  Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\geq 3$
and we let $\Sigma$ to be a closed submanifold of dimension $1 \leq k \leq N-2.
$ In this paper we study existence and non-existence of minimizers of Hardy
inequality with weight function singular on $\Sigma$ within the framework of
Brezis-Marcus-Shafrir. In particular we provide necessary and sufficient
conditions for existence of minimizers.

Analysis of PDEs