# A note on a class of Hardy-Rellich type inequalities

## 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.

## 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, for $N≥3$,

$∫ R N | ∇ u ( x ) | 2 dx≥ ( N − 2 2 ) 2 ∫ R N | u ( x ) | 2 | x | 2 dx,$
(1.1)

where $u∈ C 0 ∞ ( R N )$, the constant $( N − 2 2 ) 2$ 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 any $u∈ C 0 ∞ ( R N )$, there exists $C>0$ such that

$∥ | x | γ u ∥ L r ≤C ∥ | x | α | ∇ u | ∥ L p a ⋅ ∥ | x | β u ∥ L q 1 − a ,$
(1.2)

where

$1 r + γ N =a ( 1 p + α − 1 N ) +(1−a) ( 1 q + β N )$

and In  Costa proved the following $L 2$-case version for a class of Caffarelli-Kohn-Nirenberg inequalities with a sharp constant by an elementary method. For all $a,b∈R$ and $u∈ C 0 ∞ ( R N ∖{0})$,

$C ˆ ∫ R N | u | 2 | x | a + b + 1 dx≤ ( ∫ R N | u | 2 | x | 2 a d x ) 1 2 ( ∫ R N | ∇ u | 2 | x | 2 b d x ) 1 2 ,$
(1.3)

where the constant $C ˆ = C ˆ (a,b):= | N − ( a + b + 1 ) | 2$ 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 in $R N$ states that for $N≥5$ and $u∈ C 0 ∞ ( R N ∖{0})$,

$∫ R N | Δ u ( x ) | 2 dx≥ ( N ( N − 4 ) 4 ) 2 ∫ R N | u ( x ) | 2 | x | 4 dx.$
(1.4)

The constant $N 2 ( N − 4 ) 2 16$ is sharp and never achieved. In  Tetikas and Zographopoulos obtained a corresponding stronger versions of the Rellich inequality which reads

$( N 2 ) 2 ∫ R N | ∇ u | 2 | x | 2 dx≤ ∫ R N | △ u | 2 dx$
(1.5)

for all $u∈ C 0 ∞$ and $N≥3$. In  Costa obtained a new class of Hardy-Rellich type inequalities which contain (1.5) as a special case. If $a+b+3≤N$, then

$C ˆ ∫ R N | ∇ u | 2 | x | a + b + 1 dx≤ ( ∫ R N | △ u | 2 | x | 2 b d x ) 1 2 ( ∫ R N | ∇ u | 2 | x | 2 a d x ) 1 2 ,$
(1.6)

where the constant $C ˆ = C ˆ (a,b):=| N + a + b − 1 2 |$ is sharp.

The goal of this paper is to extend the above (1.3) and (1.6) to the general $L p$ case for $1 by a different and direct approach.

## 2 Main results

In this section, we will give the proof of the main theorems.

Theorem 1 For all $a,b∈R$ and $u∈ C 0 ∞ ( R N ∖{0})$, one has

$C ∫ R N | u | p | x | a + b + 1 dx≤ ( ∫ R N | ∇ u | p | x | a p d x ) 1 p ( ∫ R N | u | p | x | b p p − 1 d x ) p − 1 p ,$
(2.1)

where $1 and the constant $C=| N − ( a + b + 1 ) p |$ is sharp.

Proof Let $u∈ C 0 ∞ ( R N ∖{0})$, $a,b∈R$ and $λ=a+b+1$. By integration by parts and the Hölder inequality, one has

$∫ R N | u | p | x | λ d x = 1 N − λ ∫ R N | u | p div ( x | x | λ ) d x = − 1 N − λ ∫ R N p u | u | p − 2 x ⋅ ∇ u | x | λ d x ≤ | − p N − λ | ∫ R N | x ⋅ ∇ u | | x | λ | u | p − 1 d x ≤ | p N − λ | ∫ R N | ∇ u | | u | p − 1 | x | a + b d x ≤ | p N − λ | ( ∫ R N | ∇ u | p | x | a p d x ) 1 p ( ∫ R N | u | p | x | b p p − 1 d x ) p − 1 p .$

Then

$| N − λ p | ∫ R N | u | p | x | λ dx≤ ( ∫ R N | ∇ u | p | x | a p d x ) 1 p ( ∫ R N | u | p | x | b p p − 1 d x ) p − 1 p .$
(2.2)

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:

and

where $C ε$ is a positive number sequence converging to $| N − ( a + b + 1 ) p |$ as $ε→0$. By direct computation and the limit process, we know the constant $| N − ( a + b + 1 ) | p$ is sharp. □

Remark 1 When $p=2$, the inequality (2.1) covers the inequality (2.4) in .

Remark 2 When $a=0$, $b=p−1$, the inequality (2.1) is the classical $L p$ Hardy inequality:

$( N − p p ) p ∫ R N | u | p | x | p dx≤ ∫ R N | ∇ u | p dx.$
(2.3)

When we take special values for a, b, the following corollary holds.

Corollary 1 (i) When $b=(a+1)(p−1)$, the inequality (2.1) is just the weighted Hardy inequality:

$| N − p ( a + 1 ) p | p ∫ R N | u | p | x | ( a + 1 ) p dx≤ ∫ R N | ∇ u | p | x | a p dx.$
(2.4)
1. (ii)

When $a+b+1=ap$, according to the inequality (2.1), we have

$| N − a p p | ∫ R N | u | p | x | a p dx≤ ( ∫ R N | ∇ u | p | x | a p d x ) 1 p ( ∫ R N | u | p | x | a p − p p − 1 d x ) p − 1 p .$
(2.5)
2. (iii)

When $a=−p$ and $a+b+1=0$, we obtain the inequality

$N p ∫ R N | u | p dx≤ ( ∫ R N | ∇ u | p | x | p 2 d x ) 1 p ( ∫ R N | u | p | x | p d x ) p − 1 p .$
(2.6)

By a similar method, we can prove the following $L p$ case Hardy-Rellich type inequality.

Theorem 2 Let $1, $p − N p − 1 ≤a+b+1≤0$. Then, for any $u∈ C 0 ∞ ( R N ∖{0})$, the following holds:

$C ˆ ∫ R N | ∇ u | p | x | a + b + 1 dx≤ ( ∫ R N | △ p u | p | x | a p d x ) 1 p ( ∫ R N | ∇ u | q | x | b q d x ) 1 q ,$
(2.7)

where $1 p + 1 q =1$, $C ˆ =( N − p + ( p − 1 ) ( a + b + 1 ) p )$ and $△ p u=div( | ∇ u | p − 2 ∇u)$ is the p-Laplacian operator.

Proof Set $λ=a+b+1$, it is easy to see

$∫ R N | ∇ u | p | x | λ d x = 1 N − λ ∫ R N | ∇ u | p div ( x | x | λ ) d x = − 1 N − λ ∫ R N p 2 | ∇ u | p − 2 x | x | λ ⋅ ∇ ( | ∇ u | 2 ) d x = p 2 ( λ − N ) ∫ R N | ∇ u | p − 2 x ⋅ ∇ ( | ∇ u | 2 ) | x | λ d x .$
(2.8)

On the other hand,

$∫ R N △ p u x ⋅ ∇ u | x | λ d x = ∫ R N div ( | ∇ u | p − 2 ∇ u ) x ⋅ ∇ u | x | λ d x = − ∫ R N | ∇ u | p − 2 ∇ u ⋅ ∇ ( x ⋅ ∇ u | x | λ ) d x = − ∫ R N | ∇ u | p − 2 ( | ∇ u | 2 | x | λ + 1 2 x ⋅ ∇ ( | ∇ u | 2 ) | x | λ − λ ( x ⋅ ∇ u ) 2 | x | λ + 2 ) d x ,$

which means (2.9)

Then, we can deduce from (2.8) and (2.9) (2.10)

That is,

$N − p − λ p ∫ R N | ∇ u | p | x | λ dx+λ ∫ R N | ∇ u | p − 2 ( x ⋅ ∇ u ) 2 | x | λ + 2 dx= ∫ R N △ p u x ⋅ ∇ u | x | λ dx.$
(2.11)

By the Hölder inequality,

$∫ R N △ p u x ⋅ ∇ u | x | λ + 2 dx≤ ( ∫ R N | △ p u | q | x | a q ) 1 q ( ∫ R N | ∇ u | p | x | b p ) 1 p ,$
(2.12)

note that $p − N p − 1 ≤λ≤0$. Thus

$N − p + ( p − 1 ) λ p ∫ R N | ∇ u | p | x | λ dx≤ ( ∫ R N | △ p u | p | x | a p ) 1 p ( ∫ R N | ∇ u | q | x | b q ) 1 q .$
(2.13)

We mention that we do not know whether the constant $( N − p + ( p − 1 ) ( a + b + 1 ) p )$ in (2.7) is optimal or not. □

Corollary 2 When $a+b+1=0$, we have the following inequalities:

1. (i)

when $a=−1$, $b=0$, the inequality (2.7) is equivalent to the inequality

$( N − p p ) p ∫ R N | ∇ u | p dx≤ ∫ R N | △ p u | p | x | p dx.$
(2.14)
2. (ii)

When $a=1$, $b=−2$, we obtain the inequality

$( N − p p ) ∫ R N | ∇ u | p dx≤ ( ∫ R N | △ p u | p | x | p d x ) 1 p ( ∫ R N | ∇ u | q | x | 2 q d x ) 1 q .$
(2.15)
3. (iii)

When $a=0$, $b=−1$, we get

$( N − p p ) ∫ R N | ∇ u | p dx≤ ( ∫ R N | △ p u | p d x ) 1 p ( ∫ R N | ∇ u | q | x | q d x ) 1 q .$
(2.16)

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## Acknowledgements

This work is supported by NNSF of China (11001240), ZJNSF (LQ12A01023) and the foundation of the Zhejiang University of the Technology (20100229).

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Correspondence to Yongyang Jin.

### 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.

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Di, Y., Jiang, L., Shen, S. et al. A note on a class of Hardy-Rellich type inequalities. J Inequal Appl 2013, 84 (2013). https://doi.org/10.1186/1029-242X-2013-84 