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On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 149 / Issue 5 / October 2019
- Published online by Cambridge University Press:
- 15 January 2019, pp. 1163-1173
- Print publication:
- October 2019
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- Article
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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.
