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

Given a smooth positive function K on the standard sphere (Sⁿ,g₀), 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 g₀ and whose scalar curvature is given by the function K provided that the function is sufficiently close to the scalar curvature of g₀. 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.