# Existence at least four solutions for a Schr\"odinger equation with magnetic potential involving sign-changing weight function

## TL;DR

This paper proves the existence of at least four solutions for a class of elliptic Schrödinger equations with magnetic potential and sign-changing weights, using advanced variational methods.

## Contribution

It introduces new multiplicity results for Schrödinger equations with magnetic potential and sign-changing weights, expanding the understanding of solution existence in such complex settings.

## Key findings

- Existence of at least four solutions established
- Application of Bahri Li argument to magnetic Schrödinger equations
- Handling of sign-changing weight functions in solution analysis

## Abstract

In this paper we consider the following class of elliptic problems $$- \Delta_A u + u = a_{\lambda}(x) |u|^{q-2}u+b_{\mu}(x) |u|^{p-2}u ,$$ for $x \in \mathbb{R}^N$, $1<q<2<p<2^*-1= \frac{N+2}{N-2}$, $a_{\lambda}(x)$ is a sign-changing weight function, $b_{\mu}(x)$ has 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 Bahri Li argument and some preliminar results we will discuss the existence of four solution to the problem in question.