Bifurcation from infinity for elliptic problems on $R^N$

TL;DR

This paper investigates the bifurcation of solutions from infinity for a class of semilinear Schrödinger equations on the existence of bifurcations is linked to eigenvalues of the associated Hamiltonian, using topological and resonance methods.

Contribution

It establishes conditions under which bifurcation from infinity occurs in Schrödinger equations with Kato-Rellich potentials, connecting bifurcating solutions to bound states.

Findings

Bifurcation from infinity occurs at eigenvalues below the potential's asymptotic bottom.

Uses Conley index and resonance assumptions to analyze bifurcation.

Relates bifurcating solutions to bound states of the Schrödinger equation.

Abstract

In the paper the asymptotic bifurcation of solutions to a parameterized stationary semilinear Schr\"odinger equation involving a potential of the Kato-Rellich type is studied. It is shown that the bifurcation from infinity occurs if the parameter is an eigenvalue of the hamiltonian lying below the asymptotic bottom of the bounded part of the potential. Thus the bifurcating solution are related to bound states of the corresponding Schr\"odinger equation. The argument relies on the use of the (generalized) Conley index due to Rybakowski and resonance assumptions of the Landesman-Lazer or sign-condition type.