  We give sufficient and "almost" necessary conditions for the prescribed
scalar curvature problems within the conformal class of a Riemannian metric $ g
$ for both closed manifolds and compact manifolds with boundary, including the
interesting cases $ \mathbb{S}^{n} $ or some quotient of $ \mathbb{S}^{n} $, in
dimensions $ n \geqslant 3 $, provided that the first eigenvalues of conformal
Laplacian (with appropriate boundary conditions if necessary) are positive.
When the manifold is not some quotient of $ \mathbb{S}^{n} $, we show that, on
one hand, any smooth function that is a positive constant within some open
subset of the manifold with arbitrary positive measure, and has no restriction
on the rest of the manifold, is a prescribed scalar curvature function of some
metric under conformal change; on the other hand, any smooth function $ S $ is
almost a prescribed scalar curvature function of Yamabe metric within the
conformal class $ [g] $ in the sense that an appropriate perturbation of $ S $
that defers with $ S $ within an arbitrarily small open subset is a prescribed
scalar curvature function of Yamabe metric. When the manifold is either $
\mathbb{S}^{n} $ or $ \mathbb{S}^n / \Gamma $ with Kleinian group $ \Gamma $ we
show that any positive function that satisfies a technical analytical
condition, called CONDITION B, can be realized as a prescribed scalar curvature
functions on these manifolds.
