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On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

Part of: Elliptic equations and systems Partial differential equations Optimality conditions

Published online by Cambridge University Press:  15 January 2019

Vladimir Bobkov  and
Sergey Kolonitskii
Vladimir Bobkov
Affiliation:
Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, Plzeñ 306 14, Czech Republic (bobkov@kma.zcu.cz)
Sergey Kolonitskii
Affiliation:
St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia (s.kolonitsky@spbu.ru)
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Abstract

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In this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta _p u = f(u)$ in bounded Steiner symmetric domains $ \Omega \subset {{\open R}^N} $ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

Keywords

p-Laplaciansuperlinearsecond eigenvalueleast energy nodal solutionnodal setPayne conjecturepolarization

MSC classification

Primary: 35J92: Quasilinear elliptic equations with $p$-Laplacian
Secondary: 35B06: Symmetries, invariants, etc. 49K30: Optimal solutions belonging to restricted classes
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1163 - 1173
Copyright
Copyright © Royal Society of Edinburgh 2019 

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