On non-radially symmetric bifurcation in the annulus

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} 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.