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

TL;DR

This paper establishes Liouville type theorems for stable solutions of weighted fractional Lane-Emden systems, extending previous results for the Laplacian case to fractional operators and providing classification results for solutions.

Contribution

It generalizes Liouville theorems to fractional Laplacian systems with weights, improving upon prior work and covering a broader class of solutions.

Findings

Liouville theorems for stable solutions in fractional systems

Extension of results from Laplacian to fractional operators

Classification of solutions for weighted fractional Lane-Emden equations

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 $\displaystyle{\liminf_{|x|\to \infty}}\frac{h(x)}{|x|^\ell} > 0$ with $\ell > 0.$ Our results generalize the results established in \cite{HHM16} for the Laplacian case (correspond to $s=1$) and improve the previous work \cite{TuanHoang21}. 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 $\mathbb{R}^N$.