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

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*} (-Δ_$\mathbb{H}$)^su - λu &= |u|^{Q^_s -2}u $\text{ in }$ Ω,\ u &= 0 $\text{ in }$ $\mathbb{H}$^N \setminus Ω, \end{align} where $\mathrm{\Omega }\subseteq {\mathbb{H}}^N$ is bounded and has continuous boundary, $(-{\mathrm{\Delta }}_{\mathbb{H}}{)}^s$ is the horizontal fractional Laplacian, $s\in (0,1),\lambda >0,$ and $Q_s^{\ast }=\frac{2Q}{Q-2s}$ is the Sobolev critical exponent. This problem is motivated by the celebrated Brezis-Nirenberg problem \cite{brezis1983positive}.