Some remarks on Bardeen-AdS black hole surrounded by a fluid of strings

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} 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=-12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = \nicefrac {-1}{2}$$\end{document}, 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.