Abstract

The equation - Δu = λeu posed in the unit ball B ⊆ℝN , with homogeneous Dirichlet condition u|∂B = 0, has the singular solution U = log 1/x2 when λ = 2(N -2). If N ≥ 4 we show that under small deformations of the ball there is a singular solution (u, λ) close to (U,2(N -2)). In dimension JV ≥ 11 it corresponds to the extremal solution -the one associated to the largest λ for which existence holds. In contrast, we prove that if the deformation is sufficiently large then even when N > 10, the extremal solution remains bounded in many cases. © World Scientific Publishing Company.