Solitons and black holes in Einstein-Born-Infeld-dilaton theory
  Static spherically symmetric asymptotically flat solutions to the U(1) Born-Infeld theory coupled to gravity and to a dilaton are investigated. The dilaton enters in such a way that the theory is $SL(2,R)$- symmetric for a unit value of the dilaton coupling constant. We find globally regular solutions for any value of the effective coupling which is the ratio of the gravitational and dilaton couplings; they form a continuous sequence labeled by the sole free parameter of the local solution near the origin. The allowed values of this parameter are bounded from below, and, as the limiting value is approached, the mass and the dilaton charge rise indefinitely and tend to a strict equality (in suitable units). Together with the electric charge they saturate the BPS bound of the corresponding linear U(1) theory. Beyond this boundary the solutions become compact (singular), while the limiting solution at the exact boundary value is non-compact and non-asymptotically flat. The black holes in this theory exist for any value of the event horizon radius and also form a sequence labelled by a continuously varying parameter restricted to a finite interval. The singularity inside the black hole exhibits a power-law mass inflation. 
Gerard Clement and Dmitri Gal'tsov