Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights

Abstract

We establish local boundedness and local Hölder continuity of weak solutions to the following prototype problem: $$ -\operatorname{div}\left(|x|^{-2 β}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-Δ)_{p(\cdot, \cdot), β}^{s(\cdot, \cdot)} u=0 \quad \text { in } \quad Ω, $$ where $Ω\subset \mathbb{R}^n, n \geq 2$, is a bounded domain. The nonlocal operator is defined by $$ (-Δ)_{p(\cdot, \cdot), β}^{s(\cdot, \cdot)} u(x):=\mathrm{P} . \mathrm{V} . \int_Ω \frac{|u(x)-u(y)|^{p(x, y)-2}(u(x)-u(y))}{|x-y|^{n+s(x, y) p(x, y)}} \frac{1}{|x|^β|y|^β} \mathrm{d} y $$ Here, $p: Ω\times Ω\rightarrow(1, \infty)$ and $s: Ω\times Ω\rightarrow(0,1)$ are measurable functions, $\mathbf{q}:=\operatorname{ess}_{Ω\times Ω} p$, and $0 \leq β<n$. Our approach is analytic and relies on an adaptation of the De Giorgi-Nash-Moser theory to a mixed local-nonlocal framework with variable exponents and weights.