In this paper, we consider the following prescribed scalar curvature problem (the Nirenberg problem)
\(\matrix{{ - \Delta u = K(x){u^{{{n + 2} \over {n - 2}}}},} & {u > 0} & {\text{in}\,{\mathbb{R}^n},} & {u \in {D^{1,2}}} \cr} ({{\mathbb{R}^n}}),\)
where K(x) is a volcano-like positive function such that
\(K(x)=K(r_{0})-c_{0}\Vert x\vert - r_{0}\vert^{m}+O(\Vert x\vert-r_{0}\vert^{m+\theta}),\quad r_{0}-\delta<\vert x\vert<r_{0}+\delta\)
with K(r0), c0, δ > 0, θ > 2 and \(\min \{{{{n - 2} \over 2},2}\} < m < n - 2\) . We first prove the existence of infinitely many positive solutions. A consequence of our proof yields that the infinitely many solutions constructed by Wei and Yan (2010) are non-degenerate in the whole D1,2(ℝn) space. To our knowledge, this is the first existence result of infinitely many solutions of the prescribed scalar curvature problem when the potential function K(x) is not radial. Our non-degeneracy results improve the results by Guo et al. (2020) and Musso and Wei (2015).
