Abstract

Let λ* > O denote the largest possible value of λ such that {Δ 2u = λe u in B, u = ∂u/∂n = 0 on ∂B} has a solution, where B is the unit ball in &Rdbl; N and n is the exterior unit normal vector. We show that for λ = λ* this problem possesses a unique weak solution u*. We prove that u* is smooth if N ≤ 12 and singular when N ≥ 13, in which case u*(r) = - 4logr + log(8(N -2)(N 4)/λ*) + o(1) as r → 0. We also consider the problem with general constant Dirichlet boundary conditions. © 2007 Society for Industrial and Applied Mathematics.