RESEARCH Open Access
Weak lower semicontinuity of variational
functionals with variable growth
Fu Yongqiang
Correspondence: fuyqhagd@yahoo.
cn
Department of Mathematics,
Harbin Institute of Technology,
Harbin 150001, China
Abstract
In this paper, we establish the weak lower semicontinuity of variational functionals
with variable growth in variable exponent Sobolev spaces. The weak lower
semicontinuity is interesting by itself and can be applied to obtain the existence of
an equilibrium solution in nonlinear elasticity.
2000 Mathematics Subject Classification: 49A45
Keywords: lower semicontinuity, variational functional, variable growth
1 Introduction
The main purpose of this paper is to study the weak lower semicontinuity of the func-
tional
F( u)=


f (x, u, ∇u)d
x
where Ω is a bounded C
1
domain in R
n
and f : R
n
× R

(
|ζ |
p
+ |ξ|
p
)
where p ≥ 1, C ≥ 0anda(x) ≥ 0 are locally integrable, Ac erbi and Fusco [5] proved
that F is weakly l ower semicontinuous in W
1,p
(Ω, R
m
) if and only if f is quasiconvex in
the last variable. Later, Kalamajska [6] gave a shorter proof of the result in [5].
Since Kovacik and Rakosnik [7] first discussed variable exponent Lebesgue spaces
and variable exponent Sobolev spaces, the field of variable exponent function spaces
has witnessed an explosive growth in recent years and now there have been a large
number of papers concerning these kinds of variable exponent spaces, see the
Yongqiang Journal of Inequalities and Applications 2011, 2011:19
/>© 2011 Yongqiang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution
License ( which permits unrestricted use, distribution, and repr oduction in any medium,
provided the original work is properly cited.
monograph by Diening et al. [8] and the references therein. So we want to extend the
result of Acerbi and Fusco to the case that f satisfies variable growth conditions.
Thi s paper is organized as the following: In Section 2, we present some preliminary
fact s; in Section 3, we discuss the weak lower semicontinuity of variational functionals
with variable growth; in Section 4, we give an example to show that the result obtained
in Section 3 can be applied to study the existence of an equilibrium solution in non-
linear elasticity.
2 Preliminary
In this section, we first recall some facts on variable exponent spaces L

) ≤ 1
}
(2:2)
where Ω
∞
={x Î Ω: p(x)=∞}. The variable exponent Lebesgue space L
p(x)
(Ω)isthe
class of all functions f such that r
p
(l f) <∞ for some l = l(f) >0. L
p(x)
(Ω)isaBanach
space endowed with the norm (2.2). r
p
(f) is called the modular of f in L
p(x)
(Ω).
For a given p(x) Î P(Ω), we define the conjugate function p’(x) as:
p

(x)=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
∞,ifx ∈ 

and Ω only.
Theorem 2.2. The topolog y of the Banach space L
p(x)
(Ω) endowed by the norm (2.2)
coincides with the topology of modular convergence if and only if p Î L
∞
(Ω).
Theorem 2.3.The dual space to L
p(x)
(Ω) is L
p’ (x)
(Ω) if and only if p Î L
∞
(Ω).The
space L
p(x)
(Ω) is reflexive if and only if
1 < inf

p(x) ≤ sup

p(x) < ∞
.
(2:3)
Next, we assume that Ω ⊂ R
n
is a nonempty open set, p Î P(Ω), and k is a given
natural number.
Given a multi-index a =(a
1

that D
a
f Î L
p(x)
(Ω) for every multi-index a with |a| ≤ k endowed with the norm
Yongqiang Journal of Inequalities and Applications 2011, 2011:19
/>Page 2 of 9
||f ||
k,p
=

|
α
|
≤k
||D
α
f ||
p
.
(2:4)
By
W
k,p(x)
0
(
)
, we denote the subspace of W
k,p(x)
(Ω)whichistheclosureof

unique system of functions {g
a
Î L
p’(x)
(Ω): |a| ≤ k} such that
G(f )=

|α|≤k


D
α
f (x)g
α
(x)dx, f ∈ W
k,p(x)
0
()
.
The norm of
W
−k,p

(x)
0
(
)
is defined as
|
|G||

∈

{p
∗
(x) − q(x)} >
0
, then there is a compact embedding W
1,p(x)
(Ω) ® L
q(x)
(Ω).
Theorem 2.7. Let Ω be a domain with cone property. I f
p
:
¯
 →
R
is Lipschitz con-
tinuous and satisfies (1.2), and q(x) Î P(Ω) satisfies p(x) ≤ q(x) ≤ p*(x) a. e. on
¯

, then
there is a continuous embedding W
1,p(x)
(Ω) ® L
q(x)
(Ω).
Theorem 2.8. If p is continuous on
¯


(u) ≤||u||
p
2
p
.
(2) If ||u||
p
<1, then
||u||
p
2
p
≤ ρ
p
(u) ≤||u||
p
1
p
.
Lemma 2.10. Suppose
{f
n
}
∞
n
=
1
is bounded in L
p(x)
(Ω) and f

∈ R
n
m
, any open set Ω ⊂ R
n
and
z
∈ C
1
0
(, R
m
)
Yongqiang Journal of Inequalities and Applications 2011, 2011:19
/>Page 3 of 9
f (
˜
ξ)meas ≤


f (
˜
ξ + ∇z(x))dx
.
This section will establish the following result:
Theorem 3.1. Let Ω be a bounded C
1
domain in R
n
.f: R

u ∈ C
∞
0
(R
n
)
, we define
(M
∗
u)(x)=Mu(x)+
n

i
=1
(MD
i
u)(x
)
where
(Mu)(x)=sup
r>0
1
measB
r
(x)

B
r
(x)
|u(y)|dy

n
∈ C
k
(
¯

)
and
ψ
1
, , ψ
n
∈ C
k
(
¯
G
)
, we call F ak-smooth transformation.
For a measurable function u on Ω, we define a measurable function on G by Au(y)=
u(Ψ(y)).
Lemma 3.1. If F: Ω ® Gisk-smooth transformation, k ≥ 1,thenAisabounded
transformation from W
k,p(x)
(Ω) onto W
k,p(Ψ(y))
(G) and the inverse transformation of A is
bounded as well.
Proof. We need only to show
|

k,p
(
x
)
,
.
As C
∞
(Ω)isdenseinW
k,p(x)
(Ω) (see [10]), for each u Î W
k,p(x)
(Ω), there exists a
sequence {u
n
} ⊂ C
∞
(Ω) such that u
k
® u in W
k,p(x)
(Ω). For u
n
, we have
D
α
(Au
n
)(y)=


φ)((x))| det 

(x)|dx =

|β|≤|α|

G
D
β
u(x)M
αβ
((x))| det 

(x)|φdx
,
this is to say,
D
α
(Au)(y)=

|
β
|≤|α|
M
αβ
[A(D
β
u)](y
)
is satisfied in the weak sense.

|β|≤|α|
1
⎞
⎠
p
2
max
|β|≤|α|
⎛
⎝
(sup
y∈G
|M
αβ
(y)|
p
2
+1)



(D
β
u)((y))
C||u||
k,p,

p((y))
dy
⎞

k,p,

p(x)
dx.
Taking
C = C
2
⎛
⎝

|β|≤|α|
1
⎞
⎠
p
2
max
|β|≤|α|
(sup
y∈G
|M
αβ
(y)|
p
2
+1)
,
we have
||
A

n
)
1) Eu(x)=u(x) a. e. on Ω,
2)
||Eu||
k,
p
,R
n
≤ C||u||
k,
p
,

where C = C(k, p) is a constant,
then we call E a simple (k, p(x)) extension operator of Ω.
Lemma 3.2. Let Ω be a bounded C
k
domain. Then there exists a simple (k, p(x))
extension operator of Ω.
Proof. First let Ω be the half space
R
n
+
= {x ∈ R
n
: x
n
> 0
}

λ
j
u(x
1
, , x
n−1
, −jx
n
), x
n
≤ 0.
E
α
u(x)=
⎧
⎪
⎨
⎪
⎩
u(x), x
n
> 0
;
k+1

j=1
(−1)
α
n
λ

+
)
, then Eu Î C
k
(R
n
) and D
a
Eu(x)=E
a
D
a
u(x). As

R
n

D
α
Eu(x)
C||u||
k,p,R
n
+

p(x)
dx
=

R

, , x
n−1
, −jx
n
)
C||u||
k,p,R
n
+

p(x)
d
x
≤ C(k, p, α)

R
n
+

D
α
u(x)
C||u||
k,p,R
n
+

p(x)
dx,
we have

nm
® R satisfies:
(1) f is a Caratheodory function;
(2) f is quasiconvex with respect to ξ;
(3) 0 ≤ f(x, ζ, ξ) ≤ a(x)+C(|ζ|
p(x)
+|ξ|
p(x)
) for x Î R
n
, ζ Î R
n
, ξ Î R
m
,
where C is a nonnegative constant, a(x) is nonnegative and locally int egrable, and p
(x) is Lipchitz continuous and satisfies 1 ≤ p
1
≤ p(x) ≤ p
2
<+∞, then for each open sub-
set Ω ⊂ R
n
,
F( u, )=


f (x, u, ∇u)d
x
is weakly lower semicontinuous on W

→∞
F( u + z
k
, )
.
By Lemma 3.2, we ca n suppose that z
k
is defined on R
n
,and
||z
k
||
1,p
(
x
)
,R
n
is uniformly
bounded with respect to k.As
C
∞
0
(R
n
, R
m
)
is dense in W

n
<
1
k
,
|
F( u + ω
k
, ) − F(u + z
k
, )| <
1
k
.
Therefore, we can further suppose that {z
k
}isin
C
∞
0
(R
n
, R
m
)
and bounded in W
1,p(x)
(R
n
, R

k
)
p(x)
dx <ε,1≤ i ≤ m
,
for all k and B ⊂ Ω \ A
ε
with measB<δ and there exists sufficiently large l >0 such
that for all i, k,
meas{x ∈ R
n
:(M
∗
z
(i)
k
)(x) ≥ λ} < min(ε, δ)
.
(3:2)
Denote
H
λ
i,k
= {x ∈ R
n
:(M
∗
z
(i)
k

) < m min{ε, δ}
.
We can exten d
z
(
i
)
k
out of
H
λ
k
to become a Lipschitz function
g
(
i
)
k
with the Lipschitz
constant not bigger than C(n)l.As
F(u + z
k
, ) ≥ F(u + g
k
,(\A
ε
) ∩ H
λ
k
)=F(u + g

)\H
λ
k
))
≤ 2
p
2
−1
(η( mε)+C(n, )
m

i=1

(\A
ε
)\H
λ
i,k
(M
∗
z
(i)
k
)
p(x)
dx
)
≤ 2
p
2

.Iff: R
n
×R× R
n
® R satisfies
(1) f is measurable with respect to x and continuous with respect to (ζ, ξ);
(2) 0 ≤ f(x, ζ, ξ) ≤ a(x)+C(|ζ|
p(x)
+|ξ|
p(x)
) where a(x) is nonnegative and locally
integrable, and p(x) is Lipchitz continuous and satisfies 1 ≤ p
1
≤ p(x) ≤ p
2
<+∞.
Then
F( u)=


f (x, u, ∇u
)
is weakly lower semicontinuous in W
1,p(x)
(Ω) if and only if f
(x, ζ, ξ) is convex with respect to ξ.
It is immediate in view of the fact that in the case m = 1, quasiconvexity is equiva-
lent to convexity.
4 Application
We adopt the variational approach to prove the existence of an equilibrium solution in

(1) f is a Caratheodory function,
(2) b(x)+c(|ζ|
p(x)
+|ξ|
p(x)
) ≤ f(x, ζ, ξ) ≤ a(x)+C(|ζ|
p(x)
+|ξ|
p(x)
) where c, C ≥ 0,a
(x), b(x) ≥ 0 are locally integrable, p(x) is Lipchitz continuous and s atisfies 1 <p
1
≤
p(x) ≤ p
2
<+∞;
(3) f(x, ζ, ξ) is quasiconvex with respect to ξ.
Then, the variational problem
inf{F(u):u ∈ W
1,p(x)
0
(, R
m
)
}
has a solution.
Proof.Asf(x, u, ∇u) ≥ 0, F(u) is bounded below. Because
c



x
)
0
(, R
m
)
such that
lim
n→∞
F( u
n
)=inf{F(u):u ∈ W
1,p
(
x
)
0
(, R
m
)}
.
As F(u)iscoercive,{u
n
}isboundedin
W
1,p
(
x
)
0

)
0
(, R
m
)
}
. □
Competing interests
The author declares that he has no competing interests.
Received: 7 March 2011 Accepted: 19 July 2011 Published: 19 July 2011
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