High and low perturbations of the critical Choquard equation on the Heisenberg group

We study the following critical Choquard equation on the Heisenberg group: \begin{equation*} \begin{cases} \displaystyle {-Δ_H u }=μ |u|^{q-2}u+\int_Ω \frac{|u(η)|^{Q_λ^{\ast}}} {|η^{-1}ξ|^λ} dη|u|^{Q_λ^{\ast}-2}u &\mbox{in }\ Ω, u=0 &\mbox{on }\ \partialΩ, \end{cases} \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{H}}^N$ is a smooth bounded domain, ${\mathrm{\Delta }}_H$ is the Kohn-Laplacian on the Heisenberg group ${\mathbb{H}}^N$, $1<q<2$ or $2<q<Q_{\lambda }^{\ast }$, $\mu >0$, $0<\lambda <Q=2N+2$, and $Q_{\lambda }^{\ast }=\frac{2Q-\lambda }{Q-2}$ is the critical exponent. Using the concentration compactness principle and the critical point theory, we prove that the above problem has the least two positive solutions for $1<q<2$ in the case of low perturbations (small values of $\mu$), and has a nontrivial solution for $2<q<Q_{\lambda }^{\ast }$ in the case of high perturbations (large values of $\mu$). Moreover, for $1<q<2$, we also show that there is a positive ground state solution, and for $2<q<Q_{\lambda }^{\ast }$, there are at least $n$ pairs of nontrivial weak solutions.