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On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations. (English) Zbl 1429.35103
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 \(\Omega \subset{{\mathbb R}^N}\) 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 \(\Omega \). We show that the nodal set of any least energy sign-changing solution intersects the boundary of \(\Omega \). The proof is based on a moving polarization argument.

MSC:
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J40 Boundary value problems for higher-order elliptic equations
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