Wolff potentials and nonlocal equations of Lane-Emden type

Quoc-Hung Nguyen, Jihoon Ok, Kyeong Song

Published 2024-05-20Version 1

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 $1<p<n/s$. 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 $1<p\le2-s/n$. 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 \[ (-\Delta_{p})^{s}u = P(u) + \mu \quad \text{in }\Omega, \] where $P(\cdot)$ is either a power function or an exponential function.