A new Kirchhoff-Schrödinger-Poisson type system on the Heisenberg group

Abstract

In this paper, we are interested in a new Kirchhoff-Schrödinger-Poisson type system on the Heisenberg group of the form: \begin{equation*} \begin{cases} \displaystyle - \Big ( a-b\int_{\Omega}|\nabla_{H}u|^{2}dx \Big ) \Delta_{H}u +\mu\phi u=\lambda|u|^{q-2}u, & \mbox{in}\quad \Omega,\\ -\Delta_{H}\phi=u^2 & \mbox{in}\quad \Omega,\\ u=\phi=0 & \mbox{on}\quad \partial\Omega, \end{cases} \end{equation*} where $a, b > 0$, $\Delta_H$ is the Kohn-Laplacian on the first Heisenberg group $\mathbb{H}^1$, and $\Omega\subset \mathbb{H}^1$ is a smooth bounded domain, $\mu, \lambda > 0$ are some real parameters and $2 < q < 4$. According to the different ranges of these parameters, we discuss the existence and multiplicity of solutions for the above equation using variational methods under suitable conditions. Our result is new even in the Euclidean case