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Symmetry-breaking bifurcation for the Moore–Nehari differential equation

  • Ryuji Kajikiya
  • Inbo Sim
  • Satoshi Tanaka
Article
  • 48 Downloads

Abstract

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

Keywords

Bifurcation Positive solution Symmetry-breaking Morse index 

Mathematics Subject Classification

34B09 34B18 34C23 34L30 

Notes

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Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and EngineeringSaga UniversitySagaJapan
  2. 2.Department of MathematicsUniversity of UlsanUlsanRepublic of Korea
  3. 3.Department of Applied Mathematics, Faculty of ScienceOkayama University of ScienceOkayamaJapan

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