Uniqueness of solutions for a nonlocal elliptic eigenvalue problem

We examine equations of the form {eqnarray*} \{{array}{lcl} \hfill \HA u &=& λg(x) f(u) \qquad \text{in}\ Ω\hfill u&=& 0 \qquad \qquad \qquad \text{on}\ \pOm, {array}. {eqnarray*} where $λ>0$ is a parameter and $Ω$ 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 $λ^*$ is the unique solution. We also show that when $f$ is suitably supercritical and $Ω$ satisfies certain geometrical conditions then there is a unique solution for small positive $λ$.

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