# Existence and multiplicity results for a class of non-linear Schr\"odinger equations with magnetic potential involving sign-changing non linearity

## TL;DR

This paper investigates the existence and multiplicity of solutions for a class of nonlinear Schrödinger equations with magnetic potential and sign-changing nonlinearities, employing variational methods to establish infinite solutions under various conditions.

## Contribution

It introduces new existence and multiplicity results for nonlinear Schrödinger equations with magnetic potential and sign-changing nonlinearities, using the Nehari method and other analytical techniques.

## Key findings

- Proved existence of multiple solutions under certain conditions.
- Established infinite solutions using variational methods.
- Analyzed effects of sign-changing weights on solution multiplicity.

## Abstract

In this work we consider the following class of elliptic problems $$- \Delta_A u + u = a(x) |u|^{q-2}u+b(x) |u|^{p-2}u , \mbox{ in } \mathbb{R}^N, $$ $u\in H^1_A (\mathbb{R}^N)$, with $2<q<p<2^*= \frac{2N}{N-2}$, $a(x)$ and $b(x)$ are functions that can change signal and satisfy some additional conditions; $u \in H^1_A(\mathbb{R}^N)$ and $A:\mathbb{R}^N \rightarrow \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinite solutions to the problem in question, varying the assumptions about the weight functions.