Prescribing Morse scalar curvatures: subcritical blowing-up solutions

Abstract

Prescribing conformally the scalar curvature of a Riemannian manifold as a given function consists in solving an elliptic PDE involving the critical Sobolev exponent. One way of attacking this problem consist in using subcritical approximations for the equation, gaining compactness properties. Together with the results in \cite{MM1}, we completely describe the blow-up phenomenon in case of uniformly bounded energy and zero weak limit in positive Yamabe class. In particular, for dimension greater or equal to five, Morse functions and with non-zero Laplacian at each critical point, we show that subsets of critical points with negative Laplacian are in one-to-one correspondence with such subcritical blowing-up solutions.