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 $$-\Delta u=K(x)u^{\frac{n+2}{n-2}}\text{in}{B}_1\setminus \{0\}$$ 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 $$-\mathrm{div}(|x{|}^{-2a}\nabla u)=|x{|}^{-bp}{u}^{p-1}\text{in}{B}_1\setminus \{0\},$$ 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.

40 pages