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 - Δ u = K(x) uⁿ + 2/n - 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 up - 1 ~~~~~~ in ~ B₁ \ 0 \, where 0 ≤ a < n - 2/2, a ≤ b < a + 1 and p = 2 n/n - 2 + 2 (b - a). This equation is arising from the celebrated Caffarelli-Kohn-Nirenberg inequality.