A CONSIDERABLE amount of work has been devoted, in recent years, to stationary free boundary problems. In particular, obstacle problems have been the subject of extensive work concerning existence, estimates and regularity of the solution or of the coincidence set, and occurrence of multiple solutions. In [6, 8, 161 the authors consider obstacle problems with nonlinear forcing terms, leading to multiple solutions, hence to turning points when a real parameter appears in the forcing term. A local study of the branches of solutions is set up; the results obtained are the extension, in a weaker form, of those well known for equations [ll, 12). However, some important classical questions remain open for obstacle problems: