On Weak and Viscosity Solutions of Nonlocal Double Phase Equations

Abstract We consider the nonlocal double phase equation $$\begin{align*} \textrm{P.V.} &\int_{\mathbb{R}^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{sp}(x,y)\,\textrm{d}y\\ &+\textrm{P.V.} \int_{\mathbb{R}^n} a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{tq}(x,y)\,\textrm{d}y=0, \end{align*}$$where $1<p\leq q$ and modulating coefficient $a(\cdot ,\cdot )\geq 0$. Under some suitable hypotheses, we first use De Giorgi–Nash–Moser methods to derive local Hölder continuity for bounded weak solutions then establish relationship between viscosity such equations.