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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: In this paper we prove symmetry results for solutions of semilinear elliptic equations in a ball or in an annulus in $ \mathbb{R}^N$, $ N \ge 2$, in the case where the nonlinearity has a convex first derivative. More precisely we prove that solutions having Morse index $ j \le N$ 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 $ j \le N$ 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

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

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