Electron. J. Diff. Eqns., Vol. 1999(1999), No. 43, pp. 116.
Bifurcations for semilinear elliptic equations with convex nonlinearity
J. Karatson & P. L. Simon
Abstract:
We investigate the exact number of positive solutions of the semilinear
Dirichlet boundary value problem
on a ball in R^{n}
where f is a strictly convex C^{2}
function on
.
For the onedimensional 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 timemap 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 timemap.
Submitted June 22, 1999. Published October 18, 1999.
Math Subject Classifications: 35J60.
Key Words: semilinear elliptic equations, timemap, bifurcation diagram.
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Janos Karatson
Department of Applied Analysis
Eotvos Lorand University
Budapest, Hungary
email: karatson@cs.elte.hu 

Peter L. Simon
Department of Applied Analysis
Eotvos Lorand University
Budapest, Hungary
email: simonp@cs.elte.hu 
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