Existence Theory for a class of semilinear mixed local and nonlocal equations involving variable singularities and singular measures

TL;DR

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

Contribution

It introduces a novel approach to handle variable singular exponents and measure-valued data in mixed local-nonlocal equations, including singular measures.

Findings

Existence of weak solutions established for equations with measure data.

Handling of variable singular exponents is developed.

Inclusion of singular and non-singular measure sources is demonstrated.

Abstract

This article establishes the existence of weak solutions for a class of mixed local-nonlocal problems with pure and perturbed singular nonlinearities. A key novelty is the treatment of variable singular exponents alongside measure-valued data. Notably, both source terms may be measures, with the singular component modeled by both a singular and non-singular measure. Our main focus is on the singular measure data, which appears to be new, even for constant exponents.