Electronic Journal of Differential Equations,
Vol. 1999(1999), No. 43, pp. 1-16.

Title: Bifurcations for semilinear elliptic equations with convex nonlinearity

Authors: J. Karatson (Eotvos Lorand Univ., Budapest, Hungary)
 P. L. Simon (Eotvos Lorand Univ., Budapest, Hungary)

Abstract: 
 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.


Submitted June 22, 1999. Published October 18, 1999.
Math Subject Classifications: 35J60.
Key Words: semilinear elliptic equations; time-map; bifurcation diagram.