On the second minimax level for the scalar field equation

2012/08/06 by Kanishka Perera, Perera, Kanishka, Cyril Tintarev +1
Computer Science · Mathematics · #35J20 (Primary) 47J10 (Secondary) #35J61 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1208.1139

openalex publication_date 2012/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The paper studies eigenfunctions for the scalar field equation on \R^N at the second minimax level λ₂. Similarly to the well-studied case of the ground state, there is a threshold level λ^# such that λ₂≤ λ^#, and a critical point at the level λ₂ exists if the inequality is strict. Unlike the case of the ground state, the level λ₂ 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^{- ε |x|}. The paper also considers questions about the nodal character and the symmetry breaking for solutions at the level λ₂.

Citations

• Uniqueness of positive solutions of Δu−u+up=0 in Rn
• Existence of solitary waves in higher dimensions

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