Positive solutions to singular second and third order differential equations for quantum fluids

We analyze a quantum trajectory model given by a steady-state hydrodynamic system fur quantum fluids with positive constant temperature in bounded dom ains for arbitrary large data. The momentum equation can be written as a di spersive third-order equation for the particle density where viscous effect s are incorporated. The phenomena that admit positivity of the solutions ar e studied. The cases, one space dimensional dispersive or non-dispersive, v iscous or non-viscous, are thoroughly analyzed with respect to positivity a nd existence or non-existence of solutions, all depending on the constituti ve relation for the pressure law. We distinguish between isothermal (linear ) and isentropic (power law) pressure functions of the density. it is prove d that in the dispersive, non-viscous model, a classical positive solution only exists for "small" (positive) particle current densities, both for the isentropic and isothermal case. Uniqueness is also shown in the isentropic subsonic case, when the pressure law is strictly convex. However, we prove that no weak isentropic solution can exist fur "large" current densities. The dispersive, viscous problem admits a classical positive solution for al l current densities, both for the isentropic and isothermal case, with an " ultra-diffusion" condition. The proofs are based on a reformulation of the equations as a singular elli ptic second-order problem and on a variant of the Stampacchia truncation te chnique. Some of the results are extended to general third-order equations in any space dimension.