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

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