Liouville type theorems for stable solutions of the weighted fractional Lane-Emden system

Hatem Hajlaoui

doi:10.3934/dcds.2022118

Article Contents

2022, Volume 42, Issue 12: 5665-5681. Doi: 10.3934/dcds.2022118

This issue Previous Article Dynamics and entropy of $ \mathcal{S} $-graph shifts Next Article Asymptotic lattices, good labellings, and the rotation number for quantum integrable systems

Liouville type theorems for stable solutions of the weighted fractional Lane-Emden system

Hatem Hajlaoui^1, ,

1.
Institut Supérieur des Mathématiques Appliquées et de l'Informatique, , Université de Kairouan, Tunisie

^*Corresponding author: Hatem Hajlaoui
^*Corresponding author: Hatem Hajlaoui

Received: March 2022

Revised: July 2022

Early access: August 2022

Published: December 2022

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Abstract

In this paper, we prove Liouville type theorems for stable solutions to the weighted fractional Lane-Emden system

$ \begin{align*} (-\Delta)^s u = h(x)v^p,\quad (-\Delta)^s v = h(x)u^q, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N, \end{align*} $

where $ 1<q\leq p $ and $ h $ is a positive continuous function in $ \mathbb{R}^N $ satisfying $ {\liminf_{|x|\to \infty}}\frac{h(x)}{|x|^\ell} > 0 $ with $ \ell > 0. $ Our results generalize the results established in [23] for the Laplacian case (correspond to $ s = 1 $) and improve the previous work [12]. As a consequence, we prove classification result for stable solutions to the weighted fractional Lane-Emden equation $ (-\Delta)^s u = h(x)u^p $ in $ \mathbb{R}^N $.

Keywords:

Liouville type theorems,
classification results,
fractional Laplacian,
stable solutions,
weighted fractional Lane-Emden system and equation.

Mathematics Subject Classification: Primary: 35J47, 35J15; Secondary: 35B53, 35B35.

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References

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