Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains

We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain $\Omega_0$ of the domain of definition $\Omega$ of the energy balance equation and of the Poisson equation. Here $\Omega_0$ corresponds to the region of semiconducting material, $\Omega\setminus\Omega_0$ represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a $W^{1,p}$-regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem