A variational approach to the quaternionic Monge-Ampere equation

In this paper, we use the variational method to solve the quaternionic Monge-Ampere equation when the right-hand side is a positive measure of finite energy. We introduce finite energy classes of quaternionic plurisubharmonic functions of Cegrell type and define the quaternionic Monge-Ampere operator on some Cegrell's classes, the functions of which are not necessarily bounded. By using the theory of quaternionic closed positive current, we show that integration by parts and comparison principle are valid on some classes. This opens the door to prove results in the quaternionic pluripotential theory as in the seminal framework by Cegrell (Acta Math 180(2):187-217, 1998; Ann Inst Fourier (Grenoble) 54(1):159-179, 2004; Ann Polon Math 94(2):131-147, 2008) for the complex Monge-Ampere case.