An improved Hardy-Sobolev inequality and its application

For Omega subset of R-n, n greater than or equal to 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-mu(u) := (div(\ delu \ (p-2)delu) + mu/\x \ (p)\u \ (p-2)u) as mu increases to (n-p/p)(p) for 1 < p < n.