含权的Sobolev-Hardy不等式的最佳常数

在讨论含正常数C的Sobolev-Hardy不等式时，主要困难是处理β=0的情况的方法不适用于β≠0的情况．当β=0时，可利用Schwarz对称化的方法；然而，当β≠0时，无法断言在Schwarz对称化的情况下，含权的L^p模是递减的，含权的L^p的模是递增的．因此，必须寻求另外的方法．文中采用Bliss引理，证明存在一个最佳常数C使Sobolev-Hardy不等式成立．

Best Constant in Weighed Sobolev-Hardy Inequality

To the weighed Sobolev-Hardy inequality containing a positive constant C,the main obstacle is that the method used to the case β =0 does not works anywhere when β≠0.That is to say,when β =0,the method of Schwarz symmetrization can be employed,but there is no reason to believe that the Schwarz symmetrization still diminishes the weighed Lp-gradient or increases the weighed Lp -norm of a function.So another approach must be found to the Sobolev-Hardy inequality.In this paper,by Bliss lemma,it is proved that there exists a best constant C such that the weighed Sobolev-Hardy inequality is right.