Asymptotic expansions for conformal scalar curvature equations near isolated singularities

In this paper, we study asymptotic expansions of positive solutions of the conformal scalar curvature equation $$-\mathrm{\Delta }u=K(x)u^{\frac{n+2}{n-2}}\text{ in }{B}_1\setminus \{0\}$$ with an isolated singularity at the origin. Under certain flatness conditions on $K$, we establish a higher-order expansion of solutions near the origin. In particular, we give the refined second-order asymptotic expansion of solutions when $n\ge 6$. Moreover, we also obtain an arbitrary-order expansion of singular positive solutions of the anisotropic elliptic equation $$-\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }x{\mathrm{\mid }}^{-2a}\mathrm{\nabla }u)=\mathrm{\mid }x{\mathrm{\mid }}^{-bp}{u}^{p-1}\text{ in }{B}_1\setminus \{0\},$$ where $0\le a<\frac{n-2}{2}$, $a\le b<a+1$ and $p=\frac{2n}{n-2+2(b-a)}$. This equation is arising from the celebrated Caffarelli-Kohn-Nirenberg inequality.