On the critical points of solutions of Robin boundary problems

In this paper we prove the uniqueness of the critical point for stable solutions of the Robin problem $$\{\begin{array}{ll}-\mathrm{\Delta }u=f(u)& \text{in }\mathrm{\Omega }\\ u>0& \text{in }\mathrm{\Omega }\\ {\mathrm{\partial }}_{\nu }u+\beta u=0& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subseteq {\mathbb{R}}^2$ is a smooth and bounded domain with strictly positive curvature of the boundary, $f\ge 0$ is a smooth function and $\beta >0$. Moreover, for $\beta$ large the result fails as soon as the domain is no more convex, even if it is very close to be: indeed, in this case it is possible to find solutions with an arbitrary large number of critical points.