Uniqueness of the critical point for semi-stable solutions in R2

In this paper we show the uniqueness of the critical point for semi-stable solutions of the problem {-Delta u = f(u) in Omega u > 0 in Omega u = 0 on partial derivative Omega, where Omega subset of R-2 is a smooth bounded domain whose boundary has nonnegative curvature and f(0) >= 0. It extends a result by Cabre-Chanillo to the case where the curvature of partial derivative Omega vanishes.

Automated summary

This paper demonstrates the uniqueness of the critical point for semi-stable solutions under certain conditions, extending a previous result to the case where the curvature of the boundary vanishes.