  In this paper we extend Hardy-Littlewood-Sobolev inequalities on compact
Riemannian manifolds for dimension $n\ne 2$. As one application, we solve a
generalized Yamabe problem on locally conforamlly flat manifolds via a new
designed energy functional and a new variational approach. Even for the classic
Yamabe problem on locally conformally flat manifolds, our approach provides a
new and relatively simpler solution.
