# Existence, multiplicity and regularity for a Schr\"odinger equation with magnetic potential involving sign-changing weight function

## TL;DR

This paper investigates the existence, multiplicity, and regularity of solutions to a Schrödinger equation with magnetic potential and sign-changing weights, using variational methods related to the Nehari manifold and fibering maps.

## Contribution

It introduces new analytical techniques to handle sign-changing weights in magnetic Schrödinger equations, establishing existence and multiplicity results.

## Key findings

- Proved existence of solutions under certain conditions.
- Established multiplicity of solutions using variational methods.
- Analyzed regularity properties of solutions.

## Abstract

In this paper we consider the following class of elliptic problems $$- \Delta_A u + u = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u ,\,\, x\in \mathbb{R}^N$$ where $1<q<2<p<2^*-1= \frac{N+2}{N-2}$, $a_{\lambda}(x)$ is a sign-changing weight function, $b_{\mu}(x)$ have some aditional conditions, $u \in H^1_A(\mathbb{R}^N)$ and $A:\mathbb{R}^N \rightarrow\mathbb{R}^N$ is a magnetic potential. Exploring the relationship between the Nehari manifold and fibering maps, we will discuss the existence, multiplicity and regularity of solutions.