Exact black hole solutions with nonlinear 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. These solutions differ significantly from the Reissner-Nordstrom-de Sitter solution in Einstein-Maxwell gravity with a cosmological constant, due to the presence of coupling constant. For example, some of them are endowed with a topological defect. on angle theta and the electric charge of some can be much larger or smaller than their mass by varying the coupling constant. On the other hand, these spacetimes are all asymptotically de Sitter (or antide Sitter). As a result, their causal structure is similar to the Reissner Nordstrom de Sitter spacetime. Finally, the investigations on the thermodynamics reveal that the coupling constant except for solution-4 has the opposite effect as temperature on the phase, structure of black holes. Concretely, the phase-space changes from single phase to three phases with the decrease of temperature. On the contrary, it changes from three phases to a single phase with the decrease of coupling constant.