Abstract
We consider a class of sharp Hardy-Sobolev inequality, where the weights are functions of the distance from a surface. It is proved that the Hardy-Sobolev inequality can be successively improved by adding to the right-hand side a lower-order term with optimal weight and constant.
Highlights
The classical Hardy inequality says RN |u|p |x|p dx ≤ pp N−p |∇u|pdx,u ∈ C0∞ RN \ {0}, 1.1 where the constant |p/ N − p |p is optimal but never attained; see, for example, 1–4
U ∈ C0∞ RN \ {0}, 1.1 where the constant |p/ N − p |p is optimal but never attained; see, for example, 1–4. This suggests that one might look for an error term
Brezis and Vazques 5 showed that if Ω is a bounded domain in RN, N ≥ 3, with 0 ∈ Ω, there exists a positive constant λΩ such that
Summary
U ∈ C0∞ RN \ {0} , 1.1 where the constant |p/ N − p |p is optimal but never attained; see, for example, 1–4. Brezis and Vazques 5 showed that if Ω is a bounded domain in RN, N ≥ 3, with 0 ∈ Ω, there exists a positive constant λΩ such that. Assume that C∗ holds, 1 there exists a positive constant D0 D0 k, p > supx∈Ωd x such that for all D ≥ D0 and u ∈ W, there holds φ d |∇u|pdx −. Theorem 1.4 extends the inequality 1.5 to Sobolev space with general weight φ d. It includes the results of 18, 19. This is the result of Theorem B in 15 if α 0
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