An improved Hardy-Sobolev inequality and its application

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 Lμu:= - (div(|∇ u|ᵖ⁻²∇ u) + μ |x|ᵖ |u|ᵖ⁻²u ) as μ increases to (n-p/p )ᵖ for $1< p < n$.