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Results of Ambrosetti-Prodi type for non-selfadjoint elliptic operators. (English) Zbl 06976930

Summary: 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 M. S. Berger and E. Podolak [Indiana Univ. Math. J. 24, 837–846 (1975; Zbl 0329.35026)], 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.

MSC:

47Jxx Equations and inequalities involving nonlinear operators
47Bxx Special classes of linear operators
47Hxx Nonlinear operators and their properties
54Hxx Connections of general topology with other structures, applications
35Jxx Elliptic equations and elliptic systems

Citations:

Zbl 0329.35026
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References:

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