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Nonuniqueness for the Dirichlet Problem for Fully Nonlinear Elliptic Operators and the Ambrosetti–Prodi Phenomenon

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Analysis and Topology in Nonlinear Differential Equations

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 85))

Abstract

We study the uniformly elliptic fully nonlinear PDE F(D2u,Du,u,x)=f(x) in Ω where F is a convex positively 1-homogeneous operator and \( \Omega \subset \mathbb{R}^{N} \) 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.

Mathematics Subject Classification (2010). 35J25, 35B34, 35P30.

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Authors and Affiliations

  1. Departamento de Matemática, Pontificia Universidade Católica do Rio de Janeiro (PUC-Rio), Rua Marques de São Vicente, 225, Gávea, Rio de Janeiro, RJ, CEP 22451-900, Brasil

    Boyan Sirakov

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  1. Boyan Sirakov
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Editors and Affiliations

  1. IMECC-UNICAMP, Cidade Universitaria Instituto de Matematica, Estatistica e C, Campinas, São Paulo, Brazil

    Djairo G de Figueiredo

  2. Departamento de Matemática - CCEN, Universidade Federal da Paraíba, João Pessoa, Paraíba, Brazil

    João Marcos do Ó

  3. Departamento de Matemática, Pontifícia Universidade Católica do Rio, Rio de Janeiro, Rio de Janeiro, Brazil

    Carlos Tomei

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Sirakov, B. (2014). Nonuniqueness for the Dirichlet Problem for Fully Nonlinear Elliptic Operators and the Ambrosetti–Prodi Phenomenon. In: de Figueiredo, D., do Ó, J., Tomei, C. (eds) Analysis and Topology in Nonlinear Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 85. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-04214-5_24

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