New static spherically symmetric solutions of the Einstein-Maxwell gravity with the dilaton field are found. The solutions correspond to black holes and naked singularities. In addition to mass and electric charge these solutions are labeled by a new parameter, the dilaton charge of the black hole. Depending on the values of electric and dilaton charges there are different types of solutions. The solutions exhibit a new type of a symmetry. Namely one solution transforms into another when the mass and the dilaton charge are interchanged. We also found that there is a finite interval of values of electric charge for which no black hole can exist. This gap separates two different types of solutions. Inside the gap the solution can exist only if the dilaton charge is exactly equal to the negative of the mass. The behavior of the solution in the extremal regime is also analyzed.
Rakhmanov, Malik. “Dilaton Black Holes with Electric Charge.” Physical Review D, vol. 50, no. 8, American Physical Society, Oct. 1994, pp. 5155–63, doi:10.1103/PhysRevD.50.5155.
Physical Review D