Geodesic
Quantum Euler-Poisson systems: Existence of stationary states
Archivum mathematicum, Tome 40 (2004) no. 4, pp. 435-456.

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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.
Classification : 35Q55, 35Q60, 76Y05, 82C10, 82D37
Keywords: quantum hydrodynamics; existence and uniqueness of solutions; non-monotone pressure; semiconductors 
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     author = {J\"ungel, Ansgar and Li, Hailiang},
     title = {Quantum {Euler-Poisson} systems: {Existence} of stationary states},
     journal = {Archivum mathematicum},
     pages = {435--456},
     publisher = {mathdoc},
     volume = {40},
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     zbl = {1122.35140},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2004__40_4_a10/}
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Jüngel, Ansgar; Li, Hailiang. Quantum Euler-Poisson systems: Existence of stationary states. Archivum mathematicum, Tome 40 (2004) no. 4, pp. 435-456. http://geodesic.mathdoc.fr/item/ARM_2004__40_4_a10/