On the Nirenberg problem on spheres: Arbitrarily many solutions in a perturbative setting

Given a smooth positive function K on the standard sphere ( S n , g 0 ) , we use Morse theoretical methods and counting index formulae to prove that, under generic conditions on the function K , there are arbitrarily many metrics g conformally equivalent to g 0 and whose scalar curvature is given by the function K provided that the function is sufficiently close to the scalar curvature of g 0 . Our approach leverages a comprehensive characterization of blowing-up solutions of a subcritical approximation, along with various Morse relations involving their indices. Notably, this multiplicity result is achieved without relying on any symmetry or periodicity assumptions about the function K .