Nonlocal Harnack inequalities for nonlocal double phase equations I ; with positive bounded modulating coefficient with no H\"older condition

Abstract

In this paper, by applying the De Giorgi-Nash-Moser theory we prove nonlocal Harnack inequalities for (locally nonnegative in $\Omega$) weak solutions to nolocal double phase equations \begin{equation*}\begin{cases}\cL u =0 & \text{ in $\Omega$,} \\ u=g & \text{ in $\BR^n\s\Omega$ } \end{cases}\end{equation*} where $\Omega\subset\BR^n$ ($n\ge 2$) is a bounded domain with Lipschitz boundary, $\cL$ is the nonlocal double phase operator $\cL$ given by \begin{equation*}\begin{split}\cL u(x)=&\pv\int_{\BR^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{ps}(x,y)\,dy \\ &+\pv\int_{\BR^n}\fa(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{qt}(x,y)\,dy, \end{split} \end{equation*} $0<\fa(x,y) = \fa(y,x) \le \|\fa\|_{L^\iy(\BR^n\times\BR^n)} < \iy$ and $ps\ge qt$ for $0<s,t<1<p\le q<\iy$. In addition, we get local boundedness with explicit formula and weak Harnack inequalities for their weak supersolutions.