Wolff potentials and nonlocal equations of Lane-Emden type

TL;DR

This paper extends the theory of nonlocal p-Laplace equations involving Wolff potentials to include strongly singular cases and establishes conditions for the existence of solutions with measure data, using advanced potential and capacity techniques.

Contribution

It generalizes existence, regularity, and potential estimates for nonlocal equations to the strongly singular regime and characterizes solution existence via Wolff potentials and capacities.

Findings

Extended Wolff potential estimates to the case 1<p≤2−s/n.

Provided necessary and sufficient conditions for SOLA existence with measure data.

Applied Orlicz capacities to characterize solution existence.

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 $1<p<n/s$. 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.