Brezis-Nirenberg type problem for fractional sub-Laplacian on the Heisenberg group

TL;DR

This paper proves the existence of solutions for a fractional sub-Laplacian equation on the Heisenberg group involving critical Sobolev exponents, extending classical results to a non-commutative setting.

Contribution

It establishes the existence of weak solutions for a fractional sub-Laplace equation with critical exponent on the Heisenberg group, a non-Euclidean setting.

Findings

Existence of weak solutions proven

Addresses critical Sobolev exponent case

Extends Brezis-Nirenberg problem to Heisenberg group

Abstract

In this paper, we show the existence of a weak solution for a fractional sub-Laplace equation involving a term with the critical Sobolev exponent, namely, \begin{align*} (-\Delta_\mathbb{H})^su - \lambda u &= |u|^{Q^*_s -2}u \text{ in } \Omega,\\ u &= 0 \text{ in } \mathbb{H}^N \setminus \Omega, \end{align*} where $\Omega \subseteq \mathbb{H}^N$ is bounded and has continuous boundary, $(-\Delta_\mathbb{H})^s$ is the horizontal fractional Laplacian, $s \in (0,1), \lambda > 0,$ and $Q^*_s=\frac{2Q}{Q-2s}$ is the Sobolev critical exponent. This problem is motivated by the celebrated Brezis-Nirenberg problem \cite{brezis1983positive}.