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

TL;DR

This paper proves a conjecture about the boundary-intersecting property of the nodal set of least energy sign-changing solutions for certain quasilinear elliptic equations in symmetric domains, using a polarization method.

Contribution

It establishes the boundary intersection property of the nodal set for least energy sign-changing solutions, extending Payne's conjecture to the p-Laplacian case with superlinear or resonant nonlinearities.

Findings

Nodal set intersects the boundary of the domain.

Proof uses a moving polarization argument.

Results apply to superlinear and resonant nonlinearities.

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.