Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

On a class of elliptic problems in R2: symmetry and uniqueness results

J. Prajapata1 and G. Tarantelloa2

a1 Indian Statistical Institute, Bangalore Centre, 8th Mile, Mysore Road, R.V. Post, Bangalore 560 059, India

a2 Universita'di Roma ‘Tor Vergata’, Dipartimento di Matematica, Via Della Ricerca Scientifica, 00133 Rome, Italy

Abstract

In the plane R2, we classify all solutions for an elliptic problem of Liouville type involving a (radial) weight function. As a consequence, we clarify 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 than radial symmetry, the solutions always inherit the invariance of the problem under inversion with respect to suitable circles. This symmetry result is derived with the help of a 'shrinking-sphere' method.

(Received April 15 2000)

(Accepted October 04 2000)