Exact black hole solutions with non-linear electrodynamic field
We construct exact black hole solutions to Einstein gravity with nonlinear electrodynamic field. In these solutions, there are in general four parameters. They are physical mass, electric charge, cosmological constant and the coupling constant. Due to the presence of coupling constant, these solutions differ significantly from the Reissner-Nordstrom-de Sitter solution in Einstein-Maxwell gravity with a cosmological constant. In the first place, some of them are endowed with a topological defect on angle $θ$. Secondly, for the extreme charged black holes, the electric charge can be much larger or much smaller than the mass by varying the coupling constant. Thirdly, for very large coupling constant, the radius of event horizon for some black holes approaches $2M$ regardless of the electric charge. Lastly, some black holes are asymptotically neither flat nor de Sitter (or anti-de Sitter).