Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, p. 1023-1052

We study the existence of spatial periodic solutions for nonlinear elliptic equations $-\Delta u\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}g\left(x,u\left(x\right)\right)=0,\phantom{\rule{0.277778em}{0ex}}x\in {ℝ}^{N}$ 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.

DOI : https://doi.org/10.1051/m2an:2006039
Classification:  35A05,  35B35
Keywords: nonlinear elliptic equations, periodic solutions, existence and uniqueness, electron beam focusing system
@article{M2AN_2006__40_6_1023_0,
author = {Bostan, Mihai and Sonnendr\"ucker, Eric},
title = {Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {40},
number = {6},
year = {2006},
pages = {1023-1052},
doi = {10.1051/m2an:2006039},
zbl = {1133.78307},
mrnumber = {2297103},
language = {en},
url = {http://www.numdam.org/item/M2AN_2006__40_6_1023_0}
}

Bostan, Mihai; 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, Volume 40 (2006) no. 6, pp. 1023-1052. doi : 10.1051/m2an:2006039. http://www.numdam.org/item/M2AN_2006__40_6_1023_0/

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