Asymptotic expansions for conformal scalar curvature equations near isolated singularities

Abstract

In this paper, we study asymptotic expansions of positive solutions of the conformal scalar curvature equation $$ - \Delta u = K(x) u^\frac{n + 2}{n - 2} ~~~~~~ \textmd{in} ~ B_1 \setminus \{ 0 \} $$ 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 $n \geq 6$. Moreover, we also obtain an arbitrary-order expansion of singular positive solutions of the anisotropic elliptic equation $$ - \,{\rm div} (|x|^{- 2 a} \nabla u) = |x|^{- b p} u^{p - 1} ~~~~~~ \textmd{in} ~ B_1 \setminus \{ 0 \}, $$ where $0 \leq a < \frac{n - 2}{2}$, $a \leq b < a + 1$ and $p = \frac{2 n}{n - 2 + 2 (b - a)}$. This equation is arising from the celebrated Caffarelli-Kohn-Nirenberg inequality.