Elliptic Equations Governing Extremely Charged Cosmological Dust Coupled With A Dilaton

In this paper, we study the coupled Einstein and Maxwell equations which alsocouple to a dilaton field in the framework of general relativity, which obtained a gravita-tional system. We will use the methods of the sub-supersolution and the energy methodto establish a series of existence theory for this important gravitational system. Finally,we also present some asymptotic estimates for the solutions of this system.