Critical points of solutions to elliptic problems in planar domains
Given a planar domain \Omega$ and an analytic function $f$ we describe the set of critical points for the solution $u$ of the semilinear elliptic problem \Delta u f(u) in \Omega$ $u=0$ on \partial\Omega$ For simply connected domains we establish that the set of critical points is finite while for non simply connected domains we show that this set is made up of finitely many isolated points and finitely many analytic Jordan curves. Further results are given in the case that \Omega$ is an annular domain whose border has nonzero curvature.