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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 N3,

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, [17]. In [8] 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 ) +(1a) ( 1 q + β N )

and

In [9] 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,bR 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 N5 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 [10] 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 N3. In [11] Costa obtained a new class of Hardy-Rellich type inequalities which contain (1.5) as a special case. If a+b+3N, 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<p< 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,bR 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<p< and the constant C=| N ( a + b + 1 ) p | is sharp.

Proof Let u C 0 ( R N {0}), a,bR 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:

u ε (x)= e C ε β | x | β ,when β=a b p 1 +10

and

u ε (x)= 1 | x | C ε ,when β=a b p 1 +1=0,

where C ε is a positive number sequence converging to | N ( a + b + 1 ) p | as ε0. 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 [9].

Remark 2 When a=0, b=p1, 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)(p1), 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 N p 1 a+b+10. 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 λ0. 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.

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The authors declare that they have no competing interests.

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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

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Keywords

  • Hardy inequality
  • Hardy-Rellich inequality
  • Caffarelli-Kohn-Nirenberg inequality