In this paper we show the uniqueness of the critical point for semi-stable solutions of the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=f(u)&{}\quad \text {in }\Omega \\ u>0&{}\quad \text {in }\Omega \\ u=0&{}\quad \text {on }\partial \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \subset \mathbb {R}^2\) is a smooth bounded domain whose boundary has nonnegative curvature and \(f(0)\ge 0\). It extends a result by Cabré-Chanillo to the case where the curvature of \(\partial \Omega \) vanishes.

中文翻译：

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$$ \ mathbb {R} ^ 2 $$ R 2中半稳定解的临界点的唯一性

在本文中，我们显示了问题的半稳定解的临界点的唯一性

$$ \ begin {aligned} {\ left \ {\ begin {array} {ll}-\ Delta u = f（u）＆{} \ quad \ text {in} \ Omega \\ u> 0＆{} \ quad \ text {in} \ Omega \\ u = 0＆{} \ quad \ text {on} \ partial \ Omega，\ end {array} \ right。} \ end {aligned} $$

其中\（\ Omega \ subset \ mathbb {R} ^ 2 \）是一个光滑的有界域，其边界具有非负曲率和\（f（0）\ ge 0 \）。它将Cabré-Chanillo的结果扩展到\（\ partial \ Omega \）的曲率消失的情况。