Multiple positive solutions for Schrödinger-Poisson system with singularity on the Heisenberg group

In this work, we study the following Schrödinger-Poisson system $\{\begin{array}{cc}-{\mathrm{\Delta }}_{H}u+\mu \varphi u=\lambda u^{-\gamma },\hfill & \text{in }\mathrm{\Omega },\hfill \\ -{\mathrm{\Delta }}_H\varphi =u^2,\hfill & \text{in }\mathrm{\Omega },\hfill \\ u>0,\hfill & \text{in }\mathrm{\Omega },\hfill \\ u=\varphi =0,\hfill & \text{on }\partial \mathrm{\Omega },\hfill \end{array}$ \( \textstyle\begin{cases} -\Delta _{H}u+\mu \phi u=\lambda u^{-\gamma}, &\text{in } \Omega , \\ -\Delta _{H}\phi =u^{2}, &\text{in } \Omega , \\ u>0, &\text{in } \Omega , \\ u=\phi =0, &\text{on } \partial \Omega , \end{cases} \) where ${\mathrm{\Delta }}_H$ $\Delta _{H}$ is the Kohn-Laplacian on the first Heisenberg group ${\mathbb{H}}^1$ $\mathbb{H}^{1}$ , and $\mathrm{\Omega }\subset {\mathbb{H}}^1$ $\Omega \subset \mathbb{H}^{1}$ is a smooth bounded domain, $\mu =\pm 1$ $\mu =\pm 1$ , $0<\gamma <1$ $0<\gamma <1$ , and $\lambda >0$ $\lambda >0$ are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for $\mu =1$ $\mu =1$ and each $\lambda >0$ $\lambda >0$ . Multiple solutions of the system are also considered for $\mu =-1$ $\mu =-1$ and $\lambda >0$ $\lambda >0$ small enough using the critical point theory for nonsmooth functional.