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

Published online by Cambridge University Press:  12 July 2007

J. Prajapat  and
G. Tarantello
J. Prajapat
Affiliation:
Indian Statistical Institute, Bangalore Centre, 8th Mile, Mysore Road, R.V. Post, Bangalore 560 059, India
G. Tarantello
Affiliation:
Universita'di Roma ‘Tor Vergata’, Dipartimento di Matematica, Via Della Ricerca Scientifica, 00133 Rome, Italy
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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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 131 , Issue 4 , August 2001 , pp. 967 - 985
Copyright
Copyright © Royal Society of Edinburgh 2001

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