# The Existence of Solutions to the NonhomogeneousA-Harmonic Equations with Variable Exponent

Authors: Haiyu Wen
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We first discuss the existence and uniqueness of weak solution for the obstacle problem of the nonhomogeneousA-harmonic equation with variable exponent, and then we obtain the existence of the solutions of the equation d
⋆
A
(
x
,
d
ω
)
=
B
(
x
,
d
ω
) in the weighted variable exponent Sobolev space W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
.


---

## Body

## 1. Introduction

In [1–5], the nonhomogeneous A-harmonic equation d
⋆
A
(
x
,
d
ω
)
=
B
(
x
,
d
ω
) for differential forms has received much investigation. In [6], the obstacle problem of the A-harmonic equation for differential forms has been discussed. However, most of these results are developed in the L
p
(
Ω
,
Λ
l
) space or W
1
,
p
(
Ω
,
Λ
l
) space. Meanwhile, in the past few years the subject of variable exponent space has undergone a vast development; see [7–11]. For example, [8–10] discuss the weighted L
p
(
x
) and W
k
,
p
(
x
) spaces and the weak solution for obstacle problem with variable growth has been studied in [10, 11].In this paper, we are interested in the following obstacle problem:(1)
∫
Ω
‍
(
A
(
x
,
d
u
)
·
d
(
v
-
u
)
+
B
(
x
,
d
u
)
·
(
v
-
u
)
)
d
x
≥
0
for v belonging to
(2)
𝔎
ψ
,
θ
=
{
v
∈
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
:
v
≥
ψ
,
a
.
e
.
x
∈
Ω
,
v
-
θ
∈
W
0
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
}
,
where ψ
(
x
)
=
∑
‍
ψ
I
(
x
)
d
x
I
∈
Λ
l
(
ℝ
n
), v
(
x
)
=
∑
‍
v
I
(
x
)
d
x
I
∈
Λ
l
(
ℝ
n
), v
I, ψ
I
:
Ω
→
[
-
∞
,
+
∞
]; v
≥
ψ
,
a
.
e
.
x
∈
Ω means that, for any I, we have v
I
≥
ψ
I
,
a
.
e
.
x
∈
Ω
;
θ
∈
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
,
l
=
0,1
,
…
,
n
-
1, and the variable exponent p
(
x
)
∈
𝒫
(
Ω
) satisfies
(3)
1
<
p
-
≤
p
(
x
)
≤
p
+
<
∞
for
a
.
e
.
x
∈
Ω
.The operatorsA
(
x
,
ξ
)
:
Ω
×
Λ
l
(
ℝ
n
)
→
Λ
l
(
ℝ
) and B
(
x
,
ξ
)
:
Ω
×
Λ
l
(
ℝ
n
)
→
Λ
l
-
1
(
ℝ
) satisfy the following growth conditions on a bounded domain Ω:(H1)
A
(
x
,
ξ
) and B
(
x
,
ξ
) are measurable for all ξ with respect to x and continuous for a
.
e
.
x
∈
Ω with respect to ξ,
(H2)
|
A
(
x
,
ξ
)
|
≤
C
1
w
(
x
)
|
ξ
|
p
(
x
)
-
1,
(H3)
A
(
x
,
ξ
)
·
ξ
≥
C
2
w
(
x
)
|
ξ
|
p
(
x
),
(H4)
|
B
(
x
,
ξ
)
|
≤
C
3
w
(
x
)
|
ξ
|
p
(
x
)
-
1,
(H5)
B
(
x
,
d
ξ
)
·
ξ
≥
C
4
w
(
x
)
|
ξ
|
p
(
x
),
(H6)
(
A
(
x
,
d
ξ
)
-
A
(
x
,
d
η
)
)
·
(
d
ξ
-
d
η
)
+
(
B
(
x
,
d
ξ
)
-
B
(
x
,
d
η
)
)
·
(
ξ
-
η
)
≥
0 for ξ
≠
η,whereC
1
,
C
2
,
C
3
,
and
C
4 are nonnegative constants.  w
(
x
)
∈
L
1
(
Ω
) nonnegative and w
(
x
)
1
/
(
p
(
x
)
-
1
)
∈
L
1
(
Ω
). We will discuss the existence and uniqueness of the solution u
∈
𝔎
ψ
,
θ for the abovementioned obstacle problem.Now, we introduce the existing results and related definitions.Throughout this paper, we assume thatΩ is a bounded domain in ℝ
n. Let Λ
l
=
Λ
l
(
ℝ
n
) be the set of all l-forms in ℝ
n. A differential l-form u
(
x
) is generated by {
d
x
i
1
∧
d
x
i
2
∧
⋯
∧
d
x
i
l
}
,
l
=
1,2
,
…
,
n; that is, u
(
x
)
=
∑
I
‍
u
I
(
x
)
d
x
I
=
∑
‍
u
i
1
,
i
2
,
…
,
i
l
(
x
)
d
x
i
1
∧
d
x
i
2
∧
⋯
∧
d
x
i
l, where u
I
(
x
) is differential function, I
=
(
i
1
,
i
2
,
…
,
i
l
), and 1
≤
i
1
<
i
2
<
⋯
<
i
l
≤
n. Let D
′
(
Ω
,
Λ
l
) be the space of all differential l-forms on Ω. For α
(
x
)
=
Σ
α
I
(
x
)
d
x
I
∈
Λ
l and β
(
x
)
=
Σ
β
I
(
x
)
d
x
I
∈
Λ
l, then the inner product is obtained by α
·
β
=
⋆
(
α
∧
⋆
β
)
=
∑
I
‍
α
I
(
x
)
β
I
(
x
). We write |
u
|
=
(
u
·
u
)
1
/
2
=
(
∑
I
‍
|
u
I
(
x
)
|
2
)
1
/
2. We denote the exterior derivative by d
u
=
∑
i
=
1
n
‍
∑
I
‍
(
∂
u
I
(
x
)
/
∂
x
i
)
d
x
I
∧
d
x
i
:
D
′
(
Ω
,
Λ
l
)
→
D
′
(
Ω
,
Λ
l
+
1
) for l
=
0,1
,
…
,
n
-
1. Its formal adjoint operator d
⋆
:
D
′
(
Ω
,
Λ
l
+
1
)
→
D
′
(
Ω
,
Λ
l
) is given by d
⋆
=
(
-
1
)
n
l
+
1
⋆
d
⋆, l
=
0,1
,
2
,
…
,
n
-
1; here ⋆ is the well-known Hodge star operator. Denote the class of infinitely differential l-forms on Ω by C
∞
(
Ω
,
Λ
l
). A differential l-form u
∈
D
′
(
Ω
,
Λ
l
) is called a closed form if d
u
=
0 in Ω.Next we will introduce some basic properties of weighted variable exponent Lebesgue spacesL
p
(
x
)
(
Ω
,
μ
) and weighted variable exponent Sobolev spaces W
1
,
p
(
x
)
(
Ω
,
μ
), and we define 𝒫
(
Ω
) to be the set of all n-dimensioned Lebesgue measurable functions p
:
Ω
→
[
1
,
∞
]. Functions p
∈
𝒫
(
Ω
) are called variable exponents on Ω. We define p
-
:
=
ess
in
f
x
∈
Ω
p
(
x
), p
+
:
=
ess
su
p
x
∈
Ω
p
(
x
). If p
+
<
∞, then we call p a bounded variable exponent. If p
∈
𝒫
(
Ω
), then we define p
′
∈
𝒫
(
Ω
) by (
1
/
p
(
x
)
)
+
(
1
/
p
′
(
x
)
)
=
1, where 1
/
∞
:
=
0. The function p
′ is called the dual variable exponent of p. We denote w as a weight if w
∈
L
loc
⁡
1
(
ℝ
n
) and w
>
0 a.e.; also in general d
μ
=
w
d
x. From [7, 10], we know that if p
∈
𝒫
(
Ω
) satisfies (3), the weighted variable exponent Lebesgue spaces L
p
(
x
)
(
Ω
,
μ
)
=
{
f
:
∫
Ω
‍
|
λ
f
(
x
)
|
p
(
x
)
d
μ
<
∞
,
λ
>
0
} with the norm ∥
f
∥
L
p
(
x
)
(
Ω
,
μ
)
=
inf
⁡
{
λ
>
0
:
∫
Ω
‍
|
f
(
x
)
/
λ
|
p
(
x
)
d
μ
≤
1
} and the weighted variable exponent Sobolev spaces W
1
,
p
(
x
)
(
Ω
,
μ
)
=
{
f
∈
L
p
(
x
)
(
Ω
,
μ
)
:
∇
f
∈
L
p
(
x
)
(
Ω
,
μ
)
} with the norm ∥
f
∥
W
1
,
p
(
x
)
(
Ω
,
μ
)
=
∥
f
∥
L
p
(
x
)
(
Ω
,
μ
)
+
∥
∇
f
∥
L
p
(
x
)
(
Ω
,
μ
) are Banach space and reflexive and uniformly convex. On the set of all differential forms on Ω, we define the weighted variable exponent Lebesgue spaces of differential l-forms L
p
(
x
)
(
Ω
,
Λ
l
,
μ
) and the weighted variable exponent Sobolev spaces of differential forms W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
).Definition 1.
We denote the weighted variable exponent Lebesgue spaces of differentiall-forms by L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
=
{
u
=
∑
I
‍
u
I
(
x
)
d
x
I
∈
Λ
l
:
u
I
(
x
)
∈
L
p
(
x
)
(
Ω
,
μ
)
}
l
=
0,1
,
2
,
…
,
n and we endow L
p
(
x
)
(
Ω
,
Λ
l
,
μ
) with the following norm:
(4)
∥
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
=
inf
⁡
{
λ
>
0
:
∫
Ω
‍
|
u
λ
|
p
(
x
)
d
μ
≤
1
}
.
And the spacesW
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
=
{
u
∈
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
:
d
u
∈
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
)
} with the norm
(5)
∥
u
∥
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
=
∥
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
+
∥
d
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
)
are the weighted variable exponent Sobolev spaces of differential l-forms; l
=
0,1
,
2
,
…
,
n
-
1. W
0
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
) is the completion of C
0
∞
(
Ω
,
Λ
l
,
μ
) in W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
). We need the following Hölder inequalities; see [7, 10].Proposition 2.
Letp, q
∈
𝒫
(
Ω
) be such that 1
=
(
1
/
p
(
x
)
)
+
(
1
/
q
(
x
)
) for μ-almost every x
∈
Ω. Then
(6)
∫
Ω
‍
|
f
g
|
d
μ
≤
∫
Ω
‍
1
p
(
x
)
|
f
|
p
(
x
)
d
μ
+
∫
Ω
‍
1
q
(
x
)
|
g
|
q
(
x
)
d
μ
,
(7)
∫
Ω
‍
|
f
g
|
d
μ
≤
(
(
1
p
)
+
+
(
1
q
)
+
)
∥
f
∥
L
p
(
x
)
(
Ω
,
μ
)
∥
g
∥
L
q
(
x
)
(
Ω
,
μ
)
,
for all f
∈
L
p
(
x
)
(
Ω
,
μ
) and g
∈
L
q
(
x
)
(
Ω
,
μ
).Lemma 3 (see [7]).
Let(
D
,
∑
‍
,
μ
) be a σ-finite, complete measure space; if f
∈
L
p
(
·
)
(
D
,
μ
), g
∈
L
0
(
D
,
μ
), and 0
≤
|
g
|
≤
|
f
|
μ-almost everywhere, then g
∈
L
p
(
·
)
(
D
,
μ
) and ∥
g
∥
L
p
(
x
)
(
D
,
μ
)
≤
∥
f
∥
L
p
(
x
)
(
D
,
μ
).By the inequality(
∑
i
=
1
n
‍
(
a
i
)
2
)
1
/
2
≤
∑
i
=
1
n
‍
a
i
≤
n
1
/
2
(
∑
i
=
1
n
‍
(
a
i
)
2
)
1
/
2 for any a
i
≥
0, using Lemma 3, we can easily have the following lemma.Lemma 4.
Ifu
=
Σ
I
u
I
(
x
)
d
x
I
∈
D
′
(
Ω
,
Λ
l
) and |
u
|
=
(
∑
I
‍
|
u
I
|
2
)
1
/
2, then u
∈
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
) and |
u
|
∈
L
p
(
x
)
(
Ω
,
μ
) are equivalent, and ∥
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
=
∥
|
u
|
∥
L
p
(
x
)
(
Ω
,
μ
).


## 2. Main Results

In this section, we will obtain the existence and uniqueness of weak solution for obstacle problem of the nonhomogeneousA-harmonic equation in space W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
).Theorem 5.
Suppose𝔎
ψ
,
θ is not empty, under conditions (H1)–(H6), and there exists a unique solution u to the obstacle problem (1)-(2). That is, there is a differential form u in 𝔎
ψ
,
θ such that
(8)
∫
Ω
‍
(
A
(
x
,
d
u
)
·
d
(
v
-
u
)
+
B
(
x
,
d
u
)
·
(
v
-
u
)
)
d
x
≥
0
,
whenever v
∈
𝔎
ψ
,
θ.We deduce Theorem5 from a proposition of Kinderlehrer and Stampacchia.Proposition 6 (see [12]).
LetK be a nonempty closed convex subset of X and let 𝔄
:
K
→
X
′ be monotone, coercive, and weakly continuous on K. Then there exists an element u in K such that 〈
𝔄
u
,
v
-
u
〉
≥
0 whenever v
∈
K.Now letX
=
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
) and 〈
·
,
·
〉 be the usual pairing between X and X
′, 〈
f
,
g
〉
=
∫
Ω
‍
f
·
g
d
μ, where g is in X and f in X
′
=
W
d
p
′
(
x
)
(
Ω
,
Λ
l
,
μ
). We will take 𝔎
ψ
,
θ as K. We define a mapping 𝔄
:
𝔎
ψ
,
θ
→
X
′ by
(9)
〈
𝔄
v
,
u
〉
=
∫
Ω
‍
(
A
(
x
,
d
v
)
·
d
u
+
B
(
x
,
d
v
)
·
u
)
d
x
for u
∈
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
).Lemma 7.
Ifp
(
x
) satisfies (3), then spaces L
p
(
x
)
(
Ω
,
Λ
l
,
μ
) and W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
) are complete and convex.Proof.
From [7], we know that if p satisfies (3) and w
(
x
)
1
/
(
p
(
x
)
-
1
)
∈
L
1
(
Ω
), then let L
p
(
x
)
(
Ω
,
μ
) be Banach space and uniformly convex. If ω
1 and ω
2 are two l-forms: ω
1
=
∑
I
‍
a
I
d
x
I and ω
2
=
∑
I
‍
b
I
d
x
I, we can easily have ω
1
+
ω
2
=
∑
I
‍
(
a
I
+
b
I
)
d
x
I and d
(
ω
1
+
ω
2
)
=
d
ω
1
+
d
ω
2, so we can immediately obtain the convexity of spaces L
p
(
x
)
(
Ω
,
Λ
l
,
μ
) and W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
).
Letu
j
=
∑
I
‍
u
j
I
d
x
I
∈
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
) be a Cauchy sequence in W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
). Then for any I, u
j
I
(
x
) converges in L
p
(
x
)
(
Ω
,
μ
). Suppose that u
j
I
(
x
)
→
u
I
(
x
) in L
p
(
x
)
(
Ω
,
μ
). Now let u
=
∑
I
‍
u
I
d
x
I
∈
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
), we have
(10)
|
u
j
-
u
|
=
(
∑
I
‍
|
u
j
I
-
u
I
|
2
)
1
/
2
≤
∑
I
‍
|
u
j
I
-
u
I
|
,
using Lemmas 3 and 4, and we know the sequence u
j converges to u in L
p
(
x
)
(
Ω
,
Λ
l
,
μ
).
For the sequence{
d
u
j
}, we suppose d
u
j
→
v in L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
), and then v
∈
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
). So (
u
j
,
d
u
j
) converges to (
u
,
v
) in the normed space L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
×
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
). From d
u
j
-
d
u
=
∑
I
‍
(
d
u
j
I
-
d
u
I
)
∧
d
x
I, we have
(11)
|
d
u
j
-
d
u
|
=
|
∑
I
‍
∑
i
=
1
n
‍
∂
u
j
I
-
∂
u
I
∂
x
i
d
x
i
∧
d
x
I
|
≤
(
∑
I
‍
∑
i
=
1
n
‍
|
∂
u
j
I
-
∂
u
I
∂
x
i
|
2
)
1
/
2
≤
∑
I
‍
∑
i
=
1
n
‍
|
∂
u
j
I
-
∂
u
I
∂
x
i
|
≤
∑
I
‍
n
1
/
2
(
∑
i
=
1
n
‍
|
∂
u
j
I
-
∂
u
I
∂
x
i
|
2
)
1
/
2
=
n
1
/
2
∑
I
‍
|
∇
u
j
I
-
∇
u
I
|
.
In view of [10], for any I, we know that u
j
I
→
u
I in L
p
(
x
)
(
Ω
,
μ
) and ∇
u
j
I
→
v
I in L
p
(
x
)
(
Ω
,
μ
), and then ∇
u
I
=
v
I
∈
L
p
(
x
)
(
Ω
,
μ
). Using Lemmas 3 and 4,   we get the sequence  d
u
j that converges to d
u in L
p
(
x
)
(
Ω
,
Λ
l
,
μ
); it follows that v
=
d
u
∈
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
). So, we prove W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
) is a closed subspace of L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
×
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
). This ends the proof of Lemma 7.Using Lemma7, we can immediately obtain the following lemma.Lemma 8.
𝔎
ψ
,
θ is a closed convex set.Lemma 9.
For eachv
∈
𝔎
ψ
,
θ
,
𝔄
v
∈
[
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
]
′.Proof.
Using Hölder inequality (7) with 1
=
(
1
/
p
(
x
)
)
+
(
1
/
p
′
(
x
)
) and (H2), we get
(12)
|
∫
Ω
‍
A
(
x
,
v
(
x
)
)
·
u
(
x
)
d
x
|
≤
C
1
∫
Ω
‍
|
v
(
x
)
|
p
(
x
)
-
1
|
u
(
x
)
|
w
(
x
)
d
x
≤
C
1
(
1
p
-
+
1
p
′
-
)
∥
|
v
(
x
)
|
p
(
x
)
-
1
∥
L
p
′
(
x
)
(
Ω
,
Λ
l
,
μ
)
∥
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
≤
2
C
1
∥
v
(
x
)
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
p
(
x
)
-
1
∥
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
.
Similarly, for the operatorB
(
x
,
ξ
), we have
(13)
|
∫
Ω
‍
B
(
x
,
v
(
x
)
)
·
u
(
x
)
d
x
|
≤
2
C
3
∥
v
(
x
)
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
p
(
x
)
-
1
∥
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
.
Using (12) and (13) we can easily prove
(14)
|
〈
𝔄
v
,
u
〉
|
=
|
∫
Ω
‍
(
A
(
x
,
d
v
)
·
d
u
+
B
(
x
,
d
v
)
·
u
)
d
x
|
≤
2
C
1
∥
d
v
∥
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
)
p
(
x
)
-
1
∥
d
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
)
+
2
C
3
∥
d
v
∥
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
)
p
(
x
)
-
1
∥
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
≤
C
5
∥
d
v
∥
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
)
p
(
x
)
-
1
(
∥
d
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
)
+
∥
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
)
≤
C
5
∥
v
(
x
)
∥
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
p
(
x
)
-
1
∥
u
∥
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
.
So we get𝔄
v
∈
[
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
]
′, whenever v
∈
𝔎
ψ
,
θ, u
∈
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
), and
C
1 and C
3 are constants fixed to (H2) and (H4).Lemma 10.
𝔄 is monotone and coercive on 𝔎
ψ
,
θ.Proof.
It follows from (H6) that𝔄 is monotone. To show that 𝔄 is coercive on 𝔎
ψ
,
θ, fix φ
∈
𝔎
ψ
,
θ and using the conditions (H2)–(H5), (12), (13), and (6), then
(15)
〈
𝔄
u
-
𝔄
φ
,
u
-
φ
〉
=
∫
Ω
‍
(
(
A
(
x
,
d
u
)
-
A
(
x
,
d
φ
)
)
·
(
d
u
-
d
φ
)
+
(
B
(
x
,
d
u
)
-
B
(
x
,
d
φ
)
)
·
(
u
-
φ
)
)
d
x
=
∫
Ω
‍
(
A
(
x
,
d
u
)
·
d
u
)
d
x
+
∫
Ω
‍
(
A
(
x
,
d
φ
)
·
d
φ
)
d
x
-
∫
Ω
‍
(
A
(
x
,
d
u
)
·
d
φ
)
d
x
-
∫
Ω
‍
(
A
(
x
,
d
φ
)
·
d
u
)
d
x
+
∫
Ω
‍
(
B
(
x
,
d
u
)
·
u
)
d
x
+
∫
Ω
‍
(
B
(
x
,
d
φ
)
·
φ
)
d
x
-
∫
Ω
‍
(
B
(
x
,
d
u
)
·
φ
)
d
x
-
∫
Ω
‍
(
B
(
x
,
d
φ
)
·
u
)
d
x
≥
C
2
(
∫
Ω
‍
|
d
u
|
p
(
x
)
w
(
x
)
d
x
+
∫
Ω
‍
|
d
φ
|
p
(
x
)
w
(
x
)
d
x
)
-
C
1
∫
Ω
‍
|
d
u
|
p
(
x
)
-
1
|
d
φ
|
w
(
x
)
d
x
-
2
C
1
∥
d
φ
∥
p
(
x
)
-
1
∥
d
u
∥
+
C
4
(
∫
Ω
‍
|
u
|
p
(
x
)
w
(
x
)
d
x
+
∫
Ω
‍
|
φ
|
p
(
x
)
w
(
x
)
d
x
)
-
C
3
∫
Ω
‍
|
d
u
|
p
(
x
)
-
1
|
φ
|
w
(
x
)
d
x
-
2
C
3
∥
d
φ
∥
p
(
x
)
-
1
∥
u
∥
≥
C
2
∫
Ω
‍
|
d
u
|
p
(
x
)
d
μ
-
C
1
∫
Ω
‍
1
p
′
(
x
)
ɛ
|
d
u
|
p
(
x
)
d
μ
-
C
1
∫
Ω
‍
1
p
(
x
)
ɛ
1
-
p
(
x
)
|
d
φ
|
p
(
x
)
d
μ
-
C
3
∫
Ω
‍
1
p
′
(
x
)
ɛ
|
d
u
|
p
(
x
)
d
μ
-
C
3
∫
Ω
‍
1
p
(
x
)
ɛ
1
-
p
(
x
)
|
φ
|
p
(
x
)
d
μ
+
C
4
∫
Ω
‍
|
u
|
p
(
x
)
d
μ
-
2
max
⁡
(
C
1
,
C
3
)
∥
d
φ
∥
p
(
x
)
-
1
(
∥
d
u
∥
+
∥
u
∥
)
+
C
(
φ
,
p
(
x
)
,
C
2
,
C
4
)
,
where ∥
·
∥ is the L
p
(
x
)
(
Ω
,
Λ
l
,
μ
) norm; taking ɛ
=
C
2
(
p
′
)
-
/
2
(
C
1
+
C
3
), we have
(16)
〈
𝔄
u
-
𝔄
φ
,
u
-
φ
〉
≥
C
2
2
∫
Ω
‍
|
d
u
|
p
(
x
)
d
μ
+
C
4
∫
Ω
‍
|
u
|
p
(
x
)
d
μ
-
C
5
∥
d
φ
∥
p
(
x
)
-
1
(
∥
d
u
∥
+
∥
u
∥
)
+
C
6
≥
C
2
2
∫
Ω
‍
(
|
d
u
-
d
φ
|
p
(
x
)
+
|
d
φ
|
p
(
x
)
)
d
μ
+
C
4
∫
Ω
‍
(
|
u
-
φ
|
p
(
x
)
+
|
φ
|
p
(
x
)
)
d
μ
-
C
5
∥
d
φ
∥
p
(
x
)
-
1
(
∥
d
u
-
d
φ
∥
+
∥
d
φ
∥
+
∥
u
-
φ
∥
+
∥
φ
∥
)
+
C
6
≥
C
7
(
∫
Ω
‍
|
d
u
-
d
φ
|
p
(
x
)
d
μ
+
∫
Ω
‍
|
u
-
φ
|
p
(
x
)
d
μ
)
-
C
5
∥
d
φ
∥
p
(
x
)
-
1
(
∥
d
u
-
d
φ
∥
+
∥
u
-
φ
∥
)
+
C
8
.
Letρ
p
(
·
)
(
t
)
=
∫
Ω
‍
|
t
|
p
(
x
)
d
μ; from [7, pages 24, 73], we know that if the variable exponent p
∈
𝒫
(
Ω
) satisfied p
+
<
∞, then
(17)
min
⁡
{
(
ϱ
p
(
·
)
(
f
)
)
1
/
p
-
,
(
ϱ
p
(
·
)
(
f
)
)
1
/
p
+
}
≤
∥
f
∥
L
p
(
x
)
(
ω
,
μ
)
≤
max
⁡
{
(
ϱ
p
(
·
)
(
f
)
)
1
/
p
-
,
(
ϱ
p
(
·
)
(
f
)
)
1
/
p
+
}
holds. Whenever ∥
u
∥
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
→
∞, we have ∥
d
u
-
d
φ
∥
L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
)
>
1 or ∥
u
-
φ
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
>
1, and using (17) we have
(18)
∫
Ω
‍
|
d
u
-
d
φ
|
p
(
x
)
d
μ
+
∫
Ω
‍
|
u
-
φ
|
p
(
x
)
d
μ
≥
max
⁡
{
∥
d
u
-
d
φ
∥
p
-
,
∥
d
u
-
d
φ
∥
p
+
}
+
max
⁡
{
∥
u
-
φ
∥
p
-
,
∥
u
-
φ
∥
p
+
}
.
Substituting (18) in (16) we obtain
(19)
〈
𝔄
u
-
𝔄
φ
,
u
-
φ
〉
≥
C
7
(
∫
Ω
‍
|
d
u
-
d
φ
|
p
(
x
)
d
μ
+
∫
Ω
‍
|
u
-
φ
|
p
(
x
)
d
μ
)
-
C
5
∥
d
φ
∥
p
(
x
)
-
1
(
∥
d
u
-
d
φ
∥
+
∥
u
-
φ
∥
)
+
C
8
≥
C
9
(
max
⁡
{
∥
d
u
-
d
φ
∥
p
-
,
∥
d
u
-
d
φ
∥
p
+
}
+
max
⁡
{
∥
u
-
φ
∥
p
-
,
∥
u
-
φ
∥
p
+
}
)
-
C
5
∥
d
φ
∥
p
(
x
)
-
1
(
∥
d
u
-
d
φ
∥
+
∥
u
-
φ
∥
)
+
C
8
.
Then it is easy to obtain(20)
〈
𝔄
u
-
𝔄
φ
,
u
-
φ
〉
∥
u
-
φ
∥
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
⟶
∞
as ∥
u
-
φ
∥
W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
→
∞. It follows that 𝔄 is coercive on 𝔎
ψ
,
θ. This completes the proof of Lemma 10.Lemma 11.
𝔄 is weakly continuous on 𝔎
ψ
,
θ.Proof.
Letu
i
∈
𝔎
ψ
,
θ be a sequence that converges to an element u
∈
𝔎
ψ
,
θ in W
d
p
(
x
)
(
Ω
,
Λ
l
,
μ
). Pick a subsequence u
i
j such that u
i
j
→
u a.e. in Ω. Since the mapping ξ
→
A
(
x
,
ξ
) and ξ
→
B
(
x
,
ξ
) are continuous for a.e. x in Ω, we have A
(
x
,
u
i
j
(
x
)
)
w
-
1
/
p
(
x
)
→
A
(
x
,
u
(
x
)
)
w
-
1
/
p
(
x
) a.e. in Ω. Under the conditions (H2) and (H4), we know that A
(
x
,
d
u
i
j
)
w
-
1
/
p
(
x
) and B
(
x
,
d
u
i
j
)
w
-
1
/
p
(
x
) are uniformly bounded in L
p
(
x
)
(
Ω
,
Λ
l
+
1
), and we have A
(
x
,
d
u
i
j
)
w
-
1
/
p
(
x
)
→
A
(
x
,
d
u
)
w
-
1
/
p
(
x
) weakly in L
p
(
x
)
(
Ω
,
Λ
l
+
1
,
μ
) and B
(
x
,
d
u
i
j
)
w
-
1
/
p
(
x
)
→
B
(
x
,
d
u
i
j
)
w
-
1
/
p
(
x
) weakly in L
p
(
x
)
(
Ω
,
Λ
l
,
μ
).
Since the weak limit is independent of the choice of the subsequence, it follows that(21)
A
(
x
,
d
u
i
)
w
-
1
/
p
(
x
)
⟶
A
(
x
,
d
u
)
w
-
1
/
p
(
x
)
,
B
(
x
,
d
u
i
)
w
-
1
/
p
(
x
)
⟶
B
(
x
,
d
u
)
w
-
1
/
p
(
x
)
for all u
∈
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
,
d
u
w
1
/
p
(
x
)
∈
L
p
(
x
)
(
Ω
,
Λ
l
+
1
).
Then we have(22)
〈
𝔄
u
i
,
v
〉
=
∫
Ω
‍
(
A
(
x
,
d
u
i
)
w
-
1
/
p
(
x
)
·
d
v
w
1
/
p
(
x
)
)
d
x
+
∫
Ω
‍
(
B
(
x
,
d
u
i
)
w
-
1
/
p
(
x
)
·
v
w
1
/
p
(
x
)
)
d
x
⟶
∫
Ω
‍
(
A
(
x
,
d
u
)
w
-
1
/
p
(
x
)
·
d
v
w
1
/
p
(
x
)
)
d
x
+
∫
Ω
‍
(
B
(
x
,
d
u
)
w
-
1
/
p
(
x
)
·
v
w
1
/
p
(
x
)
)
d
x
=
〈
𝔄
u
,
v
〉
.
Hence𝔄 is weakly continuous on 𝔎
ψ
,
θ. This ends the proof of Lemma 11.Proof of Theorem5.
We can apply Proposition6 and the above lemmas to obtain the existence. If there are two weak solutions u
1, u
2
∈
𝔎
ψ
,
θ to obstacle problem (1)-(2), then we have
(23)
∫
Ω
‍
(
A
(
x
,
d
u
1
)
·
d
(
u
2
-
u
1
)
+
B
(
x
,
d
u
1
)
·
(
u
2
-
u
1
)
)
d
x
≥
0
,
∫
Ω
‍
(
A
(
x
,
d
u
2
)
·
d
(
u
1
-
u
2
)
+
B
(
x
,
d
u
2
)
·
(
u
1
-
u
2
)
)
d
x
≥
0
,
so
(24)
∫
Ω
‍
(
(
A
(
x
,
d
u
2
)
-
A
(
x
,
d
u
1
)
)
·
d
(
u
2
-
u
1
)
+
(
B
(
x
,
d
u
2
)
-
B
(
x
,
d
u
1
)
)
·
(
u
2
-
u
1
)
)
d
x
≤
0
.
In view of (H6), we can further infer that(25)
∫
Ω
‍
(
(
A
(
x
,
d
u
2
)
-
A
(
x
,
d
u
1
)
)
·
d
(
u
2
-
u
1
)
+
(
B
(
x
,
d
u
2
)
-
B
(
x
,
d
u
1
)
)
·
(
u
2
-
u
1
)
)
d
x
=
0
a
.
e
.
on
⁡
Ω
,
which means that u
1
=
u
2
a
.
e
.
on
⁡
Ω, and now we complete the proof.Corollary 12.
LetΩ be a bounded domain and θ
∈
W
d
p
(
x
)
(
Ω
,
Λ
l
-
1
,
μ
). Under the conditions (H1)–(H6), there is a differential form u
∈
W
d
p
(
x
)
(
Ω
,
Λ
l
-
1
,
μ
) with u
-
θ
∈
W
d
p
(
x
)
(
Ω
,
Λ
l
-
1
,
μ
) such that
(26)
d
⋆
A
(
x
,
d
u
)
=
B
(
x
,
d
u
)
,
w
e
a
k
l
y
i
n
Ω
;
that is, ∫
Ω
‍
(
A
(
x
,
d
u
)
·
d
φ
+
B
(
x
,
d
u
)
·
φ
)
d
x
=
0, whenever φ
∈
W
0
d
p
(
x
)
(
Ω
,
Λ
l
-
1
,
μ
)
,
l
=
1,2
,
…
,
n.Proof.
Chooseψ
I
≡
-
∞ and let u be the solution to the obstacle problem (1)-(2) in 𝔎
ψ
,
θ. For any φ
∈
W
0
d
p
(
x
)
(
Ω
,
Λ
l
-
1
,
μ
), since u
+
φ and u
-
φ both belong to 𝔎
ψ
,
θ, we have
(27)
-
(
∫
Ω
‍
(
A
(
x
,
d
u
)
·
d
φ
+
B
(
x
,
d
u
)
·
φ
)
d
x
)
≥
0
,
∫
Ω
‍
(
A
(
x
,
d
u
)
·
d
φ
+
B
(
x
,
d
u
)
·
φ
)
d
x
≥
0
.
Thus(28)
∫
Ω
‍
(
A
(
x
,
d
u
)
·
d
φ
+
B
(
x
,
d
u
)
·
φ
)
d
x
=
0
,
as desired.Remark 13.
Ifp
(
x
)
=
p, then ∥
u
∥
L
p
(
x
)
(
Ω
,
Λ
l
,
μ
)
=
∥
u
∥
L
p
(
Ω
,
Λ
l
,
μ
)
=
∥
u
∥
p
,
Ω
,
μ
=
(
∫
Ω
‍
|
u
|
p
d
μ
)
1
/
p and the Luxemburg norm reduces to the L
p norm. So (26) is the extension of the equation in [2–5].



---

*Source: 280214-2014-02-24.xml*