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

## TL;DR

This paper extends the Ambrosetti-Prodi theorem to non-selfadjoint elliptic operators, demonstrating that the associated semilinear operator forms a global fold and establishing a new multiplicity result for such equations.

## Contribution

It generalizes the Ambrosetti-Prodi theorem to non-selfadjoint operators and proves the semilinear operator is a global fold, providing the first exact multiplicity result in non-divergence form.

## Key findings

- Semilinear operator is a global fold
- First exact multiplicity result for non-divergence elliptic equations
- Techniques based on the maximum principle

## Abstract

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.