On $p$-Laplacian Kirchhoff-Schr\"{o}dinger-Poisson type systems with critical growth on the Heisenberg group

TL;DR

This paper studies a class of nonlinear Kirchhoff-Schrödinger-Poisson systems on the Heisenberg group involving critical growth, establishing existence and multiplicity of solutions under certain parameter conditions.

Contribution

It extends previous results by analyzing a complex system with critical growth on the Heisenberg group, providing new existence and multiplicity results.

Findings

Established existence of solutions under specific parameter conditions.

Proved multiplicity of solutions for the system.

Generalized prior results to a broader class of systems on the Heisenberg group.

Abstract

In this article, we investigate the Kirchhoff-Schr\"{o}dinger-Poisson type systems on the Heisenberg group of the following form: \begin{equation*} \left\{ \begin{array}{lll} {-(a+b\int_{\Omega}|\nabla_{H} u|^{p}d\xi)\Delta_{H,p}u-\mu\phi |u|^{p-2}u}=\lambda |u|^{q-2}u+|u|^{Q^{\ast}-2}u &\mbox{in}\ \Omega, \\ -\Delta_{H}\phi=|u|^{p} &\mbox{in}\ \Omega, \\ u=\phi=0 &\mbox{on}\ \partial\Omega, \end{array} \right. \end{equation*} where $a,b$ are positive real numbers, $\Omega\subset \mathbb{H}^N$ is a bounded region with smooth boundary, $1<p<Q$, $Q = 2N + 2$ is the homogeneous dimension of the Heisenberg group $\mathbb{H}^N$, $Q^{\ast}=\frac{pQ}{Q-p}$, $q\in(2p, Q^{\ast})$, and $\Delta_{H,p}u=\mbox{div}(|\nabla_{H} u|^{p-2}\nabla_{H} u)$ is the $p$-horizontal Laplacian. Under some appropriate conditions for the parameters $\mu$ and $\lambda$, we establish existence and multiplicity results for the system above. To some extent, we generalize the results of An and Liu (Israel J. Math., 2020) and Liu et al. (Adv. Nonlinear Anal., 2022).