Mixed local-nonlocal $p$-Laplace equation with variable singular nonlinearity in the Heisenberg group

TL;DR

This paper studies a mixed local-nonlocal p-Laplace equation with variable singular nonlinearity on the Heisenberg group, establishing fundamental properties of solutions in a non-commutative setting.

Contribution

It introduces the first analysis of mixed local-nonlocal problems with variable singular nonlinearities in the Heisenberg group, including existence, uniqueness, and regularity results.

Findings

Proved existence of weak solutions

Established uniqueness under structural conditions

Demonstrated regularity of solutions

Abstract

We investigate a mixed local-nonlocal $p$-Laplace equation on the Heisenberg group, where the nonlinear term features a variable singular exponent. Our analysis establishes the existence, uniqueness, and regularity of weak solutions under suitable structural assumptions. To the best of our knowledge, this work provides the first treatment of such mixed local-nonlocal problems in a non-commutative setting, even in the linear case $p=2$ with a constant singular exponent.