Bifurcation from infinity for an asymptotically linear Schrödinger equation

2014/12/03 by Wojciech Kryszewski, Kryszewski, Wojciech, Andrzej Szulkin +1
Computer Science · Mathematics · #35J20 #35J91 #58E05 #58E07 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods for differential equations #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.AP #msc:35J20 #msc:35J91 #msc:58E05 #msc:58E07

paper · pdf · doi:10.48550/arxiv.1412.1310

19 pages

arxiv created 2014/12/03 · openalex publication_date 2014/12/03 · arxiv updated 2014/12/04 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

We consider an asymptotically linear Schrödinger equation -Δu + V(x)u = λu + f(x,u), x∈ R^N, and show that if λ₀ is an isolated eigenvalue for the linearization at infinity, then under some additional conditions there exists a sequence (u_n,λ_n) of solutions such that ‖u_n‖→∞ and λ_n→λ₀. Our results extend some recent work by Stuart. We use degree theory if the multiplicity of λ₀ is odd and Morse theory (or more specifically, Gromoll-Meyer theory) if it is not.

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