Wolff potentials and nonlocal equations of Lane-Emden type

Abstract

We consider nonlocal equations of the type \[ (-Δ_{p})^{s}u = μ\quad \text{in}\;\; Ω, \] 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 \[ (-Δ_{p})^{s}u = P(u) + μ\quad \text{in}\;\; Ω, \] where $P(\cdot )$ is either a power function or an exponential function.

Trans. Amer. Math. Soc., to appear