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

In this paper we show the uniqueness of the critical point for \emph{semi-stable} solutions of the problem $\begin{cases} -Δu=f(u)&\text{in }Ω\\ u>0&\text{in }Ω\\ u=0&\text{on } \partialΩ,\end{cases}$ where $Ω\subset\mathbb{R}^2$ is a smooth bounded domain whose boundary has \emph{nonnegative} curvature and $f(0)\ge0$. It extends a result by Cabré-Chanillo to the case where the curvature of $\partialΩ$ vanishes.

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