Asymptotic expansions for conformal scalar curvature equations near isolated singularities

2024/02/26 by Du, Xusheng, Yang, Hui
#35C20 #35J61 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.16597

Abstract

In this paper, we study asymptotic expansions of positive solutions of the conformal scalar curvature equation - Δu = K(x) u^(n + 2)/(n - 2) ~~~~~~ \textmdin ~ 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 - \rm div (|x|^{- 2 a} ∇ u) = |x|^{- b p} u^{p - 1} ~~~~~~ \textmdin ~ 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.

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