PERTURBING SINGULAR SOLUTIONS OF THE GELFAND PROBLEM

Abstract

The equation -Δu = λeuposed in the unit ball B ⊆ ℝN, with homogeneous Dirichlet condition u|∂B= 0, has the singular solution [Formula: see text] 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 N ≥ 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.