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 of 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.