Classification of finite energy solutions to the fractional Lane-Emden-Fowler equations with slightly subcritical exponents

TL;DR

This paper classifies the asymptotic behavior of finite energy solutions to fractional Lane-Emden-Fowler equations with slightly subcritical exponents, using a unified approach applicable to both spectral and integral fractional Laplacians.

Contribution

It provides a comprehensive classification of solutions' asymptotics for fractional Lane-Emden-Fowler equations, extending to classical cases and employing a simplified, unified method.

Findings

Classified asymptotic behavior of solutions.

Unified approach for spectral and integral fractional Laplacians.

Applicable to classical Lane-Emden-Fowler equations in higher dimensions.

Abstract

We study qualitative properties of solutions to the fractional Lane-Emden-Fowler equations with slightly subcritical exponents where the associated fractional Laplacian is defined in terms of either the spectra of Dirichlet Laplacian or the integral representation. As a consequence, we classify the asymptotic behavior of all finite energy solutions. Our method also provides a simple and unified approach to deal with the classical (local) Lane-Emden-Fowler equation for any dimension greater than 2.