An improved Hardy-Sobolev inequality and its application

Abstract

Abstract. For Ω ⊂ R n , n ≥ 2, a bounded domain, and for 1 < p < n, we improve the Hardy-Sobolev inequality by adding a term with a singular weight of the type () 2 . We show that this weight function is optimal in the sense that the inequality fails for any other weight function more singular than this one. Moreover, we show that a series of finite terms can be added to improve the Hardy-Sobolev inequality, which answers a question of BrezisVazquez. Finally, we use this result to analyze the behaviour of the first eigenvalue of the operator Lµu := −(div(|∇u| p−2 ∇u) +

A unified approach to weighted Hardy type inequalities on Carnot groups

“…Remark 3. In the Abelian case, when G = R n , with the ordinary dilations, one has G = V 1 = R n so that Q = n. Now it is clear that the above inequality with the homogeneous norm N (x) = |x| and α = 0 recovers the inequality (1.4) proved by Adimurthi et al in [1].…”

“…Introduction. Hardy inequalities are of fundamental importance for studying a wide range of problems in various branches of mathematics as well as in other areas of science, and have been comprehensively studied since their discovery, see for example [5], [8], [18], [1], [13], [6], [9], [19], [20], [23], [21] and the references therein.…”

Untitled

“…• The condition λ ≤ (N − 2) 2 /4 can be seen as related to the Hardy inequality (see, for example, [8] and [1]). More precisely, if Ω 1 is a bounded open subset of R N (N ≥ 3) and 0 ∈ Ω 1 , then for R > 0 sufficiently large, there exists a positive constant C, depending on N and R, such that…”

An improved Hardy type inequality on Heisenberg group

“…Hardy inequality in ℝ N reads, for all u ∈ C ∞ 0 (R N ) and N ≥ 3, Vázquez [1] have improved it by establishing that for u ∈ C ∞ 0 ( ), 2) where ω N and |Ω| denote the volume of the unit ball and Ω, respectively, and z 0 = 2.4048... denotes the first zero of the Bessel function J 0 (z). Inequality (1.2) is optimal in case Ω is a ball centered at zero.…”

“…Triggered by the work of Brezis and Vázquez (1.2), several Hardy inequalities have been established in recent years. In particular, Adimurthi et al( [2]) proved that, for u ∈ C ∞ 0 ( ), there exists a constant C n,k such that proved that, for u ∈ C ∞ 0 ( ), there holds 4) where D ≥ sup x Ω |x|,…”