Positive versus free boundary solutions to a singular elliptic equation

The equation -Deltau = chi({u>0}) (-1/ubeta + lambdaf (x,u)) in Omega with Dirichlet boundary condition on partial derivativeOmega has a maximal solution u(lambda) greater than or equal to 0 for every lambda > 0. For lambda less than a constant lambda*, the solution vanishes inside the domain; and for lambda > lambda*, the solution is positive. We obtain optimal regularity of u(lambda) even in the presence of the free boundary.9030333