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Stationary Solutions of Nonlinear Schrödinger Equations with an External Magnetic Field

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Partial Differential Equations and the Calculus of Variations

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 1))

Abstract

In this paper we study the existence of stationary solutions of some Schrödinger equations with an external magnetic field. We obtain these solutions by solving appropriate minimization problems for the corresponding energy-functional. These problems which are a priori not compact are solved by the use of the so-called concentration-compactness method. We also prove the existence of solutions for the generalized Hartree-Fock equations which model the interaction of electrons and static nucleii through a coulombic potential and under the action of an external magnetic field.

Résumé

Dans ce travail on étudie l’existence de solutions stationnaires pour des equations de Schrödinger non linéaires avec un champ magnétique externe. Nous obtenons ces solutions en résolvant certains problémes de minimisation pour la fonctionnelle d’énergie correspondante, et nous faisons ceci en utilisant la méthode de concentration-compacité. Nous démontrons aussi l’existence de solutions pour les équations de Hartree-Fock qui modélisent l’interaction d’électrons avec des noyaux à travers un potentiel coulombien et sous l’action d’un champ magnétique externe.

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Authors and Affiliations

  1. Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, 4, place Jussieu, F-75252, Paris cedex 05, France

    Maria J. Esteban & Pierre-Louis Lions

  2. Ceremade, Université de Paris IX, Place de Lattre de Tassigny, F-75775, Paris cedex 16, France

    Maria J. Esteban & Pierre-Louis Lions

Authors
  1. Maria J. Esteban
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  2. Pierre-Louis Lions
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Editors and Affiliations

  1. Dipartimento di Matematica, Universita di Pisa, Via F. Buonarroti 2, 56100, Pisa, Italy

    Ferruccio Colombini , Antonio Marino , Luciano Modica  & Sergio Spagnolo , ,  & 

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Dedicated to Ennio De Giorgi on his sixtieth birthday

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© 1989 Birkhauser Boston

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Esteban, M.J., Lions, PL. (1989). Stationary Solutions of Nonlinear Schrödinger Equations with an External Magnetic Field. In: Colombini, F., Marino, A., Modica, L., Spagnolo, S. (eds) Partial Differential Equations and the Calculus of Variations. Progress in Nonlinear Differential Equations and Their Applications, vol 1. Birkhäuser Boston. https://doi.org/10.1007/978-1-4615-9828-2_18

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  • DOI: https://doi.org/10.1007/978-1-4615-9828-2_18

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-1-4615-9830-5

  • Online ISBN: 978-1-4615-9828-2

  • eBook Packages: Springer Book Archive

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