Wolff potentials and nonlocal equations of Lane-Emden type

We consider nonlocal equations of the type \[ (-Δₚ)ˢu = μ in Ω, \] where Ω ⊂ Rⁿ is either a bounded domain or the whole Rⁿ, μ is a Radon measure on Ω, $0<s<1$ and $1<p<n/s$. Especially, we extend the existence, regularity and Wolff potential estimates for SOLA (Solutions Obtained as Limits of Approximations), established by Kuusi, Mingione, and Sire (Comm. Math. Phys. 337:1317--1368, 2015), to the strongly singular case 1<p≤2-s/n. Moreover, using Wolff potentials and Orlicz capacities, we present both a sufficient and a necessary conditions for the existence of SOLA to nonlocal equations of the type \[ (-Δₚ)ˢu = P(u) + μ in Ω, \] where P(·) is either a power function or an exponential function.