  Given a smooth positive function $K$ on the standard sphere
$(\mathbb{S}^n,g_0)$, 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 $K$.
