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

Mihai Bostan; Eric Sonnendrücker

- Volume: 40, Issue: 6, page 1023-1052
- ISSN: 0764-583X

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topBostan, Mihai, and Sonnendrücker, Eric. "Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 40.6 (2006): 1023-1052. <http://eudml.org/doc/244840>.

@article{Bostan2006,

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.},

author = {Bostan, Mihai, Sonnendrücker, Eric},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {nonlinear elliptic equations; periodic solutions; existence and uniqueness; electron beam focusing system},

language = {eng},

number = {6},

pages = {1023-1052},

publisher = {EDP-Sciences},

title = {Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems},

url = {http://eudml.org/doc/244840},

volume = {40},

year = {2006},

}

TY - JOUR

AU - Bostan, Mihai

AU - Sonnendrücker, Eric

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2006

PB - EDP-Sciences

VL - 40

IS - 6

SP - 1023

EP - 1052

AB - 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.

LA - eng

KW - nonlinear elliptic equations; periodic solutions; existence and uniqueness; electron beam focusing system

UR - http://eudml.org/doc/244840

ER -

## References

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