Local H\"older Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights

Abstract

We establish local boundedness and local H\"older continuity of weak solutions to the following prototype problem: $$ -\operatorname{div}\left(|x|^{-2 \beta}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-\Delta)_{p(\cdot, \cdot), \beta}^{s(\cdot, \cdot)} u=0 \quad \text { in } \quad \Omega, $$ where $\Omega \subset \mathbb{R}^n, n \geq 2$, is a bounded domain. The nonlocal operator is defined by $$ (-\Delta)_{p(\cdot, \cdot), \beta}^{s(\cdot, \cdot)} u(x):=\mathrm{P} . \mathrm{V} . \int_{\Omega} \frac{|u(x)-u(y)|^{p(x, y)-2}(u(x)-u(y))}{|x-y|^{n+s(x, y) p(x, y)}} \frac{1}{|x|^\beta|y|^\beta} \mathrm{d} y $$ Here, $p: \Omega \times \Omega \rightarrow(1, \infty)$ and $s: \Omega \times \Omega \rightarrow(0,1)$ are measurable functions, $\mathbf{q}:=\operatorname{ess}_{\Omega \times \Omega} p$, and $0 \leq \beta<n$. 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.