A note on Schrödinger--Newton systems with decaying electric potential

Abstract

We prove the existence of solutions for the singularly perturbed Schrödinger--Newton system {ll} \hbar^2 Δψ- V(x) ψ+ U ψ=0 \hbar^2 ΔU + 4πγ|ψ|^2 =0 . \hbox{in $\mathbb{R}^3$} with an electric potential (V) that decays polynomially fast at infinity. The solution $ψ$ concentrates, as $\hbar \to 0$, around (structurally stable) critical points of the electric potential. As a particular case, isolated strict extrema of (V) are allowed.

19 pages