An improved Hardy-Sobolev inequality and its application

For 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 (1 (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 u: = - (div (| u|^p-2 u) + |x|^{p} |u|^p-2u) as increases to (n-pp) ^p for 1< p < n.