Perturbing singular solutions of the Gelfand problem

Abstract

he equation $-Δu = λe^u$ posed in the unit ball $B \subseteq \R^N$, with homogeneous Dirichlet condition $u|_{\partial B} = 0$, has the singular solution $U=\log\frac1{|x|^2}$ when $λ= 2(N-2)$. If $N\ge 4$ we show that under small deformations of the ball there is a singular solution $(u,λ)$ close to $(U,2(N-2))$. In dimension $N\ge 11$ 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 $N\ge 10$, the extremal solution remains bounded in many cases.