Infinitely many solutions for the prescribed scalar curvature problem with volcano-like curvature

In this paper, we consider the following prescribed scalar curvature problem: \begin{equation*} -Δu = K(x) u^{\frac{n+2}{n-2}}, \quad u>0\quad\hbox{in}\quad $\mathbb{R}$^n, \quad u \in D^{1,2}($\mathbb{R}$^n), \end{equation*} where $K(x)$ is a volcano-like positive function such that $$K(x)=K(r_0)-c_0\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }-r_0{\mathrm{\mid }}^m+O(\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }-r_0{\mathrm{\mid }}^{m+\theta }),r_0-\delta <\mathrm{\mid }x\mathrm{\mid }<r_0+\delta$$ with $K(r_0),c_0,\delta >0,\theta >2,\min \{\frac{n-2}{2},2\}<m<n-2$. We first prove the existence of infinitely many positive solutions. A consequence of our proof yields that the infinitely many solutions constructed in \cite{WY} are non-degenerate in the whole $D^{1,2}({\mathbb{R}}^n)$ space. To our knowledge, it seems to be the first result of infinitely many solutions of prescribed scalar curvature problem when the potential function $K(x)$ is not radial. Our non-degeneracy results are also more complete and improve the result in \cite{GMPS}.