The second best constant for the Hardy–Sobolev inequality on manifolds

We consider the second best constant in Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] Sobolev inequalities. Here, establish corresponding result singular case. In addition, perform blow-up analysis solutions equations minimizing type. yields informations value related functional