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 \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 O. We show that the nodal set of any least energy sign changing solution intersects the boundary of O. The proof is based on a moving polarization argument.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics