Wolff potentials and nonlocal equations of Lane-Emden type

We consider nonlocal equations of the type $$(-{\mathrm{\Delta }}_p{)}^{s}u=\mu \text{in}\mathrm{\Omega },$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is either a bounded domain or the whole ${\mathbb{R}}^n$, $\mu$ is a Radon measure on $\mathrm{\Omega }$, $0<s<1$ and $1<p<n\mathrm{/}s$. In particular, 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(3):1317–1368, 2015), to the strongly singular case $1<p\le 2-s\mathrm{/}n$. Moreover, using Wolff potentials and Orlicz capacities, we present both a sufficient condition and a necessary condition for the existence of SOLA to nonlocal equations of the type $$(-{\mathrm{\Delta }}_p{)}^{s}u=P(u)+\mu \text{in}\mathrm{\Omega },$$ where $P(\cdot )$ is either a power function or an exponential function.