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Quasilinear problems under local Landesman–Lazer condition

  • D. Arcoya
  • M. C. M. Rezende
  • E. A. B. SilvaEmail author
Article
  • 114 Downloads

Abstract

This paper presents results on the existence and multiplicity of solutions for quasilinear problems in bounded domains involving the p-Laplacian operator under local versions of the Landesman–Lazer condition. The main results do not require any growth restriction at infinity on the nonlinear term which may change sign. The existence of solutions is established by combining variational methods, truncation arguments and approximation techniques based on a compactness result for the inverse of the p-Laplacian operator. These results also establish the intervals of the projection of the solution on the direction of the first eigenfunction of the p-Laplacian operator. This fact is used to provide the existence of multiple solutions when the local Landesman–Lazer condition is satisfied on disjoint intervals.

Mathematics Subject Classification

35J20 35J92 47J30 

Notes

Acknowledgements

This work was done while the second and third authors were visiting the Departamento de Análisis Matemático, Universidad de Granada. They would like to present their gratitude for the warm hospitaltiy of the whole members of that department.

First author is supported by FEDER-MEC (Spain) PGC2018-096422-B-I00 and Junta de Andalucía FQM-116. Third author is supported by CNPq (Brazil) 311808/2014-0 and 312060/2018-1.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • D. Arcoya
    • 1
  • M. C. M. Rezende
    • 2
  • E. A. B. Silva
    • 2
    Email author
  1. 1.Departamento de Análisis MatemáticoUniversidad de GranadaGranadaSpain
  2. 2.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil

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