Before revision:  Under the additional uniform ellipticity condition, we make use of the available description of the minimal solutions to such equations, along with some basic facts from the potential theory and convex analysis, to obtain an explicit integral representation of all the positive solutions.
After revision:  Under the additional uniform ellipticity condition, we make use of the available description of the minimal solutions to such equations, along with some basic facts from potential theory and convex analysis, to obtain an explicit integral representation of all the positive solutions.