  We examine equations of the form {eqnarray*} \{{array}{lcl} \hfill \HA u &=&
\lambda g(x) f(u) \qquad \text{in}\ \Omega \hfill u&=& 0 \qquad \qquad \qquad
\text{on}\ \pOm, {array}.
  {eqnarray*} where $ \lambda >0$ is a parameter and $ \Omega$ is a smooth
bounded domain in $ \IR^N$, $ N \ge 2$. Here $ g$ is a positive function and $
f$ is an increasing, convex function with $ f(0)=1$ and either $ f$ blows up at
1 or $ f$ is superlinear at infinity. We show that the extremal solution $u^*$
associated with the extremal parameter $ \lambda^*$ is the unique solution. We
also show that when $f$ is suitably supercritical and $ \Omega$ satisfies
certain geometrical conditions then there is a unique solution for small
positive $ \lambda$.
