Existence of solution of a non-linear multiband kp model with transparent boundary conditions

The existence of the solution of a multiband kp-model that describes the quantum mechanical motion of a system of particles in a crystal is discussed. The model includes the non-parabolicity effects of the high energetic particles in the valence band and the electrostatic potential of the charges. We consider a bounded domain and the ohmic contacts are modeled by using transparent boundary conditions. We include some numerical tests that illustrate the behavior of the solution in the proximity of the resonant states. This point constitutes one of the major difficulties of the mathematical analysis of the system.