Sharp remainder terms of Hardy-Sobolev inequalities

Abstract

In this paper we shall prove the existence of sharp remainder terms involving singular weight (logR/|x|)^-2 for Hardy-Sobolev inequalities of the following type:
∫_Ω|∇u(x)|²dx≥(n-2/2)²∫_Ω|u(x)|²/|(x)|²dx for any u∈W^{1, 2}₀(Ω), Ω is a bounded domain in Rⁿ, n>2, with 0∈Ω. Here the number of remainder terms depends on the choice of R.