Symmetry-breaking bifurcation for the Moore–Nehari differential equation

Ryuji Kajikiya, Inbo Sim, Satoshi Tanaka

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

Abstract

We study the bifurcation problem of positive solutions for the Moore-Nehari differential equation, u′ ′+ h(x, λ) up= 0 , u> 0 in (- 1 , 1) with u(- 1) = u(1) = 0 , where p> 1 , h(x, λ) = 0 for | x| < λ and h(x, λ) = 1 for λ≤ | x| ≤ 1 and λ∈ (0 , 1) is a bifurcation parameter. We shall show that the problem has a unique even positive solution U(x, λ) for each λ∈ (0 , 1). We shall prove that there exists a unique λ∈ (0 , 1) such that a non-even positive solution bifurcates at λ from the curve (λ, U(x, λ)) , where λ is explicitly represented as a function of p.

Original languageEnglish
Article number54
JournalNonlinear Differential Equations and Applications
Volume25
Issue number6
DOIs
Publication statusPublished - 2018 Dec 1

Keywords

  • Bifurcation
  • Morse index
  • Positive solution
  • Symmetry-breaking

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