  In this note we prove uniqueness of the critical point for positive solutions
of elliptic problems in bounded planar domains: we first examine the Poisson
problem - Delta u = f(x,y) finding a geometric condition involving the
curvature of the boundary and the normal derivative of f on the boundary to
ensure uniqueness of the critical point. In the second part we consider stable
solutions of the nonlinear problem -Delta u = f(u) in perturbation of convex
domains.
