Harnack inequality for mixed local-nonlocal weighted homogeneous equations

Abstract

We consider the following class of mixed local-nonlocal equations: \begin{equation}\tag{P}\label{eq:P} -\Delta_p u + (-\Delta)_p^s u = V |u|^{p-2}u \text{ in } \Omega, \end{equation} where $s \in (0,1), p \in (1, \infty)$, and the weight function $V$ lies in scaling subcritical Lebesgue space $L^q(\Omega)$ where $q>d/p$ when $d>p$ and $q>1$ when $d \le p$. We establish the Harnack inequality for a weak solution and the weak Harnack inequality for a weak supersolution to \eqref{eq:P}. Our approach is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. Our results also apply to integro-differential operators, with the prototype given by $(-\Delta)_p^s$. This work generalizes some regularity results of Garain-Kinnunen (Trans. Am. Math. Soc., 375(8), 2022) and Garain (Nonlinear Anal., 256, 2025) to the setting of general weight functions.