  In this article, the new black hole solutions to the
Einstein-power-Maxwell-dilaton gravity theory have been investigated in a
four-dimensional space-time. The coupled scalar, electromagnetic and
gravitational field equations have been solved in a static and spherically
symmetric geometry. It has been shown that dilatonic potential, as the solution
to the scalar field equation, can be written in the form of a generalized
Liouville potential. Also, three classes of novel charged dilaton black hole
solutions, in the presence of power law nonlinear electrodynamics, have been
constructed out which are asymptotically non-flat and non-AdS. The conserved
and thermodynamic quantities have been calculated from geometrical and
thermodynamical approaches, separately. Since the results of these two
alternative approaches are identical one can argue that the first law of black
hole thermodynamics is valid for all of the new black hole solutions. The
thermodynamic stability or phase transition of the black holes have been
studied, making use of the canonical ensemble method. The points of type-1 and
type-2 phase transitions as well as the ranges at which the black holes are
stable have been indicated by considering the heat capacity of the new black
hole solutions. The global stability of the black holes have been studied
through the grand canonical ensemble method. Regarding the Gibbs free energy of
the black holes, the points of Hawking-Page phase transition and ranges of the
horizon radii at which the black holes are globally stable have been
determined.
