Foliated Schwarz Symmetry for the Nodal Solution at the Second Minimax Level

We study the second minimax level \(\lambda _2\) of the eigenvalue problem for the scalar field equation in \({\mathbb {R}}^N\). By using the tool of polarization, we prove that every nodal solution at the second minimax level is foliated Schwarz symmetric. As a consequence, we prove an open problem of Perera and Tintarev (Annali di Matematica Pura ed Applicata (4) 194(1):131–144, 2015).