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