Hé non-Lane-Emden conjecture for the polyharmonic system

Let $m,p,q\geq 1$, $pq\neq 1$, and $a,b\geq 0$. For nonnegative functions $u,v$ on $\mathbb{R}^n$, consider

$$\left\{\begin{array}{lcl} (-\Delta)^m u=|x|^a v^p &\text{in}& \mathbb{R}^n,\\ (-\Delta)^m v=|x|^b u^q &\text{in}& \mathbb{R}^n. \end{array}\right.$$

The pair $(p,q)$ is under the critical hyperbola when

$$\frac{n+a}{p+1}+\frac{n+b}{q+1}>n-2m.$$

Hé non-Lane-Emden conjecture. If $(u,v)$ is a nonnegative solution of the system and $(p,q)$ is under the critical hyperbola, then $u=v=0$. This is a Liouville-type conjecture for the weighted polyharmonic system. The supplied text states that it holds for bounded solutions in dimension $n=2m+1$ and for radial solutions in arbitrary dimensions, while the general nonradial case remains open.