An integral type Brezis-Nirenberg problem on the Heisenberg group

TL;DR

This paper investigates a new class of nonlinear integral equations on the Heisenberg group, related to CR Yamabe problems, establishing existence and nonexistence results using advanced analytical techniques.

Contribution

It introduces a novel integral type Brezis-Nirenberg problem on the Heisenberg group and provides the first existence and nonexistence results for these equations.

Findings

Existence of solutions under certain conditions

Nonexistence results via Pohozaev identity

Application of Hardy-Littlewood-Sobolev inequalities

Abstract

This paper is devoted to study a class of integral type Brezis-Nirenbreg problem on the Heisenberg group. It is a class of new nonlinear integral equations on the bounded domains of Heisenberg group and related to the CR Yamabe problems on the CR manifold. Based on the sharp Hardy-Littlewood-Sobolev inequalities, the nonexistence and existence results are obtained by Pohozaev type identity, variational method and blow-up analysis, respectively.