Classification conjecture for positive singular solutions of critical poly-harmonic equations

Let $m\in\mathbb N^+$, set $N=2m$, and let $n>N$. Consider positive singular solutions $u\in C^N(\mathbb R^n\setminus\{0\})$ of

$$(-\Delta)^m u=c_{n,N}u^{\frac{n+N}{n-N}} \quad \text{in} \quad \mathbb R^n\setminus\{0\},$$

where $c_{n,N}=2^N\Gamma\left(\frac{n+N}{4}\right)^2\Gamma\left(\frac{n-N}{4}\right)^{-2}$. Let $v_a$ denote the unique periodic bounded positive solution of the associated higher-order initial-value problem with coefficients $K_{2j,N}(n)$ and initial data $a_j$, and let $a^*_{n,N}=K_0^{\frac{n-N}{2N}}$. Classification conjecture. Every such $u$ is radially symmetric about the origin and monotonically decreasing. Moreover, there exist $a_0\in(0,a^*_{n,N})$ and $T\in[0,T_{a_0}]$ such that

$$u(x)=|x|^{\frac{N-n}{2}}v_a(\ln|x|+T).$$

This conjecture extends the paper’s sixth-order classification theorem to the critical even-order poly-harmonic family; the full classification is believed to hold, while the necessary higher-order ODE analysis remains open.