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

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.

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