Regularity and existence for semilinear mixed local-nonlocal equations with variable singularities and measure data

TL;DR

This paper establishes the existence and regularity of weak solutions for mixed local-nonlocal equations with variable singularities and measure data, advancing understanding of complex nonlinear PDEs with singular and measure-valued sources.

Contribution

It introduces the first analysis of mixed local-nonlocal equations with variable singular exponents and measure data, including measure-valued source terms in both singular and perturbed problems.

Findings

Proved existence of weak solutions under measure data.

Established regularity results for solutions with variable singularities.

Demonstrated the novelty of measure-valued sources in both problem components.

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.