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 \(K\leq 0\). 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.