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

Abstract

In this paper, we consider the following prescribed scalar curvature problem: \begin{equation*} -\Delta u = 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 $$ K(x)= K(r_0)- c_0 | |x|- r_0|^m + O( | |x|- r_0|^{m+\theta}),\quad r_0- \delta <|x| <r_0+\delta$$ 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}.