Liouville type theorems for fractional and higher-order fractional H\'enon-Lane-Emden systems

In this paper, we first establish decay estimates for the fractional and higher-order fractional H'enon-Lane-Emden systems by using a nonlocal average and integral estimates, which deduce a result of non-existence. Next, we apply the method of scaling spheres introduced in \cite{DQ2} to derive a Liouville type theorem. We also construct an interesting example on super $\frac{\alpha }{2}$-harmonic functions (Proposition 1.2).