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

Abstract

We study the following critical Choquard equation on the Heisenberg group: \begin{equation*} \begin{cases} \displaystyle {-\Delta_H u }={\mu} |u|^{q-2}u+\int_{\Omega} \frac{|u(\eta)|^{Q_{\lambda}^{\ast}}} {|\eta^{-1}\xi|^{\lambda}} d\eta|u|^{Q_{\lambda}^{\ast}-2}u &\mbox{in }\ \Omega, u=0 &\mbox{on }\ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{H}^N$ is a smooth bounded domain, $\Delta_H$ 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.