Existence and multiplicity of solutions for critical Kirchhoff-Choquard equations involving the fractional $p$-Laplacian on the Heisenberg group

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

In this paper, we study existence and multiplicity of solutions for the following Kirchhoff-Choquard type equation involving the fractional $p$-Laplacian on the Heisenberg group: $M(\|u\|_\mu^{p})(\mu(-\Delta)^{s}_{p}u+V(\xi)|u|^{p-2}u)= f(\xi,u)+\int_{\mathbb{H}^N}\frac{|u(\eta)|^{Q_\lambda^{\ast}}}{|\eta^{-1}\xi|^\lambda}d\eta|u|^{Q_\lambda^{\ast}-2}u$ in $\mathbb{H}^N$, where $(-\Delta)^{s}_{p}$ is the fractional $p$-Laplacian on the Heisenberg group $\mathbb{H}^N$, $M$ is the Kirchhoff function, $V(\xi)$ is the potential function, $0 < s < 1$, $1 < p < \frac{N}{s}$, $\mu > 0$, $f(\xi,u)$ is the nonlinear function, $0 < \lambda < Q$, $Q=2N+2$, and $Q_\lambda^{\ast}=\frac{2Q-\lambda}{Q-2}$ is the Sobolev critical exponent. Using the Krasnoselskii genus theorem, the existence of infinitely many solutions is obtained if $\mu$ is sufficiently large. In addition, using the fractional version of the concentrated compactness principle, we prove that problem has $m$ pairs of solutions if $\mu$ is sufficiently small. As far as we know, the results of our study are new even in the Euclidean case.

• Language: English
• Keywords: fractional concentration-compactness principle, Krasnoselskii genus, Kirchhoff-Choquard type equations, Heisenberg group
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Author Accepted Manuscript
• Year: 2024
• Number of pages: Str. 143-166
• Numbering: Vol. 8, no. 1
• PID: 20.500.12556/RUL-154014
• UDC: 517.9
• ISSN on article: 2560-6778
• DOI: 10.23952/jnva.8.2024.1.08
• COBISS.SI-ID: 181483523

Copyright

Za shranitev recenziranega rokopisa v Repozitorij Univerze v Ljubljani je bilo pridobljeno založnikovo dovoljenje. (Datum opombe: 19. 1. 2024)

• Publication date in RUL: 19.01.2024

Record is a part of a journal

• Title: Journal of nonlinear and variational analysis
• Shortened title: J. nonlinear var. anal.
• Publisher: Biemdas
• ISSN: 2560-6778
• COBISS.SI-ID: 18657625