Bifurcation from infinity with applications to reaction-diffusion systems

Chihiro Aida; Chao-Nien Chen; Kousuke Kuto; Hirokazu Ninomiya

doi:10.3934/dcds.2020053

Article Contents

2020, Volume 40, Issue 6: 3031-3055. Doi: 10.3934/dcds.2020053 open access

This issue Previous Article Preface: DCDS-A special issue to honor Wei-Ming Ni's 70th birthday Next Article Evolution equations involving nonlinear truncated Laplacian operators

Bifurcation from infinity with applications to reaction-diffusion systems

1.
Nakano Junior and Senior High School Attached to Meiji University, 3-3-4 Higashi-Nakano, Nakano-ku, 164-0003, Japan
2.
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan
3.
Department of Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
4.
School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan

^* Corresponding author: Kousuke Kuto
^* Corresponding author: Kousuke Kuto

Received: March 31, 2019

Revised: April 30, 2019

Published: March 2020

The second author was supported in part by the Ministry of Science and Technology of Taiwan, grant 105-2115-M-007-009-MY3. The third author was partially supported by JSPS KAKENHI Grant-in-Aid Grant Numbers 15K04948 and 19K03581. The fourth author would like to thank the Mathematics Division of NCTS (Taipei Office) for the warm hospitality and the support of the fourth author's visit to Taiwan. The fourth author was partially supported by JSPS KAKENHI Grant Numbers JP26287024, JP15K04963, JP16K13778 and JP16KT0022

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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.

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Mathematics Subject Classification: Primary: 35B32, 35B36; Secondary: 35B44.

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References

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