Regularizing effects for an elliptic system of singular equations

A system of two singular semi–linear elliptic equations, patterned after the Schrödinger-Maxwell system, is considered. If the reaction term of the first equation contains a datum $f\in L^m$, existence of positive solutions with finite energy is established for suitable ranges of m. In particular, the results from the theory of single, singular equations are improved. At the same time, thanks to an approach based on approximation schemes and a priori estimates on the approximated sequences of solutions, it is shown that the integrability assumptions on the datum produce higher integrability of the solutions.