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

Bai, Shujie (avtor), Repovš, Dušan (avtor), Song, Yueqiang (avtor)

We study the following critical Choquard equation on the Heisenberg group: $\begin{cases} {-\Delta_H u }={\mu} |u|^{q-2}u+\int_{\Omega} \frac{|u(\eta)|^{Q_{\lambda}^{\ast}}} {|\eta^{-1}\xi|^{\lambda}} d\eta|u|^{Q_{\lambda}^{\ast}-2}u & \mbox{in}\ \Omega, \\ u=0 & \mbox{on}\ \partial\Omega, \end{cases}$ 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 < $ 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.

• Jezik: Angleški jezik
• Ključne besede: perturbations, critical Choquard equation, Heisenberg group
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: PEF - Pedagoška fakulteta; FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Recenzirani rokopis
• Datum objave: 01.01.2024
• Leto izida: 2024
• Št. strani: Str. 153-178
• Številčenje: Vol. 29, no. 3/4
• PID: 20.500.12556/RUL-182709
• UDK: 517.9
• ISSN pri članku: 1079-9389
• DOI: 10.57262/ade029-0304-153
• COBISS.SI-ID: 173858051
• Datum objave v RUL: 21.05.2026

Gradivo je del revije

• Naslov: Advances in differential equations
• Skrajšan naslov: Adv. differ. equ.
• Založnik: Khayyam Pub.
• ISSN: 1079-9389
• COBISS.SI-ID: 15253081