Perturbing singular solutions of the Gelfand problem

Abstract

he equation $-\Delta u=\lambda e^u$ posed in the unit ball $B\subseteq {\mathbb{R}}^N$, with homogeneous Dirichlet condition $u{|}_{\partial B}=0$, has the singular solution $U=\log \frac{1}{|x{|}^2}$ when $\lambda =2(N-2)$. If $N\ge 4$ we show that under small deformations of the ball there is a singular solution $(u,\lambda )$ close to $(U,2(N-2))$. In dimension $N\ge 11$ it corresponds to the extremal solution -- the one associated to the largest $\lambda$ 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.