On a class of elliptic problems in R-2: symmetry and uniqueness results

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.