On Viscosity and Weak Solutions of Nonlocal Equations with Variable Powers

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发表日期：

2025年10月22日

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Mengna Yang,
Zhanbin Yuan,
Yufeng Nie

摘要

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 $\alpha (x,y)$ $α (x, y)$ , 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 $\Omega $ $Ω$ . Additionally, by the comparison principle, we also verify that weak solutions are viscosity solutions under a series of lemmas.