关键词： 椭圆方程, Sobolev-Hardy不等式, 山路引理

Abstract: The authors we prove an  inequality of Sobole-Hardy type and prove that the best constant is attained in the inequality.Furthermore,using the Sobole-Hardy inequality and mountain pass theorem,the existence of a nontrivial solution for a nonlinear elliptic equation involving critical singularity is proved.

Key words: elliptic equation, Sobolev-Hardy inequality, mountain pass theorem