Brezis-Nirenberg type problem for fractional sub-Laplacian on the 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*} (-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 $\Omega \subseteq {\mathbb{H}}^N$ is bounded and has continuous boundary, $(-{\Delta }_{\mathbb{H}}{)}^s$ is the horizontal fractional Laplacian, $s\in (0,1),\lambda >0,$ and $Q_s^{\ast }=\frac{2Q}{Q-2s}$ is the Sobolev critical exponent. This problem is motivated by the celebrated Brezis-Nirenberg problem \cite{brezis1983positive}.

18 pages, Accepted in Fractional Calculus and Applied Analysis