Best constant in weighted Sobolev inequality with weights being powers of distance from the origin
type: Item
instance of: scholarly article
title: Best constant in weighted Sobolev inequality with weights being powers of distance from the origin
zbMATH Open document ID: 0899.35034
DOI: 10.1155/S1025583497000180
author: Toshio Horiuchi
published in: Journal of Inequalities and Applications
publication date: 1998-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/119604
review text: The paper is devoted to the study of the best constant in the imbedding theorems for weighted Sobolev spaces with weight functions being powers of the distance from the origin in \(\mathbb{R}^n\). For instance, the author considers the variational problem: \[ S(p,q,\alpha,\beta,n)=\inf \Biggl[\int_{\mathbb{R}^n}|\nabla u|^p| x|^{p\alpha} dx : u \in W_{\alpha,\beta}^{1,p}(\mathbb{R}^n),\| u\|_{L_\beta^q}= 1\Biggr]. \] The invariance of \(\mathbb{R}^n\) by the group of dilatations may cause the loss of compactness, so the existence of extremals essentially depends on parameters. The author proves that under suitable relations between the parameters the extremals exist and this permits to get the best constant in the Sobolev inequality: \[ \int_{\mathbb{R}^n}|\nabla u|^p| x|^{p\alpha} dx\geq S(p,q,\alpha,\beta,n) \Biggl(\int_{\mathbb{R}^n}| u|^q| x|^{q\beta} dx\Biggr)^{p/q}, \] where \(u\) is any function in \(C_0^1(\mathbb{R}^n)\). The obtained theorems generalize some earlier results of Talenti, Maz'ya and Egnell.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 1139490
zbMATH Keywords: weighted Sobolev spaces
zbMATH Keywords: imbedding theorems
zbMATH Keywords: loss of compactness
reviewed by: Zdzisław Denkowski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1139490