Title: On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations,
Abstract: 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.