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The well-known Ambrosetti-Prodi theorem considers perturbations of theDirichlet Laplacian by a nonlinear function whose derivative jumps over theprincipal eigenvalue of the operator. Various extensions of this landmarkresult were obtained for self-adjoint operators, in particular by Berger andPodolak, who gave a geometrical description of the solution set. In this textwe show that similar theorems are valid for non self-adjoint operators. Inparticular, we prove that the semilinear operator is a global fold. As aconsequence, we obtain what appears to be the first exact multiplicity resultfor elliptic equations in non-divergence form. We employ techniques based onthe maximum principle.

Results of Ambrosetti-Prodi type for non-selfadjoint elliptic operators