We consider the running coupling constant in holographic models supported by Einstein-dilaton-Maxwell action for heavy and light quarks. To obtain the dependence of the running coupling constant α on temperature and chemical potential we impose boundary conditions on the dilaton field that depend on the position of the horizon. We use two types of boundary conditions: a simple boundary condition with the dilaton field vanishing at the horizon and a boundary condition that ensures an agreement with lattice calculations of string tension between quarks at zero chemical potential. The location of the first order phase transitions in the (mu,T)-plane does not depend on the dilaton boundary conditions for light and heavy quarks. At these phase transitions, the function α undergoes jumps depending on temperature and chemical potential. We also show that for the second boundary conditions the running coupling decreases with a temperature increase, and the dependence on temperature and chemical potential both for light and heavy quarks is actually specified in quark-gluon plasma (QGP) phase by functions of one variable, demonstrating in this sense automodel behavior.