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 $N\ge 3$,

$${\int }_{R^N}{|\mathrm{\nabla }u(x)|}^{2}dx\ge {\left(\frac{N-2}{2}\right)}^2{\int }_{R^N}\frac{{|u(x)|}^2}{{|x|}^2}dx,$$

(1.1)

where $u\in C_0^{\mathrm{\infty }}(R^N)$, 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 $u\in C_0^{\mathrm{\infty }}(R^N)$, there exists $C>0$ such that

$${\parallel {|x|}^{\gamma }u\parallel }_{L^r}\le C{\parallel {|x|}^{\alpha }|\mathrm{\nabla }u|\parallel }_{L^p}^a\cdot {\parallel {|x|}^{\beta }u\parallel }_{L^q}^{1-a},$$

(1.2)

where

$$\frac{1}{r}+\frac{\gamma }{N}=a(\frac{1}{p}+\frac{\alpha -1}{N})+(1-a)(\frac{1}{q}+\frac{\beta }{N})$$

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 $a,b\in R$ and $u\in C_0^{\mathrm{\infty }}(R^N\mathrm{\setminus }\{0\})$,

$$\hat{C}{\int }_{R^N}\frac{{|u|}^2}{{|x|}^{a+b+1}}dx\le {\left({\int }_{R^N}\frac{{|u|}^2}{{|x|}^{2a}}dx\right)}^{\frac{1}{2}}{\left({\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^2}{{|x|}^{2b}}dx\right)}^{\frac{1}{2}},$$

(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 $N\ge 5$ and $u\in C_0^{\mathrm{\infty }}(R^N\setminus \{0\})$,

$${\int }_{R^N}{|\mathrm{\Delta }u(x)|}^{2}dx\ge {\left(\frac{N(N-4)}{4}\right)}^2{\int }_{R^N}\frac{{|u(x)|}^2}{{|x|}^4}dx.$$

(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

$${\left(\frac{N}{2}\right)}^2{\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^2}{{|x|}^2}dx\le {\int }_{R^N}{|\mathrm{△}u|}^{2}dx$$

(1.5)

for all $u\in C_0^{\mathrm{\infty }}$ and $N\ge 3$. In [11] Costa obtained a new class of Hardy-Rellich type inequalities which contain (1.5) as a special case. If $a+b+3\le N$, then

$$\hat{C}{\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^2}{{|x|}^{a+b+1}}dx\le {\left({\int }_{R^N}\frac{{|\mathrm{△}u|}^2}{{|x|}^{2b}}dx\right)}^{\frac{1}{2}}{\left({\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^2}{{|x|}^{2a}}dx\right)}^{\frac{1}{2}},$$

(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 $1<p<\mathrm{\infty }$ 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 $u\in C_0^{\mathrm{\infty }}(R^N\setminus \{0\})$, one has

$$C{\int }_{R^N}\frac{{|u|}^p}{{|x|}^{a+b+1}}dx\le {\left({\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^p}{{|x|}^{ap}}dx\right)}^{\frac{1}{p}}{\left({\int }_{R^N}\frac{{|u|}^p}{{|x|}^{b\frac{p}{p-1}}}dx\right)}^{\frac{p-1}{p}},$$

(2.1)

where $1<p<\mathrm{\infty }$ and the constant $C=|\frac{N-(a+b+1)}{p}|$ is sharp.

Proof Let $u\in C_0^{\mathrm{\infty }}(R^N\setminus \{0\})$, $a,b\in R$ and $\lambda =a+b+1$. By integration by parts and the Hölder inequality, one has

$$\begin{array}{rcl}{\int }_{R^N}\frac{{|u|}^p}{{|x|}^{\lambda }}dx& =& \frac{1}{N-\lambda }{\int }_{R^N}{|u|}^{p}div\left(\frac{x}{{|x|}^{\lambda }}\right)dx\\ =& -\frac{1}{N-\lambda }{\int }_{R^N}pu{|u|}^{p-2}\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda }}dx\\ \le & \left|\frac{-p}{N-\lambda }\right|{\int }_{R^N}\frac{|x\cdot \mathrm{\nabla }u|}{{|x|}^{\lambda }}{|u|}^{p-1}dx\\ \le & \left|\frac{p}{N-\lambda }\right|{\int }_{R^N}\frac{|\mathrm{\nabla }u|{|u|}^{p-1}}{{|x|}^{a+b}}dx\\ \le & \left|\frac{p}{N-\lambda }\right|{\left({\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^p}{{|x|}^{ap}}dx\right)}^{\frac{1}{p}}{\left({\int }_{R^N}\frac{{|u|}^p}{{|x|}^{b\frac{p}{p-1}}}dx\right)}^{\frac{p-1}{p}}.\end{array}$$

Then

$$\left|\frac{N-\lambda }{p}\right|{\int }_{R^N}\frac{{|u|}^p}{{|x|}^{\lambda }}dx\le {\left({\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^p}{{|x|}^{ap}}dx\right)}^{\frac{1}{p}}{\left({\int }_{R^N}\frac{{|u|}^p}{{|x|}^{b\frac{p}{p-1}}}dx\right)}^{\frac{p-1}{p}}.$$

(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:

$$u_{\epsilon }(x)=e^{-\frac{C_{\epsilon }}{\beta }{|x|}^{\beta }},\text{when }\beta =a-\frac{b}{p-1}+1\ne 0$$

and

$$u_{\epsilon }(x)=\frac{1}{{|x|}^{C_{\epsilon }}},\text{when }\beta =a-\frac{b}{p-1}+1=0,$$

where $C_{\epsilon }$ is a positive number sequence converging to $|\frac{N-(a+b+1)}{p}|$ as $\epsilon \to 0$. 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:

$${\left(\frac{N-p}{p}\right)}^p{\int }_{R^N}\frac{{|u|}^p}{{|x|}^p}dx\le {\int }_{R^N}{|\mathrm{\nabla }u|}^{p}dx.$$

(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:

$$|\frac{N-p(a+1)}{p}{|}^p{\int }_{R^N}\frac{{|u|}^p}{{|x|}^{(a+1)p}}dx\le {\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^p}{{|x|}^{ap}}dx.$$

(2.4)

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

  $$\left|\frac{N-ap}{p}\right|{\int }_{R^N}\frac{{|u|}^p}{{|x|}^{ap}}dx\le {\left({\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^p}{{|x|}^{ap}}dx\right)}^{\frac{1}{p}}{\left({\int }_{R^N}\frac{{|u|}^p}{{|x|}^{ap-\frac{p}{p-1}}}dx\right)}^{\frac{p-1}{p}}.$$

  (2.5)

• (iii) When $a=-p$ and $a+b+1=0$, we obtain the inequality

  $$\frac{N}{p}{\int }_{R^N}{|u|}^{p}dx\le {\left({\int }_{R^N}{|\mathrm{\nabla }u|}^p{|x|}^{p^2}dx\right)}^{\frac{1}{p}}{\left({\int }_{R^N}\frac{{|u|}^p}{{|x|}^p}dx\right)}^{\frac{p-1}{p}}.$$

  (2.6)

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

Theorem 2 Let $1<p<N$, $\frac{p-N}{p-1}\le a+b+1\le 0$. Then, for any $u\in C_0^{\mathrm{\infty }}(R^N\setminus \{0\})$, the following holds:

$$\hat{C}{\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^p}{{|x|}^{a+b+1}}dx\le {\left({\int }_{R^N}\frac{{|{\mathrm{△}}_{p}u|}^p}{{|x|}^{ap}}dx\right)}^{\frac{1}{p}}{\left({\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^q}{{|x|}^{bq}}dx\right)}^{\frac{1}{q}},$$

(2.7)

where $\frac{1}{p}+\frac{1}{q}=1$, $\hat{C}=(\frac{N-p+(p-1)(a+b+1)}{p})$ and ${\mathrm{△}}_{p}u=div({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)$ is the p-Laplacian operator.

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

$$\begin{array}{rl}{\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^p}{{|x|}^{\lambda }}dx& =\frac{1}{N-\lambda }{\int }_{R^N}{|\mathrm{\nabla }u|}^{p}div\left(\frac{x}{{|x|}^{\lambda }}\right)dx\\ =-\frac{1}{N-\lambda }{\int }_{R^N}\frac{p}{2}{|\mathrm{\nabla }u|}^{p-2}\frac{x}{{|x|}^{\lambda }}\cdot \mathrm{\nabla }\left({|\mathrm{\nabla }u|}^2\right)dx\\ =\frac{p}{2(\lambda -N)}{\int }_{R^N}{|\mathrm{\nabla }u|}^{p-2}\frac{x\cdot \mathrm{\nabla }({|\mathrm{\nabla }u|}^2)}{{|x|}^{\lambda }}dx.\end{array}$$

(2.8)

On the other hand,

$$\begin{array}{rcl}{\int }_{R^N}{\mathrm{△}}_{p}u\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda }}dx& =& {\int }_{R^N}div\left({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\right)\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda }}dx\\ =& -{\int }_{R^N}{|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\cdot \mathrm{\nabla }\left(\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda }}\right)dx\\ =& -{\int }_{R^N}{|\mathrm{\nabla }u|}^{p-2}(\frac{{|\mathrm{\nabla }u|}^2}{{|x|}^{\lambda }}+\frac{\frac{1}{2}x\cdot \mathrm{\nabla }({|\mathrm{\nabla }u|}^2)}{{|x|}^{\lambda }}-\lambda \frac{{(x\cdot \mathrm{\nabla }u)}^2}{{|x|}^{\lambda +2}})dx,\end{array}$$

which means

(2.9)

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

(2.10)

That is,

$$\frac{N-p-\lambda }{p}{\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^p}{{|x|}^{\lambda }}dx+\lambda {\int }_{R^N}{|\mathrm{\nabla }u|}^{p-2}\frac{{(x\cdot \mathrm{\nabla }u)}^2}{{|x|}^{\lambda +2}}dx={\int }_{R^N}{\mathrm{△}}_{p}u\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda }}dx.$$

(2.11)

By the Hölder inequality,

$${\int }_{R^N}{\mathrm{△}}_{p}u\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda +2}}dx\le {\left({\int }_{R^N}\frac{{|{\mathrm{△}}_{p}u|}^q}{{|x|}^{aq}}\right)}^{\frac{1}{q}}{\left({\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^p}{{|x|}^{bp}}\right)}^{\frac{1}{p}},$$

(2.12)

note that $\frac{p-N}{p-1}\le \lambda \le 0$. Thus

$$\frac{N-p+(p-1)\lambda }{p}{\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^p}{{|x|}^{\lambda }}dx\le {\left({\int }_{R^N}\frac{{|{\mathrm{△}}_{p}u|}^p}{{|x|}^{ap}}\right)}^{\frac{1}{p}}{\left({\int }_{R^N}\frac{{|\mathrm{\nabla }u|}^q}{{|x|}^{bq}}\right)}^{\frac{1}{q}}.$$

(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

  $${\left(\frac{N-p}{p}\right)}^p{\int }_{R^N}{|\mathrm{\nabla }u|}^{p}dx\le {\int }_{R^N}{|{\mathrm{△}}_{p}u|}^p{|x|}^{p}dx.$$

  (2.14)

• (ii) When $a=1$, $b=-2$, we obtain the inequality

  $$\left(\frac{N-p}{p}\right){\int }_{R^N}{|\mathrm{\nabla }u|}^{p}dx\le {\left({\int }_{R^N}\frac{{|{\mathrm{△}}_{p}u|}^p}{{|x|}^p}dx\right)}^{\frac{1}{p}}{\left({\int }_{R^N}{|\mathrm{\nabla }u|}^q{|x|}^{2q}dx\right)}^{\frac{1}{q}}.$$

  (2.15)

• (iii) When $a=0$, $b=-1$, we get

  $$\left(\frac{N-p}{p}\right){\int }_{R^N}{|\mathrm{\nabla }u|}^{p}dx\le {\left({\int }_{R^N}{|{\mathrm{△}}_{p}u|}^{p}dx\right)}^{\frac{1}{p}}{\left({\int }_{R^N}{|\mathrm{\nabla }u|}^q{|x|}^{q}dx\right)}^{\frac{1}{q}}.$$

  (2.16)