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.