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Winter 2012 Missing terms in the weighted Hardy–Sobolev inequalities and its application
Hiroshi Ando, Toshio Horiuchi
Kyoto J. Math. 52(4): 759-796 (Winter 2012). DOI: 10.1215/21562261-1728857

Abstract

Let Ω be a bounded domain of Rn (n1) containing the origin. In the present paper we establish the weighted Hardy–Sobolev inequalities with sharp remainders. For example, when α=1n/p and 1<p<+ hold, we establish the following inequality.

There exist positive numbers Λn,p,α,C, and R such that we have

(0.1) Ω|u|p|x|αpdxΛn,p,αΩ|u(x)|p|x|nA1(|x|)pdx+CΩ|u(x)|p|x|nA1(|x|)pA2(|x|)2dx

for any uWα,01,p(Ω). Here A1(|x|)=logR|x|, and A2(|x|)=logA1(|x|). This is called the critical Hardy–Sobolev inequality with a sharp remainder involving a singular weight A1(|x|)pA2(|x|)2, in the sense that the improved inequality holds for this weight but fails for any weight more singular than this one. Here Λn,p,α is a sharp constant independent of each function u. Further we establish the Hardy–Sobolev inequalities in the subcritical case (α>1n/p) and the supercritical case (α<1n/p).

As an application, we use our improved inequality to determine exactly when the first eigenvalues of the weighted eigenvalue problems for the operators represented by div(|x|αp|u|p2u)μ/|x|nA1(|x|)p|u|p2u (the critical case) will tend to zero as μ increases to Λn,p,α. This also gives us sufficient conditions for the operators to have the positive first eigenvalue in a certain nontrivial functional framework, and we study the eigenvalue problem in the borderline case.

Citation

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Hiroshi Ando. Toshio Horiuchi. "Missing terms in the weighted Hardy–Sobolev inequalities and its application." Kyoto J. Math. 52 (4) 759 - 796, Winter 2012. https://doi.org/10.1215/21562261-1728857

Information

Published: Winter 2012
First available in Project Euclid: 15 November 2012

zbMATH: 1276.35008
MathSciNet: MR2998910
Digital Object Identifier: 10.1215/21562261-1728857

Subjects:
Primary: 35J70
Secondary: 35J60

Rights: Copyright © 2012 Kyoto University

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Vol.52 • No. 4 • Winter 2012
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