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 $$\begin{cases} -\Delta u=f(u)&\text{in }\Omega\\ u>0&\text{in }\Omega\\ u=0&\text{on } \partial\Omega,\end{cases}$$ where $\Omega\subset\mathbb{R}^2$ is a smooth bounded domain whose boundary has \emph{nonnegative} curvature and $f(0)\ge0$. It extends a result by Cabr\'e-Chanillo to the case where the curvature of $\partial\Omega$ vanishes.