Bifurcations for semilinear elliptic equations with convex nonlinearity

We investigate the exact number of positive solutions of the semilinear Dirichlet boundary value problem $Delta u+f(u) = 0$ on a ball in ${mathbb R}^n$ where $f$ is a strictly convex $C^2$ function on $[0,infty)$. For the one-dimensional case we classify all strictly convex $C^2$ functions according to the shape of the bifurcation diagram. The exact number of positive solutions may be 2, 1, or 0, depending on the radius. This full classification is due to our main lemma, which implies that the time-map cannot have a minimum. For the case $n>1$ we prove that for sublinear functions there exists a unique solution for all $R$. For other convex functions estimates are given for the number of positive solutions depending on $R$. The proof of our results relies on the characterization of the shape of the time-map.