Uniqueness of solutions for a nonlocal elliptic eigenvalue problem

We examine equations of the form\[\begin{cases}\HA u =\lambda g(x) f(u) & \text{in}\ \Omega \\u=0 & \text{on}\ \pOm,\end{cases}\]where $\lambda>0$ is a parameter and $\Omega$ is a smoothbounded domain in $\IR^N$, $N \ge 2$. Here $g$ is apositive function and $f$ is an increasing, convex functionwith $f(0)=1$ and either $f$ blows up at $1$ or $f$ issuperlinear at infinity. We show that the extremal solution$u^*$ associated with the extremal parameter $\lambda^*$ isthe unique solution. We also show that when $f$ is suitablysupercritical and $\Omega$ satisfies certain geometricalconditions then there is a unique solution for smallpositive $\lambda$.

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Published 8 November 2012