## Abstract

We discuss the radially symmetric solutions and the non-radially symmetric bifurcation of the semilinear elliptic equation Δu + 2δe^{u} = 0 in Ω and u = 0 on ∂Ω, where Ω = {xε{lunate} R^{2}: a^{2} < |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 language | English |
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Pages (from-to) | 251-279 |

Number of pages | 29 |

Journal | Journal of Differential Equations |

Volume | 80 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 1989 |