Non uniqueness for the Dirichlet problem for fully nonlinear elliptic operators and the Ambrosetti-Prodi phenomenon

Boyan Sirakov (1)

1 Departamento de Matematica

We study the uniformly elliptic fully nonlinear PDE F (D 2 u, Du, u, x) = f (x) in Ω, where F is a convex positively 1-homogeneous operator and Ω ⊂ R N is a regular bounded domain. We prove non-existence and multiplicity results for the Dirichlet problem, when the two principal eigenvalues of F are of different sign. Our results extend to more general cases, for instance, when F is not convex, and explain in a new light the classical results of Ambrosetti-Prodi type in elliptic PDE.

Domaines: Equations aux dérivées partielles [math.AP]

Boyan Sirakov. Non uniqueness for the Dirichlet problem for fully nonlinear elliptic operators and the Ambrosetti-Prodi phenomenon. Workshop on Nonlinear Differential Equations, Sep 2012, Joao Pessoa, Brazil, France. pp.16, ⟨10.1007/978-3-319-04214-5_24⟩. ⟨hal-00079186v3⟩