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A monotonicity result under symmetry and Morse index constraints in the plane

Part of: Partial differential equations

Published online by Cambridge University Press:  21 July 2020

Francesca Gladiali
Francesca Gladiali*
Affiliation:
Dipartimento di Chimica e Farmacia, Università di Sassari, via Piandanna 4, 07100Sassari, Italy (fgladiali@uniss.it)
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Abstract

This paper deals with solutions of semilinear elliptic equations of the type

\[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \]
where Ω is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions u that are invariant by rotations of a certain angle θ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or u is radial, or, else, there exists a direction $e\in \mathcal {S}$ such that u is symmetric with respect to e and it is strictly monotone in the angular variable in a sector of angle θ/2. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.

Keywords

Semilinear elliptic equationssymmetry and monotonicity resultsbounded and unbounded domainslinearized equationfirst eigenvalue and first eigenfunctionMorse index

MSC classification

Primary: 35B06: Symmetries, invariants, etc.
Secondary: 35B07: Axially symmetric solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 3 , June 2021 , pp. 885 - 915
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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