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

摘要

In this paper, we are concerned with the classification of positive supersolutions of the fractional Lane-Emden system{(-Delta)(s)u = v(p) in R-N (-Delta)(s)v = u(q) in R-N,where p, q is an element of Rand 0 s 1. We prove that this system has no positive supersolution provided that p = 0 or q = 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 = 1 or p 0, q 0, pq 1 and max {2s(p+1)/pq-1, 2s(q+1)/pq-1} N - 2S.