On the second minimax level for the scalar field equation

Abstract

The paper studies eigenfunctions for the scalar field equation on ${\mathbb{R}}^N$ at the second minimax level ${\lambda }_2$. Similarly to the well-studied case of the ground state, there is a threshold level $λ^#$ such that $λ_2\le λ^#$, and a critical point at the level ${\lambda }_2$ exists if the inequality is strict. Unlike the case of the ground state, the level ${\lambda }_2$ is not attained in autonomous problems, and the existence is shown when the potential near infinity approaches the constant level from below not faster than $e^{-\epsilon |x|}$. The paper also considers questions about the nodal character and the symmetry breaking for solutions at the level ${\lambda }_2$.

Minor revision - correction of typos. arXiv admin note: substantial text overlap with arXiv:1204.5332