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Journal of Inequalities and Applications

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Open Access 01.12.2013 | Research

A note on a class of Hardy-Rellich type inequalities

verfasst von: Yanmei Di, Liya Jiang, Shoufeng Shen, Yongyang Jin

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2013

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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.

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.

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 \geq 3

\int_{R^{N}} {| \nabla u (x) |}^{2} d x \geq {(\frac{N - 2}{2})}^{2} \int_{R^{N}} \frac{{| u (x) |}^{2}}{{| x |}^{2}} d x,

(1.1)

where

u \in C_{0}^{\infty} (R^{N})

, the constant

{(\frac{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 \in C_{0}^{\infty} (R^{N})

, there exists

C > 0

such that

{∥ {| x |}^{γ} u ∥}_{L^{r}} \leq C {∥ {| x |}^{α} | \nabla u | ∥}_{L^{p}}^{a} \cdot {∥ {| x |}^{β} u ∥}_{L^{q}}^{1 - a},

(1.2)

where

\frac{1}{r} + \frac{γ}{N} = a (\frac{1}{p} + \frac{α - 1}{N}) + (1 - a) (\frac{1}{q} + \frac{β}{N})

and

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-84/MediaObjects/13660_2012_Article_530_Equb_HTML.gif

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 \in R

and

u \in C_{0}^{\infty} (R^{N} ∖ {0})

\hat{C} \int_{R^{N}} \frac{{| u |}^{2}}{{| x |}^{a + b + 1}} d x \leq {(\int_{R^{N}} \frac{{| u |}^{2}}{{| x |}^{2 a}} d x)}^{\frac{1}{2}} {(\int_{R^{N}} \frac{{| \nabla u |}^{2}}{{| x |}^{2 b}} d x)}^{\frac{1}{2}},

(1.3)

where the constant

\hat{C} = \hat{C} (a, b) : = \frac{| 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 \geq 5

and

u \in C_{0}^{\infty} (R^{N} ∖ {0})

\int_{R^{N}} {| Δ u (x) |}^{2} d x \geq {(\frac{N (N - 4)}{4})}^{2} \int_{R^{N}} \frac{{| u (x) |}^{2}}{{| x |}^{4}} d x .

(1.4)

The constant

\frac{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

{(\frac{N}{2})}^{2} \int_{R^{N}} \frac{{| \nabla u |}^{2}}{{| x |}^{2}} d x \leq \int_{R^{N}} {| △ u |}^{2} d x

(1.5)

for all

u \in C_{0}^{\infty}

and

N \geq 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 \leq N

, then

\hat{C} \int_{R^{N}} \frac{{| \nabla u |}^{2}}{{| x |}^{a + b + 1}} d x \leq {(\int_{R^{N}} \frac{{| △ u |}^{2}}{{| x |}^{2 b}} d x)}^{\frac{1}{2}} {(\int_{R^{N}} \frac{{| \nabla u |}^{2}}{{| x |}^{2 a}} d x)}^{\frac{1}{2}},

(1.6)

where the constant

\hat{C} = \hat{C} (a, b) : = | \frac{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 < \infty

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 \in R

and

u \in C_{0}^{\infty} (R^{N} ∖ {0})

, one has

C \int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{a + b + 1}} d x \leq {(\int_{R^{N}} \frac{{| \nabla u |}^{p}}{{| x |}^{a p}} d x)}^{\frac{1}{p}} {(\int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{b \frac{p}{p - 1}}} d x)}^{\frac{p - 1}{p}},

(2.1)

where

1 < p < \infty

and the constant

C = | \frac{N - (a + b + 1)}{p} |

is sharp.

Proof Let

u \in C_{0}^{\infty} (R^{N} ∖ {0})

a, b \in R

and

λ = a + b + 1

. By integration by parts and the Hölder inequality, one has

\begin{array}{rcl} \int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{λ}} d x & = & \frac{1}{N - λ} \int_{R^{N}} {| u |}^{p} div (\frac{x}{{| x |}^{λ}}) d x \\ = & - \frac{1}{N - λ} \int_{R^{N}} p u {| u |}^{p - 2} \frac{x \cdot \nabla u}{{| x |}^{λ}} d x \\ \leq & | \frac{- p}{N - λ} | \int_{R^{N}} \frac{| x \cdot \nabla u |}{{| x |}^{λ}} {| u |}^{p - 1} d x \\ \leq & | \frac{p}{N - λ} | \int_{R^{N}} \frac{| \nabla u | {| u |}^{p - 1}}{{| x |}^{a + b}} d x \\ \leq & | \frac{p}{N - λ} | {(\int_{R^{N}} \frac{{| \nabla u |}^{p}}{{| x |}^{a p}} d x)}^{\frac{1}{p}} {(\int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{b \frac{p}{p - 1}}} d x)}^{\frac{p - 1}{p}} . \end{array}

Then

| \frac{N - λ}{p} | \int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{λ}} d x \leq {(\int_{R^{N}} \frac{{| \nabla u |}^{p}}{{| x |}^{a p}} d x)}^{\frac{1}{p}} {(\int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{b \frac{p}{p - 1}}} d x)}^{\frac{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^{- \frac{C_{ε}}{β} {| x |}^{β}}, when β = a - \frac{b}{p - 1} + 1 \neq 0

and

u_{ε} (x) = \frac{1}{{| x |}^{C_{ε}}}, when β = a - \frac{b}{p - 1} + 1 = 0,

where

C_{ε}

is a positive number sequence converging to

| \frac{N - (a + b + 1)}{p} |

ε \to 0

. By direct computation and the limit process, we know the constant

\frac{| 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:

{(\frac{N - p}{p})}^{p} \int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{p}} d x \leq \int_{R^{N}} {| \nabla u |}^{p} d x .

(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:

| \frac{N - p (a + 1)}{p} |^{p} \int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{(a + 1) p}} d x \leq \int_{R^{N}} \frac{{| \nabla u |}^{p}}{{| x |}^{a p}} d x .

(2.4)

(ii)

When

a + b + 1 = a p

, according to the inequality (2.1), we have

| \frac{N - a p}{p} | \int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{a p}} d x \leq {(\int_{R^{N}} \frac{{| \nabla u |}^{p}}{{| x |}^{a p}} d x)}^{\frac{1}{p}} {(\int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{a p - \frac{p}{p - 1}}} d x)}^{\frac{p - 1}{p}} .

(2.5)

(iii)

When

a = - p

and

a + b + 1 = 0

, we obtain the inequality

\frac{N}{p} \int_{R^{N}} {| u |}^{p} d x \leq {(\int_{R^{N}} {| \nabla u |}^{p} {| x |}^{p^{2}} d x)}^{\frac{1}{p}} {(\int_{R^{N}} \frac{{| u |}^{p}}{{| x |}^{p}} d x)}^{\frac{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

\frac{p - N}{p - 1} \leq a + b + 1 \leq 0

. Then, for any

u \in C_{0}^{\infty} (R^{N} ∖ {0})

, the following holds:

\hat{C} \int_{R^{N}} \frac{{| \nabla u |}^{p}}{{| x |}^{a + b + 1}} d x \leq {(\int_{R^{N}} \frac{{| △_{p} u |}^{p}}{{| x |}^{a p}} d x)}^{\frac{1}{p}} {(\int_{R^{N}} \frac{{| \nabla u |}^{q}}{{| x |}^{b q}} d x)}^{\frac{1}{q}},

(2.7)

where

\frac{1}{p} + \frac{1}{q} = 1

\hat{C} = (\frac{N - p + (p - 1) (a + b + 1)}{p})

and

△_{p} u = div ({| \nabla u |}^{p - 2} \nabla u)

is the p-Laplacian operator.

Proof Set

λ = a + b + 1

, it is easy to see

\begin{aligned} \int_{R^{N}} \frac{{| \nabla u |}^{p}}{{| x |}^{λ}} d x & = \frac{1}{N - λ} \int_{R^{N}} {| \nabla u |}^{p} div (\frac{x}{{| x |}^{λ}}) d x \\ = - \frac{1}{N - λ} \int_{R^{N}} \frac{p}{2} {| \nabla u |}^{p - 2} \frac{x}{{| x |}^{λ}} \cdot \nabla ({| \nabla u |}^{2}) d x \\ = \frac{p}{2 (λ - N)} \int_{R^{N}} {| \nabla u |}^{p - 2} \frac{x \cdot \nabla ({| \nabla u |}^{2})}{{| x |}^{λ}} d x . \end{aligned}

(2.8)

On the other hand,

\begin{array}{rcl} \int_{R^{N}} △_{p} u \frac{x \cdot \nabla u}{{| x |}^{λ}} d x & = & \int_{R^{N}} div ({| \nabla u |}^{p - 2} \nabla u) \frac{x \cdot \nabla u}{{| x |}^{λ}} d x \\ = & - \int_{R^{N}} {| \nabla u |}^{p - 2} \nabla u \cdot \nabla (\frac{x \cdot \nabla u}{{| x |}^{λ}}) d x \\ = & - \int_{R^{N}} {| \nabla u |}^{p - 2} (\frac{{| \nabla u |}^{2}}{{| x |}^{λ}} + \frac{\frac{1}{2} x \cdot \nabla ({| \nabla u |}^{2})}{{| x |}^{λ}} - λ \frac{{(x \cdot \nabla u)}^{2}}{{| x |}^{λ + 2}}) d x, \end{array}

which means

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-84/MediaObjects/13660_2012_Article_530_Equ15_HTML.gif

(2.9)

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

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-84/MediaObjects/13660_2012_Article_530_Equ16_HTML.gif

(2.10)

That is,

\frac{N - p - λ}{p} \int_{R^{N}} \frac{{| \nabla u |}^{p}}{{| x |}^{λ}} d x + λ \int_{R^{N}} {| \nabla u |}^{p - 2} \frac{{(x \cdot \nabla u)}^{2}}{{| x |}^{λ + 2}} d x = \int_{R^{N}} △_{p} u \frac{x \cdot \nabla u}{{| x |}^{λ}} d x .

(2.11)

By the Hölder inequality,

\int_{R^{N}} △_{p} u \frac{x \cdot \nabla u}{{| x |}^{λ + 2}} d x \leq {(\int_{R^{N}} \frac{{| △_{p} u |}^{q}}{{| x |}^{a q}})}^{\frac{1}{q}} {(\int_{R^{N}} \frac{{| \nabla u |}^{p}}{{| x |}^{b p}})}^{\frac{1}{p}},

(2.12)

note that

\frac{p - N}{p - 1} \leq λ \leq 0

. Thus

\frac{N - p + (p - 1) λ}{p} \int_{R^{N}} \frac{{| \nabla u |}^{p}}{{| x |}^{λ}} d x \leq {(\int_{R^{N}} \frac{{| △_{p} u |}^{p}}{{| x |}^{a p}})}^{\frac{1}{p}} {(\int_{R^{N}} \frac{{| \nabla u |}^{q}}{{| x |}^{b q}})}^{\frac{1}{q}} .

(2.13)

We mention that we do not know whether the constant

(\frac{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:

(i)

when

a = - 1

b = 0

, the inequality (2.7) is equivalent to the inequality

{(\frac{N - p}{p})}^{p} \int_{R^{N}} {| \nabla u |}^{p} d x \leq \int_{R^{N}} {| △_{p} u |}^{p} {| x |}^{p} d x .

(2.14)

(ii)

When

a = 1

b = - 2

, we obtain the inequality

(\frac{N - p}{p}) \int_{R^{N}} {| \nabla u |}^{p} d x \leq {(\int_{R^{N}} \frac{{| △_{p} u |}^{p}}{{| x |}^{p}} d x)}^{\frac{1}{p}} {(\int_{R^{N}} {| \nabla u |}^{q} {| x |}^{2 q} d x)}^{\frac{1}{q}} .

(2.15)

(iii)

When

a = 0

b = - 1

, we get

(\frac{N - p}{p}) \int_{R^{N}} {| \nabla u |}^{p} d x \leq {(\int_{R^{N}} {| △_{p} u |}^{p} d x)}^{\frac{1}{p}} {(\int_{R^{N}} {| \nabla u |}^{q} {| x |}^{q} d x)}^{\frac{1}{q}} .

(2.16)

Acknowledgements

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

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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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Titel: A note on a class of Hardy-Rellich type inequalities
verfasst von: Yanmei Di
Liya Jiang
Shoufeng Shen
Yongyang Jin
Publikationsdatum: 01.12.2013
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2013
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2013-84

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A note on a class of Hardy-Rellich type inequalities

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Main results

Acknowledgements

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Authors’ contributions

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Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Main results

Acknowledgements

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