Bifurcation from infinity for an asymptotically linear Schr\"odinger equation

TL;DR

This paper investigates bifurcation phenomena from infinity in an asymptotically linear Schrödinger equation, establishing the existence of unbounded solution sequences near isolated eigenvalues using topological and Morse theoretical methods.

Contribution

It extends previous results by demonstrating bifurcation from infinity for Schrödinger equations at isolated eigenvalues, employing degree and Morse theory depending on eigenvalue multiplicity.

Findings

Existence of solution sequences with unbounded norms near eigenvalues

Application of degree theory for odd multiplicity eigenvalues

Use of Morse and Gromoll-Meyer theory for non-odd multiplicity cases

Abstract

We consider an asymptotically linear Schr\"odinger equation $-\Delta u + V(x)u = \lambda u + f(x,u), \ x\in R^N$, and show that if $\lambda_0$ is an isolated eigenvalue for the linearization at infinity, then under some additional conditions there exists a sequence $(u_n,\lambda_n)$ of solutions such that $\|u_n\|\to\infty$ and $\lambda_n\to\lambda_0$. Our results extend some recent work by Stuart. We use degree theory if the multiplicity of $\lambda_0$ is odd and Morse theory (or more specifically, Gromoll-Meyer theory) if it is not.