A note on the number of nodal solutions of an elliptic equation with symmetry

We consider the semilinear problem $-\Delta u+\lambda u={\left|u\right|}^{p-2}u$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega \subset {\mathbb{R}}^N$ is a bounded smooth domain and $2<p<2^1=2N/(N-2)$. We show that if $\Omega$ is invariant under a nontrivial orthogonal involution then, for $\lambda >0$ sufficiently large, the equivariant topology of $\Omega$ is related to the number of solutions which change sign exactly once.