Uniqueness of solutions for a nonlocal elliptic eigenvalue problem

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 $\lambda$.