Asymptotic classification conjecture for positive singular solutions of poly-harmonic equations

Let $N=2m$, let $n\geqslant 2m$, and let $B_R^*=B_R\setminus\\{0\\}\subset\mathbb{R}^n$. Consider a positive singular solution $u\in C^{2m}(\mathbb{R}^n\setminus\\{0\\})$ of

$$(-\Delta)^m u=|u|^{p-1}u\quad\text{in }B_R^*,$$

where $p\in[2_{m,*},2_m^*)$, $2_{m,*}=\frac{n}{n-2m}$, and $2_m^*=\frac{n+2m}{n-2m}$. Write $\overline{u}$ for the radial asymptotic profile and let $K_0(n,p)$ and $\widehat{K}_{N,0}(n)$ be the constants appearing in the asserted asymptotics. Assume that $(-\Delta u)^j\geqslant0$ for every $j=1,\dots,m-1$.

Asymptotic classification conjecture. If $p\in(1,2_m^*-1)$, then

$$u(x)=(1+\mathcal{O}(|x|))\overline{u}(x)\quad\text{as }x\to0.$$

Moreover, (a) if $p\in(1,2_m^*)$, then $u(x)\simeq |x|^{N-n}$ as $x\to0$; (b) if $p=2_{m,*}$, then

$$u(x)=(1+\mathrm{o}(1))\widehat{K}_{0}(n)^{\frac{n-N}{N}}|x|^{N-n}(\ln|x|)^{\frac{N-N}{N}}\quad\text{as }x\to0,$$

where

$$\widehat{K}_{N,0}(n)=\frac{2^{m-2}(m-1)!}{m}\prod_{j=0}^{m-1}(n-2j)(n-N)^2;$$

and (c) if $p\in(2_{m,*},2_m^*-1)$, then

$$u(x)=(1+\mathrm{o}(1))K_0(n,p)^{\frac{1}{p-1}}|x|^{-\frac{N}{p-1}}\quad\text{as }x\to0.$$

This conjecture seeks a complete local asymptotic classification near an isolated singularity for positive singular solutions of subcritical even-order poly-harmonic equations. The parser reports that the super poly-harmonic condition can be removed by an independent proof of X. Huang, Y. Li, and H. Yang, so the stated result is solved.