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Multiplicity Results for some Perturbed and Unperturbed “Zero Mass” Elliptic Problems in Unbounded Cylinders

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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 following nonlinear elliptic problem \( \left\{ \begin{array}{clclclcllc}{{-\Delta u}=g(x,u)+f(x) \;\;\; \rm {in} \; \Omega} \\ {\quad u=0 \qquad \qquad \qquad \; \rm {on} \; \partial \Omega}\end{array}\right. \) on unbounded cylinders \( \Omega = \tilde{\Omega}\times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}, N-m \geq 2, m \geq 1, \) under suitable conditions on g and f. In the unperturbed case \( f(x) \equiv 0, \) by means of the Principle of Symmetric Criticality by Palais and some compact imbeddings in spherically symmetric spaces, existence and multiplicity results are proved by applying Mountain Pass Theorem and its Symmetric version. Multiplicity results are also proved in the perturbed case \( f(x) \equiv 0, \) f(x)≢0 by using Bolle’s Perturbation Methods and suitable growth estimates on min-max critical levels. To this aim, we improve a classical estimate of the number N_(-∆ + V) of the negative eigenvalues of the operator -∆+V(x) when the potential V is partially spherically symmetric.

Mathematics Subject Classification (2010). 35J20; 35J60; 46E35.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Bari “Aldo Moro”, Via E. Orabona 4, I-70125, Bari, Italy

    Sara Barile & Addolorata Salvatore

Authors
  1. Sara Barile
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  2. Addolorata Salvatore
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Correspondence to Sara Barile .

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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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Barile, S., Salvatore, A. (2014). Multiplicity Results for some Perturbed and Unperturbed “Zero Mass” Elliptic Problems in Unbounded Cylinders. 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_3

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