Equivalence of solutions to fractional $p$-Laplace type equations

TL;DR

This paper proves that for bounded solutions of fractional p-Laplace equations, various solution definitions such as weak, viscosity, and comparison-based solutions are equivalent, unifying different approaches.

Contribution

It establishes the equivalence of multiple solution concepts for bounded solutions of fractional p-Laplace equations, clarifying their relationship.

Findings

Different notions of solutions coincide for bounded solutions

Unification of solution concepts enhances understanding of fractional p-Laplace equations

Provides a foundation for further analysis using any of the solution definitions

Abstract

In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional $p$-Laplace type $${\rm P.V.} \int_{\mathbb R^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dy = 0.$$ Solutions are defined via integration by parts with test functions, as viscosity solutions or via comparison. Our main result states that for bounded solutions, the three different notions coincide.