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.