Bifurcation from infinity with applications to reaction-diffusion systems

Chihiro Aida, Chao Nien Chen, Kousuke Kuto, Hirokazu Ninomiya

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Abstract

The bifurcation method is one of powerful tools to study the existence of a continuous branch of solutions. However without further analysis, the local theory only ensures the existence of solutions within a small neighborhood of bifurcation point. In this paper we extend the theory of bifurcation from infinity, initiated by Rabinowitz [11] and Stuart [13], to find solutions of elliptic partial differential equations with large amplitude. For the applications to the reaction-diffusion systems, we are able to relax the conditions to obtain the bifurcation from infinity for the following nonlinear terms; (i) nonlinear terms satisfying conditions similar to [11] (all directions), (ii) nonlinear terms satisfying similar conditions only on the strip domain along the direction determined by the eigenfunction, (iii) p-homogeneous nonlinear terms with degenerate conditions.

Original languageEnglish
Pages (from-to)3031-3055
Number of pages25
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume40
Issue number6
DOIs
Publication statusPublished - 1 Jan 2020

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Keywords

  • Bifurcation from infinity
  • P-homogeneity
  • Reaction-diffusion systems

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