Regularity and existence for semilinear mixed local-nonlocal equations with variable singularities and measure data
Sanjit Biswas, Prashanta Garain
TL;DR
This work develops an existence and regularity theory for a broad class of mixed local-nonlocal equations with measure data and a variable singular exponent, extending beyond the classical constant-exponent setting. It employs an approximation framework with truncations and fixed-point arguments, plus capacity-based measure data, to handle singular nonlinearities and measures as sources. The main contributions include existence and regularity results for variable $\delta(x)$ under a natural compatibility condition, extensions to constant $\delta$ with precise data requirements, and a comprehensive set of a priori estimates that yield $L^\infty$ and high-integrability results under sharp thresholds. The general operator $\mathcal{M}$ encompasses both local and nonlocal components, making the results applicable to a wide range of problems, with potential implications for physics and geometry where mixed diffusion processes arise.
Abstract
This article proves the existence and regularity of weak solutions for a class of mixed local-nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central contribution of this work is the inclusion of a variable singular exponent in the context of measure-valued data. Another notable feature is that the source terms in both the purely singular and perturbed components can simultaneously take the form of measures. To the best of our knowledge, this phenomenon is new, even in the case of a constant singular exponent.