Finitely many solutions for the prescribed scalar curvature problem at the critical flatness order

We study the following prescribed scalar curvature problem $$ -Δu=K(y)u^{2^*-1},\ \ u>0 \,\ \ \text{in}\ \ \mathbb{R}^N, \tag*{$(*)$} $$ where $2^*=\frac{2N}{N-2}$. For $N\geq4$, we construct a positive $K(y)$ with maximum set $\{|y|=1\}$ and critical order of flatness $m=N-2$ at $|y|=1$, such t