On non-radially symmetric bifurcation in the annulus

Song-Sun Lin*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

35 Scopus citations

Abstract

We discuss the radially symmetric solutions and the non-radially symmetric bifurcation of the semilinear elliptic equation Δu + 2δeu = 0 in Ω and u = 0 on ∂Ω, where Ω = {xε{lunate} R2: a2 < |x| < 1. We prove that, for each a ε{lunate} (0, 1), there exists a decreasing sequence δ*(k, a)k = 0 with δ*(k, a) → 0 as k → ∞ such that the equation has exactly two radial solutions for δ ε{lunate} (0, δ*(0, a)), exactly one for δ = δ*(0, a), and none for δ > δ*(0, a). The upper branch of radial solutions has a non-radially symmetric bifurcation (symmetry breaking) at each δ*(k, a), k ≥ 1. As a → 0, the radial solutions will tend to the radial solutions on the disk and δ*(0, a) → δ* = 1, the critical number on the disk.

Original languageEnglish
Pages (from-to)251-279
Number of pages29
JournalJournal of Differential Equations
Volume80
Issue number2
DOIs
StatePublished - 1 Jan 1989

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