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On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains

Abstract

The goal of this note is to study the asymptotic behavior of positive solutions for a class of semilinear elliptic equations which can be realized as minimizers of their energy functionals. This class includes the Fisher-KPP and Allen–Cahn nonlinearities. We consider the asymptotic behavior in domains becoming infinite in some directions. We are in particular able to establish an exponential rate of convergence for this kind of problems.

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Acknowledgments

This work has been performed during a visit of the first author at the Universidad de Chile in Santiago and at the SBAI at the Sapienza Università di Roma. He would like to thank both institutions for their kind hospitality. The second and third authors have been supported by Grants Fondecyt 1130360, 1150066, Fondo Basal CMM and Millenium Nucleus CAPDE NC130017.

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Correspondence to Manuel del Pino.

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Chipot, M., Dávila, J. & del Pino, M. On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains. J. Fixed Point Theory Appl. 19, 205–213 (2017). https://doi.org/10.1007/s11784-016-0349-1

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  • DOI: https://doi.org/10.1007/s11784-016-0349-1

Keywords

  • Semilinear elliptic equation
  • Cylindrical domain
  • Asymptotic behavior

Mathematics Subject Classification

  • 35J15
  • 35J25
  • 35J61