A note on Schr\"odinger--Newton systems with decaying electric potential

TL;DR

This paper proves the existence of solutions to a Schr"odinger--Newton system with a decaying electric potential, showing solutions concentrate around critical points as the semiclassical parameter approaches zero.

Contribution

It establishes the existence of solutions for Schr"odinger--Newton systems with polynomially decaying potentials, including cases with isolated strict extrema.

Findings

Solutions concentrate around critical points of the potential as 7 b7 b7 0.

Existence of solutions is proven for potentials with polynomial decay at infinity.

The results include cases with isolated strict extrema of the potential.

Abstract

We prove the existence of solutions for the singularly perturbed Schr\"odinger--Newton system {ll} \hbar^2 \Delta \psi - V(x) \psi + U \psi =0 \hbar^2 \Delta U + 4\pi \gamma |\psi|^2 =0 . \hbox{in $\mathbb{R}^3$} with an electric potential (V) that decays polynomially fast at infinity. The solution $\psi$ 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.