Liouville-type Theorems for Stable Solutions of the H\'enon-Lane-Emden System

We investigate the H\'enon-Lane-Emden system defined by $- \Delta u=|x|^a |v|^{p-1}v$ and $- \Delta v=|x|^b |u|^{q-1}u$ in $\mathbb{R}^N \!\setminus\! \{0\}$. We begin by establishing a general Liouville-type theorem for the subcritical case. Then we prove that the H\'{e}non-Lane-Emden conjecture is valid for solutions stable outside a compact set, provided that $0 < \min\,\{p, q\} < 1$, or $0 \leq a - b \leq (N-2)(p - q)$, or $N \leq \frac{2(p+q+2)}{pq-1} + 10$. Additional Liouville-type theorems for the subcritical case are also obtained. Furthermore, we address the supercritical case. To our knowledge, these results constitute the first Liouville-type theorems for this class of solutions in the H\'{e}non-Lane-Emden system. As a by-product, several existing results in the literature are refined.