# An Approximation of Hedberg’s Type in Sobolev Spaces with Variable Exponent and Application

Authors: Abdelmoujib Benkirane; Mostafa El Moumni; Aziz Fri
Journal: Chinese Journal of Mathematics
 (2014)

---

## Abstract

The aim of this paper is to extend the usual framework of PDE withA
u
=
-
d
i
v
a
x
,
u
,
∇
u to include a large class of cases with A
u
=
∑
β
≤
α
-
1
β
D
β
A
β
x
,
u
,
∇
u
,
…
,
∇
α
u, whose coefficient A
β satisfies conditions (including growth conditions) which guarantee the solvability of the problem A
u
=
f. This new framework is conceptually more involved than the classical one includes many more fundamental examples. Thus our main result can be applied to various types of PDEs such as reaction-diffusion equations, Burgers type equation, Navier-Stokes equation, and p-Laplace equation.


---

## Body

## 1. Introduction

This paper is motivated by the study of the unilateral problem associated with the following equation:(1)
A
(
u
)
+
g
(
x
,
u
)
=
f
.We show the existence of variational solutions of this elliptic boundary value problem for strongly elliptic systems of order2
m on a domain Ω in ℝ
N in generalized divergence form as follows:
(2)
A
(
u
)
=
∑
|
β
|
≤
α
‍
(
-
1
)
|
β
|
D
β
A
β
(
u
,
∇
u
,
…
,
∇
α
u
)
.The functiong satisfies a sign condition but has otherwise completely unrestricted growth with respect to u.Equations of type (1) were first considered by Browder [1] as an application to the theory of not everywhere defined mapping of monotone type. For α
=
1, that is, A of second order, their solvability under fairly general and natural assumptions was proved by Hess [2]. The treatment of the case α
>
1 is more involved due to the lack of a simple truncation operator in higher order Sobolev spaces. Webb [3] observed that rather delicate approximation procedure introduced in nonlinear potential theory by Hedberg [4] could be used in place of truncation. This yielded the solvability of (1) for α
>
1. Brezis and Browder [5] then used this approximation procedure to solve a question which they had considered earlier [6] about the action of some distribution. They also showed that their result on the action of some distributions could itself be used in place of truncation in the study the problem (1). In a more general case, Boccardo et al. studied inequations associated with (1), see [7].The functional setting in all the results mentioned above is that of the usual Sobolev spacesW
α
,
p
(
ℝ
N
), and the functions A
β in (2) are supposed to satisfy polynomial growth conditions with respect to u and its derivatives. Benkirane and Gossez established this result in the Orlicz-Sobolev spaces W
α
L
A
(
ℝ
N
), see [8–10].It is our purpose in this paper to study these problems in this setting of Sobolev spaces with variable exponentW
α
,
p
(
·
)
(
ℝ
N
) of the harder higher order case α
>
1. We consider problem (1) as well as Hedberg’s approximation theorem and Brezis-Browder’s question on the action of some distributions.The paper is structured as follows. After some necessary preliminaries, in Section3, we give the proof of the approximation theorem. In addition, Section 4 forms a useful supplement to some applications of (1).


## 2. Preliminaries

In this section we list briefly some definitions and well known facts about Sobolev spaces with variable exponent and Bessel potential spaces with variable exponent. Standard references are [11, 12].LetΩ be an open subset of ℝ
N, by the symbol 𝒫
(
Ω
), we denote the family of all measurable functions p
(
·
)
:
Ω
→
[
1
,
∞
].Forp
(
·
)
∈
𝒫
(
Ω
), put
(3)
p
+
=
ess
sup
⁡
x
∈
Ω
p
(
x
)
,
p
-
=
ess
inf
⁡
x
∈
Ω
p
(
x
)
.
Furthermore, we introduce a class ℬ
(
Ω
) by
(4)
ℬ
(
Ω
)
:
=
{
p
∈
𝒫
(
Ω
)
;
1
<
p
-
≤
p
+
<
∞
}
.
Let p
(
·
)
∈
ℬ
(
Ω
), and consider the functional
(5)
ϱ
p
(
·
)
(
f
)
=
∫
Ω
‍
|
f
(
x
)
|
p
(
x
)
d
x
,
on all measurable function f on Ω. The Lebesgue space with variable exponent L
p
(
·
)
(
Ω
) is defined as the set of all measurable functions f on Ω such that, for some λ
>
0,
(6)
ϱ
p
(
·
)
(
f
λ
)
<
∞
,
equipped with the norm
(7)
∥
f
∥
p
(
·
)
=
inf
⁡
{
λ
>
0
;
ϱ
p
(
·
)
(
f
λ
)
≤
1
}
.
The space (
L
p
(
·
)
(
Ω
)
,
∥
·
∥
p
(
·
)
) is a separable Banach space. Moreover, if 1
<
p
-
≤
p
+
<
+
∞, then L
p
(
·
)
(
Ω
) is uniformly convex, hence, reflexive, and its dual space is isomorphic to L
p
′
(
·
)
(
Ω
), where 1
/
p
(
x
)
+
1
/
p
′
(
x
)
=
1.Finally, we have the Hölder type inequality as follows:(8)
|
∫
Ω
‍
u
v
d
x
|
≤
(
1
p
-
+
1
p
-
′
)
∥
u
∥
p
(
·
)
∥
v
∥
p
′
(
·
)
for all u
∈
L
p
(
·
)
(
Ω
) and v
∈
L
p
′
(
·
)
(
Ω
).The Hardy-Little-wood maximal operatorM is defined on locally integrable functions f on Ω by the following formula:
(9)
M
f
(
x
)
=
sup
⁡
R
>
0
1
|
B
(
x
,
R
)
|
∫
B
(
x
,
R
)
‍
|
f
(
y
)
|
d
y
,
where B
(
x
,
R
) denote the open ball in Ω with center x and radius R, and |
B
(
x
,
R
)
| denotes the volume of B
(
x
,
R
).Definition 1.
Byℳ
(
Ω
), denote the class of all functions p
∈
ℬ
(
Ω
) for which the operator M is bounded on L
p
(
·
)
(
Ω
); that is,
(10)
∥
M
f
∥
p
(
·
)
≤
C
∥
f
∥
p
(
·
)
,
with a positive constant C independent of f.Remark 2.
For example,p
(
·
)
∈
ℳ
(
Ω
) if the following two conditions are satisfied:
(11)
|
p
(
x
)
-
p
(
y
)
|
≤
c
-
log
⁡
⁡
(
|
x
-
y
|
)
,
|
x
-
y
|
≤
1
2
,
|
p
(
x
)
-
p
(
y
)
|
≤
c
log
⁡
(
e
+
|
x
|
)
,
|
y
|
>
|
x
|
.
For more details, see [13–16], where various sufficient conditions for p
(
·
)
∈
ℳ
(
Ω
) can be found.
Letp
(
·
)
∈
ℬ
(
Ω
) and α
∈
ℕ; we define the Sobolev space with variable exponent by
(12)
W
α
,
p
(
·
)
(
Ω
)
=
{
u
,
D
β
u
∈
L
p
(
·
)
(
Ω
)
if
|
β
|
≤
α
}
,
equipped with the norm
(13)
∥
u
∥
W
α
,
p
(
·
)
=
∑
|
β
|
≤
α
‍
∥
D
β
u
∥
L
p
(
·
)
,
where β
∈
ℕ
0
N is a multi-index, |
β
|
=
β
1
+
⋯
+
β
N, and D
β
=
∂
|
β
|
/
∂
β
1
x
1
⋯
∂
β
N
x
N.
Next, we defineW
0
α
,
p
(
·
)
(
Ω
) as the closure of 𝒟
(
Ω
) in W
α
,
p
(
·
)
(
Ω
) and W
-
α
,
p
′
(
·
)
(
Ω
) the dual space of W
α
,
p
(
·
)
(
Ω
), where 1
/
p
(
·
)
+
1
/
p
′
(
·
)
=
1.
The Bessel kernelG
α (see [17]) of order α
>
0 is defined by
(14)
G
α
(
x
)
=
π
N
/
2
Γ
(
α
/
2
)
∫
0
∞
‍
e
-
s
-
π
2
|
x
|
2
/
s
s
(
α
-
N
)
/
2
d
s
s
,
ggggggggggggggggggggggg
x
∈
ℝ
N
.
The Riesz kernel I
α (see [18]) of order α
>
0 is defined by
(15)
I
α
(
x
)
=
γ
α
|
x
|
N
-
α
,
where γ
α is a certain constant, whose exact value is
(16)
γ
α
=
Γ
(
(
N
-
α
)
/
2
)
(
π
N
/
2
2
α
Γ
(
α
/
2
)
)
.
It follows easily that
(17)
G
α
(
x
)
~
I
α
(
x
)
,
|
x
|
⟶
0
,
0
<
α
<
N
,
and an examination of (14) shows without much effort that for any c
<
1,
(18)
G
α
(
x
)
=
o
(
e
-
c
|
x
|
)
,
|
x
|
⟶
∞
,
N
>
α
>
0
,
∫
|
x
-
y
|
<
δ
‍
f
(
y
)
d
y
|
x
-
y
|
N
-
α
≤
A
δ
α
M
f
(
x
)
.
Writing G
α
(
x
)
=
G
α
(
r
), r
=
|
x
|, this implies that, for N
>
α
>
1,
(19)
G
α
′
(
r
)
~
-
(
N
-
α
)
G
α
-
1
(
r
)
,
r
⟶
0
,
N
>
α
>
0
,
G
α
′
(
r
)
~
-
a
α
r
(
α
-
N
-
1
)
/
2
e
-
r
~
-
c
G
α
(
r
)
,
r
⟶
∞
,
with c
=
2
(
α
+
N
-
3
)
/
2
π
(
N
-
α
-
1
)
/
2 and 1
/
a
α
=
(
4
π
)
(
α
)
/
2
Γ
(
(
α
)
/
2
).


## 3. Main Results

First, we give the following results which will be used in our main result.


### 3.1. Useful Results

Letp
(
·
)
∈
ℬ
(
ℝ
N
) and α
>
0. The Bessel potential space with variable exponent L
α
,
p
(
·
)
(
ℝ
N
) is defined, for N
>
α
>
0, by
(20)
L
α
,
p
(
·
)
(
ℝ
N
)
=
{
u
=
G
α
*
f
;
f
∈
L
p
(
·
)
(
ℝ
N
)
}
,
and is equipped with the norm
(21)
∥
u
∥
α
,
p
(
·
)
=
∥
f
∥
p
(
·
)
.Lemma 3 (see [12]).
Ifp
(
·
)
∈
ℳ
(
ℝ
N
) and α
∈
ℕ, then
(22)
L
α
,
p
(
·
)
(
ℝ
N
)
≅
W
α
,
p
(
·
)
(
ℝ
N
)
,
and the corresponding norms are equivalent.Lemma 4 (see [12]).
Suppose thatp
(
·
)
∈
ℳ
(
ℝ
N
) and α
≥
0. Then there exists a positive constant C such that
(23)
∥
G
α
*
f
∥
p
(
·
)
≤
C
∥
f
∥
p
(
·
)
,
f
o
r
f
∈
L
p
(
·
)
(
ℝ
N
)
.We will now verify thatu
n satisfies all the required properties in Proposition 7. The argument relies heavily on the following Lemmas 5 and 6. In the sequel, we need the following two technical lemmas.Lemma 5.
If1
<
p
-
≤
p
(
x
)
≤
p
+
<
∞ and for any multi-index |
ξ
|
<
α
<
N, there exists a constant A such that for any f
∈
L
p
(
·
)
(
ℝ
N
),
(24)
|
D
ξ
(
G
α
*
f
(
x
)
)
|
≤
A
(
M
f
(
x
)
)
|
ξ
|
/
α
(
G
α
*
|
f
|
(
x
)
)
1
-
|
ξ
|
/
α
,
f
o
r
a
l
m
o
s
t
e
v
e
r
y
x
∈
ℝ
N
.Proof.
We assume thatM
f
(
x
)
<
∞; otherwise, there is nothing to prove. We then observe that there is a constant A
1 such that G
α
*
|
f
|
(
x
)
≤
A
1
M
f
(
x
). In fact, by (17), (18), and ([11], Lemma 6.1.4), there exists a constant A such that(25)
G
α
*
|
f
|
(
x
)
≤
A
∫
|
x
-
y
|
<
1
‍
|
f
(
y
)
|
d
y
|
x
-
y
|
N
-
α
+
A
∫
|
x
-
y
|
≥
1
‍
|
f
(
y
)
|
e
-
|
x
-
y
|
/
2
d
y
≤
A
M
f
(
x
)
+
A
∑
i
=
1
∞
‍
e
-
i
/
2
∫
i
+
1
>
|
x
-
y
|
≥
i
‍
|
f
(
y
)
|
d
y
≤
A
M
f
(
x
)
+
A
M
f
(
x
)
∑
i
=
1
∞
‍
(
i
+
1
)
N
e
-
i
/
2
=
A
1
M
f
(
x
)
.
Then, by (19) and (18), for any δ
≤
1, we have
(26)
|
D
ξ
(
G
α
*
|
f
|
(
x
)
)
|
≤
A
∫
|
x
-
y
|
<
δ
‍
|
f
(
y
)
|
d
y
|
x
-
y
|
N
-
α
+
|
ξ
|
+
A
∫
δ
≤
|
x
-
y
|
<
1
‍
|
f
(
y
)
|
d
y
|
x
-
y
|
N
-
α
+
|
ξ
|
+
A
∫
|
x
-
y
|
≥
1
‍
G
α
(
x
-
y
)
|
f
(
y
)
|
d
y
≤
A
(
δ
α
-
|
ξ
|
M
f
(
x
)
+
δ
-
|
ξ
|
(
G
α
*
|
f
|
(
x
)
)
+
(
G
α
*
|
f
|
(
x
)
)
δ
α
-
|
ξ
|
M
f
(
x
)
)
.
Now choosing δ
α
=
G
α
*
|
f
|
(
x
)
/
A
1
M
f
(
x
); then, δ
≤
1 and the result follows.Lemma 6.
Suppose that1
<
p
-
≤
p
(
x
)
≤
p
+
<
∞ and 0
<
α
<
N. Let H
∈
C
k
(
ℝ
+
) for some k
≥
α, if H satisfies
(27)
sup
⁡
t
>
0
|
t
i
-
1
H
(
i
-
1
)
(
t
)
|
≤
L
<
∞
,
w
i
t
h
i
=
1,2
,
…
,
k
.
Then H
(
G
α
*
f
)
∈
L
α
,
p
(
·
)
(
ℝ
N
) for every f
∈
L
+
p
(
·
)
(
ℝ
N
), and there exists a constant A, depending only on α and N, such that
(28)
∥
H
(
G
α
*
f
)
∥
α
,
p
(
·
)
≤
A
L
∥
G
α
*
f
∥
α
,
p
(
·
)
=
A
L
∥
f
∥
p
(
·
)
.Proof.
Forα integer. Assume that f
∈
C
0
∞
(
ℝ
N
) and f
≥
0. Set u
=
G
α
*
f and notice that u
(
x
)
>
0 for all x, so that H
(
u
) is defined. If ξ is a multi-indix with |
ξ
|
=
α, we find by the chain rule that
(29)
D
ξ
(
H
(
u
)
)
=
∑
i
=
1
α
‍
H
(
i
)
(
u
)
∑
‍
c
ξ
D
ξ
1
u
⋯
D
ξ
i
u
,
where the interior sum is over all ordered i-tuples of multi-indices {
ξ
1
,
…
,
ξ
i
} such that ξ
1
+
⋯
+
ξ
i
=
ξ, and all |
ξ
j
|
≥
1. The c
ξ are coefficients, whose exact value is of no consequence to us. Thus, by assumption of Lemma 3.4 we get
(30)
|
D
ξ
(
H
(
u
)
)
|
≤
A
L
∑
i
=
1
α
‍
u
1
-
i
∑
‍
|
D
ξ
1
u
⋯
D
ξ
i
u
|
.
For i
>
1, we estimate these derivatives by means of Lemma 5. By the positivity of f, we have
(31)
|
D
ξ
j
u
|
≤
A
(
M
f
(
x
)
)
|
ξ
j
|
/
α
u
1
-
|
ξ
j
|
/
α
.
Thus, since ∑
j
=
1
i
‍
(
1
-
|
ξ
j
|
/
α
)
=
i
-
|
ξ
|
/
α
=
i
-
1,
(32)
∑
i
=
2
α
‍
u
1
-
i
∑
‍
|
D
ξ
1
u
⋯
D
ξ
i
u
|
≤
A
∑
i
=
2
α
‍
u
1
-
i
M
f
u
i
-
1
=
A
M
f
.
Taking the term with i
=
1 into account, we obtain
(33)
|
D
ξ
(
H
(
u
)
)
|
≤
A
L
(
|
M
f
|
+
|
D
ξ
u
|
)
.
But we already know from (10) that ∥
M
f
∥
p
(
·
)
≤
A
∥
f
∥
p
(
·
) and that ∥
D
ξ
(
G
α
*
f
)
∥
p
(
·
)
≤
A
∥
f
∥
p
(
·
) for |
ξ
|
=
α.
This finishes the proof for smoothf.
Now we pass to the general case and letf be an arbitrary function in L
+
p
(
·
)
(
ℝ
N
). Then there are nonnegative functions f
i
∈
C
0
∞
(
ℝ
N
), i
=
1,2
,
…, such that
(34)
lim
⁡
i
→
∞
∥
f
i
-
f
∥
p
(
·
)
=
0
.
By the first part of the proof,
(35)
∥
H
(
(
G
α
*
f
i
)
)
∥
α
,
p
(
·
)
≤
A
L
∥
f
∥
p
(
·
)
,
for all sufficiently large i.
Thus, settingH
(
(
G
α
*
f
i
)
)
=
G
α
*
g
i, we can assume that {
g
i
}
1
∞ converges weakly in L
p
(
·
)
(
ℝ
N
) to an element g, with ∥
g
∥
p
(
·
)
≤
A
L
∥
f
∥
p
(
·
).
We have to prove thatG
α
*
g
=
H
(
(
G
α
*
f
)
).
The strong convergence of{
f
i
}
1
∞ and the fact that G
α
∈
L
1
(
ℝ
N
) imply, by Lemma 4, that {
G
α
*
f
i
}
1
∞ converges strongly in L
p
(
·
)
(
ℝ
N
) to G
α
*
f. After extraction of a subsequence, we can assume that
(36)
lim
⁡
i
→
∞
G
α
*
f
i
(
x
)
=
G
α
*
f
(
x
)
a
.
e
.
But H is continuous, so it follows that
(37)
lim
⁡
i
→
∞
G
α
*
g
i
(
x
)
=
lim
⁡
i
→
∞
H
(
(
G
α
*
f
i
)
)
(
x
)
=
H
(
(
G
α
*
f
)
)
(
x
)
a
.
e
.
On the other hand, the weak convergence of{
g
i
}
1
∞ implies that the pointwise limit of {
G
α
*
g
i
}
1
∞ (which is now known to exist a.e.) is G
α
*
g. In fact, setting g
i
-
g
=
h
i, for an arbitrary ϵ
>
0,
(38)
G
α
*
h
i
(
x
)
=
∫
|
x
-
y
|
≤
ϵ
‍
G
α
(
x
-
y
)
h
i
(
y
)
d
y
+
∫
|
x
-
y
|
>
ϵ
‍
G
α
(
x
-
y
)
h
i
(
y
)
d
y
.
By weak convergence, the last term tends to zero, since G
α is in L
p
′
(
·
)
(
ℝ
N
) away from the origin. Consider the following:
(39)
Let
λ
>
liminf
⁡
i
→
∞
∥
∫
|
y
|
≤
ϵ
‍
G
α
(
y
)
h
i
(
·
-
y
)
d
y
∥
p
(
·
)
.
We deduce that, for all j, λ
>
inf
⁡
i
≥
j
∥

∫
|
y
|
≤
ϵ
‍
G
α
(
y
)
h
i
(
·
-
y
)
d
y
∥
p
(
·
), which implies that for all j, there exists i
≥
j, such that
(40)
λ
>
∥
∫
|
y
|
≤
ϵ
‍
G
α
(
y
)
h
i
(
·
-
y
)
d
y
∥
p
(
·
)
.
Then,
(41)
∫
|
y
|
≤
ϵ
‍
(
liminf
⁡
i
→
∞
⁡
G
α
(
y
)
h
i
(
x
-
y
)
λ
)
p
(
x
)
d
x
≤
liminf
⁡
i
→
∞
∫
|
y
|
≤
ϵ
‍
(
G
α
(
y
)
h
i
(
·
-
y
)
λ
)
p
(
x
)
d
x
≤
1
.
Since λ
≥
∥
liminf
⁡
i
→
∞
G
α
*
h
i
∥
p
(
·
), then we have
(42)
∥
lim
⁡
i
→
∞
G
α
*
h
i
∥
p
(
·
)
≤
liminf
⁡
i
→
∞
∥
∫
|
y
|
≤
ϵ
‍
G
α
(
y
)
h
i
(
·
-
y
)
d
y
∥
p
(
·
)
≤
sup
⁡
i
∥
h
i
∥
p
(
·
)
∫
|
y
|
≤
ϵ
‍
G
α
(
y
)
d
y
,
which is an arbitrary small number, and thus for a.e. x,
(43)
G
α
*
g
(
x
)
=
lim
⁡
i
→
∞
G
α
*
g
i
(
x
)
=
lim
⁡
i
→
∞
H
(
(
G
α
*
f
i
)
)
(
x
)
=
H
(
(
G
α
*
f
)
)
(
x
)
.
This completes the proof of Lemma 5.Proposition 7.
Letu
∈
W
α
,
p
(
·
)
(
ℝ
N
), there exist a sequence u
n, such that (i)
u
n
∈
W
α
,
p
(
·
)
(
ℝ
N
)
∩
L
∞
(
ℝ
N),
supp
⁡
u
n
is compact;
(ii)
|
u
n
(
x
)
|
≤
|
u
(
x
)
|
and
u
n
(
x
)
u
(
x
)
≥
0
a.e. in
ℝ
N
;
(iii)
u
n
→
u
in
W
α
,
p
(
·
)
(
ℝ
N
)
as
n
→
∞
.Proof.
The proof of Proposition7 is done in two steps as follows.
Step  1
(
c
a
s
e
α
p
^
-
>
N
). Let ξ
∈
C
0
∞
(
ℝ
N
) be a fixed function such that 0
≤
ξ
≤ 1 and ξ
(
x
)
=
1 in a neighborhood of the origin.
Letξ
n
(
x
) = ξ
(
x
/
n
); then, u
n
(
x
) = ξ
n
(
x
)
u
(
x
) satisfies all the required properties, using the fact that u
∈
L
∞
(
Ω
) by Sobolev’s theorem [19].
Step  2 (
c
a
s
e
α
p
-
≤
N
). We assume that u has compact support, if necessary by multiplying with a suitable ξ
n. We represent u as a Bessel potential, u
=
G
α
*
f, so that ∥
f
∥
p
(
·
)
≤
A
∥
u
∥
W
α
;
p
(
·
). Set
(44)
v
=
G
α
*
|
f
|
,
and let T
∈
C
∞
(
ℝ
) be a function such that 0
≤
T
≤
1, T
(
t
)
=
1 for 0
≤
t
≤
1
/
2 and T
(
t
)
=
0 for t
≥
1. Then, set
(45)
u
n
(
x
)
=
T
(
v
(
x
)
n
)
u
(
x
)
,
n
=
1
,
2
,
3
,
…
.
We First observe that, u
n
(
x
)
=
0 on the set {
x
:
v
(
x
)
≥
n
}, which includes {
x
:
|
u
(
x
)
|
≥
n
}, and so we have |
u
n
(
x
)
|
<
n a.e., and thus u
n
(
x
)
u
(
x
)
≥
0. It remains to prove that u
n
∈
W
α
,
p
(
·
)
(
ℝ
N
) and that u
n converges to u as n tends to ∞.
Letη be any multi-index with 0
<
|
η
|
=
β
≤
α. If η = η
1
+
⋯
+
η
i, i
>
1, and all |
η
j
|
≥
1, we find by the same arguments as in the proof of Lemma 6 that
(46)
|
D
η
T
(
v
(
x
)
n
)
|
≤
A
∑
i
=
1
β
‍
n
-
i
∑
‍
|
D
η
1
v
(
x
)
⋯
D
η
i
v
(
x
)
|
.
By Lemma 5, we have, for any multi-index η with 0
<
|
η
|
<
α,
(47)
|
D
η
v
(
x
)
|
≤
A
M
f
(
x
)
|
η
|
/
α
(
v
(
x
)
)
1
-
|
η
|
/
α
.
On the open set {
x
:
v
(
x
)
>
n
}, using the fact that D
η
T
(
v
(
x
)
/
n
)
=
0; then,
(48)
|
D
η
H
(
v
(
x
)
n
)
|
≤
A
∑
i
=
1
β
‍
n
-
i
n
i
-
|
η
|
/
α
M
f
(
x
)
|
η
|
/
α
≤
A
n
-
|
η
|
/
α
M
f
(
x
)
|
η
|
/
α
,
for |
η
|
≤
α
-
1, and
(49)
|
D
η
T
(
v
(
x
)
n
)
|
≤
A
n
-
1
(
M
f
(
x
)
+
|
D
η
v
(
x
)
|
)
,
for |
η
| = α.
By using Leibniz’s formula, we have for|
ξ
| = α, if v
(
x
)
≤
n, again using Lemma 5, that
(50)
|
D
ξ
u
n
(
x
)
-
D
ξ
u
(
x
)
|
=
|
(
1
-
T
(
v
(
x
)
n
)
)
D
ξ
u
(
x
)
|
,
|
D
ξ
u
n
(
x
)
-
D
ξ
u
(
x
)
|
≤
A
∑
0
<
|
η
|
<
α
‍
|
D
η
T
(
v
(
x
)
n
)
|
|
D
ξ
-
η
u
(
x
)
|
+
|
D
ξ
H
(
v
(
x
)
n
)
|
|
u
(
x
)
|
≤
A
∑
0
<
β
<
α
‍
n
-
β
/
α
M
f
(
x
)
β
/
α
M
f
(
x
)
1
-
β
/
α
v
(
x
)
β
/
α
+
A
n
-
1
(
M
f
(
x
)
+
|
D
ξ
v
(
x
)
|
)
v
(
x
)
≤
A
(
v
(
x
)
n
)
1
/
α
M
f
(
x
)
+
A
(
v
(
x
)
n
)
(
M
f
(
x
)
+
|
D
ξ
v
(
x
)
|
)
.
If v
(
x
)
>
n, we have D
ξ
u
n
(
x
)
=
0. It follows that a.e.
(51)
lim
⁡
n
→
∞
|
D
ξ
u
n
(
x
)
-
D
ξ
u
(
x
)
|
=
0
,
|
D
ξ
u
n
(
x
)
-
D
ξ
u
(
x
)
|
≤
|
D
ξ
u
(
x
)
|
+
A
(
M
f
(
x
)
+
|
D
ξ
v
(
x
)
|
)
.
The functions on the right hand side belong toL
p
(
·
), so the theorem follows by applying the dominated convergence.Remark 8.
The sequenceu
n constructed above satisfies
(52)
∥
u
n
∥
W
α
,
p
(
·
)
≤
C
∥
u
∥
W
α
,
p
(
·
)
,
with a constant C depending only on α and N.


### 3.2. Existence Result

This subsection is devoted to establish the following existence theorem.Theorem 9.
LetT
∈
L
l
o
c
1
(
ℝ
N
)
∩
W
-
α
,
p
′
(
·
)
(
ℝ
N
) and u
∈
W
α
,
p
(
·
)
(
ℝ
N
). Assume that T
(
x
)
u
(
x
)
≥
h
(
x
) a.e. in ℝ
N, for some h
∈
L
1
(
ℝ
N
). Then,
(53)
T
u
∈
L
1
(
ℝ
N
)
,
∫
ℝ
N
‍
T
(
x
)
u
(
x
)
d
x
=
〈
T
,
u
〉
.Proof.
We first deduce Theorem9 as a simple consequence of Proposition 7. Let u
n be a sequence defined in Proposition 7. It follows easily from (i) in Proposition 7 (using convolution with mollifiers and according to [11]) that
(54)
∫
ℝ
N
‍
T
(
x
)
u
n
(
x
)
d
x
=
〈
T
,
u
n
〉
.
By Proposition 7, the right hand side of (54) converges as n
→
∞ to 〈
T
,
u
〉. On the other hand, we have T
u
n
≥
-
|
h
| a.e. We deduce from Fatou’s lemma that T
u
∈
L
1. We conclude by dominated convergence that
(55)
∫
‍
T
(
x
)
u
n
(
x
)
d
x
⟶
∫
‍
T
(
x
)
u
(
x
)
d
x
and
thus
∫
‍
T
(
x
)
u
(
x
)
d
x
=
〈
T
,
u
〉
.Theorem 10.
LetT
∈
L
l
o
c
1
(
Ω
)
∩
W
-
α
,
p
′
(
·
)
(
Ω
) be such that
(56)
∫
Ω
∩
B
(
0
,
R
)
‍
|
T
(
x
)
|
d
x
<
∞
f
o
r
e
v
e
r
y
R
<
∞
.
Assume that u
∈
W
0
α
,
p
(
·
)
(
Ω
) and T
(
x
)
u
(
x
)
≥
h
(
x
) a.e. in Ω, for some h
∈
L
1
(
Ω
). Then,
(57)
T
u
∈
L
1
(
Ω
)
,
∫
Ω
‍
T
(
x
)
u
(
x
)
d
x
=
〈
T
,
u
〉
.Proof.
The proof is straightforward whenα
p
-
>
N; therefore, we may assume that α
p
-
≤
N. Using ξ
n
u in place of u, we may always reduce to the case where supp
⁡
u is bounded. Set(58)
u
¯
=
{
u
(
x
)
if
x
∈
Ω
,
0
if
x
∈
ℝ
N
∖
Ω
.
Then u
¯
∈
W
α
,
p
(
·
)
(
ℝ
N
). This allows us to write
(59)
u
¯
=
G
α
*
f
,
for some f in L
p
(
·
)
(
ℝ
N
). As in the proof of Proposition 7, set
(60)
v
=
G
α
*
|
f
|
,
u
n
=
H
(
v
n
)
u
.
Since u
∈
W
0
α
,
p
(
·
)
(
Ω
), there exists a sequence u
j
∈
C
0
∞
(
Ω
) such that u
j
→
u in W
α
,
p
(
·
)
(
Ω
) and a.e. (see [19]). For each j, we perform the above construction and we set
(61)
u
¯
j
=
G
α
*
f
j
,
v
j
=
G
α
*
|
f
j
|
,
u
n
j
=
H
(
v
j
n
)
u
j
.
Fix ξ
∈
C
0
∞
(
ℝ
N
). We clearly have
(62)
∫
Ω
‍
T
ξ
u
n
j
d
x
=
〈
T
,
ξ
u
n
j
〉
.
As we keep n fixed and let j
→
∞, we see that
(63)
∫
Ω
‍
T
ξ
u
n
j
d
x
⟶
∫
Ω
‍
T
ξ
u
n
d
x
,
by dominated convergence and (56).
On the other hand, by Remark8, we obtain
(64)
∥
u
n
j
∥
W
α
,
p
(
·
)
(
Ω
)
≤
∥
u
¯
j
∥
W
α
,
p
(
·
)
(
ℝ
N
)
≤
C
,
where C does not depend on j and n.
Therefore,u
n
j converges weakly to u
n in W
0
α
,
p
(
·
)
(
Ω
) as j
→
∞ and thus ξ
u
n
j
⇀
ξ
u
n as j
→
∞. Passing to the limit in (62) as j
→
∞, we find
(65)
∫
Ω
‍
T
ξ
u
n
=
〈
T
,
ξ
u
n
〉
.
We conclude easily (by the argument as in the proof of Theorem 9) that
(66)
T
u
∈
L
1
(
Ω
)
,
∫
Ω
‍
T
(
x
)
u
(
x
)
d
x
=
〈
T
,
u
〉
.


## 3.1. Useful Results

Letp
(
·
)
∈
ℬ
(
ℝ
N
) and α
>
0. The Bessel potential space with variable exponent L
α
,
p
(
·
)
(
ℝ
N
) is defined, for N
>
α
>
0, by
(20)
L
α
,
p
(
·
)
(
ℝ
N
)
=
{
u
=
G
α
*
f
;
f
∈
L
p
(
·
)
(
ℝ
N
)
}
,
and is equipped with the norm
(21)
∥
u
∥
α
,
p
(
·
)
=
∥
f
∥
p
(
·
)
.Lemma 3 (see [12]).
Ifp
(
·
)
∈
ℳ
(
ℝ
N
) and α
∈
ℕ, then
(22)
L
α
,
p
(
·
)
(
ℝ
N
)
≅
W
α
,
p
(
·
)
(
ℝ
N
)
,
and the corresponding norms are equivalent.Lemma 4 (see [12]).
Suppose thatp
(
·
)
∈
ℳ
(
ℝ
N
) and α
≥
0. Then there exists a positive constant C such that
(23)
∥
G
α
*
f
∥
p
(
·
)
≤
C
∥
f
∥
p
(
·
)
,
f
o
r
f
∈
L
p
(
·
)
(
ℝ
N
)
.We will now verify thatu
n satisfies all the required properties in Proposition 7. The argument relies heavily on the following Lemmas 5 and 6. In the sequel, we need the following two technical lemmas.Lemma 5.
If1
<
p
-
≤
p
(
x
)
≤
p
+
<
∞ and for any multi-index |
ξ
|
<
α
<
N, there exists a constant A such that for any f
∈
L
p
(
·
)
(
ℝ
N
),
(24)
|
D
ξ
(
G
α
*
f
(
x
)
)
|
≤
A
(
M
f
(
x
)
)
|
ξ
|
/
α
(
G
α
*
|
f
|
(
x
)
)
1
-
|
ξ
|
/
α
,
f
o
r
a
l
m
o
s
t
e
v
e
r
y
x
∈
ℝ
N
.Proof.
We assume thatM
f
(
x
)
<
∞; otherwise, there is nothing to prove. We then observe that there is a constant A
1 such that G
α
*
|
f
|
(
x
)
≤
A
1
M
f
(
x
). In fact, by (17), (18), and ([11], Lemma 6.1.4), there exists a constant A such that(25)
G
α
*
|
f
|
(
x
)
≤
A
∫
|
x
-
y
|
<
1
‍
|
f
(
y
)
|
d
y
|
x
-
y
|
N
-
α
+
A
∫
|
x
-
y
|
≥
1
‍
|
f
(
y
)
|
e
-
|
x
-
y
|
/
2
d
y
≤
A
M
f
(
x
)
+
A
∑
i
=
1
∞
‍
e
-
i
/
2
∫
i
+
1
>
|
x
-
y
|
≥
i
‍
|
f
(
y
)
|
d
y
≤
A
M
f
(
x
)
+
A
M
f
(
x
)
∑
i
=
1
∞
‍
(
i
+
1
)
N
e
-
i
/
2
=
A
1
M
f
(
x
)
.
Then, by (19) and (18), for any δ
≤
1, we have
(26)
|
D
ξ
(
G
α
*
|
f
|
(
x
)
)
|
≤
A
∫
|
x
-
y
|
<
δ
‍
|
f
(
y
)
|
d
y
|
x
-
y
|
N
-
α
+
|
ξ
|
+
A
∫
δ
≤
|
x
-
y
|
<
1
‍
|
f
(
y
)
|
d
y
|
x
-
y
|
N
-
α
+
|
ξ
|
+
A
∫
|
x
-
y
|
≥
1
‍
G
α
(
x
-
y
)
|
f
(
y
)
|
d
y
≤
A
(
δ
α
-
|
ξ
|
M
f
(
x
)
+
δ
-
|
ξ
|
(
G
α
*
|
f
|
(
x
)
)
+
(
G
α
*
|
f
|
(
x
)
)
δ
α
-
|
ξ
|
M
f
(
x
)
)
.
Now choosing δ
α
=
G
α
*
|
f
|
(
x
)
/
A
1
M
f
(
x
); then, δ
≤
1 and the result follows.Lemma 6.
Suppose that1
<
p
-
≤
p
(
x
)
≤
p
+
<
∞ and 0
<
α
<
N. Let H
∈
C
k
(
ℝ
+
) for some k
≥
α, if H satisfies
(27)
sup
⁡
t
>
0
|
t
i
-
1
H
(
i
-
1
)
(
t
)
|
≤
L
<
∞
,
w
i
t
h
i
=
1,2
,
…
,
k
.
Then H
(
G
α
*
f
)
∈
L
α
,
p
(
·
)
(
ℝ
N
) for every f
∈
L
+
p
(
·
)
(
ℝ
N
), and there exists a constant A, depending only on α and N, such that
(28)
∥
H
(
G
α
*
f
)
∥
α
,
p
(
·
)
≤
A
L
∥
G
α
*
f
∥
α
,
p
(
·
)
=
A
L
∥
f
∥
p
(
·
)
.Proof.
Forα integer. Assume that f
∈
C
0
∞
(
ℝ
N
) and f
≥
0. Set u
=
G
α
*
f and notice that u
(
x
)
>
0 for all x, so that H
(
u
) is defined. If ξ is a multi-indix with |
ξ
|
=
α, we find by the chain rule that
(29)
D
ξ
(
H
(
u
)
)
=
∑
i
=
1
α
‍
H
(
i
)
(
u
)
∑
‍
c
ξ
D
ξ
1
u
⋯
D
ξ
i
u
,
where the interior sum is over all ordered i-tuples of multi-indices {
ξ
1
,
…
,
ξ
i
} such that ξ
1
+
⋯
+
ξ
i
=
ξ, and all |
ξ
j
|
≥
1. The c
ξ are coefficients, whose exact value is of no consequence to us. Thus, by assumption of Lemma 3.4 we get
(30)
|
D
ξ
(
H
(
u
)
)
|
≤
A
L
∑
i
=
1
α
‍
u
1
-
i
∑
‍
|
D
ξ
1
u
⋯
D
ξ
i
u
|
.
For i
>
1, we estimate these derivatives by means of Lemma 5. By the positivity of f, we have
(31)
|
D
ξ
j
u
|
≤
A
(
M
f
(
x
)
)
|
ξ
j
|
/
α
u
1
-
|
ξ
j
|
/
α
.
Thus, since ∑
j
=
1
i
‍
(
1
-
|
ξ
j
|
/
α
)
=
i
-
|
ξ
|
/
α
=
i
-
1,
(32)
∑
i
=
2
α
‍
u
1
-
i
∑
‍
|
D
ξ
1
u
⋯
D
ξ
i
u
|
≤
A
∑
i
=
2
α
‍
u
1
-
i
M
f
u
i
-
1
=
A
M
f
.
Taking the term with i
=
1 into account, we obtain
(33)
|
D
ξ
(
H
(
u
)
)
|
≤
A
L
(
|
M
f
|
+
|
D
ξ
u
|
)
.
But we already know from (10) that ∥
M
f
∥
p
(
·
)
≤
A
∥
f
∥
p
(
·
) and that ∥
D
ξ
(
G
α
*
f
)
∥
p
(
·
)
≤
A
∥
f
∥
p
(
·
) for |
ξ
|
=
α.
This finishes the proof for smoothf.
Now we pass to the general case and letf be an arbitrary function in L
+
p
(
·
)
(
ℝ
N
). Then there are nonnegative functions f
i
∈
C
0
∞
(
ℝ
N
), i
=
1,2
,
…, such that
(34)
lim
⁡
i
→
∞
∥
f
i
-
f
∥
p
(
·
)
=
0
.
By the first part of the proof,
(35)
∥
H
(
(
G
α
*
f
i
)
)
∥
α
,
p
(
·
)
≤
A
L
∥
f
∥
p
(
·
)
,
for all sufficiently large i.
Thus, settingH
(
(
G
α
*
f
i
)
)
=
G
α
*
g
i, we can assume that {
g
i
}
1
∞ converges weakly in L
p
(
·
)
(
ℝ
N
) to an element g, with ∥
g
∥
p
(
·
)
≤
A
L
∥
f
∥
p
(
·
).
We have to prove thatG
α
*
g
=
H
(
(
G
α
*
f
)
).
The strong convergence of{
f
i
}
1
∞ and the fact that G
α
∈
L
1
(
ℝ
N
) imply, by Lemma 4, that {
G
α
*
f
i
}
1
∞ converges strongly in L
p
(
·
)
(
ℝ
N
) to G
α
*
f. After extraction of a subsequence, we can assume that
(36)
lim
⁡
i
→
∞
G
α
*
f
i
(
x
)
=
G
α
*
f
(
x
)
a
.
e
.
But H is continuous, so it follows that
(37)
lim
⁡
i
→
∞
G
α
*
g
i
(
x
)
=
lim
⁡
i
→
∞
H
(
(
G
α
*
f
i
)
)
(
x
)
=
H
(
(
G
α
*
f
)
)
(
x
)
a
.
e
.
On the other hand, the weak convergence of{
g
i
}
1
∞ implies that the pointwise limit of {
G
α
*
g
i
}
1
∞ (which is now known to exist a.e.) is G
α
*
g. In fact, setting g
i
-
g
=
h
i, for an arbitrary ϵ
>
0,
(38)
G
α
*
h
i
(
x
)
=
∫
|
x
-
y
|
≤
ϵ
‍
G
α
(
x
-
y
)
h
i
(
y
)
d
y
+
∫
|
x
-
y
|
>
ϵ
‍
G
α
(
x
-
y
)
h
i
(
y
)
d
y
.
By weak convergence, the last term tends to zero, since G
α is in L
p
′
(
·
)
(
ℝ
N
) away from the origin. Consider the following:
(39)
Let
λ
>
liminf
⁡
i
→
∞
∥
∫
|
y
|
≤
ϵ
‍
G
α
(
y
)
h
i
(
·
-
y
)
d
y
∥
p
(
·
)
.
We deduce that, for all j, λ
>
inf
⁡
i
≥
j
∥

∫
|
y
|
≤
ϵ
‍
G
α
(
y
)
h
i
(
·
-
y
)
d
y
∥
p
(
·
), which implies that for all j, there exists i
≥
j, such that
(40)
λ
>
∥
∫
|
y
|
≤
ϵ
‍
G
α
(
y
)
h
i
(
·
-
y
)
d
y
∥
p
(
·
)
.
Then,
(41)
∫
|
y
|
≤
ϵ
‍
(
liminf
⁡
i
→
∞
⁡
G
α
(
y
)
h
i
(
x
-
y
)
λ
)
p
(
x
)
d
x
≤
liminf
⁡
i
→
∞
∫
|
y
|
≤
ϵ
‍
(
G
α
(
y
)
h
i
(
·
-
y
)
λ
)
p
(
x
)
d
x
≤
1
.
Since λ
≥
∥
liminf
⁡
i
→
∞
G
α
*
h
i
∥
p
(
·
), then we have
(42)
∥
lim
⁡
i
→
∞
G
α
*
h
i
∥
p
(
·
)
≤
liminf
⁡
i
→
∞
∥
∫
|
y
|
≤
ϵ
‍
G
α
(
y
)
h
i
(
·
-
y
)
d
y
∥
p
(
·
)
≤
sup
⁡
i
∥
h
i
∥
p
(
·
)
∫
|
y
|
≤
ϵ
‍
G
α
(
y
)
d
y
,
which is an arbitrary small number, and thus for a.e. x,
(43)
G
α
*
g
(
x
)
=
lim
⁡
i
→
∞
G
α
*
g
i
(
x
)
=
lim
⁡
i
→
∞
H
(
(
G
α
*
f
i
)
)
(
x
)
=
H
(
(
G
α
*
f
)
)
(
x
)
.
This completes the proof of Lemma 5.Proposition 7.
Letu
∈
W
α
,
p
(
·
)
(
ℝ
N
), there exist a sequence u
n, such that (i)
u
n
∈
W
α
,
p
(
·
)
(
ℝ
N
)
∩
L
∞
(
ℝ
N),
supp
⁡
u
n
is compact;
(ii)
|
u
n
(
x
)
|
≤
|
u
(
x
)
|
and
u
n
(
x
)
u
(
x
)
≥
0
a.e. in
ℝ
N
;
(iii)
u
n
→
u
in
W
α
,
p
(
·
)
(
ℝ
N
)
as
n
→
∞
.Proof.
The proof of Proposition7 is done in two steps as follows.
Step  1
(
c
a
s
e
α
p
^
-
>
N
). Let ξ
∈
C
0
∞
(
ℝ
N
) be a fixed function such that 0
≤
ξ
≤ 1 and ξ
(
x
)
=
1 in a neighborhood of the origin.
Letξ
n
(
x
) = ξ
(
x
/
n
); then, u
n
(
x
) = ξ
n
(
x
)
u
(
x
) satisfies all the required properties, using the fact that u
∈
L
∞
(
Ω
) by Sobolev’s theorem [19].
Step  2 (
c
a
s
e
α
p
-
≤
N
). We assume that u has compact support, if necessary by multiplying with a suitable ξ
n. We represent u as a Bessel potential, u
=
G
α
*
f, so that ∥
f
∥
p
(
·
)
≤
A
∥
u
∥
W
α
;
p
(
·
). Set
(44)
v
=
G
α
*
|
f
|
,
and let T
∈
C
∞
(
ℝ
) be a function such that 0
≤
T
≤
1, T
(
t
)
=
1 for 0
≤
t
≤
1
/
2 and T
(
t
)
=
0 for t
≥
1. Then, set
(45)
u
n
(
x
)
=
T
(
v
(
x
)
n
)
u
(
x
)
,
n
=
1
,
2
,
3
,
…
.
We First observe that, u
n
(
x
)
=
0 on the set {
x
:
v
(
x
)
≥
n
}, which includes {
x
:
|
u
(
x
)
|
≥
n
}, and so we have |
u
n
(
x
)
|
<
n a.e., and thus u
n
(
x
)
u
(
x
)
≥
0. It remains to prove that u
n
∈
W
α
,
p
(
·
)
(
ℝ
N
) and that u
n converges to u as n tends to ∞.
Letη be any multi-index with 0
<
|
η
|
=
β
≤
α. If η = η
1
+
⋯
+
η
i, i
>
1, and all |
η
j
|
≥
1, we find by the same arguments as in the proof of Lemma 6 that
(46)
|
D
η
T
(
v
(
x
)
n
)
|
≤
A
∑
i
=
1
β
‍
n
-
i
∑
‍
|
D
η
1
v
(
x
)
⋯
D
η
i
v
(
x
)
|
.
By Lemma 5, we have, for any multi-index η with 0
<
|
η
|
<
α,
(47)
|
D
η
v
(
x
)
|
≤
A
M
f
(
x
)
|
η
|
/
α
(
v
(
x
)
)
1
-
|
η
|
/
α
.
On the open set {
x
:
v
(
x
)
>
n
}, using the fact that D
η
T
(
v
(
x
)
/
n
)
=
0; then,
(48)
|
D
η
H
(
v
(
x
)
n
)
|
≤
A
∑
i
=
1
β
‍
n
-
i
n
i
-
|
η
|
/
α
M
f
(
x
)
|
η
|
/
α
≤
A
n
-
|
η
|
/
α
M
f
(
x
)
|
η
|
/
α
,
for |
η
|
≤
α
-
1, and
(49)
|
D
η
T
(
v
(
x
)
n
)
|
≤
A
n
-
1
(
M
f
(
x
)
+
|
D
η
v
(
x
)
|
)
,
for |
η
| = α.
By using Leibniz’s formula, we have for|
ξ
| = α, if v
(
x
)
≤
n, again using Lemma 5, that
(50)
|
D
ξ
u
n
(
x
)
-
D
ξ
u
(
x
)
|
=
|
(
1
-
T
(
v
(
x
)
n
)
)
D
ξ
u
(
x
)
|
,
|
D
ξ
u
n
(
x
)
-
D
ξ
u
(
x
)
|
≤
A
∑
0
<
|
η
|
<
α
‍
|
D
η
T
(
v
(
x
)
n
)
|
|
D
ξ
-
η
u
(
x
)
|
+
|
D
ξ
H
(
v
(
x
)
n
)
|
|
u
(
x
)
|
≤
A
∑
0
<
β
<
α
‍
n
-
β
/
α
M
f
(
x
)
β
/
α
M
f
(
x
)
1
-
β
/
α
v
(
x
)
β
/
α
+
A
n
-
1
(
M
f
(
x
)
+
|
D
ξ
v
(
x
)
|
)
v
(
x
)
≤
A
(
v
(
x
)
n
)
1
/
α
M
f
(
x
)
+
A
(
v
(
x
)
n
)
(
M
f
(
x
)
+
|
D
ξ
v
(
x
)
|
)
.
If v
(
x
)
>
n, we have D
ξ
u
n
(
x
)
=
0. It follows that a.e.
(51)
lim
⁡
n
→
∞
|
D
ξ
u
n
(
x
)
-
D
ξ
u
(
x
)
|
=
0
,
|
D
ξ
u
n
(
x
)
-
D
ξ
u
(
x
)
|
≤
|
D
ξ
u
(
x
)
|
+
A
(
M
f
(
x
)
+
|
D
ξ
v
(
x
)
|
)
.
The functions on the right hand side belong toL
p
(
·
), so the theorem follows by applying the dominated convergence.Remark 8.
The sequenceu
n constructed above satisfies
(52)
∥
u
n
∥
W
α
,
p
(
·
)
≤
C
∥
u
∥
W
α
,
p
(
·
)
,
with a constant C depending only on α and N.


## 3.2. Existence Result

This subsection is devoted to establish the following existence theorem.Theorem 9.
LetT
∈
L
l
o
c
1
(
ℝ
N
)
∩
W
-
α
,
p
′
(
·
)
(
ℝ
N
) and u
∈
W
α
,
p
(
·
)
(
ℝ
N
). Assume that T
(
x
)
u
(
x
)
≥
h
(
x
) a.e. in ℝ
N, for some h
∈
L
1
(
ℝ
N
). Then,
(53)
T
u
∈
L
1
(
ℝ
N
)
,
∫
ℝ
N
‍
T
(
x
)
u
(
x
)
d
x
=
〈
T
,
u
〉
.Proof.
We first deduce Theorem9 as a simple consequence of Proposition 7. Let u
n be a sequence defined in Proposition 7. It follows easily from (i) in Proposition 7 (using convolution with mollifiers and according to [11]) that
(54)
∫
ℝ
N
‍
T
(
x
)
u
n
(
x
)
d
x
=
〈
T
,
u
n
〉
.
By Proposition 7, the right hand side of (54) converges as n
→
∞ to 〈
T
,
u
〉. On the other hand, we have T
u
n
≥
-
|
h
| a.e. We deduce from Fatou’s lemma that T
u
∈
L
1. We conclude by dominated convergence that
(55)
∫
‍
T
(
x
)
u
n
(
x
)
d
x
⟶
∫
‍
T
(
x
)
u
(
x
)
d
x
and
thus
∫
‍
T
(
x
)
u
(
x
)
d
x
=
〈
T
,
u
〉
.Theorem 10.
LetT
∈
L
l
o
c
1
(
Ω
)
∩
W
-
α
,
p
′
(
·
)
(
Ω
) be such that
(56)
∫
Ω
∩
B
(
0
,
R
)
‍
|
T
(
x
)
|
d
x
<
∞
f
o
r
e
v
e
r
y
R
<
∞
.
Assume that u
∈
W
0
α
,
p
(
·
)
(
Ω
) and T
(
x
)
u
(
x
)
≥
h
(
x
) a.e. in Ω, for some h
∈
L
1
(
Ω
). Then,
(57)
T
u
∈
L
1
(
Ω
)
,
∫
Ω
‍
T
(
x
)
u
(
x
)
d
x
=
〈
T
,
u
〉
.Proof.
The proof is straightforward whenα
p
-
>
N; therefore, we may assume that α
p
-
≤
N. Using ξ
n
u in place of u, we may always reduce to the case where supp
⁡
u is bounded. Set(58)
u
¯
=
{
u
(
x
)
if
x
∈
Ω
,
0
if
x
∈
ℝ
N
∖
Ω
.
Then u
¯
∈
W
α
,
p
(
·
)
(
ℝ
N
). This allows us to write
(59)
u
¯
=
G
α
*
f
,
for some f in L
p
(
·
)
(
ℝ
N
). As in the proof of Proposition 7, set
(60)
v
=
G
α
*
|
f
|
,
u
n
=
H
(
v
n
)
u
.
Since u
∈
W
0
α
,
p
(
·
)
(
Ω
), there exists a sequence u
j
∈
C
0
∞
(
Ω
) such that u
j
→
u in W
α
,
p
(
·
)
(
Ω
) and a.e. (see [19]). For each j, we perform the above construction and we set
(61)
u
¯
j
=
G
α
*
f
j
,
v
j
=
G
α
*
|
f
j
|
,
u
n
j
=
H
(
v
j
n
)
u
j
.
Fix ξ
∈
C
0
∞
(
ℝ
N
). We clearly have
(62)
∫
Ω
‍
T
ξ
u
n
j
d
x
=
〈
T
,
ξ
u
n
j
〉
.
As we keep n fixed and let j
→
∞, we see that
(63)
∫
Ω
‍
T
ξ
u
n
j
d
x
⟶
∫
Ω
‍
T
ξ
u
n
d
x
,
by dominated convergence and (56).
On the other hand, by Remark8, we obtain
(64)
∥
u
n
j
∥
W
α
,
p
(
·
)
(
Ω
)
≤
∥
u
¯
j
∥
W
α
,
p
(
·
)
(
ℝ
N
)
≤
C
,
where C does not depend on j and n.
Therefore,u
n
j converges weakly to u
n in W
0
α
,
p
(
·
)
(
Ω
) as j
→
∞ and thus ξ
u
n
j
⇀
ξ
u
n as j
→
∞. Passing to the limit in (62) as j
→
∞, we find
(65)
∫
Ω
‍
T
ξ
u
n
=
〈
T
,
ξ
u
n
〉
.
We conclude easily (by the argument as in the proof of Theorem 9) that
(66)
T
u
∈
L
1
(
Ω
)
,
∫
Ω
‍
T
(
x
)
u
(
x
)
d
x
=
〈
T
,
u
〉
.


## 4. An Application to a Strongly Nonlinear Elliptic Equation

LetΩ be an open set on ℝ
N, and assume that A
:
W
0
α
,
p
(
·
)
(
Ω
)
→
W
-
α
,
p
′
(
·
)
(
Ω
) is a pseudomonotone operator which maps bounded sets into bounded sets and which is coercive. And let g
(
x
,
s
)
:
Ω
×
ℝ
→
ℝ be a Carathéodory function satisfying the sign condition (Ω
×
ℝ) and for each t
>
0, there exists h
t
∈
L
1
(
Ω
) as follows:
(67)
sup
⁡
|
s
|
≤
t
|
g
(
x
,
s
)
|
≤
h
t
(
x
)
.Theorem 11.
For everyf
∈
W
-
α
,
p
′
(
·
)
(
Ω
), there exists u
∈
W
0
α
,
p
(
·
)
(
Ω
) such that
(68)
g
(
x
,
u
)
∈
L
1
(
Ω
)
,
g
(
x
,
u
)
u
∈
L
1
(
Ω
)
,
〈
A
u
,
v
〉
+
∫
Ω
‍
g
(
x
,
u
)
v
d
x
=
〈
f
,
v
〉
,
∀
v
∈
W
0
α
,
p
(
·
)
(
Ω
)
∩
L
∞
(
Ω
)
.
Furthermore, if g is nondecreasing in u and if u
1and u
2 are two solutions corresponding to f
1 and f
2, respectively, then
(69)
〈
A
u
1
-
A
u
2
,
u
1
-
u
2
〉
+
∫
Ω
‍
[
g
(
x
,
u
1
)
-
g
(
x
,
u
2
)
]
(
u
1
-
u
2
)
d
x
=
〈
f
1
-
f
2
,
u
1
-
u
2
〉
.Proof.
Let(70)
g
n
(
x
,
s
)
=
{
η
(
x
n
)
g
(
x
,
s
)
if
|
g
(
x
,
s
)
|
≤
n
η
(
x
n
)
n
sign
⁡
(
g
(
x
,
s
)
)
if
|
g
(
x
,
s
)
|
>
n
,
where η
∈
𝒟
(
ℝ
N
) with 0
≤
η
≤
1 and η
(
x
)
=
1 near x
=
0. It follows easily from the theory of pseudomonotone operators that there exists u
n
∈
W
0
α
,
p
(
·
)
(
Ω
) such that
(71)
A
u
n
+
g
n
(
x
,
u
n
)
=
f
.
In addition,
(72)
∥
u
n
∥
W
0
α
,
p
(
·
)
(
Ω
)
≤
C
,
∫
Ω
‍
g
n
(
x
,
u
n
)
u
n
≤
C
.
Then we can assume that
(73)
u
n
⇀
u
weakly
in
W
0
α
,
p
(
·
)
(
Ω
)
,
a.e
in
Ω
,
A
u
n
⇀
χ
weakly
in
W
-
α
,
p
′
(
·
)
(
Ω
)
.
Moreover,
(74)
|
g
n
(
x
,
u
n
)
|
≤
sup
⁡
|
t
|
≤
s
|
g
(
x
,
t
)
|
+
1
s
g
n
(
x
,
u
n
)
u
n
,
which implies that, for any measurable subset E
⊂
Ω,
(75)
∫
E
‍
|
g
n
(
x
,
u
n
)
|
≤
∫
E
‍
h
s
(
t
)
+
C
s
.
Then, thanks to Vitali’s theorem, we deduce that
(76)
g
n
(
x
,
u
n
)
⟶
g
(
x
,
u
)
.
By Fatou’s lemma, it is easy to see that
(77)
limsup
⁡
n
〈
A
u
n
,
u
n
〉
≤
〈
f
,
u
〉
-
∫
Ω
‍
g
(
x
,
u
)
d
x
.
Set T
=
g
(
x
,
u
)
=
f
-
χ, we have T
∈
L
1
(
Ω
)
∩
W
-
α
,
p
′
(
·
)
(
Ω
) and as a consequence of Theorem 10, we conclude that
(78)
∫
Ω
‍
g
(
x
,
u
)
u
d
x
=
〈
f
-
χ
,
u
〉
.
Therefore,
(79)
limsup
⁡
n
〈
A
u
n
,
u
n
〉
≤
〈
χ
,
u
〉
and
consequently
A
u
=
χ
.
The conclusion follows readily. (68) is again a direct consequence of Theorem 11.



---

*Source: 549051-2014-04-29.xml*