Existence and concentration of ground state solutions for doubly critical Schrödinger–Poisson-type systems

In this paper, we are concerned with the existence and concentration of ground state solutions for the following nonlinear Schrödinger–Poisson-type system with doubly critical growth - ε 2 Δ u + V ( x ) u - ϕ | u | 3 u = | u | 4 u + f ( u ) , in R 3 , - ε 2 Δ ϕ = | u | 5 , in R 3 , where ε > 0 is a small parameter. By employing the concentration-compactness principle and mountain pass theorem, we prove the existence of positive ground state solutions v ε with exponential decay at infinity for ε sufficiently small under some suitable assumptions on the potential V and nonlinearity f . Moreover, as ε → 0 + , v ε concentrates around a global minimum point of V .