On a class of singular critical elliptic systems on compact riemannian manifolds

We study a strongly coupled system of elliptic equations on compact Riemannian manifolds, involving critical Hardy-Sobolev exponents and Hardy-type potentials. Through minimization methods, we study the existence of nontrivial solutions to the system in dimension $ n\ge4 $, when the potentials are compared with the scalar curvature at the singular point. In dimension $ n = 3 $, we employ a global argument to establish existence when the mass of the operator $ -\Delta_{g} + h $ is positive at this singular point, where $ h $ is a combination of the involved potentials.