Nonlinear Schr{\"o}dinger equation: concentration on circles driven by an external magnetic field

TL;DR

This paper investigates the semiclassical limit of the magnetic nonlinear Schrödinger equation, demonstrating that solutions concentrate around a circle influenced by magnetic and electric potentials, revealing the magnetic field's role in solution localization.

Contribution

The study introduces a refined penalization method to prove the existence of cylindrically symmetric solutions concentrating on circles, highlighting magnetic field effects in the semiclassical limit.

Findings

Solutions concentrate around a circle as 6 6 0.

Magnetic and electric potentials influence the concentration locus.

Concentration occurs in the semiclassical limit with magnetic field impact.

Abstract

In this paper, we study the semiclassical limit for the stationary magnetic nonlinear Schr\"odinger equation \begin{align}\label{eq:initialabstract}\left( i \hbar \nabla + A(x) \right)^2 u + V(x) u = |u|^{p-2} u, \quad x\in \mathbb{R}^{3},\end{align}where $p\textgreater{}2$, $A$ is a vector potential associated to a given magnetic field $B$, i.e $\nabla \times A =B$ and $V$ is a nonnegative, scalar (electric) potential which can be singular at the origin and vanish at infinity or outside a compact set.We assume that $A$ and $V$ satisfy a cylindrical symmetry. By a refined penalization argument, we prove the existence of semiclassical cylindrically symmetric solutions of upper equation whose moduli concentrate, as $\hbar \to 0$, around a circle. We emphasize that the concentration is driven by the magnetic and the electric potentials. Our result thus shows that in the semiclassical limit, the magnetic field also influences the location of the solutions of $(\ref{eq:initialabstract})$ if their concentration occurs around a locus, not a single point.