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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. [email protected]
Eric Sonnendrücker
Affiliation:
IRMA, Université Louis Pasteur, rue René Descartes, 67084 Strasbourg Cedex, France. [email protected]
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Abstract

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

Keywords

Nonlinear elliptic equationsperiodic solutionsexistence anduniquenesselectron 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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