(p, 2)-Equations with a Crossing Nonlinearity and Concave Terms

Abstract

We consider a parametric Dirichlet problem driven by the sum of a p-Laplacian (\(p>2\)) and a Laplacian (a (p, 2)-equation). The reaction consists of an asymmetric \((p-1)\)-linear term which is resonant as \(x \rightarrow - \infty \), plus a concave term. However, in this case the concave term enters with a negative sign. Using variational tools together with suitable truncation techniques and Morse theory (critical groups), we show that when the parameter is small the problem has at least three nontrivial smooth solutions.

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Correspondence to Calogero Vetro.

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Papageorgiou, N.S., Vetro, C. & Vetro, F. (p, 2)-Equations with a Crossing Nonlinearity and Concave Terms. Appl Math Optim 81, 221–251 (2020). https://doi.org/10.1007/s00245-018-9482-0

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Keywords

  • p-Laplacian
  • Concave term
  • Crossing nonlinearity
  • Nonlinear regularity
  • Nonlinear maximum principle
  • Critical groups
  • Multiple smooth solutions

Mathematics Subject Classification

  • 35J20
  • 35J60
  • 58E05