Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights
Juan Pablo Alcon Apaza
TL;DR
This work proves local boundedness and local Hölder continuity for weak solutions of a mixed local–nonlocal quasilinear elliptic equation with weights and variable exponents, governed by a weighted local $q$-Laplacian and a weighted regional fractional $p(\cdot,\cdot)$-Laplacian. The authors develop an analytic De Giorgi–Nash–Moser framework tailored to the combined local and nonlocal structure, leveraging energy estimates, Sobolev inequalities in weighted variable-exponent spaces, andTail-type controls to handle the nonlocal part. The main results provide precise oscillation bounds and tail-inclusive estimates, with explicit Hölder exponents and constants under a set of structural conditions coupling $p$, $s$, and $\beta$. The analysis extends the regularity theory to a broad class of operators with nonstandard growth, variable orders, and multi-scale interactions, offering tools applicable to models with spatially heterogeneous diffusion and weights. Overall, the paper advances interior regularity theory for mixed operators in weighted, nonuniform growth contexts and deepens understanding of how local and nonlocal effects combine to yield Hölder continuity.
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.