In this work we investigate existence as well as multiplicity of conformal metrics with prescribed scalar curvature on the standard n-dimensional Sphere. Due to the existence of critical point at infinity, the standard variational methods cannot be applied. To overcome this difficulty, we prove that in a neighborhood of critical points at infinity, a Morse lemmas at infinity reduction holds, then develop a whole Morse theory of this non-compact variational problem and provide compactness and existence results. In particular we establish, under generic condition Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem in terms of the total contribution of the critical point at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai