  Given a closed manifold of positive Yamabe invariant and for instance
positive Morse functions upon it, the conformally prescribed scalar curvature
problem raises the question, whether or not such functions can by conformally
changing the metric be realised as the scalar curvature of this manifold. As we
shall quantify depending on the shape and structure of such functions, every
lack of a solution for some candidate function leads to existence of
energetically uniformly bounded solutions for entire classes of related
candidate functions.
