On the second minimax level of the scalar field equation and symmetry breaking

Abstract

We study the second minimax level λ2 of the eigenvalue problem for the scalar field equation in RN. We prove the existence of an eigenfunction at the level λ2 when the potential near infinity approaches the constant level from below no faster than (Formula presented.). We also consider questions about the nodality of eigenfunctions at this level and establish symmetry breaking at the levels (Formula presented.).