Multiple solutions for a NLS equation with critical growth and magnetic field

TL;DR

This paper investigates multiple solutions for a nonlinear Schrödinger equation with critical growth and magnetic field, establishing a link between the number of solutions and the domain's topology using Lusternik-Schnirelman theory.

Contribution

It introduces a novel application of Lusternik-Schnirelman theory to count solutions of a complex NLS equation with critical growth and magnetic field.

Findings

Multiple solutions are proven to exist for the equation.

The number of solutions correlates with the topological complexity of the domain.

The approach extends previous methods to include magnetic fields and critical nonlinearities.

Abstract

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$ (-i\nabla - A(\mu x))^{2}u= \mu |u|^{q-2}u + |u|^{2^{*}-2}u \ \mbox{in} \ \Omega, \ \ \ \ u \in H^{1}_{0}(\Omega,\mathbb{C}), $$ where $\Omega \subset \mathbb{R}^{N} (N \geq 4)$ is a bounded domain with smooth boundary. Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of $\Omega$.