Uniqueness of solutions for a nonlocal elliptic eigenvalue problem

Abstract: 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$.