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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 191649, 12 pages
doi:10.1155/2009/191649
Research Article
Multiple Solutions for a Class of
px-Laplacian Systems
Yongqiang Fu and Xia Zhang
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Xia Zhang,
Received 19 November 2008; Accepted 11 February 2009
Recommended by Ondrej Dosly
We study the multiplicity of solutions for a class of Hamiltonian systems with the px-Laplacian.
Under suitable assumptions, we obtain a sequence of solutions associated with a sequence of
positive energies going toward infinity.
Copyright q 2009 Y. Fu and X. Zhang. 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 and Main Results
Since the space L
px
and W
1,px
were thoroughly studied by Kov
´
a
ˇ
cik and R
´
akosn
´
ık 1,
variable exponent Sobolev spaces have been used in the last decades to model various
phenomena. In 2,R
˚
u
ˇ
zi
ˇ
cka presented the mathematical theory for the application of variable
exponent spaces in electro-rheological fluids.
In recent years, the differential equations and variational problems with px-growth
conditions have been studied extensively; see for example 3–6.In7, De Figueiredo and
Ding discussed the multiple solutions for a kind of elliptic systems on a smooth bounded
domain. Motivated by their work, we will consider the following sort of px-Laplacian
systems with “concave and convex nonlinearity”:
−div
∇u
px−2
∇u
|u|
px−2
u H
u
x, u, v,x∈ Ω,
−div
∇v
px−2
∇v
|v|
px−2
v −H
v
x, u, v,x∈ Ω,
uxvx0,x∈ ∂Ω,
1.1
where Ω ⊂ R
N
is a bounded domain, p is continuous on Ω and satisfies 1 <p
−
≤ px ≤ p
<
N,andH :
Ω × R
2
→ R is a C
1
function. In this paper, we are mainly interested in the class
2 Journal of Inequalities and Applications
of Hamiltonians H such that
Hx, u, v
|u|
αx
αx
|v|
βx
βx
Fx, u, v, 1.2
where 1 <α
−
≤ αx ≤ px,px βx p
∗
x. Here we denote
p
sup
x∈Ω
px,p
−
inf
x∈Ω
px, 1.3
and denote by px βx the fact that inf
x∈Ω
βx − px > 0. Throughout this paper,
Fx, u, v satisfies the following conditions:
H1 F ∈ C
1
Ω × R
2
, R. Writing z u, v,Fx, 0 ≡ 0,F
z
x, 0 ≡ 0;
H2 there exist px <q
1
x p
∗
x, 1 <q
2−
≤ q
2
x <px such that
F
u
x, u, v
,
F
v
x, u, v
≤ a
0
1 |u|
q
1
x−1
|v|
q
2
x−1
, 1.4
where a
0
is positive constant;
H3 there exist μx,νx ∈ C
1
Ω with px μx p
∗
x, 1 <ν
−
≤ νx ≤ px,
and R
0
> 0 such that
1
μx
F
u
x, u, vu
1
νx
F
v
x, u, vv ≥ Fx, u, v > 0, 1.5
when |u, v|≥R
0
.
As 8, Lemma 1.1, from assumption H3, there exist b
0
,b
1
> 0 such that
Fx, u, v ≥ b
0
|u|
μx
|v|
νx
− b
1
, 1.6
for any x, u, v ∈
Ω × R
2
. We can also get that there exists b
2
> 0 such that
1
μx
F
u
x, u, vu
1
νx
F
v
x, u, vv b
2
≥ Fx, u, v, 1.7
for any x, u, v ∈
Ω × R
2
. In this paper, we will prove the following result.
Theorem 1.1. Assume that hypotheses (H1)–(H3) are fulfilled. If Fx, z is even in z, then problem
1.1 has a sequence of solutions {z
n
} such that
I
z
n
Ω
∇u
n
px
u
n
px
px
−
∇v
n
px
v
n
px
px
− H
x, z
n
dx −→ ∞ , 1.8
as n →∞.
Journal of Inequalities and Applications 3
2. Preliminaries
First we recall some basic properties of variable exponent spaces L
px
Ω and variable
exponent Sobolev spaces W
1,px
Ω, where Ω ⊂ R
N
is a domain. For a deeper treatment
on these spaces, we refer to 1, 9–11.
Let PΩ be the set of all Lebesgue measurable functions p : Ω → 1, ∞ and
|u|
px
inf
λ>0:
Ω
u
λ
px
dx ≤ 1
. 2.1
The variable exponent space L
px
Ω is the class of all functions u such that
Ω
|ux|
px
dx <
∞. Under the assumption that p
< ∞,L
px
Ω is a Banach space equipped with the norm
2.1.
The variable exponent Sobolev space W
1,px
Ω is the class of all functions u ∈
L
px
Ω such that |∇u|∈L
px
Ω and it can be equipped with the norm
u
1,px
|u|
px
|∇u|
px
. 2.2
For u ∈ W
1,px
Ω, if we define
|||u||| inf
λ>0:
Ω
|u|
px
|∇u|
px
λ
px
dx ≤ 1
, 2.3
then |||u||| and u
1,px
are equivalent norms on W
1,px
Ω.
By W
1,px
0
Ω we denote the subspace of W
1,px
Ω which is the closure of C
∞
0
Ω
with respect to the norm 2.2 and denote the dual space of W
1,px
0
Ω by W
−1,p
x
Ω. We
know that if Ω ⊂ R
N
is a bounded domain, ||u||
1,px
and |∇u|
px
are equivalent norms on
W
1,px
0
Ω.
Under the condition 1 <p
−
≤ p
< ∞,W
1,px
0
Ω is a separable and reflexive Banach
space, then there exist {e
n
}
∞
n1
⊂ W
1,px
0
Ω and {f
m
}
∞
m1
⊂ W
−1,p
x
Ω such that
f
m
e
n
1ifn m,
0ifn
/
m,
W
1,px
0
Ω span
e
i
: i 1, ,n,
,
W
−1,p
x
Ω span
f
j
: j 1, ,m,
.
2.4
In the following, we will denote that E E
1
⊕ E
2
, where
E
1
{0}×W
1,px
0
Ω,E
2
W
1,px
0
Ω ×{0}. 2.5
4 Journal of Inequalities and Applications
For any z ∈ E, define the norm ||z|| ||u, v|| |||u||| |||v|||. For any n ∈ N, set e
1
n
0,e
n
,e
2
n
e
n
, 0 and
X
n
span
e
1
1
, ,e
1
n
⊕ E
2
,X
n
E
1
⊕ span
e
2
1
, ,e
2
n
, 2.6
denote the complement of X
n
in E by X
n
⊥
span{e
2
n1
,e
2
n2
, }.
3. The Proof of Theorem 1.1
Definition 3.1. We say that z
0
u
0
,v
0
∈ E is a weak solution of problem 1.1,thatis,
Ω
∇u
0
px−2
∇u
0
∇u
u
0
px−2
u
0
u −
∇v
0
px−2
∇v
0
∇v
−
v
0
px−2
v
0
v − H
u
x, u
0
,v
0
u − H
v
x, u
0
,v
0
v
dx 0, ∀z ∈ E.
3.1
In this section, we denote that V
m
span{e
i
: i 1, ,m}, for any m ∈ N,andc
i
is
positive constant, for any i 0, 1, 2
Lemma 3.2. Any PS sequence {z
n
}⊂E, that is, |Iz
n
|≤c and I
z
n
→ 0, as n →∞, is
bounded.
Proof. Let s>0besufficiently small such that l
1
inf
x∈Ω
1/px − 1 s/μx > 0,l
2
inf
x∈Ω
1 s/νx − 1/px > 0,l
3
sup
x∈Ω
1/αx − 1 s/μx > 0,l
4
sup
x∈Ω
1
s/νx − 1/βx > 0.
Let {z
n
}⊂E be such that |Iz
n
|≤c and I
z
n
→ 0, as n →∞. We get
I
z
n
−
I
z
n
,
1 s
μx
u
n
,
1 s
νx
v
n
Ω
1
px
−
1 s
μx
∇u
n
px
u
n
px
1 su
n
μx
2
∇u
n
px−2
∇u
n
∇μ
1 s
νx
−
1
px
∇v
n
px
v
n
px
−
1 sv
n
νx
2
∇v
n
px−2
∇v
n
∇ν
1 s
μx
F
u
x, u
n
,v
n
u
n
1 s
νx
F
v
x, u
n
,v
n
v
n
− F
x, u
n
,v
n
1 s
μx
−
1
αx
u
n
αx
1 s
νx
−
1
βx
v
n
βx
dx
≥
Ω
l
1
∇u
n
px
l
2
∇v
n
px
sF
x, u
n
,v
n
− l
3
u
n
αx
l
4
v
n
βx
1 su
n
μx
2
∇u
n
px−2
∇u
n
∇μ −
1 sv
n
νx
2
∇v
n
px−2
∇v
n
∇ν − 1 sb
2
dx.
3.2
Journal of Inequalities and Applications 5
As μx,νx ∈ C
1
Ω, by the Young inequality, we can get that for any ε
1
,ε
2
∈ 0, 1,
1 su
n
μx
2
∇u
n
px−2
∇u
n
∇μ
≤ c
0
∇u
n
px−1
u
n
≤ c
0
ε
1
px − 1
px
∇u
n
px
ε
1−px
1
px
u
n
px
≤ c
0
ε
1
∇u
n
px
ε
1−p
1
u
n
px
,
1 sv
n
νx
2
∇v
n
px−2
∇v
n
∇ν
≤ c
1
ε
2
∇v
n
px
ε
1−p
2
v
n
px
.
3.3
Let ε
1
,ε
2
be sufficiently small such that
c
0
ε
1
≤
l
1
2
,c
1
ε
2
≤
l
2
2
, 3.4
then
I
z
n
−
I
z
n
,
1 s
μx
u
n
,
1 s
νx
v
n
≥
Ω
l
1
2
∇u
n
px
l
2
2
∇v
n
px
s
b
0
u
n
μx
b
0
v
n
νx
− b
1
−
l
3
u
n
αx
c
0
ε
1−p
1
u
n
px
l
4
v
n
βx
− c
1
ε
1−p
2
v
n
px
− 1 sb
2
dx.
3.5
Note that αx ≤ px μx,px βx, by the Young inequality, for any ε
3
,ε
4
,ε
5
∈ 0, 1,
we get
u
n
αx
≤
ε
3
αx
u
n
μx
μx
μx − αx
μx
ε
αx/αx−μx
3
≤ ε
3
u
n
μx
ε
−α
/μ−α
−
3
,
u
n
px
≤
ε
4
px
μx
u
n
μx
μx − px
μx
ε
px/px−μx
4
≤ ε
4
u
n
μx
ε
−p/μ−p
−
4
,
v
n
px
≤
ε
5
px
βx
v
n
βx
βx − px
βx
ε
px/ px−βx
5
≤ ε
5
v
n
βx
ε
−p/ β−p
−
5
.
3.6
6 Journal of Inequalities and Applications
Let ε
3
,ε
4
,ε
5
be sufficiently small such that l
3
ε
3
c
0
ε
1−p
1
ε
4
≤ sb
0
and c
1
ε
1−p
2
ε
5
≤ l
4
, then we
get
I
z
n
−
I
z
n
,
1 s
μx
u
n
,
1 s
νx
v
n
≥
Ω
l
1
2
∇u
n
px
l
2
2
∇v
n
px
− c
2
dx
. 3.7
Note that
I
z
n
,
1 s
μx
u
n
,
1 s
νx
v
n
≤
I
z
n
·
1 s
μx
u
n
1 s
νx
v
n
≤ c
3
I
z
n
·
∇
1 s
μx
u
n
px
∇
1 s
νx
v
n
px
≤ c
4
I
z
n
·
∇u
n
px
∇v
n
px
,
3.8
and for n ∈ N being large enough, we have
c
4
I
z
n
≤ min
l
1
4
,
l
2
4
. 3.9
It is easy to know that if |∇u
n
|
px
≥ 1and|∇v
n
|
px
≥ 1,
∇u
n
px
≤
Ω
∇u
n
px
dx,
∇v
n
px
≤
Ω
∇v
n
px
dx, 3.10
thus we get
I
z
n
≥
Ω
l
1
4
∇u
n
px
l
2
4
∇v
n
px
− c
2
dx, 3.11
then |∇u
n
|
px
, |∇v
n
|
px
are bounded. Similarly, if |∇u
n
|
px
< 1or|∇v
n
|
px
< 1, we can also
get that |∇u
n
|
px
, |∇v
n
|
px
are bounded. It is immediate to get that {z
n
} is bounded in E.
Lemma 3.3. Any PS sequence contains a convergent subsequence.
Proof. Let {z
n
}⊂E be a PS sequence. By Lemma 3.2,weobtainthat{z
n
} is bounded in E.
As E is reflexive, passing to a subsequence, still denoted by {z
n
}, we may assume that there
Journal of Inequalities and Applications 7
exists z ∈ E such that z
n
→ z weakly in E. Then we can get u
n
→ u weakly in W
1,px
0
Ω.
Note that
I
z
n
− I
z,
u
n
− u, 0
Ω
∇u
n
px−2
∇u
n
−
∇u
px−2
∇u
∇
u
n
− u
u
n
px−2
u
n
−|u|
px−2
u
u
n
− u
−
u
n
αx−2
u
n
−|u|
αx−2
u
u
n
− u
−
F
u
x, u
n
,v
n
− F
u
x, u, v
u
n
− u
dx.
3.12
It is easy to get that
I
z
n
− I
z,
u
n
− u, 0
−→ 0,
Ω
F
u
x, u, v
u
n
− u
dx −→ 0,
3.13
and u
n
→ u in L
px
Ω,u
n
→ u in L
αx
Ω, as n →∞. Then
Ω
u
n
px−2
u
n
−|u|
px−2
u
u
n
− u
dx −→ 0,
Ω
u
n
αx−2
u
n
−|u|
αx−2
u
u
n
− u
dx −→ 0,
3.14
as n →∞. By condition H2,weobtain
Ω
F
u
x, u
n
,v
n
u
n
− u
dx
≤
Ω
a
0
1
u
n
q
1
x−1
v
n
q
2
x−1
u
n
− u
dx
≤ a
1
u
n
− u
1
u
n
q
1
x−1
q
1
x
·
u
n
− u
q
1
x
v
n
q
2
x−1
q
2
x
·
u
n
− u
q
2
x
.
3.15
It is immediate to get that |u
n
− u|
1
→ 0, ||u
n
|
q
1
x−1
|
q
1
x
, ||v
n
|
q
2
x−1
|
q
2
x
are bounded and
|u
n
− u|
q
1
x
→ 0, |u
n
− u|
q
2
x
→ 0, then we get
Ω
F
u
x, u
n
,v
n
u
n
− u
dx −→ 0,
Ω
∇u
n
px−2
∇u
n
−
∇u
px−2
∇u
∇
u
n
− u
dx −→ 0,
3.16
as n →∞. Similar to 3, 4, Theorem 3.1, we divide Ω into two parts:
Ω
1
{x ∈ Ω : px < 2}, Ω
2
{x ∈ Ω : px ≥ 2}. 3.17
8 Journal of Inequalities and Applications
On Ω
1
, we have
Ω
1
∇u
n
−∇u
px
dx
≤ c
5
Ω
1
∇u
n
px−2
∇u
n
−
∇u
px−2
∇u
∇u
n
−∇u
px/ 2
×
∇u
n
px
∇u
px
2−px/2
dx
≤ c
6
∇u
n
px−2
∇u
n
−
∇u
px−2
∇u
∇u
n
−∇u
px / 2
2/px ,Ω
1
×
∇u
n
px
∇u
px
2−px/2
2/2−px,Ω
1
,
3.18
then
Ω
1
|∇u
n
−∇u|
px
dx → 0. On Ω
2
, we have
Ω
2
|∇u
n
−∇u|
px
dx ≤ c
7
Ω
2
|∇u
n
|
px−2
∇u
n
−|∇u|
px−2
∇u∇u
n
−∇udx −→ 0.
3.19
Thus we get
Ω
|∇u
n
−∇u|
px
dx → 0. Then u
n
→ u in W
1,px
0
Ω, as n →∞. Similarly,
v
n
→ v in W
1,px
0
Ω.
Lemma 3.4. There exists R
m
> 0 such that Iz ≤ 0 for all z ∈ X
m
with ||z|| ≥ R
m
.
Proof. For any z u, v ∈ X
m
,u∈ V
m
, we have
Iz ≤
Ω
∇u
px
|u|
px
px
−
∇v
px
|v|
px
px
− Fx, u, v
dx
≤
Ω
∇u
px
|u|
px
p
−
−
∇v
px
|v|
px
p
− b
0
|u|
μx
b
1
dx.
3.20
In the following, we will consider
Ω
|∇u|
px
|u|
px
/p
−
− b
0
|u|
μx
dx.
i If |||u||| ≤ 1. We have
Ω
∇u
px
|u|
px
p
−
− b
0
|u|
μx
dx ≤
1
p
−
. 3.21
ii If |||u||| > 1. Note that μ, p ∈ C
Ω,px μx. For any x ∈ Ω, there exists Qx
which is an open subset of
Ω such that
p
x
sup
y∈Qx
py <μ
x
inf
y∈Qx
μy, 3.22
Journal of Inequalities and Applications 9
then {Qx}
x∈Ω
is an open covering of Ω. As Ω is compact, we can pick a finite subcovering
{Qx}
n
i1
for Ω. Thus there exists a sequence of open set {Ω
i
}
n
i1
such that Ω
n
i1
Ω
i
and
p
i
sup
x∈Ω
i
px <μ
i−
inf
x∈Ω
i
μx, 3.23
for i 1, ,n.Denote that r
i
|||u|||
Ω
i
, then we have
Ω
∇u
px
|u|
px
p
−
− b
0
|u|
μx
dx
n
i1
Ω
i
∇u
px
|u|
px
p
−
− b
0
|u|
μx
dx
r
i
>1
Ω
i
∇u
px
|u|
px
p
−
− b
0
|u|
μx
dx
r
i
≤1
Ω
i
∇u
px
|u|
px
p
−
− b
0
|u|
μx
dx
≤
r
i
>1
|||u|||
p
i
Ω
i
p
−
− b
0
k
m
i
|||u|||
μ
i−
Ω
i
n
p
−
,
3.24
where k
m
i
inf
u∈V
m
|
Ω
i
, |||u|||
Ω
i
1
Ω
i
|u|
μx
dx. As V
m
|
Ω
i
is a finite dimensional space, we have
k
m
i
> 0, for i 1, ,n.
We denote by s
i
the maximum of polynomial t
p
i
/p
−
− b
0
k
m
i
t
μ
i−
on 0, ∞, for i
1, ,n.Then there exists t
0
> 1 such that
t
p
i
p
−
− b
0
k
m
i
t
μ
i−
c
8
≤ 0, 3.25
for t>t
0
and i 1, ,n,where c
8
n
i1
s
i
n/p
−
b
1
meas Ω.
Let R
m
max{2, 2p
c
8
1/p
−
1 /p
−
, 2nt
0
}. If ||z|| ≥ R
m
, we get |||u||| ≥ R
m
/2or
|||v||| ≥ R
m
/2.
i If |||u||| ≥ R
m
/2, |||u||| ≥ nt
0
> 1. It is easy to verify that there exists at least i
0
such
that |||u|||
Ω
i
0
≥ t
0
> 1, thus
Iz ≤
|||u|||
p
i
0
Ω
i
0
p
−
− b
0
k
m
i
0
|||u|||
μ
i
0
−
Ω
i
0
c
8
≤ 0. 3.26
10 Journal of Inequalities and Applications
ii If |||v||| ≥ R
m
/2, |||v||| ≥ p
c
8
1/p
−
1/p
−
. We obtain
Iz ≤ c
8
1
p
−
−
|||v|||
p
−
p
≤ 0. 3.27
Now we get the result.
Lemma 3.5. There exist r
m
> 0 and a
m
→∞m →∞ such that Iz ≥ a
m
, for any z ∈ X
m−1
⊥
with ||z|| r
m
.
Proof. For z u, v ∈ X
m−1
⊥
,v 0. By condition H2, there exists c
9
> 0 such that
Fx, u, 0
≤ c
9
|u|
q
1
x
c
9
. 3.28
Let ||z|| ≥ 1, we get
Iz
Ω
∇u
px
|u|
px
px
−
|u|
αx
αx
− Fx, u, 0
dx
≥
Ω
∇u
px
|u|
px
p
−
|u|
αx
α
−
− c
9
|u|
q
1
x
− c
9
dx
≥
Ω
∇u
px
|u|
px
p
− c
10
|u|
q
1
x
dx − c
11
.
3.29
Denote that
θ
m
sup
u∈V
⊥
m
|||u|||≤1
Ω
|u|
q
1
x
dx, 3.30
thus
Iz ≥
|||u|||
p
−
p
− c
10
θ
m
u
q
1
− c
11
. 3.31
Let
r
m
max
1,
p
−
c
10
p
q
1
θ
m
1/q
1
−p
−
,
2c
11
p
q
1
q
1
− p
−
1/p
−
. 3.32
Journal of Inequalities and Applications 11
By 5, Lemma 3.3,wegetthatθ
m
→ 0, as m →∞, then
Iz ≥ r
p
−
m
q
1
− p
−
p
q
1
− c
11
a
m
,
3.33
when m is sufficiently large and ||z|| r
m
. It is easy to get that a
m
→∞, as m →∞.
Lemma 3.6. I is bounded from above on any bounded set of X
m
.
Proof. For z u, v ∈ X
m
. We get
Iz ≤
Ω
∇u
px
|u|
px
px
− Fx, u, v
dx. 3.34
By conditions H2 and H3, we know that if |u, v|≥R
0
,Fx, u, v ≥ 0andif|u, v| <
R
0
, |Fx, u, v|≤c
0
. Then
Iz ≤
Ω
∇u
px
|u|
px
px
c
12
dx, 3.35
and it is easy to get the result.
Proof of Theorem 1.1. By Lemmas 3.2–3.6 above, and 7, Proposition 2.1 and Remark 2.1,we
know that the functional I has a sequence of critical values c
k
→∞, as k →∞. Now we
complete the proof.
Acknowledgments
This work is supported by Science Research Foundation in Harbin Institute of Technology
HITC200702 and The Natural Science Foundation of Heilongjiang Province A2007-04.
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