Bai, Shujie; Repovš, Dušan D.; Song, Yueqiang High and low perturbations of the critical Choquard equation on the Heisenberg group. (English) Zbl 1532.35192 Adv. Differ. Equ. 29, No. 3-4, 153-178 (2024). Summary: In this paper, our aim is to study the following critical Choquard equation on the Heisenberg group: \[ \begin{cases} - \Delta_H u = \mu |u|^{q-2}u + \int_{\Omega}\frac{|u(\eta)|^{Q_{\lambda}^{\ast}}}{|\eta^{-1}\xi|^{\lambda}}\,d\eta|u|^{Q_{\lambda}^{\ast}-2} u &\text{ in } \Omega, \\ u=0 &\text{ on } \partial\Omega, \end{cases}\] where \(\Omega\subset \mathbb{H}^N\) is a smooth bounded domain, \( \Delta_H\) is the Kohn-Laplacianon 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. MSC: 35J61 Semilinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35R03 PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. 35A01 Existence problems for PDEs: global existence, local existence, non-existence Keywords:Choquard equation on the Heisenberg group; existence × Cite Format Result Cite Review PDF Full Text: DOI arXiv Link References: [1] N. 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