Multi-bump solutions for the magnetic Schrödinger–Poisson system with critical growth

In this paper, we are concerned with the following magnetic Schrödinger–Poisson system { − ( ∇ + i A ( x ) ) 2 u + ( λ V ( x ) + 1 ) u + ϕ u = α f ( | u | 2 ) u + | u | 4 u ,  in  R 3 , − Δ ϕ = u 2 ,  in  R 3 , where λ > 0 is a parameter, f is a subcritical nonlinearity, the potential V : R 3 → R is a continuous function verifying some conditions, the magnetic potential A ∈ L l o c 2 ( R 3 , R 3 ) . Assuming that the zero set of V ( x ) has several isolated connected components Ω 1 , … , Ω k such that the interior of Ω j is non-empty and ∂ Ω j is smooth, where j ∈ { 1 , … , k } , then for λ > 0 large enough, we use the variational methods to show that the above system has at least 2 k − 1 multi-bump solutions.