Existence of extremal functions for a Hardy-Sobolev inequality
type: Item
instance of: scholarly article
title: Existence of extremal functions for a Hardy-Sobolev inequality
zbMATH Open document ID: 1200.26034
published in: Journal of Functional Analysis
publication date: 2010-07-07T00:00:00Z
review text: The best constant and the extremal functions for the Improved Hardy-Sobolev inequality are obtained. Also, the author proved that under the transformation \(v(r)=u(t)\), \(t=(-\log (\frac rR))^{-\frac{1}{N-2}}\) the following inequalities are equivalent: (i) \(\int_0^Rr|v'|^2dr \geq c(\int_0^Rr^{-1}(-\log (\frac rR))^{-\frac{2(N-1)}{N-2}} |v|^{\frac{2N}{N-2}}dr)^{\frac{N-2}{N}}\), \(v\in C_0^{\infty}(0,R)\), (ii) \(\int_{\mathbb{R}^N}|\nabla u|^2dx \geq S(\int_{\mathbb{R}^N}|u|^{\frac{2N}{N-2}}dx)^{\frac{N-2}{N}}\) (Sobolev inequality).
reviewed by: Sanja Varošanec
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 5731611
zbMATH Keywords: best constants
zbMATH Keywords: extremal functions
zbMATH Keywords: Hardy-Sobolev inequality
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: Bliss' inequality
author: Nikolaos B. Zographopoulos
MaRDI profile type: Publication
full work available at URL: j.jfa.2010.03.020
OpenAlex ID: W2094742490
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DOI: 10.1016/J.JFA.2010.03.020
description: scientific article; zbMATH DE number 5731611