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We consider the following prescribed scalar curvature equations in${\mathbb{R}}^N$ $$  - \Delta u =K(|y|)u^{2^*-1},\quad u>0 \quad \mbox{in} \quad {\mathbb{R}}^N,\quad u \in D^{1, 2}({\mathbb{R}}^N), $$ where $K(r)$ is a positive function,$2^*=\frac{2N}{N-2}$. We modify the methods and improve the results in [15].

Non-degeneracy of multi-bump solutions for the prescribed scalar curvature equations and applications