A note on positive supersolutions of the fractional Lane–Emden system

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In this paper, we are concerned with the classification of positive supersolutions of the fractional Lane–Emden system $$\begin{aligned} {\left\{ \begin{array}{ll}(- \Delta )^s u= v^p \text{ in } {\mathbb {R}}^N\\ (- \Delta )^s v= u^q \text{ in } {\mathbb {R}}^N\end{array}\right. }, \end{aligned}$$ where $$p,q\in {\mathbb {R}}$$ and $$0<s<1$$ . We prove that this system has no positive supersolution provided that $$p\le 0$$ or $$q\le 0$$ . Consequently, this together with the results in Leite and Montenegro (Differ Integr Equ 30(11–12):947–974, 2017), Biswas (Nonlinearity 32(6):2246–2268, 2019) completes the classification of positive supersolutions of the system in the full range of p, q. On the other hand, we also provide a simple proof of the nonexistence of positive supersolutions of the system in the case $$p>0,q>0$$ and $$pq\le 1$$ or $$p>0,q>0,pq>1$$ and $$\max \left\{ \frac{2s(p+1)}{pq-1},\frac{2s(q+1)}{pq-1}\right\} > N-2s$$ .