R E S E A R C H
Open Access
A note on a class of Hardy-Rellich type
inequalities
Yanmei Di, Liya Jiang, Shoufeng Shen and Yongyang Jin
**Correspondence:
[email protected] Department of Mathematics, Zhejiang University of Technology, Hangzhou, P.R. China
Abstract
In this note we provide simple and short proofs for a class of Hardy-Rellich type inequalities with the best constant, which extends some recent results. MSC: 26D15; 35A23
Keywords: Hardy inequality; Hardy-Rellich inequality; Caffarelli-Kohn-Nirenberg inequality
1 Introduction
It is well known that Hardy’s inequality and its generalizations play important roles in many areas of mathematics. The classical Hardy inequality is given by, forN≥,
RN
∇u(x)
dx≥
N–
RN
|u(x)|
|x| dx, (.)
whereu∈C∞ (RN), the constant (N–
)is optimal and not attained.
Recently there has been a considerable interest in studying the Hardy-type and Rellich-type inequalities. See, for example, [–]. In [] Caffarelli, Kohn and Nirenberg proved a rather general interpolation inequality with weights. That is the following so-called Caffarelli-Kohn-Nirenberg inequality. For anyu∈C∞ (RN), there existsC> such that
|x|γuLr≤C|x|
α|∇
u|aLp·|x|
β
u–Lqa, (.)
where
r+
γ
N =a
p+
α–
N
+ ( –a)
q+
β
N
and
p,q≥, r≥, ≤a≤,
p+
α
N > ,
q+
β
N > ,
r+
γ
N > , γ =aσ+ ( –a)β.
In [] Costa proved the following L-case version for a class of Caffarelli-Kohn-Nirenberg inequalities with a sharp constant by an elementary method. For alla,b∈R
andu∈C∞(RN\{}),
ˆ C
RN
|u|
|x|a+b+dx≤
RN
|u|
|x|adx
RN
|∇u|
|x|b dx
, (.)
where the constantCˆ =Cˆ(a,b) :=|N–(a+b+)| is sharp.
On the other hand, the Rellich inequality is a generalization of the Hardy inequality to second-order derivatives, and the classical Rellich inequality inRN states that forN≥
andu∈C∞(RN\ {}),
RN
u(x)dx≥
N(N– )
RN
|u(x)|
|x| dx. (.)
The constant N(N–) is sharp and never achieved. In [] Tetikas and Zographopoulos obtained a corresponding stronger versions of the Rellich inequality which reads
N
RN
|∇u| |x| dx≤
RN|u|
dx (.)
for allu∈C∞andN≥. In [] Costa obtained a new class of Hardy-Rellich type inequal-ities which contain (.) as a special case. Ifa+b+ ≤N, then
ˆ C
RN
|∇u|
|x|a+b+dx≤
RN
|u|
|x|b dx
RN
|∇u|
|x|a dx
, (.)
where the constantCˆ =Cˆ(a,b) :=|N+a+b–|is sharp.
The goal of this paper is to extend the above (.) and (.) to the generalLp case for
<p<∞by a different and direct approach.
2 Main results
In this section, we will give the proof of the main theorems.
Theorem For all a,b∈R and u∈C∞(RN\ {}),one has
C
RN
|u|p
|x|a+b+dx≤
RN
|∇u|p
|x|ap dx
p
RN
|u|p
|x|bp–p
dx
p– p
, (.)
where <p<∞and the constant C=|N–(ap+b+)|is sharp.
Proof Letu∈C∞ (RN \ {}),a,b∈Randλ=a+b+ . By integration by parts and the
Hölder inequality, one has
RN
|u|p |x|λdx=
N–λ
RN|u| pdiv
x |x|λ
dx
= –
N–λ
RN
pu|u|p–x· ∇u |x|λ dx
≤ –p N–λ
RN
|x· ∇u| |x|λ |u|
≤ p N–λ
RN
|∇u||u|p–
|x|a+b dx
≤ p N–λ
RN
|∇u|p
|x|ap dx
p
RN
|u|p
|x|bp–p
dx
p– p
.
Then
N–λ
p
RN
|u|p
|x|λdx≤
RN
|∇u|p
|x|ap dx
p
RN
|u|p
|x|b
p p–
dx
p– p
. (.)
It remains to show the sharpness of the constant. By the condition with equality in the Hölder inequality, we consider the following family of functions:
uε(x) =e–
Cε
β|x|β, whenβ=a– b
p– + =
and
uε(x) =
|x|Cε, whenβ=a–
b
p– + = ,
whereCεis a positive number sequence converging to|N–(ap+b+)|asε→. By direct
com-putation and the limit process, we know the constant |N–(ap+b+)|is sharp.
Remark Whenp= , the inequality (.) covers the inequality (.) in [].
Remark Whena= ,b=p– , the inequality (.) is the classicalLpHardy inequality:
N–p p
p RN
|u|p
|x|pdx≤
RN|∇u|
pdx. (.)
When we take special values fora,b, the following corollary holds.
Corollary (i)When b= (a+ )(p– ),the inequality (.)is just the weighted Hardy inequality:
N–pp(a+ )pRN
|u|p |x|(a+)pdx≤
RN
|∇u|p
|x|ap dx. (.)
(ii)When a+b+ =ap,according to the inequality(.),we have
N–papRN
|u|p |x|apdx≤
RN
|∇u|p |x|ap dx
p
RN
|u|p |x|ap–p–p
dx
p– p
. (.)
(iii)When a= –p and a+b+ = ,we obtain the inequality
N p
RN|u| pdx≤
RN|∇u| p|x|pdx
p
RN
|u|p |x|p dx
p– p
By a similar method, we can prove the followingLp case Hardy-Rellich type
inequal-ity.
Theorem Let <p<N,pp––N ≤a+b+ ≤.Then,for any u∈C∞(RN\{}),the following
holds:
ˆ C
RN
|∇u|p
|x|a+b+dx≤
RN
|pu|p
|x|ap dx
p
RN
|∇u|q
|x|bq dx
q
, (.)
wherep+q= ,Cˆ= (N–p+(p–)(p a+b+))andpu=div(|∇u|p–∇u)is the p-Laplacian
opera-tor.
Proof Setλ=a+b+ , it is easy to see
RN
|∇u|p
|x|λ dx=
N–λ
RN|∇
u|pdiv
x |x|λ
dx
= –
N–λ
RN
p
|∇u|
p– x
|x|λ· ∇
|∇u|dx
= p
(λ–N)
RN|∇u|
p–x· ∇(|∇u|)
|x|λ dx. (.)
On the other hand,
RNpu
x· ∇u |x|λ dx=
RN
div|∇u|p–∇ux· ∇u |x|λ dx
= –
RN|∇u|
p–∇u· ∇
x· ∇u |x|λ
dx
= –
RN|∇u| p–
|∇
u| |x|λ +
x· ∇(|∇u| )
|x|λ –λ
(x· ∇u)
|x|λ+
dx,
which means
RN|∇u|
p–·x· ∇(|∇u|)
|x|λ dx
=
λ
RN|∇
u|p–(x· ∇u)
|x|λ dx–
RN
|∇u|p
|x|λ dx–
RNp
ux· ∇u |x|λ
. (.)
Then, we can deduce from (.) and (.)
RN
|∇u|p
|x|λ dx
= p
λ–N
λ
RN|∇u|
p–(x· ∇u)
|x|λ dx–
RN
|∇u|p
|x|λ dx–
RNpu
x· ∇u |x|λ
. (.)
That is,
N–p–λ
p
RN
|∇u|p
|x|λ dx+λ
RN|∇u|
p–(x· ∇u)
|x|λ+ dx=
RNpu
x· ∇u
By the Hölder inequality,
RNpu
x· ∇u |x|λ+ dx≤
RN
|pu|q
|x|aq
q
RN
|∇u|p
|x|bp
p
, (.)
note that pp––N ≤λ≤. Thus
N–p+ (p– )λ
p
RN
|∇u|p
|x|λ dx≤
RN
|pu|p
|x|ap
p
RN
|∇u|q
|x|bq
q
. (.)
We mention that we do not know whether the constant (N–p+(p–)(p a+b+)) in (.) is
opti-mal or not.
Corollary When a+b+ = ,we have the following inequalities: (i)when a= –,b= ,the inequality(.)is equivalent to the inequality
N–p p
p RN|∇u|
pdx≤
RN|pu|
p|x|pdx. (.)
(ii)When a= ,b= –,we obtain the inequality
N–p
p RN|∇
u|pdx≤
RN
|pu|p
|x|p dx
p
RN|∇
u|q|x|qdx
q
. (.)
(iii)When a= ,b= –,we get
N–p
p RN|∇
u|pdx≤
RN|p
u|pdx
p
RN|∇
u|q|x|qdx
q
. (.)
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors jointly worked on the results and they read and approved the final manuscript.
Acknowledgements
This work is supported by NNSF of China (11001240), ZJNSF (LQ12A01023) and the foundation of the Zhejiang University of the Technology (20100229).
Received: 31 May 2012 Accepted: 18 February 2013 Published: 4 March 2013 References
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Cite this article as:Di et al.:A note on a class of Hardy-Rellich type inequalities.Journal of Inequalities and Applications
