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.

爱因斯坦质量引力理论的场方程已在三维球对称时空中，在存在三种非线性电动力学理论的情况下分别求解。结果，在质量引力理论中获得了三类新颖的带非线性电荷的黑洞，它们是渐近的大质量BTZ黑洞。当非线性参数选择得非常大时，重力和电磁场方程的解可以在爱因斯坦-麦克斯韦质量理论中恢复其相应的量。已经计算出黑洞的守恒和热力学量，例如质量，电荷，温度，熵和电势，并且已经表明，热力学第一定律对所有新的黑洞都是有效的。黑洞的热力学稳定性和相变已通过采用典型的集合和热力学几何方法进行了分析，并注意到了热容量和热力学Ricci标量。然后比较了这些替代方法的结果。最后，利用宏正则系综并考虑了吉布斯自由能，研究了黑洞的整体稳定性和霍金-佩奇相变。