Le nombre de solutions de certains problèmes semi-linéaires elliptiques

AbstractIn this paper, we compute the exact number of solutions for certain semi-linear elliptic problems. In the first part, we derive some complementary results to a theorem of Ambrosetti and Prodi. The global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity: in the second part we consider the branch of positive solutions. Lastly, in the third part, in dimension one, for an autonomous equation, we obtain a precise description of all the solutions to these problems. The proofs rely on simple eigenvalue comparison arguments