Harnack inequality for mixed local-nonlocal weighted homogeneous equations

TL;DR

This paper proves Harnack inequalities for solutions of mixed local and nonlocal weighted equations, extending regularity results to broader weighted settings using advanced PDE techniques.

Contribution

It establishes Harnack inequalities for a class of weighted mixed local-nonlocal equations, generalizing previous regularity results to include weight functions.

Findings

Proved Harnack inequality for weak solutions.

Established weak Harnack inequality for supersolutions.

Results apply to a broad class of integro-differential operators.

Abstract

We consider the following class of mixed local-nonlocal equations: \begin{align}\label{abs}\tag{$\mathcal{P}$} -\Delta_p u + (-\Delta)_p^s u = V |u|^{p-2}u \text{ in } \Omega, \end{align} where $s \in (0,1), p \in (1, \infty)$, and the weight function $V$ lies in scaling subcritical Lebesgue space $L^q(\Omega)$ where $q>\frac{d}{p}$ when $d>p$ and $q>1$ when $d \le p$. We establish Harnack inequality for weak solution and weak Harnack inequality for weak supersolution to ($\mathcal{P}$). 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.