# Neumann fractional $p-$Laplacian: eigenvalues and existence results

## TL;DR

This paper investigates the properties of the fractional p-Laplacian with Neumann boundary conditions, establishing eigenvalue sequences, analyzing associated evolution problems, and exploring nonlinear equations without the Ambrosetti-Rabinowitz condition.

## Contribution

It introduces new properties of the fractional p-Neumann derivative, proves the existence of eigenvalues, and studies evolution and nonlinear problems without standard growth conditions.

## Key findings

- Existence of a diverging sequence of eigenvalues.
- Basic properties of solutions to the evolution problem.
- Analysis of nonlinear problems without Ambrosetti-Rabinowitz condition.

## Abstract

We develop some properties of the $p-$Neumann derivative for the fractional $p-$Laplacian in bounded domains with general $p>1$. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear problem with source in absence of the Ambrosetti-Rabinowitz condition.