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

We study the existence of spatial periodic solutions for nonlinear elliptic equations -Delta u + g( x, u( x)) = 0, x is an element of 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.