Nonlinear Schr\"odinger equations with strongly singular potentials

TL;DR

This paper establishes the existence of cylindrically symmetric standing wave solutions with angular momentum for nonlinear Schrödinger equations featuring strongly singular potentials, using a minimization approach under weak assumptions.

Contribution

It introduces a novel minimization method to find standing waves with angular momentum in equations with strongly singular potentials, including the nonlinear hydrogen atom case.

Findings

Existence of standing waves with non-zero angular momentum.

Solutions obtained via a minimization approach despite lack of compactness.

Application to nonlinear hydrogen atom equations.

Abstract

In this paper we look for standing waves for nonlinear Schr\"odinger equations $$ i\frac{\partial \psi}{\partial t}+\Delta \psi - g(|y|) \psi -W^{\prime}(| \psi |)\frac{\psi}{| \psi |}=0 $$ with cylindrically symmetric potentials $g$ vanishing at infinity and non-increasing, and a $C^1$ nonlinear term satisfying weak assumptions. In particular we show the existence of standing waves with non-vanishing angular momentum with prescribed $L^2$ norm. The solutions are obtained via a minimization argument, and the proof is given for an abstract functional which presents lack of compactness. As a particular case we prove the existence of standing waves with non-vanishing angular momentum for the nonlinear hydrogen atom equation.