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

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 Cabre-Chanillo to the case where the curvature of $$\partial \Omega $$ vanishes.