Shujie Bai, Yueqiang Song, Dušan D. Repovš, Existence and multiplicity of solutions for critical Kirchhoff-Choquard equations involving the fractional p-Laplacian on the Heisenberg group

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DOI: 10.23952/jnva.8.2024.1.08

Volume 8, Issue 1, 1 February 2024, Pages 143-166

Abstract. In this paper, we study the 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, $0\textless s \textless 1$ , $1\textless p \textless\frac{N}{s}$ , $\mu>0$ , $f(\xi,u)$ is the nonlinear function, $0 \textless \lambda \textless Q$ , $Q=2N+2$ , 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.

How to Cite this Article:
S. Bai, Y. Song, D.D. Repovš, Existence and multiplicity of solutions for critical Kirchhoff-Choquard equations involving the fractional p-Laplacian on the Heisenberg group, J. Nonlinear Var. Anal. 8 (2024), 143-166.