We obtain a class of solutions that correspond to a generalization of the Bardeen black hole solution by solving the Einstein equations coupled to a particular nonlinear electromagnetic field. The generalization is realized by considering, additionally, the presence of the cosmological constant and a source corresponding to an anisotropic fluid, namely, a fluid of strings, which surrounds the black hole. We show that the obtained class of solutions preserve or not the regularity of the original Bardeen black hole solution, depending on the values of the parameter \(\beta \) which labels the different solutions. We discuss the characteristics of the solutions, from the point of view of the singularities of the spacetime, by examining the behavior of the Kretschmann scalar as well as of the geodesics with respect to their completeness. We analyze some aspects of the thermodynamics, particularizing to one of the solutions obtained, namely, for \(\beta = \nicefrac {-1}{2}\), in which case the regularity of Bardeen black hole is preserved. We also show that there is an incompatibility between the temperature arising from the first law of the black hole thermodynamics and the one using the surface gravity. Some thermodynamic quantities are obtained and analyzed, as for example, pressure, heat capacity, and the critical points, and we show how these quantities change for different values of the parameter q associated with the original Bardeen solution, as well as with the parameter b associated with the presence of the fluid of strings. The phase transitions are also analyzed by using the equation of state and the Helmholtz free energy.

通过求解耦合到特定非线性电磁场的爱因斯坦方程，我们获得了一类与巴丁黑洞解的推广相对应的解。通过另外考虑宇宙学常数的存在和对应于各向异性流体（即围绕黑洞的弦流体）的源来实现该概括。我们表明，所获得的解类是否保留原始巴丁黑洞解的规律性，具体取决于标记不同解的参数 \(\beta \) 的值。我们从时空奇点的角度，通过检查 Kretschmann 标量以及测地线的完整性的行为来讨论解的特征。我们分析了热力学的某些方面，特别是所获得的解决方案之一，即 \(\beta = \nicefrac {-1}{2}\)，在这种情况下，巴丁黑洞的规律性得以保留。我们还表明，黑洞热力学第一定律产生的温度与利用表面重力产生的温度之间存在不相容性。获得并分析了一些热力学量，例如压力、热容和临界点，并且我们展示了这些量如何随着与原始 Bardeen 解以及参数 b 相关的参数 q 的不同值而变化与弦流体的存在有关。还使用状态方程和亥姆霍兹自由能来分析相变。