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Periodic solutions for nonlinear elliptic equations. Application to charged particle beam focusing systems. (English) Zbl 1133.78307
Summary: We study the existence of spatial periodic solutions for nonlinear elliptic equations of the form $-\Delta u+ g(x, u(x))=0,\quad x\in\mathbb{R}^N,$ where $$g$$ is a continuous function, nondecreasing with respect to $$u$$. We give necessary and sufficient conditions for the existence of periodic solutions. Some cases with nonincreasing function $$g$$ are investigated as well. As an application we analyze a mathematical model for the electron beam focusing system and prove the existence of positive periodic solutions for the envelope equation. We also present numerical simulations.
##### MSC:
 78A35 Motion of charged particles 35J60 Nonlinear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B10 Periodic solutions to PDEs 47J30 Variational methods involving nonlinear operators 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces