Prescribing scalar curvatures: loss of minimizability

Prescribing conformally the scalar curvature on a closed manifold with negative Yamabe invariant as a given function $K$ is possible under smallness assumptions on $K_{+}=\max\{K,0\}$ and in particular, when $K<0$. In addition, while solutions are unique in case $K\leq 0$, non uniqueness generally holds, when $K$ is sign changing and $K_{+}$ sufficiently small and flat around its critical points. These solutions are found variationally as minimizers. Here we study, what happens, when the relevant arguments fail to apply, describing on one hand the loss of minimizability generally, while on the other we construct a function $K$, for which saddle point solutions to the conformally prescribed scalar curvature problem still exist.