Symmetry of solutions to semilinear elliptic equations via Morse index

Authors:
Filomena Pacella and Tobias Weth

Journal:
Proc. Amer. Math. Soc. **135** (2007), 1753-1762

MSC (2000):
Primary 35J60

Published electronically:
January 31, 2007

MathSciNet review:
2286085

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove symmetry results for solutions of semilinear elliptic equations in a ball or in an annulus in , , in the case where the nonlinearity has a convex first derivative. More precisely we prove that solutions having Morse index are foliated Schwarz symmetric, i.e. they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. From this we deduce, under some additional hypotheses on the nonlinearity, that the nodal set of sign changing solutions with Morse index intersects the boundary of the domain.

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Additional Information

**Filomena Pacella**

Affiliation:
Dipartimento di Matematica, Universitá di Roma “La Sapienza”, P. le A. Moro 2, 00185 Roma, Italy

**Tobias Weth**

Affiliation:
Mathematisches Institut, Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany

DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08652-2

Received by editor(s):
September 30, 2005

Received by editor(s) in revised form:
January 31, 2006

Published electronically:
January 31, 2007

Additional Notes:
The first author’s research was supported by M.I.U.R., project ‘Variational Methods and Nonlinear Differential Equations’

Communicated by:
David S. Tartakoff

Article copyright:
© Copyright 2007
American Mathematical Society