Liouville theorems for integral systems related to fractional Lane-Emden systems in $\mathbb{R}^N_+$

Abstract

In this paper, we consider some integral systems in the half space $\mathbb R^N_+$ and obtain Liouville type theorems about the positive solutions. By moving plane method in terms of the integral form, we shall see that the positive solution $(u(x_1,...,x_N), v(x_1,...,x_N))$ of the integral systems must be independent of the first $(N-1)$-variables, i.e., $u=u(x_N),v=v(x_N)$. Then, combine with the order estimates about $x_N$, we reduce the problem to a sequence of algebraic systems. Furthermore, we discuss the relationship between the integral system and the fractional differential system related to the fractional Lane-Emden equations. By this way, we obtain two non-existence theorems for the fractional differential system.