Bifurcation from Infinity of the Schr\"odinger Equation via Invariant Manifolds

TL;DR

This paper investigates bifurcations from infinity in the nonlinear Schrödinger equation by employing invariant manifold theory, Conley index, and shape theory to establish new results on solution multiplicity under Landesman-Lazer conditions.

Contribution

It introduces a novel approach using invariant manifolds and topological methods to analyze bifurcations from infinity in unbounded domains for Schrödinger equations.

Findings

Established a global invariant manifold for the associated parabolic equation.

Derived new bifurcation and multiplicity results using Conley index and shape theory.

Provided conditions under which solutions bifurcate from infinity.

Abstract

This paper is concerned with the bifurcation from infinity of the nonlinear Schr\"odinger equation $$-\Delta u+V(x)u=\lambda u+f(x,u),\hspace{0.4cm} x\in \mathbb{R}^N.$$ We treat this problem in the framework of dynamical systems by considering the corresponding parabolic equation on unbounded domains. Firstly, we establish a global invariant manifold for the parabolic equation on $\mathbb{R}^N$. Then, we restrict the parabolic equation to this invariant manifold, which generates a system of finite dimension. Finally, we use the Conley index theory and the shape theory of attractors to establish some new results on bifurcations from infinity and multiplicity of solutions of the Schr\"odinger equation under an appropriate Landesman-Lazer type condition.