On Weak and Viscosity Solutions of Nonlocal Double Phase Equations

Abstract

AbstractWe 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 the modulating coefficient $a(\cdot ,\cdot )\geq 0$. Under some suitable hypotheses, we first use the De Giorgi–Nash–Moser methods to derive the local Hölder continuity for bounded weak solutions and then establish the relationship between weak solutions and viscosity solutions to such equations.