Prescribing scalar curvatures: on the negative Yamabe case

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 $K\le 0$. 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.