Sublinear elliptic problems with a Hardy potential

We discuss the existence and the boundary behavior of positive solutions of an elliptic equation with a Hardy potential and a sublinear nonlinearity. This problem has two particular features: the Hardy potential is singular at the boundary and the unique continuation property doesn’t hold. The Hardy potential forces the solutions either to vanish or to blow up at the boundary. The picture of the radial solutions in balls and annuli is fairly complete. At the end we present results for general domains and point out some open problems.