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Quantum Euler-Poisson systems: Existence of stationary states

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*Ansgar Juengel, Hailiang Li*

**Address.**

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

**Abstract.**

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.