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 \[ -Δ_{\mathbb{H}} u = a u+ \left(\int_Ω\frac{|u(η)|^{Q^*_λ}}{|η^{-1}ξ|^λ}dη\right)|u|^{Q^*_λ-2}u \; \text{in}\; Ω,\quad u = 0 \; \text{ on } \partial Ω, \] 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, $Q_{\lambda }^{\ast }=\frac{2Q-\lambda }{Q-2},Q=2N+2$ 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 $a$.

32 pages