Bifurcations for semilinear elliptic equations convex nonlinearity

Karátson, J. ; Simon, P.L.

Electronic Journal of Differential Equations, Tome 1999 (1999).

Résumé

Summary: We investigate the exact number of positive solutions of the semilinear Dirichlet boundary value problem $$\Delta u+f(u) = 0$$ on a ball in $R^{n}$ 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.

Détail

Classification : 35J60
Keywords: semilinear elliptic equations, time-map, bifurcation diagram