An improved Hardy-Sobolev inequality and its application

Abstract

For Ω⊂Rⁿ, 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 (1/log(1/|x|))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 Brezis-Vazquez. Finally, we use this result to analyze the behaviour of the first eigenvalue of the operator L_μu:=-(div(|∇u|^p-2∇u)+μ/|x|^p|u|^p-2u) as λ increases to (n-p/p)^p for 1<p<n.