Quantum Euler-Poisson systems: Existence of stationary states

Ansgar Juengel, Hailiang Li

  Fachbereich Mathematik und Informatik, Universitaet Mainz, Staudingerweg 9, 55099 Mainz, Germany

Department of Mathematics, Capital Normal University, Beijing 100037, P. R. China

E-mail. juengel@mathematik.uni-mainz.de, lihl@math.sci.osaka-u.ac.jp

A one-dimensional quantum Euler-Poisson system for semiconductors for the electron density and the electrostatic potential in bounded intervals is considered. The existence and uniqueness of strong solutions with positive electron density is shown for quite general (possibly non-convex or non-monotone) pressure-density functions under a ``subsonic'' condition, i.e. assuming sufficiently small current densities. The proof is based on a reformulation of the dispersive third-order equation for the electron density as a nonlinear elliptic fourth-order equation using an exponential transformation of variables.

AMSclassification. 35J40, 35J60, 76Y05.

Keywords.  Quantum hydrodynamics, existence and uniqueness of solutions, non-monotone pressure, semiconductors.