Einstein-Born-Infeld black holes with a scalar hair in three dimensions

We present black hole solutions in (2+1)-dimensional Einstein's theory of gravity coupled with Born-Infeld (BI) nonlinear electrodynamic and a massless self-interacting scalar field. The model has five free parameters: mass M, cosmological constant l, electric q and scalar r(0) charges and BI parameter beta. To attain exact solution for such a highly nonlinear system we adjust, i.e. finely tune, the parameters of the theory with the integration constants. In the limit beta -> 0, we recover the results of Einstein-Maxwell-Scalar theory, obtained before. The self-interacting potential admits finite minima apt for the vacuum contribution. Hawking temperature of the model is investigated versus properly tuned parameters. By employing this tuned-solution as basis, we obtain also a dynamic solution which in the proper limit admits the known solution in Einstein gravity coupled with self-interacting scalar field. Finally, we establish the equations of a general scalar-tensor field coupled to nonlinear electrodynamics (NED) field in 2 +1 dimensions without searching for exact solutions.