm
u, υ
px,qx
: max
u
px
υ
qx
∀u, υ ∈ W
px,qx
, 2.14
where u
px
resp., u
qx
is the norm of W
1,px
0
R
N
resp., W
1,qx
0
R
N
. The space
W
∗
px,qx
denotes the dual space of W
px,qx
and equipped with the norm ·
∗,px,qx
.Thus,
J
u, υ
∗,px,qx
D
1
Ju, υ
∗,px
D
2
Ju, υ
∗,qx
, 2.15
where W
−1,p
x
R
N
resp., W
−1,q
x
R
N
is the dual space of W
1,px
0
R
N
resp.,
W
1,qx
0
R
N
,and·
∗,px
resp., ·
∗,qx
is its norm.
For u, υ and ϕ, ψ in W
px,qx
,let
Fu, υ
R
N
Fx, ux,υxdx. 2.16
Then,
F
u, υϕ, ψD
1
Fu, υϕD
2
Fu, υψ, 2.17
where
D
1
Fu, υϕ
R
N
∂F
∂u
x, u, υϕdx,
D
2
Fu, υψ
R
N
∂F
∂υ
x, u, υψdx.
2.18
The Euler-Lagrange functional associated to P,Q is defined by
Ju, υ
R
N
1
px
|∇u|
px
dx
R
N
1
qx
|∇υ|
qx
dx − u, υ. 2.19
It is easy to verify that J ∈ C
1
W
px,qx
, R and that
J
u, υϕ, ψD
1
Ju, υϕD
2
Ju, υψ, 2.20
where
D
1
Ju, υϕ
R
N
|∇u|
px−2
∇u∇ϕdx− D
1
Fu, υϕ,
D
2
Ju, υψ
R
N
|∇υ|
qx−2
∇υ∇ψdx− D
2
Fu, υψ.
2.21
Definition 2.7. u, υ is called a weak solution of the system P,Q if
R
N
|∇u|
px−2
∇u∇ϕdx
R
N
|∇υ|
qx−2
∇υ∇ψdx
R
N
∂F
∂u
x, u, υϕdx
R
N
∂F
∂υ
x, u, υψdx,
2.22
for all ϕ, ψ ∈ W
px,qx
.
6 Journal of Inequalities and Applications
Definition 2.8. We say that J satisfies the Cerami condition C if every sequence ω
n
∈
W
px,qx
such that
|Jω
n
|≤c, 1 ω
n
J
ω
n
−→ 0 2.23
contains a convergent subsequence in the norm of W
px,qx
.
In this paper, we will use the following assumptions:
F1 F ∈ C
1
R
N
× R
2
, R and Fx, 0, 00;
F2 for all u, υ ∈ R
2
and for a.e. x ∈ R
N
∂F
∂u
x, u, υ
≤ a
1
x|u, υ|
p
−
−1
a
2
x|u, υ|
p
−1
,
∂F
∂υ
x, u, υ
≤ b
1
x|u, υ|
q
−
−1
b
2
x|u, υ|
q
−1
,
2.24
where
1 <p
−
,q
−
≤ p
,q
< p
∗
−
, q
∗
−
,
a
i
∈ L
δx
R
N
∩ L
βx
R
N
,b
i
∈ L
γx
R
N
∩ L
βx
R
N
,i 1, 2,
δx
px
px − 1
,γx
qx
qx − 1
, px
p
∗
xpx
p
∗
x − px
,
qx
q
∗
xqx
q
∗
x − qx
,βx
p
∗
xq
∗
x
p
∗
xq
∗
x − p
∗
xq
∗
x
;
2.25
F3u, υ·∇Fx, u, υ − Fx, u, υ ≤ 0 for all x, u, υ ∈ R
N
× R
2
\{0, 0}, where ∇F
∂F/∂u, ∂F/∂υ;
F4 suppose there exist two positive and bounded functions a ∈ L
N/px
R
N
and b ∈
L
N/qx
R
N
such that
lim
|u,υ|→0
sup
pxqx|Fx, u, υ|
qxax|u|
px
pxbx|υ|
qx
<λ
1
< lim
|u,υ|→∞
inf
pxqx|Fx, u, υ|
qxax|u|
px
pxbx|υ|
qx
.
2.26
Let λ
1
denote the first eigenvalue of the nonlinear eigenvalue problem in R
N
:
−Δ
px
u λax|u|
px−2
u in R
N
,
−Δ
qx
υ λbx|υ|
qx−2
υ in R
N
.
2.27
It is useful to recall the variational characterization:
λ
1
inf
R
N
1/px|∇u|
px
1/qx|∇υ|
qx
dx
R
N
ax/px|u|
px
bx/qx|υ|
qx
dx
: u, υ ∈ W
px,qx
\{0, 0}
.
2.28
We will assume that λ
1
is a positive real number for all u, υ ∈ W
px,qx
\{0, 0}. For
more details about the eigenvalue problems, we refer the reader to 17.
S. Ogras et al. 7
3. The main results
We will use the mountain pass theorem together with the following lemmas to get our main
results.
Lemma 3.1. Under the assumptions F1 and F2, the functional F is well defined, and it is of class
C
1
on W
px,qx
. Moreover, its derivative is
F
u, υω, z
R
N
∂F
∂u
x, u, υωdx
R
N
∂F
∂υ
x, u, υzdx ∀u, υ, ω, z ∈ W
px,qx
.
3.1
Proof. For all pair of real functions u, υ ∈ W
px,qx
, under the assumptions F1 and F2,
we can write
Fx, u, υ
u
0
∂F
∂s
x, s, υds Fx, 0,υ
u
0
∂F
∂s
x, s, υds
υ
0
∂F
∂s
x, 0,sds Fx, 0, 0,
Fx, u, υ ≤ c
1
a
1
x|u|
p
−
|υ|
p
−
−1
|u|a
2
x|u|
p
|υ|
p
−1
|u|b
1
x|υ|
q
−
b
2
x|υ|
q
.
3.2
Then,
R
N
Fx, u, υdx ≤ c
2
R
N
a
1
x|u|
p
−
dx
R
N
a
1
x|υ|
p
−
−1
|u|dx
R
N
a
2
x|u|
p
dx
R
N
a
2
x|υ|
p
−1
|u|dx
R
N
b
1
x|υ|
q
−
dx
R
N
b
2
x|υ|
q
dx
,
3.3
if we consider the fact that
W
1,px
0
R
N
→ L
p
−
px
R
N
⇒||u|
p
−
|
px
|u|
p
−
p
−
px
≤ cu
p
−
px
for p
−
> 1, 3.4
and if we apply Propositions 2.1, 2.3,and2.6 and take a
i
∈ L
δx
R
N
∩ L
βx
R
N
,b
i
∈
L
γx
R
N
, then we have
R
N
Fx, u, υdx ≤ 2c
1
|a
1
|
δx
||u|
p
−
|
px
|a
1
|
βx
||υ|
p
−
−1
|
q
∗
x
|u|
p
∗
x
|a
2
|
δx
||u|
p
|
px
|a
2
|
βx
||υ|
p
−1
|
q
∗
x
|u|
p
∗
x
|b
1
|
γx
||υ|
q
−
|
qx
|b
2
|
γx
||υ|
q
|
qx
2c
1
|a
1
|
δx
|u|
p
−
p
−
px
|a
1
|
βx
|υ|
p
−
−1
p
−
−1q
∗
x
|u|
p
∗
x
|a
2
|
δx
|u|
p
p
px
|a
2
|
βx
|υ|
p
−1
p
−1q
∗
x
|u|
p
∗
x
|b
1
|
γx
|υ|
q
−
q
−
qx
|b
2
|
γx
|υ|
q
q
qx
≤ c
3
|a
1
|
δx
u
p
−
px
|a
1
|
βx
υ
p
−
−1
qx
u
px
|a
2
|
δx
u
p
px
|a
2
|
βx
υ
p
−1
qx
u
px
|b
1
|
γx
υ
q
−
qx
|b
2
|
γx
υ
q
qx
< ∞.
3.5
8 Journal of Inequalities and Applications
Hence, F is well defined. Moreover, one can see easily that F
is also well defined on W
px,qx
.
Indeed, using F2 for all ω, z ∈ W
px,qx
, we have
F
u, υω, z
R
N
∂F
∂u
x, u, υωdx
R
N
∂F
∂υ
x, u, υzdx,
F
u, υω, z ≤
R
N
a
1
x|u, υ|
p
−
−1
a
2
x|u, υ|
p
−1
|ω|dx
R
N
b
1
x|u, υ|
q
−
−1
b
2
x|u, υ|
q
−1
|z|dx
≤
R
N
a
1
x|u|
p
−
−1
|ω|dx
R
N
a
1
x|υ|
p
−
−1
|ω|dx
R
N
a
2
x|u|
p
−1
|ω|dx
R
N
a
2
x|υ|
p
−1
|ω|dx
R
N
b
1
x|u|
q
−
−1
|z|dx
R
N
b
1
x|υ|
q
−
−1
|z|dx
R
N
b
2
x|u|
q
−1
|z|dx
R
N
b
2
x|υ|
q
−1
|z|dx,
3.6
and applying Propositions 2.1, 2.3,and2.6 and considering the conditions px >px and
qx >qx, it follows that
R
N
∂F
∂u
x, u, υωdx≤ 2
|a
1
|
δx
||u|
p
−
−1
|
p
∗
x
|ω|
px
|a
1
|
βx
||υ|
p
−
−1
|
q
∗
x
|ω|
p
∗
x
|a
2
|
δx
||u|
p
−1
|
p
∗
x
|ω|
px
|a
2
|
βx
||υ|
p
−1
|
q
∗
x
|ω|
p
∗
x
≤ 2
|a
1
|
δx
|u|
p
−
−1
p
−
−1p
∗
x
|ω|
px
|a
1
|
βx
|υ|
p
−
−1
p
−
−1q
∗
x
|ω|
p
∗
x
|a
2
|
δx
|u|
p
−1
p
−1p
∗
x
|ω|
px
|a
2
|
βx
|υ|
p
−1
p
−1q
∗
x
|ω|
p
∗
x
≤ c
4
|a
1
|
δx
u
p
−
−1
px
|a
1
|
βx
υ
p
−
−1
qx
|a
2
|
δx
u
p
−1
px
|a
2
|
β
υ
p
−1
qx
ω
px
< ∞,
3.7
and similarly
R
N
∂F
∂υ
x, u, υz dx
≤ c
5
|b
1
|
βx
u
q
−
−1
px
|b
1
|
γ
x
υ
q
−
−1
qx
|b
2
|
βx
u
q
−1
px
|b
2
|
γx
υ
q
−1
qx
z
qx
<∞.
3.8
Now let us show that F is differentiable in sense of Fr
´
echet, that is, for fixed u, υ ∈
W
px,qx
and given ε>0, there must be a δ ε, u, υ > 0 such that
|Fu ω, υ z −Fu, υ −F
u, υω, z|≤εω
px
z
qx
, 3.9
for all ω, z ∈ W
px,qx
with ω
px
z
qx
≤ δ.
S. Ogras et al. 9
Let B
r
be the ball of radius r which is centered at the origin of R
N
and denote B
r
R
N
− B
r
. Moreover, let us define the functional F
r
on W
1,px
0
B
r
× W
1,qx
0
B
r
as follows:
F
r
u, υ
B
r
Fx, ux,υxdx. 3.10
If we consider F1 and F2, it is easy to see that F
r
∈ C
1
W
1,px
0
B
r
× W
1,qx
0
B
r
, and in
addition for all ω, z ∈ W
1,px
0
B
r
× W
1,qx
0
B
r
, we have
F
r
u, υω, z
B
r
∂F
∂u
x, u, υωdx
B
r
∂F
∂υ
x, u, υzdx. 3.11
Also as we know, the operator F
r
: W
px,qx
→ W
∗
px,qx
is compact 3. Then, for all
u, υ, ω, z ∈ W
px,qx
, we can write
Fu ω, υ z −Fu, υ −F
u, υω, z
≤
F
r
u ω, υ z −F
r
u, υ −F
r
u, υω, z
B
r
Fx, u ω, υ z − Fx, u, υ −
∂F
∂u
x, u, υω −
∂F
∂υ
x, u, υzdx
.
3.12
By virtue of the mean-value theorem, there exist ζ
1
,ζ
2
∈ 0, 1 such that
Fx, u ω, υ z − Fx, u, υ
∂F
∂u
x, u ζ
1
ω, υω
∂F
∂υ
x, u, υ ζ
2
zz. 3.13
Using the condition F2, we have
B
r
∂F
∂u
x, u ζ
1
ω, υω
∂F
∂υ
x, u, υ ζ
2
zz −
∂F
∂u
x, u, υω −
∂F
∂υ
x, u, υz
dx
≤
B
r
a
1
x
|u ζ
1
ω|
p
−
−1
−|u|
p
−
−1
a
2
x
|u ζ
1
ω|
p
−1
−|u|
p
−1
|ω|dx
B
r
b
1
x
|υ ζ
2
z|
q
−
−1
−|υ|
q
−
−1
b
2
x
|υ ζ
2
z|
q
−1
−|υ|
q
−1
|z|dx
.
3.14
By help of the elementary inequality |a b|
s
≤ 2
s−1
|a|
s
|b|
s
for a, b ∈ R
N
, we can write
≤ 2
p
−
−1
− 1
B
r
a
1
x|u|
p
−
−1
|ω|dx ζ
1
2
p
−
−1
B
r
a
1
x|ω|
p
−
−1
|ω|dx
2
p
−1
− 1
B
r
a
2
x|u|
p
−1
|ω|dx ζ
1
2
p
−1
B
r
a
2
x|ω|
p
−1
|ω|dx
2
q
−
−1
− 1
B
r
b
1
x|υ|
q
−
−1
|z|dx ζ
2
2
q
−
−1
B
r
b
1
x|z|
q
−
−1
|z|dx
2
q
−1
− 1
B
r
b
2
x|υ|
q
−1
|z|dx ζ
2
2
q
−1
B
r
b
2
x|z|
q
−1
|z|dx,
3.15
10 Journal of Inequalities and Applications
applying Propositions 2.1, 2.3,and2.6, then we have
≤ c
6
|a
1
|
δx
||u|
p
−
−1
|
p
∗
x
|ω|
px
|a
1
|
δx
||ω|
p
−
−1
|
p
∗
x
|ω|
px
|a
2
|
δx
||u|
p
−1
|
p
∗
x
|ω|
px
|a
2
|
δx
||ω|
p
−1
|
p
∗
x
|ω|
px
|b
1
|
γx
||υ|
q
−
−1
|
q
∗
x
|z|
qx
|b
1
|
γx
||z|
q
−
−1
|
q
∗
x
|z|
qx
|b
2
|
γx
||υ|
q
−1
|
q
∗
x
|z|
qx
|b
2
|
γx
||z|
q
−1
|
q
∗
x
|z|
qx
,
≤ c
7
|a
1
|
δx
u
p
−
−1
px
|a
1
|
δx
ω
p
−
−1
px
|a
2
|
δx
u
p
−1
px
|a
2
|
δx
ω
p
−1
px
ω
px
|b
1
|
γx
υ
q
−
−1
qx
|b
1
|
γx
z
q
−
−1
qx
|b
2
|
γx
υ
q
−1
qx
|b
2
|
γx
z
q
−1
qx
z
qx
,
3.16
and by the fact that
|a
i
|
L
δx
B
r
−→ 0,
|b
i
|
L
γx
B
r
−→ 0
3.17
for i 1, 2, as r →∞, and for r sufficiently large, it follows that
B
r
Fx, u ω, υ z − Fx, u, υ −
∂F
∂u
x, u, υω −
∂F
∂υ
x, u, υ
zdx
≤ εω
px
z
qx
.
3.18
It remains only to show that F
is continuous on W
px,qx
. Let u
n
,υ
n
, u, υ ∈
W
px,qx
such that u
n
,υ
n
→ u, υ. Then, for ω, z ∈ W
px,qx
, we have
|F
u
n
,υ
n
ω, z −F
u, υω, z|≤|F
r
u
n
,υ
n
ω, z −F
r
u, υω, z|
B
r
∂F
∂u
x, u
n
,υ
n
∂F
∂u
x, u, υ
ωdx
B
r
∂F
∂υ
x, u
n
,υ
n
∂F
∂υ
x, u, υ
zdx
,
3.19
then by F2, we can write
B
r
a
1
x|u
n
|
p
−
−1
|u|
p
−
−1
|υ
n
|
p
−
−1
|υ|
p
−
−1
|ω|dx
3.20
B
r
a
2
x|u
n
|
p
−1
|u|
p
−1
|υ
n
|
p
−1
|υ|
p
−1
|ω|dx
I
1
B
r
b
1
x|u
n
|
q
−
−1
|u|
q
−
−1
|υ
n
|
q
−
−1
|υ|
q
−
−1
|z|dx
I
2
B
r
b
2
x|u
n
|
q
−1
|u|
q
−1
|υ
n
|
q
−1
|υ|
q
−1
|z|dx.
3.21
S. Ogras et al. 11
Thus,
I
1
≤
B
r
a
1
x|u
n
|
p
−
−1
|ω|dx
B
r
a
1
x|u|
p
−
−1
|ω|dx
B
r
a
1
x|υ
n
|
p
−
−1
|ω|dx
B
r
a
1
x|υ|
p
−
−1
|ω|dx
B
r
a
2
x|u
n
|
p
−1
|ω|dx
B
r
a
2
x|u|
p
−1
|ω|dx
B
r
a
2
x|υ
n
|
p
−1
|ω|dx
B
r
a
2
x|υ|
p
−1
|ω|dx
≤ c
9
|a
1
|
δx
u
n
p
−
−1
px
|a
1
|
δx
u
p
−
−1
px
|a
1
|
βx
υ
n
p
−
−1
qx
|a
1
|
βx
υ
p
−
−1
qx
|a
2
|
δx
u
n
p
−1
px
|a
2
|
δx
u
p
−1
px
|a
2
|
βx
υ
n
p
−1
qx
|a
2
|
βx
υ
p
−1
qx
ω
px.
3.22
Similarly,
I
2
≤ c
10
|b
1
|
βx
u
n
q
−
−1
px
|b
1
|
βx
u
q
−
−1
px
|b
1
|
γ
x
υ
n
q
−
−1
qx
|b
1
|
γ
x
υ
q
−
−1
qx
|b
2
|
βx
u
n
q
−1
px
|b
2
|
βx
u
q
−1
px
|b
2
|
γx
υ
n
q
−1
qx
|b
2
|
γx
υ
q
−1
qx
z
qx
.
3.23
Since F
r
is continuous on W
1,px
0
B
r
× W
1,qx
0
B
r
, then we have
|F
r
u
n
,υ
n
ω, z −F
r
u, υω, z|−→0, 3.24
as n →∞. Moreover, using 3.17, when r sufficiently large, I
1
and I
2
tend also to 0. Hence,
|F
u
n
,υ
n
ω, z −F
u, υω, z|−→0, 3.25
as u
n
,υ
n
→ u, υ, this implies F
is continuous on W
px,qx
.
Lemma 3.2. Under the assumptions F1 and F2, F
is compact from W
px,qx
to W
∗
px,qx
.
Proof. Let u
n
,υ
n
be a bounded sequence in W
px,qx
. Then, there exists a subsequence we
denote again as u
n
,υ
n
which converges weakly in W
px,qx
to a u, υ ∈ W
px,qx
. Then, if
we use the same arguments as above, we have
|F
u
n
,υ
n
ω, z −F
u, υω, z|≤|F
r
u
n
,υ
n
ω, z −F
r
u, υω, z|
B
r
∂F
∂u
x, u
n
,υ
n
−
∂F
∂u
x, u, υ
ωdx
B
r
∂F
∂υ
x, u
n
,υ
n
−
∂F
∂υ
x, u, υ
zdx
.
3.26
Since the restriction operator is continuous, then u
n
,υ
n
u, υ in W
1,px
0
B
r
× W
1,qx
0
B
r
.
Because of the compactness of F
r
, the first expression on the right-hand side of the equation
tends to 0, as n →∞, and as we did above, when r sufficiently large, I
1
and I
2
tend also to
0. This implies F
is compact from W
px,qx
to W
∗
px,qx
.
12 Journal of Inequalities and Applications
Lemma 3.3. If F1, F2, and F3 hold, then J satisfies the condition C, that is, there exists a
sequence u
n
,υ
n
∈ W
px,qx
such that
i |Ju
n
,υ
n
|≤c,
ii1 u
n
px
υ
n
qx
J
u
n
,υ
n
∗,px,qx
→ 0 as n → ∞
contains a convergent subsequence.
Proof. By the assumption ii, it is clear that J
u
n
,υ
n
ω, z ≤ ξ
n
→ 0asn →∞for all
ω, z ∈ W
px,qx
. Let us choose ω, zu
n
,υ
n
, then we have
ξ
n
≥ J
u
n
,υ
n
u
n
,υ
n
≥u
n
p
−
px
υ
n
q
−
qx
−
R
N
∂F
∂u
x, u
n
,υ
n
u
n
∂F
∂υ
x, u
n
,υ
n
υ
n
dx.
3.27
Moreover, by the assumption i, we can write
c ≥−Ju
n
,υ
n
≥−
1
p
u
n
p
−
px
−
1
q
υ
n
q
−
qx
R
N
Fx, u
n
,υ
n
dx. 3.28
Using the assumption F3, it follows that
ξ
n
c ≥ J
u
n
,υ
n
u
n
,υ
n
− Ju
n
,υ
n
≥
1 −
1
p
u
n
p
−
px
1 −
1
q
υ
n
q
−
qx
R
N
Fx, u
n
,υ
n
dx
−
R
N
∂F
∂u
x, u
n
,υ
n
u
n
∂F
∂υ
x, u
n
,υ
n
υ
n
dx
≥
1 −
1
p
u
n
p
−
px
1 −
1
q
υ
n
q
−
qx
.
3.29
Thus, the sequence u
n
,υ
n
is bounded in W
px,qx
. Then, there exists a subsequence we
denote again as u
n
,υ
n
which converges weakly in W
px,qx
.
We recall the elementary inequalities:
2
2−p
|a − b|
p
≤ a|a|
p−2
− b|b|
p−2
·a − b if p ≥ 2, 3.30
p − 1|a − b|
2
|a| |b|
p−2
≤ a|a|
p−2
− b|b|
p−2
·a − b if 1 <p<2, 3.31
for all a, b ∈ R
N
, where · denotes the standard inner product in R
N
. We will show that u
n
,υ
n
contains a Cauchy subsequence. Let us define the sets
U
p
{x ∈ R
N
: px ≥ 2}, V
p
{x ∈ R
N
:1<px < 2},
U
q
{x ∈ R
N
: qx ≥ 2}, V
q
{x ∈ R
N
:1<qx < 2}.
3.32
For all x ∈ R
N
, we put
Φ
n,k
|∇u
n
|
px−2
∇u
n
−|∇u
k
|
px−2
∇u
k
·∇u
n
−∇u
k
,
Ψ
n,k
|∇u
n
| |∇u
k
|
2−px
.
3.33
S. Ogras et al. 13
Therefore for px ≥ 2, using 3.30, we have
2
2−p
U
p
|∇u
n
−∇u
k
|
px
dx ≤
U
p
|∇u
n
|
px−2
∇u
n
−|∇u
k
|
px−2
∇u
k
·∇u
n
−∇u
k
dx
≤
R
N
Φ
n,k
dx : T
n,k
D
1
Ju
n
,υ
n
−D
1
Ju
k
,υ
k
D
1
Fu
n
,υ
n
−D
1
Fu
k
,υ
k
u
n
−u
k
≤
D
1
Ju
n
,υ
n
∗,px,qx
D
1
Ju
k
,υ
k
∗,px,qx
u
n
− u
k
px
D
1
Fu
n
,υ
n
− D
1
Fu
k
,υ
k
∗,px,qx
u
n
− u
k
px
.
3.34
When 1 <px < 2, employing 3.31 and Proposition 2.2, it follows
V
p
|∇u
n
−∇u
k
|
px
dx≤
V
p
|∇u
n
−∇u
k
|
px
|∇u
n
||∇u
k
|
pxpx−2/2
|∇u
n
||∇u
k
|
px2−px/2
dx
≤ 2
|∇u
n
−∇u
k
|
px
.Ψ
−px/2
n,k
2/px
×
Ψ
px/2
n,k
2/2−px
≤2 max
R
N
|∇u
n
−∇u
k
|
2
Ψ
−1
n,k
dx
p
−
/2
,
R
N
|∇u
n
−∇u
k
|
2
Ψ
−1
n,k
dx
p
/2
× max
R
N
Ψ
px/2−px
n,k
dx
2−p
−
/2
,
R
N
Ψ
px/2−px
n,k
dx
2−p
/2
≤ 2 max
p
−
− 1
−p
−
/2
· T
p
−
/2
n,k
, p
−
− 1
−p
/2
· T
p
/2
n,k
× max
R
N
Ψ
px/2−px
n,k
dx
2−p
−
/2
,
R
N
Ψ
px/2−px
n,k
dx
2−p
/2
.
3.35
Since T
n,k
is uniformly bounded in W
1,px
0
R
N
in accordance with n, k, and by the fact
that J
u
m
,υ
m
∗,px,qx
→ 0asm → ∞, F
is compact and by Proposition 2.4, we have
lim
n,k→∞
R
N
|∇u
n
−∇u
k
|
px
dx 0. 3.36
Applying the same arguments, we can find a subsequence of u
n
,υ
n
such that
lim
n,k→∞
R
N
|∇υ
n
−∇υ
k
|
qx
dx 0. 3.37
Therefore by Proposition 2.2, for a convenient subsequence, we have
lim
n,k→∞
u
n
,υ
n
− u
k
,υ
k
∗,px,qx
0. 3.38
Hence, u
n
,υ
n
contains a Cauchy subsequence and so contains a strongly convergent
subsequence. The proof is complete.
14 Journal of Inequalities and Applications
Lemma 3.4. Under the assumptions F1–F4, the functional J satisfies the following.
i There exists ρ,σ > 0 such that u
px
υ
qx
ρ implies Ju, υ ≥ σ>0.
ii There exists E ∈ W
px,qx
such that E
px,qx
>ρand JE ≤ 0.
Proof. By F4, we can find ρ>0 such that u
px
υ
qx
ρ, so we have
Fx, u, υ <λ
1
ax
px
|u|
px
bx
qx
|υ|
qx
,
R
N
Fx, u, υ <λ
1
R
N
ax
px
|u|
px
bx
qx
|υ|
qx
dx,
3.39
since λ
1
> 0, then we have
R
N
Fx, u, υ <
R
N
1
px
|∇u|
px
1
qx
|∇υ|
qx
dx,
0 <
R
N
1
px
|∇u|
px
1
qx
|∇υ|
qx
dx −
R
N
Fx, u, υJu, υ.
3.40
Hence, there exists σ>0 such that J ≥ σ>0.
Let τ,θ be an eigenfunction relative to λ
1
. Then, using the assumption F4, we can
obtain for >0andt sufficiently large,
Fx, t
1/px
τ,t
1/qx
θ ≥ tλ
1
ax
px
|τ|
px
bx
qx
|θ|
qx
. 3.41
Thus,
Jt
1/px
τ,t
1/qx
θt
R
N
1
px
|∇τ|
px
1
qx
|∇θ|
qx
dx
−
R
N
Fx, t
1/px
τ,t
1/qx
θdx
≤
R
N
1
px
|∇τ|
px
1
qx
|∇θ|
qx
dx
− tλ
1
R
N
ax
px
|τ|
px
dx
R
N
bx
qx
|θ|
qx
dx
,
3.42
then it follows
Jt
1/px
τ,t
1/qx
θ ≤−t
1
p
R
N
ax|τ|
px
dx
1
q
R
N
bx|θ|
qx
dx
. 3.43
So, we can conclude that lim
t→∞
Jt
1/px
τ,t
1/qx
θ−∞. Hence, for t sufficiently large,
Jt
1/px
τ,t
1/qx
θ ≤ 0. As a consequence, we can say that the functional Ju, υ has a critical
point; and as we know, the critical points of Ju, υ are the weak solutions of the system
P,Q.
Theorem 3.5. The system P,Q has at least one nontrivial solution u, υ.
Proof. By Lemmas 3.3 and 3.4, we can apply the mountain pass theorem to obtain that the
system P,Q has a nontrivial weak solution.
S. Ogras et al. 15
Acknowledgments
The authors want to express their gratitude to Professor M. G. Huidobro and to anonymous
referee for a careful reading and valuable suggestions. This research project was supported
by DUAPK -2008-59-74, Dicle University, Turkey.
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