A note on a class of Hardy-Rellich type inequalities
© Di et al.; licensee Springer 2013
Received: 31 May 2012
Accepted: 18 February 2013
Published: 4 March 2013
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.
KeywordsHardy inequality Hardy-Rellich inequality Caffarelli-Kohn-Nirenberg inequality
where , the constant is optimal and not attained.
where the constant is sharp.
where the constant is sharp.
The goal of this paper is to extend the above (1.3) and (1.6) to the general case for by a different and direct approach.
2 Main results
In this section, we will give the proof of the main theorems.
where and the constant is sharp.
where is a positive number sequence converging to as . By direct computation and the limit process, we know the constant is sharp. □
Remark 1 When , the inequality (2.1) covers the inequality (2.4) in .
When we take special values for a, b, the following corollary holds.
- (ii)When , according to the inequality (2.1), we have(2.5)
- (iii)When and , we obtain the inequality(2.6)
By a similar method, we can prove the following case Hardy-Rellich type inequality.
where , and is the p-Laplacian operator.
We mention that we do not know whether the constant in (2.7) is optimal or not. □
- (i)when , , the inequality (2.7) is equivalent to the inequality(2.14)
- (ii)When , , we obtain the inequality(2.15)
- (iii)When , , we get(2.16)
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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