In this paper, we consider the existence of minimizers of the Hardy-Sobolev type variational problem. Recently, Ghoussoub and Robert proved that the Hardy-Sobolev best constant admits its minimizers provided the bounded smooth domain has the negative mean curvature at the origin on the boundary. We generalize their results by using the idea of Brézis and Nirenberg, and as a consequence, we shall prove the existence of positive solutions to the elliptic equation involving two different kinds of Hardy-Sobolev critical exponents.
"Minimizing problems for the Hardy-Sobolev type inequality with the singularity on the boundary." Tohoku Math. J. (2) 64 (1) 79 - 103, 2012. https://doi.org/10.2748/tmj/1332767341