An improved Hardy-Sobolev inequality and its application

For $\Omega \subset \mathbb{R}^{n} , n \geq 2$, 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 $(\frac{1}{\log (1/\vert x\vert)})^{2}$. 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 $L_{\mu }u:= - (\text{div}(\vert\nabla u\vert^{p-2}\nabla u) + \frac{\mu }{\vert x\vert^{p}} \vert u\vert^{p-2}u )$ as $\mu$increases to $\left (\frac{n-p}{p}\right )^{p}$ for $1< p < n$.