  he equation $-\Delta u = \lambda e^u$ 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 $\lambda = 2(N-2)$. 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.
