This note deals with the existence of periodic solutions via shooting method for a class of semilinear diffusion equations arising from highly idealized climate models. Sub- and supersolutions come from an associated stationary diffusion problem studied by the author [The structure of the principal component for semilinear diffusion equations from energy balance climate models (to appear)]. The procedure closely follows by H. Amann [Nonlinear analysis, 1-29 (1978; Zbl 0464.35050)]. From the climatic viewpoint it is important that our approach allows us to estimate the amplitude of these periodic solutions by means of a forthcoming numerical analysis of the stationary diffusion problem which is based on the work by H. Jarausch and W. Mackens [Large scale scientific computing, Proc. Meet., Oberwolfach/FRG 1985, Proc. Sci. Comput. 7, 114-137 (1987; Zbl 0617.65048)].