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

Abstract

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

Keywords

• Schrödinger-Poisson system
• Singularity
• Positive solutions
• Heisenberg group