Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions

Silvia Cingolani; Louis Jeanjean; Simone Secchi

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 3, page 653-675
  • ISSN: 1292-8119

Abstract

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In this work we consider the magnetic NLS equation ( ℏ i ∇ - A ( x ) ) 2 u + V ( x ) u - f ( | u | 2 ) u = 0 in ℝ N ( 0 . 1 ) where N ≥ 3 , A : ℝ N → ℝ N is a magnetic potential, possibly unbounded, V : ℝ N → ℝ is a multi-well electric potential, which can vanish somewhere, f is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution u : ℝ N → ℂ to (0.1), under conditions on the nonlinearity which are nearly optimal.

How to cite

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Cingolani, Silvia, Jeanjean, Louis, and Secchi, Simone. "Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions." ESAIM: Control, Optimisation and Calculus of Variations 15.3 (2009): 653-675. <http://eudml.org/doc/244886>.

@article{Cingolani2009,
abstract = {In this work we consider the magnetic NLS equation\[\hspace*\{54.06023pt\}( \frac\{\hbar \}\{i\} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u \, = 0 \, \qquad \mbox\{ in \} \mathbb \{R\}^N\qquad \qquad (0.1)\]where $N \ge 3$, $A \colon \mathbb \{R\}^N \rightarrow \mathbb \{R\}^N$ is a magnetic potential, possibly unbounded, $V \colon \mathbb \{R\}^N \rightarrow \mathbb \{R\}$ is a multi-well electric potential, which can vanish somewhere, $f$ is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u\colon \mathbb \{R\}^N \rightarrow \mathbb \{C\}$ to (0.1), under conditions on the nonlinearity which are nearly optimal.},
author = {Cingolani, Silvia, Jeanjean, Louis, Secchi, Simone},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {nonlinear Schrödinger equations; magnetic fields; multi-peaks},
language = {eng},
number = {3},
pages = {653-675},
publisher = {EDP-Sciences},
title = {Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions},
url = {http://eudml.org/doc/244886},
volume = {15},
year = {2009},
}

TY - JOUR
AU - Cingolani, Silvia
AU - Jeanjean, Louis
AU - Secchi, Simone
TI - Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2009
PB - EDP-Sciences
VL - 15
IS - 3
SP - 653
EP - 675
AB - In this work we consider the magnetic NLS equation\[\hspace*{54.06023pt}( \frac{\hbar }{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u \, = 0 \, \qquad \mbox{ in } \mathbb {R}^N\qquad \qquad (0.1)\]where $N \ge 3$, $A \colon \mathbb {R}^N \rightarrow \mathbb {R}^N$ is a magnetic potential, possibly unbounded, $V \colon \mathbb {R}^N \rightarrow \mathbb {R}$ is a multi-well electric potential, which can vanish somewhere, $f$ is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u\colon \mathbb {R}^N \rightarrow \mathbb {C}$ to (0.1), under conditions on the nonlinearity which are nearly optimal.
LA - eng
KW - nonlinear Schrödinger equations; magnetic fields; multi-peaks
UR - http://eudml.org/doc/244886
ER -

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