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

Published online by Cambridge University Press:  20 August 2008

Silvia Cingolani ,
Louis Jeanjean  and
Simone Secchi
Silvia Cingolani
Affiliation:
Dipartimento di Matematica, Politecnico di Bari, via Orabona 4, 70125 Bari, Italy. s.cingolani@poliba.it
Louis Jeanjean
Affiliation:
Équipe de Mathématiques (UMR CNRS 6623), 16 Route de Gray, 25030 Besançon, France. louis.jeanjean@univ-fcomte.fr
Simone Secchi
Affiliation:
Dipartimento di Matematica ed Applicazioni, Università di Milano-Bicocca, via Cozzi 53, 20125 Milano, Italy. Simone.Secchi@unimib.it
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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.

Keywords

Nonlinear Schrödinger equationsmagnetic fieldsmulti-peaks
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 3 , July 2009 , pp. 653 - 675
Copyright
© EDP Sciences, SMAI, 2008

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