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

Silvia Cingolani; Louis Jeanjean; Simone Secchi

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • 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 (2008): 653-675. <http://eudml.org/doc/90931>.

@article{Cingolani2008,
abstract = { In this work we consider the magnetic NLS equation $$ ( \frac\{\hbar\}\{i\} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u \, = 0 \, \qquad \mbox\{ in \} \mathbb\{R\}^N\qquad\qquad(0.1)$$ where $N \geq 3$, $A \colon \mathbb\{R\}^N \to \mathbb\{R\}^N$ is a magnetic potential, possibly unbounded, $V \colon \mathbb\{R\}^N \to \mathbb\{R\}$ 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 $u\colon \mathbb\{R\}^N \to \mathbb\{C\}$ 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; nonlinear Schrödinger equations; multi-peaks},
language = {eng},
month = {8},
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/90931},
volume = {15},
year = {2008},
}

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
DA - 2008/8//
PB - EDP Sciences
VL - 15
IS - 3
SP - 653
EP - 675
AB - In this work we consider the magnetic NLS equation $$ ( \frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u \, = 0 \, \qquad \mbox{ in } \mathbb{R}^N\qquad\qquad(0.1)$$ where $N \geq 3$, $A \colon \mathbb{R}^N \to \mathbb{R}^N$ is a magnetic potential, possibly unbounded, $V \colon \mathbb{R}^N \to \mathbb{R}$ 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 $u\colon \mathbb{R}^N \to \mathbb{C}$ to (0.1), under conditions on the nonlinearity which are nearly optimal.
LA - eng
KW - Nonlinear Schrödinger equations; magnetic fields; multi-peaks; nonlinear Schrödinger equations; multi-peaks
UR - http://eudml.org/doc/90931
ER -

References

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