Nonlocal singular problem and associated Sobolev type inequality with extremal in the Heisenberg group

TL;DR

This paper investigates a fractional p-Laplace equation with singular nonlinearity in the Heisenberg group, establishing existence, regularity, and uniqueness of solutions, and linking them to extremals of a Sobolev-type inequality, offering new insights even for classical cases.

Contribution

It introduces new existence and regularity results for singular fractional p-Laplace equations in the Heisenberg group and characterizes extremals of related Sobolev inequalities.

Findings

Existence and regularity of weak solutions established.

Uniqueness of solutions for constant singular exponent proved.

Connection between solutions and Sobolev extremals demonstrated.

Abstract

We study a fractional $p$-Laplace equation involving a variable exponent singular nonlinearity in the framework of the Heisenberg group. We first establish the existence and regularity of weak solutions. In the case of a constant singular exponent, we further prove the uniqueness of solutions and characterize the extremals of a related Sobolev-type inequality. Additionally, we demonstrate a connection between the solutions of the singular problem and these extremals. To the best of our knowledge, these findings provide new insights even in the classical case $p=2$.