VSE knjižnice (vzajemna bibliografsko-kataložna baza podatkov COBIB.SI)
  • High and low perturbations of the critical Choquard equation on the Heisenberg group
    Bai, Shujie ; Repovš, Dušan, 1954- ; Song, Yueqiang
    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.
    Vir: Advances in differential equations. - ISSN 1079-9389 (Vol. 29, no. 3/4, 2024, str. 153-178)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2024
    Jezik - angleški
    COBISS.SI-ID - 173858051
    DOI

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