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

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 $\mathbb{R}^N$.