Asymptotically massive-BTZ black holes with nonlinear electrodynamics in massive gravity theory

The field equations of Einstein-massive gravity theory has been solved in a three-dimensional spherically symmetric spacetime, and in the presence of three alternative theories of nonlinear electrodynamics, separately. As the result, three novel classes of nonlinearly charged black holes have been obtained in massive gravity theory, which are asymptotically massive-BTZ black holes. The solutions of gravitational and electromagnetic field equations recover their corresponding quantities in the Einstein–Maxwell-massive theory, when the nonlinearity parameters are chosen very large. The conserved and thermodynamic quantities of the black holes such as mass, charge, temperature, entropy and electric potential have been calculated, and it has been shown that the thermodynamical first law is valid for all of new black holes. Thermodynamic stability and phase transition of the black holes have been analyzed by applying the canonical ensemble and thermodynamic geometry approaches, and noting the heat capacities and thermodynamic Ricci scalars. Then results of these alternative approaches have been compared. Finally, global stability and Hawking–Page phase transition of the black holes have been studied making use of the grand canonical ensemble and regarding the Gibbs free energies.