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 RN 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, mrabil@dicle.edu.tr
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 pxLaplacian. 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:
−Δpx u ∂F
x, u, υ in RN ,
∂u
∂F
x, u, υ in RN ,
−Δqx υ ∂υ
P,Q
where px and qx are two functions such that 1 < px, qx < N N ≥ 2, for every
x ∈ RN . However, F ∈ C1 RN × R2 and Δpx is the px-Laplacian operator defined by
Δpx 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 Lpx and Sobolev space
1,px
1–5 have been a subject of active research stimulated mainly by the development
W
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 Δpx 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 Δpx 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 pLaplacian problem:
−Δp u ∂F
x, u, υ
∂u
in RN ,
−Δq υ ∂F
x, u, υ
∂υ
in RN ,
1.1
where F ∈ C1 RN × R2 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
x, u, υ
∂u
− div|∇υ|qx−2 ∇υ ∂F
x, u, υ in Ω,
∂υ
u 0,
in Ω,
1.2
υ 0 on Ω,
where Ω ⊂ RN is a bounded domain with a smooth boundary ∂Ω, N ≥ 2, p, q ∈
2
CΩ , px > 1, qx > 1, for every x ∈ Ω. The function F is assumed to be continuous
in x ∈ Ω and of class C1 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 PalaisSmale 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
x, u, υ
∂u
− div|∇υ|qx−2 ∇υ |υ|qx−2 υ ∂F
x, u, υ in RN ,
∂υ
in RN ,
1.3
u, υ ∈ W 1,px RN × W 1,qx RN ,
where N ≥ 2, px, qx are functions on RN . The function F is assumed to satisfy
Caratheodory conditions and to be L∞ in x ∈ RN and C1 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 RN , the
main difficulty arises from the lack 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
1,px
RN ,
We will investigate our problem P,Q in the variable exponent Sobolev space W0
px
N
1,px
RN .
so we need to recall some theories and basic properties on spaces L R and W
Set
C RN h ∈ CRN : inf hx > 1 .
2.1
x∈RN
For every h ∈ C RN , denote
h− : inf hx,
h : sup hx.
x∈RN
2.2
x∈RN
Let us define by URN the set of all measurable real-valued functions defined on RN .
For p ∈ C RN , we denote the variable exponent Lebesgue space by
px
px
N
N
L R u ∈ UR :
|ux|
dx < ∞ ,
2.3
RN
which is equipped with the norm, so-called Luxemburg norm 1, 3, 4:
|u|px : |u|Lpx RN inf λ > 0 :
ux px
dx ≤ 1 ,
λ RN
2.4
and Lpx RN , |·|Lpx RN becomes a Banach space, we call it generalized Lebesgue space.
Define the variable exponent Sobolev space W 1,px RN by
W 1,px RN {u ∈ Lpx RN : |∇u| ∈ Lpx RN },
2.5
and it can be equipped with the norm
u1,px : uW 1,px |u|px |∇u|px
1,px
The space W0
∀u ∈ W 1,px RN .
2.6
RN is denoted by the closure of C0∞ RN in W 1,px RN and it is
1,px
equipped with the norm for all u ∈ W0
RN :
upx |∇u|px
1,px
∀u ∈ W0
RN .
1,px
If p− > 1, then the spaces Lpx RN , W 1,px RN , and W0
reflexive Banach spaces.
2.7
RN are separable and
4
Journal of Inequalities and Applications
Proposition 2.1 see 1, 3, 4. The conjugate space of Lpx RN is Lp x RN , where 1/p x 1/px 1. For any u ∈ Lpx RN and v ∈ Lp x RN , we have
≤ 1 1
uv
dx
|u|px |v|p x ≤ 2|u|px |v|p x .
N
p− p −
R
Proposition 2.2 see 1, 3, 4. Denote px u RN
2.8
|ux| px dx for all u ∈ Lpx RN , one has
p−
p−
p
p
min |u|px , |u|px ≤ px u ≤ max |u|px , |u|px .
2.9
Proposition 2.3 see 1. Let px and qx be measurable functions such that px ∈ L∞ RN 0. Then,
and 1 ≤ pxqx ≤ ∞, for a.e. x ∈ RN . Let u ∈ Lqx RN , u /
p
p−
|u|pxqx ≤ 1 ⇒ |u|pxqx ≤ |u|px qx ≤ |u|pxqx ,
|u|pxqx ≥ 1 ⇒
p−
|u|pxqx
p
≤ |u|px qx ≤ |u|pxqx .
2.10
In particular, if px p is constant, then
p
||u|p |qx |u|pqx .
2.11
Proposition 2.4 see 3, 4. If u, un ∈ Lpx RN , n 1, 2, . . . , then the following statements are
equivalent to each other:
1 limn→∞ |un − u|px 0,
2 limn→∞ un − u 0,
3 un → u in measure in RN and limn→∞ un u.
Definition 2.5. 1 < px < N and for all x ∈ RN , let define
⎧
⎪
⎨ Npx
p∗ x 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 ∈ C0,1 RN , that is, Lipschitz-continuous function defined
on RN , then there exists a positive constant c such that
|u|p∗ x ≤ cupx ,
1,px
for all u ∈ W0
2.12
RN .
In the following discussions, we will use the product space
1,px
Wpx,qx : W0
1,qx
RN × W0
RN ,
2.13
S. Ogras et al.
5
which is equipped with the norm
u, υpx,qx : max upx υqx
∀u, υ ∈ Wpx,qx ,
1,px
2.14
1,qx
where upx resp., uqx is the norm of W0
RN resp., W0
RN . The space
∗
denotes the dual space of Wpx,qx and equipped with the norm ·∗,px,qx . Thus,
Wpx,qx
J u, υ∗,px,qx D1 Ju, υ∗,px D2 Ju, υ∗,qx ,
2.15
1,px
where W −1,p x RN resp., W −1,q x RN is the dual space of W0
RN resp.,
1,qx
W0
RN ,
and ·∗,px resp., ·∗,qx is its norm.
For u, υ and ϕ, ψ in Wpx,qx , let
Fx, ux, υxdx.
Fu, υ RN
2.16
Then,
F u, υϕ, ψ D1 Fu, υϕ D2 Fu, υψ,
where
D1 Fu, υϕ RN
∂F
x, u, υϕ dx,
∂u
∂F
D2 Fu, υψ x, u, υψ dx.
RN ∂υ
The Euler-Lagrange functional associated to P,Q is defined by
1
1
Ju, υ |∇u|px dx |∇υ|qx dx − u, υ.
RN px
RN qx
2.17
2.18
2.19
It is easy to verify that J ∈ C1 Wpx,qx , R and that
J u, υϕ, ψ D1 Ju, υϕ D2 Ju, υψ,
where
2.20
D1 Ju, υϕ RN
|∇u|px−2 ∇u∇ϕ dx − D1 Fu, υϕ,
2.21
D2 Ju, υψ qx−2
RN
|∇υ|
∇υ∇ψ dx − D2 Fu, υψ.
Definition 2.7. u, υ is called a weak solution of the system P,Q if
∂F
∂F
px−2
qx−2
x, u, υϕ dx x, u, υψ dx,
|∇u|
∇u∇ϕ dx
|∇υ|
∇υ∇ψ dx ∂u
RN
RN
RN
RN ∂υ
2.22
for all ϕ, ψ ∈ Wpx,qx .
6
Journal of Inequalities and Applications
Definition 2.8. We say that J satisfies the Cerami condition C if every sequence ωn ∈
Wpx,qx such that
1 ωn J ωn −→ 0
|Jωn | ≤ c,
2.23
contains a convergent subsequence in the norm of Wpx,qx .
In this paper, we will use the following assumptions:
F1 F ∈ C1 RN × R2 , R and Fx, 0, 0 0;
F2 for all u, υ ∈ R2 and for a.e. x ∈ RN
∂F
x, u, υ ≤ a1 x|u, υ|p− −1 a2 x|u, υ|p −1 ,
∂u
∂F
x, u, υ ≤ b1 x|u, υ|q− −1 b2 x|u, υ|q −1 ,
∂υ
2.24
where
1 < p− ,
q− ≤ p ,
ai ∈ Lδx RN ∩ Lβx RN ,
δx qx
px
,
px − 1
bi ∈ Lγx RN ∩ Lβx RN ,
γx q∗ xqx
,
− qx
q∗ x
q < p∗ − , q∗ − ,
qx
,
qx − 1
βx px i 1, 2,
p∗ xpx
,
p∗ x − px
2.25
p∗ xq∗ x
;
− p∗ x q∗ x
p∗ xq∗ x
F3 u, υ·∇Fx, u, υ − Fx, u, υ ≤ 0 for all x, u, υ ∈ RN × R2 \ {0, 0}, where ∇F ∂F/∂u, ∂F/∂υ;
F4 suppose there exist two positive and bounded functions a ∈ LN/px RN and b ∈
LN/qx RN such that
lim sup
|u,υ|→0
pxqx|Fx, u, υ|
qxax|u|px pxbx|υ|qx
< λ1 <
lim
|u,υ|→∞
inf
pxqx|Fx, u, υ|
qxax|u|px pxbx|υ|qx
2.26
.
Let λ1 denote the first eigenvalue of the nonlinear eigenvalue problem in RN :
−Δpx u λax|u|px−2 u
in RN ,
−Δqx υ λbx|υ|qx−2 υ
in RN .
2.27
It is useful to recall the variational characterization:
1/px|∇u|px 1/qx|∇υ|qx dx
RN
λ1 inf : u, υ ∈ Wpx,qx \ {0, 0} .
ax/px|u|px bx/qx|υ|qx dx
RN
2.28
We will assume that λ1 is a positive real number for all u, υ ∈ Wpx,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
C1 on Wpx,qx . Moreover, its derivative is
F u, υω, z ∂F
x, u, υω dx RN ∂u
∂F
x, u, υz dx
RN ∂υ
∀u, υ, ω, z ∈ Wpx,qx .
3.1
Proof. For all pair of real functions u, υ ∈ Wpx,qx , under the assumptions F1 and F2,
we can write
Fx, u, υ u
0
∂F
x, s, υds Fx, 0, υ ∂s
−
Fx, u, υ ≤ c1 a1 x|u|p |υ|p
−
−1
u
0
∂F
x, s, υds ∂s
|u| a2 x|u|p |υ|p
−1
υ
0
∂F
x, 0, sds Fx, 0, 0,
∂s
−
|u| b1 x|υ|q b2 x|υ|q .
3.2
Then,
RN
Fx, u, υdx ≤ c2
−
RN
a1 x|u|p dx p −1
RN
a2 x|υ|
RN
a1 x|υ|p
−
−1
|u|dx |u|dx q−
RN
RN
a2 x|u|p dx
b1 x|υ| dx RN
b2 x|υ| dx ,
3.3
for p− > 1,
3.4
q
if we consider the fact that
1,px
W0
RN → Lp
−
px
p−
−
p−
RN ⇒ ||u|p |px |u|p− px ≤ cupx
and if we apply Propositions 2.1, 2.3, and 2.6 and take ai ∈ Lδx RN ∩ Lβx RN , bi ∈
Lγx RN , then we have
RN
−
−
Fx, u, υdx ≤ 2c1 |a1 |δx ||u|p |px |a1 |βx ||υ|p −1 |q∗ x |u|p∗ x |a2 |δx ||u|p |px
|a2 |βx ||υ|p
−1
−
|q∗ x |u|p∗ x |b1 |γx ||υ|q |qx |b2 |γx ||υ|q |qx
p−
p− −1
p
2c1 |a1 |δx |u|p− px |a1 |βx |υ|p− −1q∗ x |u|p∗ x |a2 |δx |u|p px
p −1
|a2 |βx |υ|p −1q∗ x |u|p∗ x
q−
|b1 |γx |υ|q− qx
q
|b2 |γx |υ|q qx
p−
p− −1
p
≤ c3 |a1 |δx upx |a1 |βx υqx upx |a2 |δx upx
p −1
q−
q
|a2 |βx υqx upx |b1 |γx υqx |b2 |γx υqx < ∞.
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 Wpx,qx .
Indeed, using F2 for all ω, z ∈ Wpx,qx , we have
∂F
∂F
x, u, υω dx x, u, υz dx,
F u, υω, z RN ∂u
RN ∂υ
−
a1 x|u, υ|p −1 a2 x|u, υ|p −1 |ω|dx
F u, υω, z ≤
RN
≤
RN
RN
b1 x|u, υ|q
a1 x|u|p
−
−1
RN
RN
RN
a2 x|u|p
−1
b2 x|u, υ|q
|ω|dx −1
q− −1
b1 x|u|
q −1
b2 x|u|
−
RN
a1 x|υ|p
−
−1
|ω|dx |z|dx RN
RN
a2 x|υ|p
RN
|z|dx
|ω|dx
−1
3.6
|ω|dx
b1 x|υ|q
−
−1
|z|dx
b2 x|υ|q
−1
|z|dx,
|z|dx −1
and applying Propositions 2.1, 2.3, and 2.6 and considering the conditions px > px and
qx
> qx, it follows that
−
−
∂F
x, u, υω dx ≤ 2 |a1 |δx ||u|p −1 |p∗ x |ω|px |a1 |βx ||υ|p −1 |q∗ x |ω|p∗ x
RN ∂u
|a2 |δx ||u|p −1 |p∗ x |ω|px |a2 |βx ||υ|p −1 |q∗ x |ω|p∗ x
p− −1
p− −1
≤ 2 |a1 |δx |u|p− −1p∗ x |ω|px |a1 |βx |υ|p− −1q∗ x |ω|p∗ x
p −1
p −1
|a2 |δx |u|p −1p∗ x |ω|px |a2 |βx |υ|p −1q∗ x |ω|p∗ x
p− −1
p− −1
p −1
p −1
≤ c4 |a1 |δx upx |a1 |βx υqx |a2 |δx upx |a2 |β υqx ωpx
< ∞,
3.7
and similarly
∂F
x, u, υz dx
RN ∂υ
q− −1
q− −1
q −1
q −1
≤ c5 |b1 |βx upx |b1 |γx υqx |b2 |βx upx |b2 |γx υqx zqx < ∞.
3.8
Now let us show that F is differentiable in sense of Fréchet, that is, for fixed u, υ ∈
Wpx,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| ≤ εωpx zqx ,
for all ω, z ∈ Wpx,qx with ωpx zqx ≤ δ.
3.9
S. Ogras et al.
9
Let Br be the ball of radius r which is centered at the origin of RN and denote Br 1,px
1,qx
Br × W0
Br as follows:
RN − Br . Moreover, let us define the functional Fr on W0
Fr u, υ 3.10
Fx, ux, υxdx.
Br
1,px
If we consider F1 and F2, it is easy to see that Fr ∈ C1 W0
1,px
addition for all ω, z ∈ W0
1,qx
Br × W0
Fr u, υω, z
1,qx
Br × W0
Br , and in
Br , we have
∂F
x, u, υω dx Br ∂u
∂F
x, u, υz dx.
Br ∂υ
3.11
∗
Also as we know, the operator Fr : Wpx,qx → Wpx,qx
is compact 3. Then, for all
u, υ, ω, z ∈ Wpx,qx , we can write
Fu ω, υ z − Fu, υ − F u, υω, z
≤ Fr u ω, υ z − Fr u, υ − Fr u, υω, z
∂F
∂F
Fx, u ω, υ z − Fx, u, υ −
x, u, υω −
x, u, υzdx.
∂u
∂υ
Br
3.12
By virtue of the mean-value theorem, there exist ζ1 , ζ2 ∈ 0, 1 such that
∂F
∂F
x, u ζ1 ω, υω x, u, υ ζ2 zz.
∂u
∂υ
Fx, u ω, υ z − Fx, u, υ 3.13
Using the condition F2, we have
∂F
∂F
∂F
∂F
x,
u
ζ
x,
u,
υ
ζ
x,
u,
υω
−
x,
u,
υz
dx
ω,
υω
zz
−
1
2
∂u
∂υ
∂u
∂υ
Br
−
−
≤ a1 x |u ζ1 ω|p −1 − |u|p −1 a2 x |u ζ1 ω|p −1 − |u|p −1 |ω|dx
Br
Br
b1 x |υ ζ2 z|
q− −1
q− −1 − |υ|
b2 x |υ ζ2 z|
q −1
3.14
|z|dx.
q −1 − |υ|
By help of the elementary inequality |a b|s ≤ 2s−1 |a|s |b|s for a, b ∈ RN , we can write
p− −1
≤ 2
2p
− 1
−1
q− −1
2
2q
−1
p− −1
Br
a1 x|u|
− 1
Br
− 1
Br
a2 x|u|p
Br
−1
q− −1
b1 x|υ|
− 1
|ω|dx ζ1 2
b2 x|υ|q
−1
p− −1
Br
|ω|dx ζ1 2p
|z|dx ζ2 2
−1
q− −1
|z|dx ζ2 2q
a1 x|ω|p
−1
−1
Br
Br
−
a2 x|ω|p
−1
q− −1
b1 x|z|
Br
|ω|dx
b2 x|z|q
−1
|ω|dx
3.15
|z|dx
|z|dx,
10
Journal of Inequalities and Applications
applying Propositions 2.1, 2.3, and 2.6, then we have
−
−
≤ c6 |a1 |δx ||u|p −1 |p∗ x |ω|px |a1 |δx ||ω|p −1 |p∗ x |ω|px
|a2 |δx ||u|p
|b1 |γx ||υ|q
−
−1
−1
|p∗ x |ω|px |a2 |δx ||ω|p
|q∗ x |z|qx
|b1 |γx ||z|q
−
−1
−1
|p∗ x |ω|px
|q∗ x |z|qx
3.16
,
|b2 |γx ||υ|q −1 |q∗ x |z|qx
|b2 |γx ||z|q −1 |q∗ x |z|qx
p− −1
p− −1
p −1
p −1
≤ c7 |a1 |δx upx |a1 |δx ωpx |a2 |δx upx |a2 |δx ωpx ωpx
q− −1
q− −1
q −1
q −1
|b1 |γx υqx |b1 |γx zqx |b2 |γx υqx |b2 |γx zqx zqx ,
and by the fact that
|ai |Lδx Br −→ 0,
3.17
|bi |Lγx Br −→ 0
for i 1, 2, as r → ∞, and for r sufficiently large, it follows that
∂F
∂F
≤ εω
x,
u,
υω
−
x,
u,
υ
zdx
Fx,
u
ω,
υ
z
−
Fx,
u,
υ
−
px zqx .
∂u
∂υ
Br
3.18
It remains only to show that F is continuous on Wpx,qx . Let un , υn , u, υ ∈
Wpx,qx such that un , υn → u, υ. Then, for ω, z ∈ Wpx,qx , we have
|F un , υn ω, z − F u, υω, z| ≤ |Fr un , υn ω, z − Fr u, υω, z|
∂F
∂F
x, un , υn x, u, υ ω dx
∂u
∂u
Br
∂F
∂F
x, un , υn x, u, υ z dx,
∂υ
∂υ
Br
3.19
then by F2, we can write
Br
a1 x|un |p
−
−1
|u|p
Br
a2 x|un |p
Br
Br
−1
|υn |p
−1
|u|p
−
−1
|υ|p
−1
|υn |p
−
−1
|ω|dx
−1
|υ|p
3.20
−1
|ω|dx
I1 b1 x|un |q
−
−1
|u|q
−
−1
|υn |q
−
−1
|υ|q
−
−1
|z|dx
I2 b2 x|un |q
−1
|u|q
−1
|υn |q
−1
|υ|q
−1
|z|dx.
3.21
−
S. Ogras et al.
11
Thus,
I1 ≤
Br
a1 x|un |p
−
−1
Br
a1 x|υ|p
−
−1
Br
|ω|dx a2 x|υn |p
Br
a1 x|u|p
−
−1
|ω|dx −1
Br
a2 x|un |p
a2 x|υ|p
|ω|dx |ω|dx Br
−1
−1
Br
a1 x|υn |p
−
−1
|ω|dx Br
a2 x|u|p
|ω|dx
−1
|ω|dx
3.22
|ω|dx
p− −1
p− −1
p− −1
p− −1
≤ c9 |a1 |δx un px |a1 |δx upx |a1 |βx υn qx |a1 |βx υqx
p −1
p −1
p −1
p −1
|a2 |δx un px |a2 |δx upx |a2 |βx υn qx |a2 |βx υqx ωpx.
Similarly,
q− −1
q− −1
q− −1
q− −1
I2 ≤ c10 |b1 |βx un px |b1 |βx upx |b1 |γx υn qx |b1 |γx υqx
q −1
q −1
q −1
q −1
|b2 |βx un px |b2 |βx upx |b2 |γx υn qx |b2 |γx υqx zqx .
1,px
Since Fr is continuous on W0
1,qx
Br × W0
3.23
Br , then we have
|Fr un , υn ω, z − Fr u, υω, z| −→ 0,
3.24
as n → ∞. Moreover, using 3.17, when r sufficiently large, I1 and I2 tend also to 0. Hence,
|F un , υn ω, z − F u, υω, z| −→ 0,
3.25
as un , υn → u, υ, this implies F is continuous on Wpx,qx .
∗
Lemma 3.2. Under the assumptions F1 and F2, F is compact from Wpx,qx to Wpx,qx
.
Proof. Let un , υn be a bounded sequence in Wpx,qx . Then, there exists a subsequence we
denote again as un , υn which converges weakly in Wpx,qx to a u, υ ∈ Wpx,qx . Then, if
we use the same arguments as above, we have
|F un , υn ω, z − F u, υω, z| ≤ |Fr un , υn ω, z − Fr u, υω, z|
∂F
∂F
x, un , υn −
x, u, υ ω dx
∂u
∂u
Br
∂F
∂F
x, un , υn −
x, u, υ z dx.
∂υ
∂υ
Br
1,px
3.26
1,qx
Since the restriction operator is continuous, then un , υn u, υ in W0
Br × W0
Br .
Because of the compactness of Fr , 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, I1 and I2 tend also to
∗
.
0. This implies F is compact from Wpx,qx to Wpx,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 un , υn ∈ Wpx,qx such that
i |Jun , υn | ≤ c,
ii 1 un px υn qx J un , υn ∗,px,qx → 0 as n → ∞
contains a convergent subsequence.
Proof. By the assumption ii, it is clear that J un , υn ω, z ≤ ξn → 0 as n → ∞ for all
ω, z ∈ Wpx,qx . Let us choose ω, z un , υn , then we have
ξn ≥ J un , υn un , υn p−
q−
≥ un px υn qx −
RN
∂F
∂F
x, un , υn un x, un , υn υn dx.
∂u
∂υ
Moreover, by the assumption i, we can write
1
1
p−
q−
c ≥ −Jun , υn ≥ − un px − υn qx p
q
3.27
RN
Fx, un , υn dx.
3.28
Using the assumption F3, it follows that
ξn c ≥ J un , υn un , υn − Jun , υn 1
1
p−
q−
Fx, un , υn dx
≥ 1 − un px 1 − υn qx p
q
RN
∂F
∂F
x, un , υn un x, un , υn υn dx
−
∂υ
RN ∂u
1
1
p−
q−
≥ 1 − un px 1 − υn qx .
p
q
3.29
Thus, the sequence un , υn is bounded in Wpx,qx . Then, there exists a subsequence we
denote again as un , υn which converges weakly in Wpx,qx .
We recall the elementary inequalities:
22−p |a − b|p ≤ a|a|p−2 − b|b|p−2 ·a − b
if p ≥ 2,
p − 1|a − b|2 |a| |b|p−2 ≤ a|a|p−2 − b|b|p−2 ·a − b
if 1 < p < 2,
3.30
3.31
for all a, b ∈ RN , where · denotes the standard inner product in RN . We will show that un , υn contains a Cauchy subsequence. Let us define the sets
Up {x ∈ RN : px ≥ 2},
Vp {x ∈ RN : 1 < px < 2},
Uq {x ∈ RN : qx ≥ 2},
Vq {x ∈ RN : 1 < qx < 2}.
3.32
For all x ∈ RN , we put
Φn,k |∇un |px−2 ∇un − |∇uk |px−2 ∇uk ·∇un − ∇uk ,
Ψn,k |∇un | |∇uk |2−px .
3.33
S. Ogras et al.
13
Therefore for px ≥ 2, using 3.30, we have
|∇un − ∇uk |px dx ≤
|∇un |px−2 ∇un − |∇uk |px−2 ∇uk ·∇un − ∇uk dx
22−p
Up
Up
≤
RN
Φn,k dx : Tn,k
D1 Jun , υn −D1 Juk , υk D1 Fun , υn −D1 Fuk , υk un −uk ≤ D1 Jun , υn ∗,px,qx D1 Juk , υk ∗,px,qx un − uk px
D1 Fun , υn − D1 Fuk , υk ∗,px,qx un − uk px .
3.34
When 1 < px < 2, employing 3.31 and Proposition 2.2, it follows
|∇un −∇uk |px dx ≤ |∇un −∇uk |px |∇un ||∇uk |pxpx−2/2 |∇un ||∇uk |px2−px/2 dx
Vp
Vp
−px/2 ≤ 2|∇un − ∇uk |px .Ψn,k
≤ 2 max
RN
2/2−px
p− /2 p /2
2 −1
|∇un −∇uk |2 Ψ−1
dx
,
|∇u
−∇u
|
Ψ
dx
n
k
n,k
n,k
× max
2/px
px/2 × Ψn,k RN
RN
2−p− /2 2−p /2 px/2−px
px/2−px
Ψn,k
dx
,
Ψn,k
dx
RN
p− /2
p /2 −p− /2
−p /2
≤ 2 max p− − 1
· Tn,k , p− − 1
· Tn,k
× max
RN
px/2−px
Ψn,k
dx
2−p− /2 2−p /2 px/2−px
,
Ψn,k
dx
.
RN
3.35
1,px
Since Tn,k is uniformly bounded in W0
RN in accordance with n, k, and by the fact
that J um , υm ∗,px,qx → 0 as m → ∞, F is compact and by Proposition 2.4, we have
lim
n,k→∞ RN
|∇un − ∇uk |px dx 0.
Applying the same arguments, we can find a subsequence of un , υn such that
lim
|∇υn − ∇υk |qx dx 0.
n,k→∞ RN
3.36
3.37
Therefore by Proposition 2.2, for a convenient subsequence, we have
lim un , υn − uk , υk ∗,px,qx 0.
n,k→∞
3.38
Hence, un , υn 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 upx υqx ρ implies Ju, υ ≥ σ > 0.
ii There exists E ∈ Wpx,qx such that Epx,qx > ρ and JE ≤ 0.
Proof. By F4, we can find ρ > 0 such that upx υqx ρ, so we have
ax px bx qx
|u|
|υ|
Fx, u, υ < λ1
,
px
qx
ax px bx qx
|u|
|υ|
Fx, u, υ < λ1
dx,
qx
RN
RN px
since λ1 > 0, then we have
1
1
px
qx
|∇u|
|∇υ|
Fx, u, υ <
dx,
qx
RN
RN px
1
1
|∇u|px |∇υ|qx dx −
Fx, u, υ Ju, υ.
0<
qx
RN px
RN
3.39
3.40
Hence, there exists σ > 0 such that J ≥ σ > 0.
Let τ, θ be an eigenfunction relative to λ1 . Then, using the assumption F4, we can
obtain for > 0 and t sufficiently large,
ax px bx qx
|τ|
|θ|
.
3.41
Fx, t1/px τ, t1/qx θ ≥ tλ1 px
qx
Thus,
Jt1/px τ, t1/qx θ t
RN
−
RN
1
1
|∇τ|px |∇θ|qx dx
px
qx
Fx, t1/px τ, t1/qx θdx
1
1
|∇τ|px |∇θ|qx dx
≤
qx
RN px
ax px
bx qx
|τ| dx |θ| dx ,
− tλ1 RN px
RN qx
then it follows
1/px
Jt
τ, t
1/qx
1
θ ≤ −t p
px
RN
ax|τ|
1
dx q
RN
bx|θ|qx dx .
3.42
3.43
So, we can conclude that limt→∞ Jt1/px τ, t1/qx θ −∞. Hence, for t sufficiently large,
Jt1/px τ, t1/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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