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 $Ω\subset \mathbb{H}^N$ is a smooth bounded domain, $Δ_H$ is the Kohn-Laplacian on the Heisenberg group $\mathbb{H}^N$, $1 0$, $0<λ<Q=2N+2$, and $Q_λ^{\ast}=\frac{2Q-λ}{Q-2}$ 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 $μ$), and has a nontrivial solution for $2<q<Q_λ^\ast$ in the case of high perturbations (large values of $μ$). Moreover, for $1<q<2$, we also show that there is a positive ground state solution, and for $2<q<Q_λ^\ast$, there are at least $n$ pairs of nontrivial weak solutions.