# Multiplicity results for fractional magnetic problems involving exponential growth

## TL;DR

This paper investigates fractional magnetic elliptic equations with exponential growth nonlinearities, establishing the existence of multiple solutions using critical point theory in a non-resonant setting.

## Contribution

It introduces new multiplicity results for fractional magnetic problems with exponential growth nonlinearities, extending previous work to include magnetic operators and critical exponential growth.

## Key findings

- Multiple solutions exist for the fractional magnetic problem.
- Solutions are obtained under critical exponential growth conditions.
- The results apply to non-resonant parameter ranges.

## Abstract

We study the following fractional elliptic equations of the type, \begin{equation*} (-\Delta)^{\frac12}_A u = \lambda u+f(|u|)u ,\;\textrm{in } \;(-1, 1),\; u=0\;\textrm{in } \;\mathbb R\setminus (-1, 1), \end{equation*} where $\lambda$ is a positive real parameter and $(-\Delta)^{\frac12}_A$ is the fractional magnetic operator with $A:\mathbb R\to \mathbb R$ being a smooth magnetic field. Using a classical critical point theorems, we prove the existence of multiple solutions in the non-resonant case when the nonlinear term $f(t)$ has a critical exponential growth in the sense of Trudinger-Moser inequality.