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

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...

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Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2019-10, Vol.149 (5), p.1163-1173
Main Authors: Bobkov, Vladimir, Kolonitskii, Sergey
Format: Article
Language:English
Subjects:
Boundary conditions
Dirichlet problem
Domains
Eigenvectors
Elliptic functions
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Summary: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.
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2018.88