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: $$-\mathrm{div}\left(|x{|}^{-2\beta }|\nabla u{|}^{\mathbf{q}-2}\nabla u\right)+(-\Delta {)}_{p(\cdot ,\cdot ),\beta }^{s(\cdot ,\cdot )}u=0\text{ in }\Omega ,$$ where $\Omega \subset {\mathbb{R}}^n,n\ge 2$, 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, $p:\Omega \times \Omega \to (1,\infty )$ and $s:\Omega \times \Omega \to (0,1)$ are measurable functions, $\mathbf{q}:={\mathrm{ess}}_{\Omega \times \Omega }p$, 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.

arXiv comment36 pages

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