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 - u = K (x) uⁿ + 2n - 2 ~~~~~~ in ~ B₁ \ 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 6. Moreover, we also obtain an arbitrary-order expansion of singular positive solutions of the anisotropic elliptic equation - \, div (|x|^- 2 a u) = |x|^- b p u^p - 1 ~~~~~~ in ~ B₁ \ 0 \, where 0 a < n - 22, a b < a + 1 and p = 2 nn - 2 + 2 (b - a). This equation is arising from the celebrated Caffarelli-Kohn-Nirenberg inequality.