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Periodic solutions for nonlinear elliptic equations.Application to charged particle beam focusing systems

Published online by Cambridge University Press:  15 February 2007

Mihai Bostan  and
Eric Sonnendrücker
Mihai Bostan
Affiliation:
Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France. mbostan@math.univ-fcomte.fr
Eric Sonnendrücker
Affiliation:
IRMA, Université Louis Pasteur, rue René Descartes, 67084 Strasbourg Cedex, France. sonnen@math.u-strasbg.fr
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Abstract

We study the existence of spatial periodic solutions for nonlinear elliptic equations $- \Delta u \, + \, g(x,u(x)) = 0, \;x \in {\mathbb R}^N$ where g is a continuous function, nondecreasing w.r.t. u. We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing functions g are investigated as well. As an application we analyze the mathematical model of electron beam focusing system and we prove the existence of positive periodic solutions for the envelope equation. We present also numerical simulations.

Keywords

Nonlinear elliptic equationsperiodic solutionsexistence and uniquenesselectron beam focusing system.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 6 , November 2006 , pp. 1023 - 1052
Copyright
© EDP Sciences, SMAI, 2007

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