Foliated Schwarz symmetry for the nodal solution at the second minimax level (Q1756474)

The authors study the second minimax level of the eigenvalue problem for the scalar field equation $$-\Delta u+ V(x)\,u= \mu|u|^{p-2}u\quad\text{in }\mathbb{R}^N$$ with $V\in L^\infty(\mathbb{R}^N)$ and $p\in(2,2^*)$, where $2^*= 2N/(N-2)$, $N\ge 3$, in the Sobolev space $H^1(\mathbb{R}^N)$. By using arguments involving polarization, they show that for nodal solutions this level is foliated Schwarz symmetric. This is applied to an open problem raised by \textit{K. Perera} and \textit{C. Tintarev} [Bull. Lond. Math. Soc. 46, No. 6, 1218--1225 (2014; Zbl 1317.35167)].