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

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 , where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 , and Ω ⊂ H 1 is a smooth bounded domain, μ = ± 1 , 0 < γ < 1 , and λ > 0 are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for μ = 1 and each λ > 0 . Multiple solutions of the system are also considered for μ = − 1 and λ > 0 small enough using the critical point theory for nonsmooth functional.