Exact asymptotically flat charged hairy black holes with a dilaton potential

We find broad classes of exact 4-dimensional asymptotically flat black hole solutions in Einstein-Maxwell theories with a non-minimally coupled dilaton and its non-trivial potential. We consider a few interesting limits, in particular, a regular generalization of the dilatonic Reissner-Nordstr{\"o}m solution and, also, smooth deformations of supersymmetric black holes. Further examples are provided for more general dilaton potentials. We discuss the thermodynamical properties and show that the first law is satisfied. In the non-extremal case the entropy depends, as expected, on the asymptotic value of the dilaton. In the extremal limit, the entropy is determined purely in terms of charges and is independent of the asymptotic value of the dilaton. The attractor mechanism can be used as a criterion for the existence of the regular solutions. Since there is a `competition' between the effective potential and dilaton potential, we also obtain regular extremal black hole solutions with just one U(1) gauge field turned on.