Payne nodal set conjecture for the fractional p-Laplacian in Steiner symmetric domains

Let $u$ be either a second eigenfunction of the fractional $p$-Laplacian or a least energy nodal solution of the equation $(-\Delta)^s_p u = f(u)$ with superhomogeneous and subcritical nonlinearity $f$, in a bounded open set $\Omega$ and under the nonlocal zero Dirichlet conditions. Assuming that $\Omega$ is Steiner symmetric, we show that the supports of positive and negative parts of $u$ intersect $\partial\Omega$, and, consequently, the nodal set of $u$ has the same property.
The proof involves the analysis of certain polarization inequalities related to positive and negative parts of $u$, and alternative characterizations of second eigenfunctions and least energy nodal solutions.
The talk is based on a joint work with S. Kolonitskii.