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

Abstract

In this paper, our aim is to study thefollowing critical Choquard equation onthe 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-Laplacianon 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 compactnessprinciple and the critical point theory,we prove that the aboveproblem has the least two positivesolutions for$1 < q < 2$ in the caseof low perturbations (small values of$\mu$), andhas a nontrivial solution for$2 < q < Q_\lambda^\ast$ in the case ofhigh perturbations (large values of$\mu$). Moreover, for$1 < q < 2$,we also show that there is a positiveground state solution, and for$2 < q < Q_\lambda^\ast$, there are at least$n$pairs of nontrivial weak solutions.