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 $(-\mathrm{\Delta }{)}_p^{s}u=f(u)$ with superhomogeneous and subcritical nonlinearity $f$, in a bounded open set $\mathrm{\Omega }$ and under the nonlocal zero Dirichlet conditions. Assuming only that $\mathrm{\Omega }$ is Steiner symmetric, we show that the supports of positive and negative parts of $u$ touch $\mathrm{\partial }\mathrm{\Omega }$. As a consequence, the nodal set of $u$ has the same property whenever $\mathrm{\Omega }$ is connected. The proof is based on the analysis of equality cases in certain polarization inequalities involving positive and negative parts of $u$, and on alternative characterizations of second eigenfunctions and least energy nodal solutions.