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

Abstract

In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional $p$-Laplace type $$P.V.{\int }_{{\mathbb{R}}^n}\frac{|u(x)-u(y){|}^{p-2}(u(x)-u(y))}{|x-y{|}^{n+sp}}dy=0.$$ 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.

arXiv comment21 pages, to appear in Journal de Mathématiques Pures et Appliquées