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

Abstract

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.