Uniqueness of the critical point for semi-stable solution in ${\mathbb{R}}^2$

Abstract

In this paper we show the uniqueness of the critical point for \emph{semi-stable} solutions of the problem $$\{\begin{array}{ll}-\Delta u=f(u)& \text{in }\Omega \\ u>0& \text{in }\Omega \\ u=0& \text{on }\partial \Omega ,\end{array}$$ where $\Omega \subset {\mathbb{R}}^2$ is a smooth bounded domain whose boundary has \emph{nonnegative} curvature and $f(0)\ge 0$. It extends a result by Cabré-Chanillo to the case where the curvature of $\partial \Omega$ vanishes.