Nonlinear Schrödinger equation: concentration on circles driven by an external magnetic field

In this paper, we study the semiclassical limit for the stationary magnetic nonlinear Schrödinger equation iħ∇+A(x)2u+V(x)u=|u|p-2u,x∈R3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( i \hbar \nabla + A(x) \right) ^2 u + V(x) u = |u|^{p-2} u, \quad x\in \mathbb {R}^{3}, \end{aligned}$$\end{document}where p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>2$$\end{document}, A is a vector potential associated to a given magnetic field B, i.e ∇×A=B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \times A =B$$\end{document} 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 (0.1) whose moduli concentrate, as ħ→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar \rightarrow 0$$\end{document}, 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 (0.1) if their concentration occurs around a locus, not a single point.