  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
$\lambda_2$.

Analysis of PDEs