The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schr\"odinger equation with magnetic field

TL;DR

This paper develops a Morse theory-based approach to estimate the number of nontrivial solutions for a nonlinear Schrödinger equation influenced by a magnetic field, providing a theoretical framework for solution multiplicity.

Contribution

It introduces an abstract Morse relations framework and applies it to prove multiple solutions for Schrödinger equations with magnetic fields, extending previous methods.

Findings

Established Morse relations for nonlinear Schrödinger equations

Proved existence of multiple solutions under magnetic field influence

Extended Morse theory applications to magnetic Schrödinger problems

Abstract

Based on some ideas introduced by Benci and Cerami \cite{benci-cerami_calvar}, we obtain an abstract result that establishes a version of the Morse relations. Afterward, we use this result to prove multiplicity of solutions for a nonlinear Schr\"odinger equation with an external magnetic field.