On Viscosity and Weak Solutions of Nonlocal Equations with Variable Powers

In this manuscript, we study properties of viscosity solutions for a class of nonlinear nonlocal equations with variable-powers fractional p(x, y)-Laplacian. More precisely, under sharp assumptions on the functions α(x,y), p(x, y) and w(x, y), we prove the local Hölder continuity of bounded viscosity solutions via Krylov-Safonov theory. Furthermore, based on properties of generalized Lebesgue spaces and inf-convolution approximation techniques, we show that viscosity solutions are weak solutions when the right-hand side f is continuous and locally bounded in Ω. Additionally, by the comparison principle, we also verify that weak solutions are viscosity solutions under a series of lemmas.