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

Abstract

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