  We consider a simplified version of the Budyko diffusive energy balance
climate model. We obtain the exact number of monotone stationary solutions of
the associated discontinuous nonlinear elliptic with absorption. We show that
the bifurcation curve, in terms of the solar constant parameter, is S-shaped.
We prove the instability of the decreasing part and the stability of the
increasing part of the bifurcation curve. In terms of the Budyko climate
problem the above results lead to an important qualitative information which is
far to be evident and which seems to be new in the mathematical literature on
climate models. We prove that if the solar constant is represented by $\lambda
\in (\lambda _{1},\lambda _{2}),$ for suitable $\lambda _{1}<\lambda _{2},$
then there are exactly two stationary solutions giving rise to a free boundary
(i.e. generating two symmetric polar ice caps: North and South ones) and a
third solution corresponding to a totally ice covered Earth. Moreover, we prove
that the solution with smaller polar ice caps is stable and the one with bigger
ice caps is unstable.
