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

2 MEPHYSTO - Quantitative methods for stochastic models in physics
Inria Lille - Nord Europe, ULB - Université Libre de Bruxelles [Bruxelles], LPP - Laboratoire Paul Painlevé - UMR 8524
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 $p>2$, $A$ 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.
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Cited literature [34 references]

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Denis Bonheure, Silvia Cingolani, Manon Nys. Nonlinear Schrödinger equation: concentration on circles driven by an external magnetic field. Calculus of Variations and Partial Differential Equations, Springer Verlag, 2016, 55, ⟨10.1007/s00526-016-1013-8⟩. ⟨hal-01182834v2⟩

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