The well-known Ambrosetti–Prodi theorem considers perturbations of the Dirichlet Laplacian by a nonlinear function whose derivative jumps over the principal eigenvalue of the operator. Various extensions of this landmark result were obtained for self-adjoint operators, in particular by Berger and Podolak, who gave a geometrical description of the solution set. In this text we show that similar theorems are valid for non-self-adjoint operators. In particular, we prove that the semilinear operator is a global fold. As a consequence, we obtain what appears to be the first exact multiplicity result for elliptic equations in non-divergence form. We employ techniques based on the maximum principle.
Cite this article
Boyan Sirakov, Carlos Tomei, André Zaccur, Results of Ambrosetti–Prodi type for non-selfadjoint elliptic operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 7, pp. 1757–1772DOI 10.1016/J.ANIHPC.2018.03.001