# Spectrum of the fractional $p-$Laplacian in $\mathbb{R}^N$ and decay estimate for positive solutions of a Schr\"odinger equation

## TL;DR

This paper investigates the spectral properties of the fractional p-Laplacian in Euclidean space, establishing eigenvalue existence, nonexistence under certain conditions, and decay estimates for eigenfunctions and solutions of related Schrödinger equations.

## Contribution

It proves the existence of an unbounded eigenvalue sequence, nonexistence results for positive integral weights, and extends decay estimates to Schrödinger equation solutions.

## Key findings

- Existence of an unbounded sequence of eigenvalues for the fractional p-Laplacian.
- Nonexistence of eigenvalues when the weight has positive integral.
- Sharp decay estimates for the first eigenfunction and solutions of Schrödinger equations.

## Abstract

In this paper, we prove the existence of unbounded sequence of eigenvalues for the fractional $p-$Laplacian with weight in $\mathbb{R}^N.$ We also show a nonexistence result when the weighthas positive integral. In addition, we show some qualitative properties of the first eigenfunction including a sharp decay estimate. Finally, we extend the decay result to the positive solutions of a Schr\"odinger type equation.