  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 only that
$\Omega$ is Steiner symmetric, we show that the supports of positive and
negative parts of $u$ touch $\partial\Omega$. As a consequence, the nodal set
of $u$ has the same property whenever $\Omega$ 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.
