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On the spectrum of a Baouendi–Grushin type operator: an Orlicz–Sobolev space setting approach

  • Mihai Mihăilescu
  • Denisa Stancu-Dumitru
  • Csaba Varga
Article
  • 127 Downloads

Abstract

We show that the spectrum of a nonhomogeneous Baouendi–Grushin type operator subject with a homogeneous Dirichlet boundary condition is exactly the interval \({(0,\infty)}\) . This is in sharp contrast with the situation when we deal with the “classical” Baouendi–Grushin operator (i.e., an operator of type \({-\Delta_x-|x|^{\xi}\Delta_y}\)) when the spectrum is an increasing and unbounded sequence of positive real numbers. Our proofs rely on a symmetric mountain-pass argument due to Kajikiya. In addition, we can show that for each eigenvalue there exists a sequence of eigenfunctions converging to zero.

Keywords

Baouendi–Grushin type operator Orlicz–Sobolev space Hypoelliptic equation Nonlinear eigenvalue problem Variational methods 

Mathematics Subject Classification

47J10 47J30 49R05 35H10 46E30 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Mihai Mihăilescu
    • 1
    • 2
  • Denisa Stancu-Dumitru
    • 3
  • Csaba Varga
    • 3
  1. 1.Department of MathematicsUniversity of CraiovaCraiovaRomania
  2. 2.Research Group of the Project PN-II-ID-PCE-2012-4-0021“Simion Stoilow” Institute of Mathematics of the Romanian AcademyBucharestRomania
  3. 3.Department of MathematicsBabeş-Bolyai UniversityCluj-NapocaRomania

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