Finite and infinite energy solutions of singular elliptic problems: Existence and uniqueness

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by

$?\mathrm{\Delta }u=h(u)f\text{in}\mathrm{\Omega },$

where f is an irregular datum, possibly a measure, and h is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.