PAYNE NODAL SET CONJECTURE FOR THE FRACTIONAL P-LAPLACIAN

We consider second eigenfunctions of the fractional p-Laplacian and least energy nodal solutions to a super-homogeneous fractional p-Laplacian equation, in a bounded open set $\Omega$ and with zero nonlocal Dirichlet conditions. Assuming only that $\Omega$ is Steiner symmetric, we show that the supports of the positive and negative parts of the considered functions touch the boundary of $\Omega$. The proof is based on the method of polarization of functions, on the analysis of equality cases in some inequalities, and on alternative characterizations of second eigenfunctions and least energy nodal solutions.