Elliptic equations governing extremely charged cosmological dust coupled with the dilaton

In this paper, we consider the coupled Einstein and Maxwell equations which are also coupled to a dilaton field in the framework of general relativity. Within the Majumdar–Papapetrou framework, for the static Einstein–Maxwell equations with charged dust as the external source of the fields, one can reduce the electrovacuum field equations into the Poisson equation in the flat space. By using the sub-supersolution method and an energy method, we establish a series of existence theorems for the solutions of this important gravitational system.