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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 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, [1–7]. 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 ) +(1−a) ( 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,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 [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 N≥3. In [11] 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<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,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<p<∞ 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:

u ε (x)= e − C ε β | x | β ,when Î²=a− b p − 1 +1≠0

and

u ε (x)= 1 | x | C ε ,when Î²=a− b p − 1 +1=0,

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 [9].

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 − 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.

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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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