Hénon–Lane–Emden conjecture for the subcritical weighted system

Let $N\geq 2$, $p,q>0$, and $a,b\in\mathbb{R}$. Consider the Hénon–Lane–Emden system

$$\left\{ \begin{aligned} -\Delta u&=|x|^a|v|^{p-1}v \\ -\Delta v&=|x|^b|u|^{q-1}u \end{aligned} \quad\text{in }\mathbb{R}^N\setminus\{0\}. \right.$$

The quadruple $(p,q,a,b)$ is subcritical when

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

Hénon–Lane–Emden conjecture. If $(p,q,a,b)$ is subcritical, then the system admits no positive solutions in $\left(C^2(\mathbb{R}^N\setminus\{0\})\cap C(\mathbb{R}^N)\right)^2$.

The conjecture concerns a weighted generalization of the Lane–Emden nonexistence problem. The source provides no resolution status for this weighted conjecture; the paper establishes it only under additional hypotheses for stable solutions outside a compact set.