Translator Disclaimer
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


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.


Download Citation

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


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

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

Primary: 35J70
Secondary: 35J60

Rights: Copyright © 2012 Kyoto University


Vol.52 • No. 4 • Winter 2012
Back to Top