A note on Schrödinger–Newton systems with decaying electric potential

We prove the existence of solutions for the singularly perturbed Schrödinger–Newton system {ħ2Δψ−V(x)ψ+Uψ=0ħ2ΔU+4πγ|ψ|2=0in R3 with an electric potential VV that decays polynomially fast at infinity. The solution ψψ concentrates, as ħ→0ħ→0, around (structurally stable) critical points of the electric potential. As a particular case, isolated strict extrema of VV are allowed.