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Quantum Euler-Poisson systems: existence of stationary states. (English) Zbl 1122.35140

Summary: 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.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
35Q55 NLS equations (nonlinear Schrödinger equations)
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
82D37 Statistical mechanics of semiconductors
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