In the plane R-2, we classify all solutions for an elliptic problem of Liou
ville type involving a (radial) weight function. As a consequence, we clari
fy the origin of the non-radially symmetric solutions for the given problem
, as established by Chanillo and Kiessling.
For a more general class of Liouville-type problems, we show that, rather t
han radial symmetry, the solutions always inherit the invariance, of the pr
oblem under inversion with respect to suitable circles. This symmetry resul
t is derived with the help of a 'shrinking-sphere' method.