On a property of the nodal set of least energy signchanging solutions for quasilinear elliptic equations
Abstract
In this note we prove the Paynetype conjecture about the behaviour of the nodal set of least energy signchanging 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 signchanging solutions are second eigenfunctions of the zero Dirichlet $p$Laplacian in $\Omega$. We show that the nodal set of any least energy signchanging solution intersects the boundary of $\Omega$. The proof is based on a moving polarization argument.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.02816
 Bibcode:
 2017arXiv170702816B
 Keywords:

 Mathematics  Analysis of PDEs;
 35J92;
 35B06;
 49K30
 EPrint:
 10 pages, 1 figure. Minor improvements according to referee's suggestions. Accepted to Proceedings of the Royal Society of Edinburgh, Section A: Mathematics