Gravitating BIon and BIon black hole with a dilaton

We construct static and spherically symmetric particlelike and black hole solutions with magnetic or electric charge in the Einstein-Born-Infeld-dilaton system, which is a generalization of the Einstein-Maxwell-dilaton (EMD) system and of the Einstein-Born-Infeld (EBI) system. They have remarkable properties which are not seen for the corresponding solutions in the EBI and the EMD system. In the electrically charged case, the electric field is neutralized at the origin by the effect of the diverging dilaton field. The extreme solution does not exist but we can take the zero horizon radius limit for any Born-Infeld (BI) parameter b. Although the solution in this limit corresponds to the "particlelike" solutions in the Einstein frame, it is not a relevant solution in the string frame. In the magnetically charged case, the extreme solution does exist for the critical BI parameter root bQ(m)=1/2. The critical BI parameter divides the solutions qualitatively. For root bQ(m)<1/2, there exists the particlelike solution for which the dilaton field is finite everywhere, while no particlelike solution exists and the solution in the r(h)-->0 limit becomes naked for root bQ(m)>1/2. Examining the internal structure of the black holes, we find that there is no inner horizon and that the global structure is the same as the Schwarzschild one in any charged case.