On the second minimax level for the scalar field equation

TL;DR

This paper investigates the second minimax level for scalar field equations on ^N, establishing conditions for the existence of eigenfunctions, analyzing their properties, and exploring symmetry aspects.

Contribution

It introduces new conditions for the existence of solutions at the second minimax level and examines their nodal and symmetry properties.

Findings

Existence of solutions at the second minimax level under specific potential decay conditions.

Identification of a threshold level ^# for the second minimax level.

Analysis of symmetry breaking and nodal characteristics of solutions.

Abstract

The paper studies eigenfunctions for the scalar field equation on $\R^N$ at the second minimax level $\lambda_2$. Similarly to the well-studied case of the ground state, there is a threshold level $\lambda^#$ such that $\lambda_2\le \lambda^#$, 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 $e^{- \varepsilon |x|}$. The paper also considers questions about the nodal character and the symmetry breaking for solutions at the level $\lambda_2$.