Wolff potentials and nonlocal equations of Lane-Emden type

Abstract

We consider nonlocal equations of the type \[ (-\Delta_{p})^{s}u = \mu \quad \text{in}\;\; \Omega, \] where $\Omega \subset \mathbb{R}^{n}$ is either a bounded domain or the whole $\mathbb{R}^{n}$, $\mu$ is a Radon measure on $\Omega$, $0 < s < 1$ and $1 < p < n/s$. 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 $1 < p \le 2-s/n$. 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 \[ (-\Delta_{p})^{s}u = P(u) + \mu \quad \text{in}\;\; \Omega, \] where $P(\cdot)$ is either a power function or an exponential function.