In this talk, we present our recent research results related to Euler-Poisson equations with sonic boundary for semiconductor models. The sonic boundary is a critical boundary, which with the effect of doping profile make the system with various possibilities for the physical steady-states and causes these solutions to possess singularities at the boundary. The main interest of this talk is to present the structural stability of subsonic steady-states when the doping profile is a small perturbation. The subsonic steady-states possess a strong singularity at the sonic bouundary. In order to overcome it, a weighted space is introduced.