  This paper is focused on the Monotonicity Principle (MP) for nonlinear
materials with piecewise growth exponent. This results are relevant because
enables the use of a fast imaging method based on MP, to the wide class of
problems with two or more materials, where at least one is nonlinear.
  The treatment is very general and allows to model a wide variety of practical
configurations such as, for instance, Superconducting (SC) or Perfect
Electrical Conducting (PEC) or Perfect Electrical Insulating (PEI) materials.
  A key role is played by the average Dirichlet-to-Neumann operator, introduced
in [Corbo Esposito et. al, Inverse Problems 2021], where the MP for a single
type of nonlinearity was treated.
  Realistic numerical examples confirm the theoretical findings.
