The steady state expansion of a low-pressure plasma generated by a given Gaussian ionization profile into a dilute neutral gas is calculated using one-fluid and two-fluid models. The inertia of ions and electrons, collisions and widening geometries are taken into account. The plasma density n(0) at the centre or a so-called escape or drain factor and the effective collision frequency nu occur as intrinsic eigenvalues. Monotonic solutions of the basic equations occur if n(0) is below and nu is above certain threshold values. They involve a small positive space charge density in the bulk and a positive boundary sheath in front of an absorbing wall. Non-monotonic solutions exist if n(0) is great and nu is small. In this case a zone with negative space charge can occur between the plasma core and the sheath if nu does not vanish. Due to the collisions the transition between the two types of solution occurs continuously.