Critical points of solutions to quasilinear elliptic problems

We describe the set of critical points (points of vanishing gradient) associated to solutions of an important class of quasilinear elliptic problems with zero Dirichlet condition in planar domains. We show that the critical set is made up of finitely many isolated points and finitely many (regular) analytic Jordan curves. Further, we generalize the well-known result of Makar-Limanov, according to which the solution to the Poisson equation Δu=1, with zero Dirichlet condition in a convex domain, has a unique critical point.