1 Introduction
The Caffarelli–Kohn–Nirenberg (CKN) inequalities were first introduced in 1984, see [3].
Theorem A.
Let u∈C0∞(RN). In what follows, p,q,r,α,β,σ and a are fixed real numbers satisfying
(1.1)
p
,
q
≥
1
,
r
>
0
,
0
≤
a
≤
1
,
(1.2)
1
p
+
α
N
>
0
,
1
q
+
β
N
>
0
,
1
r
+
γ
N
>
0
,
where γ=aσ+(1-a)β. There exists a positive constant C such that
(1.3)
∥
|
x
|
γ
u
∥
L
r
≤
C
∥
|
x
|
α
D
u
∥
L
p
a
∥
|
x
|
β
u
∥
L
q
1
-
a
for all
u
∈
C
0
∞
(
ℝ
N
)
if and only if the following relations hold (the first one being the dimensional balance):
(1.4)
1
r
+
γ
N
=
a
(
1
p
+
α
-
1
N
)
+
(
1
-
a
)
(
1
q
+
β
N
)
,
0
≤
α
-
σ
if
a
>
0
,
α
-
σ
≤
1
if
a
>
0
and
1
p
+
α
-
1
N
=
1
r
+
γ
N
.
Furthermore, on any compact set in the parameter space in which (1.1), (1.2), (1.4) and 0≤α-σ≤1 hold, the constant C is bounded.
The CKN inequality plays a crucial role in partial differential equations, geometric analysis and other branches of mathematics. It is worth noting that many well-known and important inequalities such as Gagliardo–Nirenberg inequalities, Hardy–Sobolev inequalities, Nash inequalities, etc. are just variances of the CKN inequalities.
We would like to study the following equivalent form of the CKN inequality:
(1.5)
(
∫
ℝ
N
|
u
|
r
d
x
|
x
|
s
)
1
r
≤
C
(
∫
ℝ
N
|
∇
u
|
p
d
x
|
x
|
μ
)
a
p
(
∫
ℝ
N
|
u
|
q
d
x
|
x
|
σ
)
1
-
a
q
,
where
a
=
[
(
N
-
σ
)
r
-
(
N
-
s
)
q
]
p
[
(
N
-
σ
)
p
-
(
N
-
μ
-
p
)
q
]
r
.
This form is acquired by performing the change of variables α=-μp, β=-σq and γ=-sr in (1.3).
When a=1 and s=μ=0, (1.5) becomes the well-known Sobolev inequality
(1.6)
(
∫
ℝ
N
|
u
|
p
⋆
d
x
)
1
p
⋆
≤
S
(
N
,
p
)
(
∫
ℝ
N
|
∇
u
|
p
d
x
)
1
p
,
where p⋆=NpN-p. This inequality has many applications in numerous areas and has been studied for many years. For example, Aubin [1] and Talenti [15] found the best constant S(N,p) and the minimizers for the Sobolev inequality (1.6) via the Schwarz rearrangement and the Bliss inequality [2].
By setting a=1, μ=0 and 0≤s≤p<N in (1.5), we can also obtain the Hardy–Sobolev (HS) inequality
(1.7)
(
∫
ℝ
N
|
u
|
p
⋆
(
s
)
d
x
|
x
|
s
)
1
p
⋆
(
s
)
≤
HS
(
N
,
p
,
s
)
(
∫
ℝ
N
|
∇
u
|
p
d
x
)
1
p
,
where p⋆(s):=(N-s)pN-p.
For the case p=2, Lieb [13] applied the Schwarz symmetrization method to study (1.7) and found the best constants and explicit optimizers. Moreover, Ghoussoub and Yuan [10] studied the best constant and extremal functions for inequality (1.7).
When a=1, 0<μ,s<p<N, we have the following non-interpolation CKN inequality:
(1.8)
(
∫
ℝ
N
|
u
|
p
⋆
(
s
,
μ
)
d
x
|
x
|
s
)
1
p
⋆
(
s
,
μ
)
≤
C
(
∫
ℝ
N
|
∇
u
|
p
d
x
|
x
|
μ
)
1
p
,
where p⋆(s,μ):=(N-s)pN-μ-p and C=C(N,p,s,μ).
Plenty of progress has been made on (1.8) regarding sharp constants and the existence and symmetry of maximizers, especially for p=2. For instance, Chou and Chu [5] considered the best constants and explicit optimizers for the case μ2<sr<μ+22. Further work about the existence of the ground state solution of degenerate elliptic equations was carried out by Wang and Willem [16]. In [4], Catrina and Wang investigated the case μ<0 and established the attainability/inattainability and symmetry breaking of extremal functions.
When 0<a<1, the CKN inequality has an interpolation term, which makes it more difficult to study; researchers could only write out the sharp constant and its corresponding extremal functions in few restricted regions. For example, in the special case when s=μ=σ=0, p=2, q=t+1 and r=2t, the sharp constant and the extremal functions of the CKN inequality (1.5) were given by Del Pino and Dolbeault in [6]. In an other particular case, where p=q=2, μ<N-2, σ=μ+2, μ2<sr<μ+22 and r=2Nr(N+μ)r-2s, the best constant and optimizers for the CKN inequality (1.5) as well as the weighted logarithmic inequality were investigated by Dolbeault and Esteban [7], and Dolbeault, Esteban, Tarantello and Tertikas [8]. For the radical symmetric situation, they also gave the explicit sharp constants and the form of maximizers.
In [9], the author together with Lam and Lu investigated the borderline case of the CKN inequality when N=p. Denoting Dμ,σp,q(ℝN) the completion of the space of smooth compactly supported functions with the norm
(
∫
ℝ
N
|
∇
u
|
p
d
x
|
x
|
μ
)
1
p
+
(
∫
ℝ
N
|
u
|
q
d
x
|
x
|
σ
)
1
q
,
and setting
CKN
(
N
,
s
,
q
,
r
)
=
sup
u
∈
D
0
,
s
N
,
q
(
ℝ
N
)
(
∫
ℝ
N
|
u
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
∇
u
|
N
d
x
)
1
N
(
1
-
q
r
)
(
∫
ℝ
N
|
u
|
q
d
x
|
x
|
s
)
1
r
and
GN
(
N
,
q
,
r
)
=
sup
u
∈
D
0
,
0
N
,
q
(
ℝ
N
)
(
∫
ℝ
N
|
u
|
r
d
x
)
1
r
(
∫
ℝ
N
|
∇
u
|
N
d
x
)
1
N
(
1
-
q
r
)
(
∫
ℝ
N
|
u
|
q
d
x
)
1
r
,
we established that CKN(N,s,q,r) and GN(N,q,r) can be achieved when the parameters are in the following special region:
μ
=
0
,
p
=
N
and
0
<
s
=
σ
<
N
.
Moreover, when r=pq-1p-1 or q=pr-1p-1, we gave the explicit form of the best constant and extremal function.
Lam and Lu [11] established that the sharp constant of the CKN inequality can be achieved in a wider region. More specifically, they extended the result with respect to the following case:
1
<
p
<
p
+
μ
<
N
,
σ
≤
N
μ
N
-
p
≤
s
<
N
and
1
≤
q
<
r
<
N
p
N
-
p
.
Assume Dμ,σp,q(ℝN) is as defined above and set
CKN
(
N
,
μ
,
σ
,
s
,
p
,
q
,
r
)
=
sup
u
∈
D
0
,
s
N
,
q
(
ℝ
N
)
(
∫
ℝ
N
|
u
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
∇
u
|
p
d
x
|
μ
|
)
a
p
(
∫
ℝ
N
|
u
|
q
d
x
|
x
|
σ
)
1
-
a
q
,
where
a
=
[
(
N
-
σ
)
r
-
(
N
-
s
)
q
]
p
[
(
N
-
σ
)
p
-
(
N
-
μ
-
p
)
q
]
r
.
They proved that CKN(N,μ,σ,s,p,q,r) can be achieved and all the extremal functions are radially symmetric. Moreover, when σ=s=NμN-p, for r=pq-1p-1 or q=pr-1p-1, they also gave the explicit form of all extremal functions and the exact best constants. We also note that the weighted Moser–Onofri and logarithmic Sobolev inequalities have also been obtained by Lam and Lu [12] using CKN inequalities.
Recently, Zhong and Zou [17] studied the existence of extremal functions of the CKN inequality in a wider region. They also set up the continuity and compactness on weighted Sobolev spaces. Their main result is stated as follows.
Theorem B.
Let Ω⊂RN be a cone (in particular, Ω=RN). Assume that p>1, s>0, max{σ,s}<μ+p<N, 1≤r, 1≤q<min{p⋆,p⋆(σ,μ)}, max{p(σ-s)N-μ-p+q,σ-sN-σq+q}<r<min{p⋆,p⋆(s,μ)} and
p
(
s
-
σ
)
+
q
(
μ
+
p
-
s
)
<
r
(
μ
+
p
-
σ
)
,
(
N
p
-
N
r
+
p
r
)
(
s
-
σ
)
>
(
N
μ
-
N
s
+
p
s
)
(
r
-
q
)
.
Then the sharp constant C(p,q,r,μ,σ,s) can be achieved and it is given by
C
(
p
,
q
,
r
,
μ
,
σ
,
s
)
=
(
1
ρ
)
(
μ
+
p
-
σ
)
r
+
(
p
-
q
)
(
N
-
s
)
r
[
(
N
-
σ
)
p
-
(
N
-
μ
-
p
)
q
]
,
where
ρ
:=
inf
{
∫
Ω
|
∇
u
|
p
|
x
|
μ
d
x
+
λ
⋆
∫
Ω
|
u
|
q
|
x
|
σ
d
x
:
∫
Ω
|
u
|
r
|
x
|
s
d
x
=
1
}
,
which can be attained, and
λ
⋆
:=
{
p
(
N
-
s
)
-
(
N
-
μ
-
p
)
r
(
μ
+
p
-
σ
)
r
+
(
p
-
q
)
(
N
-
s
)
}
(
μ
+
p
-
σ
)
r
+
(
p
-
q
)
(
N
-
s
)
(
N
-
s
)
p
-
(
N
-
μ
-
p
)
r
{
(
N
-
σ
)
r
-
(
N
-
s
)
q
p
(
N
-
s
)
-
(
N
-
μ
-
p
)
r
}
(
N
-
σ
)
r
-
(
N
-
s
)
q
p
(
N
-
s
)
-
(
N
-
μ
-
p
)
r
.
In 1986, Lin [14] established the higher-order derivatives of the CKN inequality, but it is worth noting that no one had studied the best constants and extremal functions of it yet. The inequality of higher-order derivatives is list below.
Theorem C.
Let u∈C0∞(RN). In what follows p,q,r,α,β,γ and a are fixed real numbers, and j≥0, m>0 are integers satisfying
p
,
q
≥
1
,
r
≠
0
,
j
m
≤
a
≤
1
,
1
p
+
α
N
>
0
,
1
q
+
β
N
>
0
,
1
r
+
γ
N
>
0
,
m
-
j
-
N
p
is not a nonnegative integer.
There exists a positive constant C such that
(1.9)
∥
|
x
|
γ
D
j
u
∥
L
r
≤
C
∥
|
x
|
α
D
m
u
∥
L
p
a
∥
|
x
|
β
u
∥
L
q
1
-
a
for all
u
∈
C
0
∞
(
ℝ
N
)
if and only if the following relations hold (the first one being the dimensional balance):
1
r
+
γ
-
j
N
=
a
(
1
p
+
α
-
m
N
)
+
(
1
-
a
)
(
1
q
+
β
N
)
.
γ
≤
a
α
+
(
1
-
a
)
β
,
a
(
α
-
m
)
+
(
1
-
a
)
β
+
j
≤
γ
if
r
>
0
and
1
q
+
β
N
=
1
p
+
α
-
m
N
,
γ
=
a
α
+
(
1
-
a
)
β
if
a
=
j
m
.
When a=1, from (1.9), we get the higher-order derivative CKN inequality without interpolation as follows:
(1.10)
∥
|
x
|
γ
D
j
u
∥
L
r
≤
C
∥
|
x
|
α
D
m
u
∥
L
p
,
where
(1.11)
1
p
+
α
N
>
0
,
1
r
+
γ
N
>
0
,
α
-
m
+
j
≤
γ
≤
α
,
(1.12)
1
r
+
γ
-
j
N
=
1
p
+
α
-
m
N
.
Similar to the first-order, we assume that α=-μp, β=-σq and γ=-sr. Then, by a simple transformation, (1.9) becomes
(1.13)
(
∫
ℝ
N
|
D
j
u
|
r
d
x
|
x
|
s
)
1
r
≤
C
(
∫
ℝ
N
|
D
m
u
|
p
d
x
|
x
|
μ
)
a
p
(
∫
ℝ
N
|
u
|
q
d
x
|
x
|
σ
)
1
-
a
q
,
where C=C(p,q,r,μ,σ,s,m,j) and
a
=
[
(
N
-
σ
)
r
-
(
N
-
s
-
j
r
)
q
]
p
[
(
N
-
σ
)
p
-
(
N
-
μ
-
m
p
)
q
]
r
.
Motivated by the results in [17], we want to obtain the existence of an extremal function for the CKN inequalities (1.13) and find the sharp constants of it. More precisely, we want to study the inequality under the following conditions on the parameters p,q,r,μ,σ,s and integers m,j:
(1.14)
{
1
<
p
<
N
,
0
≤
j
<
m
,
N
-
μ
-
m
p
>
0
,
-
∞
<
σ
<
μ
+
m
p
,
(
N
-
μ
-
m
p
)
(
r
-
q
)
+
(
N
-
σ
)
(
p
-
r
)
<
(
N
-
s
-
j
r
)
(
p
-
q
)
,
(
p
⋆
(
s
,
μ
,
m
-
j
)
-
r
)
(
N
-
μ
-
(
m
-
j
)
p
)
σ
+
(
r
⋆
(
σ
,
s
,
j
)
-
q
)
(
N
-
s
-
j
r
)
μ
<
(
p
⋆
(
σ
,
μ
,
m
)
-
q
)
(
N
-
μ
-
m
p
)
s
.
Moreover, assume that one of the following holds:
(1.15)
{
max
{
0
,
N
μ
N
-
(
m
-
j
)
p
}
≤
s
≤
μ
+
(
m
-
j
)
p
,
1
≤
r
<
p
⋆
(
s
,
μ
,
m
-
j
)
,
1
≤
q
<
r
⋆
(
σ
,
s
,
j
)
,
(1.16)
{
μ
>
0
,
0
<
s
≤
N
μ
N
-
(
m
-
j
)
p
,
1
≤
r
<
p
m
-
j
⋆
,
1
≤
q
<
r
⋆
(
σ
,
s
,
j
)
,
where
p
⋆
(
s
,
μ
,
m
)
:=
(
N
-
s
)
p
N
-
μ
-
m
p
and pm⋆:=NpN-mp for short.
Assume Ω⊂ℝN is an open Lipschitz domain (in particular, Ω=ℝN). We denote the homogeneous Sobolev space H˙μm,p(Ω) with respect to the norm
∥
u
∥
H
˙
μ
m
,
p
(
Ω
)
=
(
∫
Ω
|
D
m
u
|
p
d
x
|
x
|
μ
)
1
p
.
Then we define a new Sobolev space
(1.17)
X
μ
,
σ
p
(
m
)
,
q
(
Ω
)
:=
H
˙
μ
m
,
p
(
Ω
)
∩
L
σ
q
(
Ω
)
,
with respect to the norm
(1.18)
∥
u
∥
X
μ
,
σ
p
(
m
)
,
q
(
Ω
)
:=
(
∫
Ω
|
D
m
u
|
p
d
x
|
x
|
μ
)
1
p
+
(
∫
Ω
|
u
|
q
d
x
|
x
|
σ
)
1
q
.
When m=1, we use Xμ,σp,q for short.
For any M>0, we consider the variational problem
𝒜
=
1
M
θ
inf
{
ℐ
[
u
]
:
u
∈
X
μ
,
σ
p
(
m
)
,
q
(
ℝ
N
)
and
∫
ℝ
N
|
D
j
u
|
r
d
x
|
x
|
s
=
M
}
,
where
ℐ
[
u
]
:=
1
p
∫
ℝ
N
|
D
m
u
|
p
d
x
|
x
|
μ
+
1
q
∫
ℝ
N
|
u
|
q
d
x
|
x
|
σ
and
θ
=
(
N
-
σ
)
p
-
(
N
-
μ
-
m
p
)
q
(
μ
+
m
p
-
σ
)
r
-
(
N
-
s
-
j
r
)
(
q
-
p
)
.
Theorem 1.1.
Assume (1.14) and that either (1.15) or (1.16) is satisfied. Set the quotient
𝒬
[
u
]
:=
(
∫
ℝ
N
|
D
j
u
|
r
d
x
|
x
|
s
)
1
r
(
∫
ℝ
N
|
D
m
u
|
p
d
x
|
x
|
μ
)
a
p
(
∫
ℝ
N
|
u
|
q
d
x
|
x
|
σ
)
1
-
a
q
for all
u
∈
X
μ
,
σ
p
(
m
)
,
q
(
ℝ
N
)
.
Denote the best constant determined by the supremum of the quotient by
(1.19)
ℒ
=
Lin
(
N
,
p
,
q
,
r
,
μ
,
σ
,
s
,
m
,
j
)
:=
sup
u
∈
X
μ
,
σ
p
(
m
)
,
q
(
ℝ
N
)
𝒬
[
u
]
.
Then the sharp constant L can be achieved and it is given by
ℒ
=
K
𝒜
1
r
θ
,
where
K
=
(
[
(
N
-
σ
+
j
q
)
r
-
(
N
-
s
)
q
]
p
(
μ
+
m
p
-
σ
)
r
-
(
N
-
s
-
j
r
)
(
q
-
p
)
)
(
N
-
σ
+
j
q
)
r
-
(
N
-
s
)
q
(
N
-
σ
)
p
-
(
N
-
μ
-
m
p
)
q
(
[
(
N
-
s
)
p
-
(
N
-
μ
-
(
m
-
j
)
p
)
]
q
(
μ
+
m
p
-
σ
)
r
-
(
N
-
s
-
j
r
)
(
q
-
p
)
)
(
N
-
s
)
p
-
(
N
-
μ
-
(
m
-
j
)
)
p
(
N
-
σ
)
p
-
(
N
-
μ
-
m
p
)
q
.
Moreover, when ∫RN|Dju|rdx|x|s=1, the minimizer is a ground state solution to the following equation:
(
-
1
)
m
∇
m
(
|
D
m
u
|
p
-
2
D
m
u
|
x
|
μ
)
+
|
u
|
q
-
2
u
|
x
|
σ
=
(
-
1
)
j
r
θ
𝒜
∇
j
(
|
D
j
u
|
r
-
2
D
j
u
|
x
|
s
)
.
This paper is organized as follows. In Section 2, we will establish the compactness of the embedding on the weighted Sobolev space of higher-order derivatives. In Section 3, we will prove the existence of extremal functions.
2 Compact Embedding
In this section our goal is to prove that Xμ,σp(m),q(ℝN)↪H˙sj,r(ℝN) is a compact embedding.
Lemma 2.1.
Let Ω∈RN be an open and bounded Lipschitz domain. Suppose 1≤p<N and N-mp-μ>0. Then, for the bounded sequence {un}⊂H˙μm,p(Ω), there exists u∈H˙μm,p(Ω) such that, up to a subsequence, Dkun→Dku a.e. in Ω, where k=0,1,…,m-1.
Proof.
We will consider this lemma in two situations.
First, for the case μ=0, from the well-known Rellich–Kondrachov theorem, we have that the embedding Wm,p(Ω)↪Wk,1(Ω) is compact for all k=0,1,…,m-1, under our assumption. Therefore, there exists u∈H˙m,p(Ω) such that, up to subsequence, limn→∞∫Ω|Dkun-Dku|dx=0. Then it is easy to see that Dkun→Dku a.e. in Ω.
For the case μ≠0, we set μ¯:=μp. Direct calculations yield
D
m
(
u
n
|
x
|
μ
¯
)
=
∑
i
=
0
m
(
m
i
)
D
i
(
|
x
|
-
μ
¯
)
D
m
-
i
(
u
n
)
.
It follows that
(2.1)
|
D
m
(
u
n
|
x
|
μ
¯
)
|
≤
∑
i
=
0
m
(
m
i
)
|
D
i
(
|
x
|
-
μ
¯
)
|
|
D
m
-
i
(
u
n
)
|
=
∑
i
=
0
m
C
(
m
,
i
,
μ
¯
)
|
D
m
-
i
(
u
n
)
|
|
x
|
μ
¯
+
i
.
Consider the higher-order derivative CKN inequality without interpolation (1.10), and assume r=p and j=m-i. Then, from (1.12), we can set α=-t and γ=-t-i, and hence we have
(2.2)
∥
|
D
m
-
i
u
|
|
x
|
t
+
i
∥
L
p
≤
C
∥
|
D
m
u
|
|
x
|
t
∥
L
p
,
where N-pt-pi>0.
For p≥1, it is easy to verify |∑i=1mxi|p≤mp-1∑i=1m|xi|p. Due to our assumption N-mp-μ>0, we notice that we can apply (2.2) when t=μ¯ and 0≤i≤m. Therefore, in combination with (2.1), we have
∫
Ω
|
D
m
(
u
n
|
x
|
μ
¯
)
|
p
d
x
≤
∫
Ω
|
∑
i
=
0
m
C
(
m
,
i
,
μ
¯
)
|
D
m
-
i
(
u
n
)
|
|
x
|
μ
¯
+
i
|
p
d
x
≤
C
(
m
,
μ
,
p
)
∑
i
=
1
m
∫
Ω
|
|
D
m
-
i
(
u
n
)
|
|
x
|
μ
¯
+
i
|
p
d
x
≤
C
(
m
,
μ
,
p
)
∑
i
=
1
m
∫
Ω
|
D
m
(
u
n
)
|
x
|
μ
¯
|
p
d
x
=
C
(
m
,
μ
,
p
)
∑
i
=
1
m
∫
Ω
|
D
m
(
u
n
)
|
p
|
x
|
μ
𝑑
x
.
Hence, {un|x|μ¯} is bounded in H˙m,p, and by applying the result when μ=0, we prove that there exists u|x|μ¯∈H˙m,p such that, up to a subsequence, Dk(un|x|μ¯)→Dk(u|x|μ¯) a.e. in Ω for k=0,1,…,m-1. From Fatou’s lemma, we have u∈H˙μm,p(Ω).
When k=0, it is natural to see that un→u a.e. in Ω. Then, by induction, it is not hard to see that Dkun→Dku a.e. in Ω for k=0,1,…,m-1. Hence, the lemma is proved. ∎
Next we want to consider the local compactness for the weighted Rellich–Kondrachov theorem.
Lemma 2.2.
Assume Ω∈RN is an open and bounded Lipschitz domain. Suppose that
1
≤
p
<
N
,
N
-
μ
-
m
p
>
0
𝑎𝑛𝑑
m
-
j
>
0
.
Then the embedding H˙μm,p(Ω)↪H˙sj,q(Ω) is compact if
(2.3)
N
μ
N
-
(
m
-
j
)
p
≤
s
≤
μ
+
(
m
-
j
)
p
𝑎𝑛𝑑
1
≤
q
<
p
⋆
(
s
,
μ
,
m
-
j
)
,
or
(2.4)
s
<
N
μ
N
-
(
m
-
j
)
p
𝑎𝑛𝑑
1
≤
q
<
p
m
-
j
⋆
.
Proof.
We first assume that {un}∈H˙μm,p(Ω) is a bounded sequence. By Lemma 2.1, we may assume that there exists some u∈H˙μm,p(Ω) such that, up to a subsequence, Djun→Dju a.e. in Ω.
Consider the first case (2.3). Let α=-μp and γ=-sr in the non-interpolation CKN inequality (1.10). Combining this with assumption (1.12), we know that r=p⋆(s,μ,m-j). Also direct calculations show that (1.11) is equivalent to (2.3). Therefore, for any v∈H˙μm,p(Ω), if NμN-(m-j)p≤s≤μ+(m-j)p, we have the inequality
∫
Ω
|
D
j
v
|
p
⋆
(
s
,
μ
,
m
-
j
)
d
x
|
x
|
s
<
∞
.
Then, by Hölder’s inequality, for any subset Λ⊂Ω and for 1≤q<p⋆(s,μ,m-j), we have
∫
Λ
|
D
j
u
n
-
D
j
u
|
q
d
x
|
x
|
s
≤
(
∫
Λ
|
D
j
u
n
-
D
j
u
|
p
⋆
(
s
,
μ
,
m
-
j
)
d
x
|
x
|
s
)
q
p
⋆
(
s
,
μ
,
m
-
j
)
(
∫
Λ
d
x
|
x
|
s
)
p
⋆
(
s
,
μ
,
m
-
j
)
-
q
p
⋆
(
s
,
μ
,
m
-
j
)
≤
C
(
∫
Λ
d
x
|
x
|
s
)
p
⋆
(
s
,
μ
,
m
-
j
)
-
q
p
⋆
(
s
,
μ
,
m
-
j
)
.
Since s≤μ+(m-j)p<N, we have ∫Λdx|x|s→0 as the measure L(Λ)→0. Therefore,
(2.5)
∫
Λ
|
D
j
u
n
-
D
j
u
|
q
d
x
|
x
|
s
→
0
as
L
(
Λ
)
→
0
uniformly for all
n
.
Since Ω is bounded and Djun→Dju a.e. in Ω, by applying Egorov’s theorem, we have Djun→Dju uniformly on Ω∖E for some E such that L(E) is sufficient small. Therefore, letting L(E)→0 and applying (2.5), up to a subsequence, we have
lim
n
→
∞
∫
Ω
|
D
j
u
n
-
D
j
u
|
q
d
x
|
x
|
s
=
lim
n
→
∞
∫
Ω
∖
E
|
D
j
u
n
-
D
j
u
|
q
d
x
|
x
|
s
+
lim
n
→
∞
∫
E
|
D
j
u
n
-
D
j
u
|
q
d
x
|
x
|
s
=
∫
Ω
∖
E
lim
n
→
∞
|
D
j
u
n
-
D
j
u
|
q
d
x
|
x
|
s
=
0
.
Hence, we proved the first situation.
For the second case (2.4), we assume s0=NμN-(m-j)p. Direct calculations show that p⋆(s0,μ,m-j)=pm-j⋆. Applying the result of first case under condition (2.4), we have
u
n
→
u
strongly in
H
˙
s
0
j
,
q
(
Ω
)
.
Therefore,
∫
Ω
|
D
j
u
n
-
D
j
u
|
q
d
x
|
x
|
s
≤
C
∫
Ω
|
D
j
u
n
-
D
j
u
|
q
d
x
|
x
|
s
0
→
0
as
n
→
∞
.
∎
We will now prove that Xμ,σp(m),q(ℝN)↪H˙sj,r(ℝN) is a compact embedding.
Theorem 2.3.
Let Ω∈RN be an open Lipschitz domain (in particular, Ω=RN). Suppose that the parameters p,q,r,μ,σ,s and integers m,j satisfy condition (1.14) and either (1.15) or (1.16). Then Xμ,σp(m),q(Ω)↪H˙sj,r(Ω) is a compact embedding.
Proof.
First we want to show that Xμ,σp(m),q(Ω)↪H˙sj,r(Ω) is a continuous embedding. Let u∈Xμ,σp(m),q(Ω) and recall (1.13). Then we have the higher-order CKN inequality
(
∫
Ω
|
D
j
u
|
r
d
x
|
x
|
s
)
1
r
≤
C
(
∫
Ω
|
D
m
u
|
p
d
x
|
x
|
μ
)
a
p
(
∫
Ω
|
u
|
q
d
x
|
x
|
σ
)
1
-
a
q
for some C=C(p,q,r,μ,σ,s,m,j), with
a
=
[
(
N
-
σ
)
r
-
(
N
-
s
-
j
r
)
q
]
p
[
(
N
-
σ
)
p
-
(
N
-
μ
-
m
p
)
q
]
r
∈
(
0
,
1
)
.
Since r≥1, by Young’s inequality, we have
(2.6)
∥
u
∥
H
˙
s
j
,
r
(
Ω
)
≤
C
max
{
a
,
1
-
a
}
∥
u
∥
X
μ
,
σ
p
(
m
)
,
q
(
Ω
)
.
Thus, Xμ,σp(m),q(Ω)↪H˙sj,r(Ω) is a continuous embedding.
For the compactness, it is sufficient to prove the case when Ω=ℝN. By the definition of Xμ,σp(m),q(ℝN) in (1.17) and (1.18), we notice that Xμ,σp(m),q(ℝN) is a closed subspace of H˙μm,p(ℝN). Hence, from Lemma 2.2, we have that Xμ,σp(m),q(ℝN) is locally compactly embedded in Hsj,r(ℝN).
Let {un} be a bounded sequence in Xμ,σp(m),q(ℝN). Pick s¯<s sufficient close to s so that the assumption of our parameters still hold after replacing s by s¯. From the continuity of the embedding, we have
sup
n
∥
u
n
∥
H
˙
s
¯
j
,
r
(
ℝ
N
)
<
∞
.
Then it follows that
∫
|
x
|
>
R
|
D
j
u
n
|
r
d
x
|
x
|
s
<
R
s
¯
-
s
∥
u
n
∥
H
˙
s
¯
j
,
r
(
ℝ
N
)
→
0
uniformly for all
n
as
R
→
∞
.
Hence, we proved the boundness of {un} in Xμ,σp(m),q(ℝN) at infinity. Combining this with the local compactness, we can prove that the embedding Xμ,σp(m),q(ℝN)↪H˙sj,r(ℝN) is compact. ∎
The following compact embedding for Sobolev spaces could be easily verified as a special case of Theorem 2.3.
Corollary 2.4.
Assume 1<p<N, max{σ,s}<μ+p<N, 1≤r<min{p⋆,p⋆(s,μ)}, 1≤q<r⋆(σ,s,j) and
{
(
N
-
p
-
μ
)
(
r
-
q
)
+
(
N
-
σ
)
(
p
-
r
)
<
(
N
-
s
)
(
p
-
q
)
,
(
p
⋆
(
s
,
μ
)
-
r
)
(
N
-
μ
-
p
)
σ
+
(
r
0
⋆
(
σ
,
s
)
-
q
)
(
N
-
s
)
μ
<
(
p
⋆
(
σ
,
μ
)
-
q
)
(
N
-
μ
-
p
)
s
.
Then Xμ,σp,q(RN)↪Lsr(RN) is a compact embedding.
Proof.
For m=1 and j=0 in Theorem 2.3, we could directly achieve the result. ∎
3 Existence of Extremal Function
In this section, Theorem 1.1 will be proved via the following series of lemmas under the assumption on the parameters (1.14) and the assumption that either (1.15) or (1.16) holds. Also, in particular, we consider Ω=ℛN; the general case follows directly.
Lemma 3.1.
For all M>0, consider the variational problem
(3.1)
𝒜
=
1
M
θ
inf
{
ℐ
[
u
]
:
u
∈
X
μ
,
σ
p
(
m
)
,
q
(
ℝ
N
)
and
∫
ℝ
N
|
D
j
u
|
r
d
x
|
x
|
s
=
M
}
,
where
(3.2)
ℐ
[
u
]
:=
1
p
∫
ℝ
N
|
D
m
u
|
p
d
x
|
x
|
μ
+
1
q
∫
ℝ
N
|
u
|
q
d
x
|
x
|
σ
and
(3.3)
θ
=
(
N
-
σ
)
p
-
(
N
-
μ
-
m
p
)
q
(
μ
+
m
p
-
σ
)
r
-
(
N
-
s
-
j
r
)
(
q
-
p
)
.
Then A>0 and it is independent of M. Moreover, the sharp constant of (1.19) is
ℒ
=
K
𝒜
1
r
θ
,
where
K
=
(
[
(
N
-
σ
+
j
q
)
r
-
(
N
-
s
)
q
]
p
(
μ
+
m
p
-
σ
)
r
-
(
N
-
s
-
j
r
)
(
q
-
p
)
)
(
N
-
σ
+
j
q
)
r
-
(
N
-
s
)
q
(
N
-
σ
)
p
-
(
N
-
μ
-
m
p
)
q
(3.4)
×
(
[
(
N
-
s
)
p
-
(
N
-
μ
-
(
m
-
j
)
p
)
]
q
(
μ
+
m
p
-
σ
)
r
-
(
N
-
s
-
j
r
)
(
q
-
p
)
)
(
N
-
s
)
p
-
(
N
-
μ
-
(
m
-
j
)
)
p
(
N
-
σ
)
p
-
(
N
-
μ
-
m
p
)
q
.
Proof.
First, from (2.6) and (3.1), we have 𝒜>0.
For any x∈ℝN and λ>0, we define the scaling function by
u
λ
(
x
)
:=
λ
N
-
s
r
-
j
u
(
λ
x
)
,
so that ∥uλ∥H˙sj,r=∥u∥H˙sj,r. A change of variables gives
(3.5)
ℐ
[
u
λ
]
=
1
p
λ
(
N
-
s
)
p
r
-
(
N
-
μ
-
(
m
-
j
)
p
)
∥
u
∥
H
˙
μ
m
,
p
p
+
1
q
λ
(
N
-
s
)
q
r
-
(
N
-
σ
-
j
q
)
∥
u
∥
L
σ
q
q
.
For simplicity, we set
P
=
(
N
-
s
)
p
r
-
(
N
-
μ
-
(
m
-
j
)
p
)
and
-
Q
=
(
N
-
s
)
q
r
-
(
N
-
σ
+
j
q
)
.
If we fix u, then (3.5) can be consider as a function of the variable λ, that is,
F
(
λ
)
=
1
p
λ
P
∥
u
∥
H
˙
μ
m
,
p
p
+
1
q
λ
-
Q
∥
u
∥
L
σ
q
q
.
Solving F′(λ)=0, we notice that F(λ) obtains the optimal value when
(3.6)
λ
=
λ
0
:=
(
Q
p
P
q
∥
u
∥
L
σ
q
q
∥
u
∥
H
˙
μ
m
,
p
p
)
1
P
+
Q
.
Therefore,
min
λ
>
0
ℐ
[
u
λ
]
=
ℐ
[
u
λ
0
]
=
1
p
λ
0
P
∥
u
∥
H
˙
μ
m
,
p
p
+
1
q
λ
0
-
Q
∥
u
∥
L
σ
q
q
=
(
1
P
+
1
Q
)
(
P
p
)
Q
P
+
Q
(
Q
q
)
P
P
+
Q
∥
u
∥
H
˙
μ
m
,
p
Q
p
P
+
Q
∥
u
∥
L
σ
q
P
q
P
+
Q
.
Then direct calculations show that
θ
=
P
q
+
Q
p
r
(
P
+
Q
)
.
Also, by setting
K
′
=
K
′
(
N
,
p
,
q
,
r
,
μ
,
σ
,
s
,
m
,
j
)
:=
(
1
P
+
1
Q
)
(
P
p
)
Q
P
+
Q
(
Q
q
)
P
P
+
Q
,
we get
(3.7)
min
λ
>
0
ℐ
[
u
λ
]
=
K
′
(
∥
u
∥
H
˙
s
j
,
r
r
𝒬
r
[
u
]
)
θ
=
K
′
M
θ
(
1
𝒬
[
u
]
)
r
θ
.
Combining (1.19), (3.1) and (3.7), we have
𝒜
=
inf
u
∈
X
μ
,
σ
p
(
m
)
,
q
(
ℝ
N
)
K
′
(
1
𝒬
[
u
]
)
r
θ
=
K
′
ℒ
r
θ
.
Hence, 𝒜 is independent of M and direct calculations show ℒ=K𝒜1rθ. ∎
From Lemma 3.1, we reveal the relationship between the sharp constant of the higher-order derivatives of the CKN inequalities and the infimum value of a corresponding variational problem. According to the proof process, we also notice that if u is a minimizer of ℐ[u], then it is exactly the maximizer of 𝒬[u].
Lemma 3.2.
Consider the variational problem given by (3.1)–(3.3). Then there exists u0∈Xμ,σp(m),q(RN), with ∫RN|Dju0|rdx|x|s=M, such that u0 is a minimizer of the variational problem. Furthermore, u0 is a ground state solution to the following equation:
(3.8)
(
-
1
)
m
∇
m
(
|
D
m
u
|
p
-
2
D
m
u
|
x
|
μ
)
+
|
u
|
q
-
2
u
|
x
|
σ
=
(
-
1
)
j
r
θ
𝒜
∇
j
(
|
D
j
u
|
r
-
2
D
j
u
|
x
|
s
)
.
Proof.
Assume that, for all M>0, {un} is a minimizing sequence, with un∈Xμ,σp(m),q(ℝN) and ∫ℝN|Djun|rdx|x|s=M.
Since 𝒜>0, by (3.1) and (3.2), it is not hard to see that {un} is bounded in Xμ,σp(m),q(ℝN). Applying Theorem 2.3, we can assume that there exists u0∈Xμ,σp(m),q(ℝN) such that, up to a subsequence, un→u0 in H˙sj,r. Then, from Fatou’s lemma, we have
ℐ
(
u
0
)
≤
lim
inf
n
→
∞
ℐ
(
u
n
)
=
𝒜
.
On another hand, by the definition of 𝒜, we have ℐ(u0)≥𝒜. Therefore, u0 is a minimizer of the variational problem.
For any v:ℝN→ℝ, Dkv∈ℝNk, we define
∇
k
D
k
v
:=
∇
(
∇
(
⋯
∇
(
D
k
v
)
)
)
∈
ℝ
.
Then, by the Lagrange multiplier theorem, there exists a Lagrange multiplier λ¯ such that
(3.9)
(
-
1
)
m
∇
m
(
|
D
m
u
0
|
p
-
2
D
m
u
0
|
x
|
μ
)
+
|
u
0
|
q
-
2
u
0
|
x
|
σ
=
(
-
1
)
j
λ
¯
∇
j
(
|
D
j
u
0
|
r
-
2
D
j
u
0
|
x
|
s
)
,
Without lose of generality, we pick M=1 in (3.1), then multiply both sides of (3.9) by u and integrate to obtain
(3.10)
∥
u
0
∥
H
˙
μ
m
,
p
p
+
∥
u
0
∥
L
σ
q
q
=
λ
¯
.
By definition, we also have
(3.11)
1
p
∥
u
0
∥
H
˙
μ
m
,
p
p
+
1
q
∥
u
0
∥
L
σ
q
q
=
𝒜
.
Notice that u0 is a minimizer, hence we have λ0=1 in (3.6), which shows
(3.12)
[
(
N
-
s
)
p
-
(
N
-
μ
-
(
m
-
j
)
p
)
r
]
p
∥
u
0
∥
H
˙
μ
m
,
p
p
=
[
(
N
-
σ
+
j
q
)
r
-
(
N
-
s
)
q
]
q
∥
u
0
∥
L
σ
q
q
.
Combining (3.10)–(3.12), we obtain
λ
¯
=
r
θ
𝒜
.
Therefore, the minimizer is a ground state solution to the equation (3.8). ∎
