Multiplicity of solutions for a NLS equations with magnetic fields in $\mathbb{R}^{N}$}

TL;DR

This paper explores multiple solutions for a nonlinear Schrödinger equation with magnetic fields in Euclidean space, employing variational methods and topological tools to establish the existence of several solitary wave solutions.

Contribution

It introduces a novel application of minimax and Lusternik-Schnirelman theory to prove solution multiplicity for NLS equations with magnetic fields.

Findings

Multiple nontrivial weak solutions are proven to exist.

The methods connect the existence of solutions with topological properties of the underlying functional.

The approach advances understanding of solitary waves in magnetic field contexts.

Abstract

We investigate the multiplicity of nontrivial weak solutions for a class of complex equations. This class of problems are related with the existence of solitary waves for a nonlinear Sch\"{o}dinger equation. The main result is established by using minimax methods and Lusternik-Schnirelman theory of critical points.