Symmetry-breaking bifurcation for the one-dimensional Hénon equation

Inbo Sim, Satoshi Tanaka

Research output: Contribution to journalArticlepeer-review

Abstract

We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation u″ + |x|lup = 0, x ∈ (-1, 1), u(-1) = u(1) = 0, where l > 0 and p > 1. Moreover, employing a variant of Rabinowitz's global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz's global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz's global bifurcation) for the problem u″ + |x|l(λ)up = 0, x ∈ (-1, 1), u(-1) = u(1) = 0, where l is a specified continuous parametrization function.

Original languageEnglish
Article number1750097
JournalCommunications in Contemporary Mathematics
Volume21
Issue number1
DOIs
Publication statusPublished - 2019 Feb 1

Keywords

  • Hénon equation
  • positive solution
  • symmetry-breaking bifurcation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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