  The problem of prescribing conformally the scalar curvature on a closed
Riemannian manifold of negative Yamabe invariant is always solvable, when the
function $K$ to be prescribed is strictly negative, while sufficient and
necessary conditions are known for $K\leq 0$. For sign changing $K$ Rauzy
showed solvability, if $K$ is not too positive. We revisit this problem in a
different variational context, thereby recovering and quantifying the principle
existence result of Rauzy and show under additional assumptions, that for a
sign changing $K$ solutions to the conformally prescribed scalar curvature
problem, while existing, are not unique.
