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

Mihai Bostan; Eric Sonnendrücker

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- 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 40.6 (2007): 1023-1052. <http://eudml.org/doc/194343>.

@article{Bostan2007,

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},

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

language = {eng},

month = {2},

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/194343},

volume = {40},

year = {2007},

}

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

DA - 2007/2//

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.; nonlinear elliptic equations; existence and uniqueness; electron beam focusing system

UR - http://eudml.org/doc/194343

ER -

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- Z. Meiyue, C. Taiyoung, L. Wenbin and J. Yong, Existence of positive periodic solution for the electron beam focusing system. Math. Meth. Appl. Sci.28 (2005) 779–788. Zbl1069.34065
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