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.

10 pages, 1 figure. Minor improvements according to referee's suggestions. Accepted to Proceedings of the Royal Society of Edinburgh, Section A: Mathematics