On non-radially symmetric bifurcation in the annulus

AbstractWe 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 R2: a2 < |x| < 1. We prove that, for each a ϵ (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 (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