On the second minimax level for the scalar field equation

Abstract

The paper studies eigenfunctions for the scalar field equation on $\R^N$ at the second minimax level $λ_2$. 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 $λ_2$ exists if the inequality is strict. Unlike the case of the ground state, the level $λ_2$ 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 $e^{- \varepsilon |x|}$. The paper also considers questions about the nodal character and the symmetry breaking for solutions at the level $λ_2$.