The fractional Hénon-Lane-Emden conjecture and some critical nonexistence phenomena

We refine the proof method for the Hénon-Lane-Emden system and extend it to fractional orders $0<s<1$, proving subcritical nonexistence of positive classical solutions. And we find examples with $0<s<1\mathrm{/}2$ and fixed nonzero weights for which existence and nonexistence occur at different points of the same critical hyperbola.