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In this note we prove the Payne-type conjecture about the behaviour of thenodal 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 assumedto be either superlinear or resonant. In the latter case, least energysign-changing solutions are second eigenfunctions of the zero Dirichlet$p$-Laplacian in $\Omega$. We show that the nodal set of any least energysign-changing solution intersects the boundary of $\Omega$. The proof is basedon a moving polarization argument.

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