Brezis-Nirenberg type result for Kohn Laplacian with critical Choquard Nonlinearity

Abstract: In this article, we are study the following Dirichlet problem with Choquard type non linearity [ -\Delta_{\mathbb{H}} u = a u+ \left(\int_{\Omega}\frac{|u(\eta)|^{Q^_\lambda}}{|\eta^{-1}\xi|^{\lambda}}d\eta\right)|u|^{Q^_\lambda-2}u \; \text{in}\; \Omega,\quad u = 0 \; \text{ on } \partial \Omega , ] where $\Omega$ is a smooth bounded subset of the Heisenberg group $\mathbb{H}^N, N\in \mathbb N$ with $C^2$ boundary and $\Delta_{\mathbb{H}}$ is the Kohn Laplacian on the Heisenberg group $\mathbb{H}^N$. Here, $Q^*_\lambda=\frac{2Q-\lambda}{Q-2},\; Q= 2N+2$ and $a$ is a positive real parameter. We derive the Brezis-Nirenberg type result for the above problem. Moreover, we also prove the regularity of solutions and nonexistence of solutions depending on the range of $a$.