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.
Categories: math.AP