  We consider the problem of prescribing conformally the scalar curvature on
compact manifolds of positive Yamabe class in dimension $n \geq 5$. We prove
new existence results using Morse theory and some analysis on blowing-up
solutions, under suitable pinching conditions on the curvature function. We
also provide new non-existence results showing the sharpness of some of our
assumptions, both in terms of the dimension and of the Morse structure of the
prescribed function.
