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On nonlinear parametric problems for p-Laplacian-like operators

Sobre los problemas paramétricos no lineales con operadores de tipo p-Laplaciano

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Abstract

We study a nonlinear parametric problem driven by a p-Laplacian-like operator (which need not be homogeneous) and with a ( p - 1)-superlinear nonlinearity which satisfy weaker conditions than the Ambrosetti-Rabinowitz condition. Using critical point theory, we show that for every λ > 0, the nonlinear parametric problem has a nontrivial solution. Then, by strengthening the conditions on the operator and the nonlinearity, and using variational methods together with suitable truncation techniques and tools from Morse theory, we show that, for every λ > 0, the nonlinear parametric problem has three nontrivial smooth solutions.

Resumen

En este artículo estudiamos un problema paramétrico no lineal que involucra al operador de tipo p-Laplaciano (que en general no és homogéneo) y donde la derivada del potencial es una función ( p - 1)-superlinear que verifica una condición más débil que la conocida condición de Ambrosetti-Rabinowitz. Utilizando métodos variacionales, mostramos que, para todo λ > 0, el problema paramétrico no lineal tiene una solución no trivial. Entonces, fortaleciendo las condiciones y usando herramientas de la teoría de Morse junto con adecuadas técnicas de truncación, mostramos que, para cada λ > 0, el problema tiene tres soluciones suaves.

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Correspondence to Nikolaos S. Papageorgiou.

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Papageorgiou, N.S., Rocha, E.M. On nonlinear parametric problems for p-Laplacian-like operators. Rev. R. Acad. Cien. Serie A. Mat. 103, 177–200 (2009). https://doi.org/10.1007/BF03191850

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Keywords

  • p-Laplacian-like operator
  • superlinear nonlinearity
  • strong deformation retract
  • critical groups and Morse theory

Mathematics Subject Classifications

  • 35J25
  • 35J80
  • 58E05