We study the stationary problem of viscous quantum hydrodynamics as it appears in 

the modelling of semiconductor devices, in particular, the resonant tunnel diode. 

This model features a mixed order structure

of the derivatives of the unknown functions. The velocity of the electrons

is partly subsonic, and supersonic at other parts of the device. Moreover, 

because of the barrier potential, only weak solutions can exist. Using variational 

arguments, we prove the existence of such solutions, without any smallness 

assumptions on the given data.