A Liouville type theorem for Lane–Emden systems involving the fractional Laplacian

We establish a Liouville type theorem for the fractional Lane–Emden system:

where $\alpha \in (0,1)$, $N>2\alpha$ and p, q are positive real numbers and in an appropriate new range. To prove our result we will use the local realization of fractional Laplacian, which can be constructed as a Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre (2007 Commun. PDE 32 1245–60). Our proof is based on a monotonicity argument for suitable transformed functions and the method of moving planes in a half infinite cylinder (${IR}\times S_{+}^{N}$, where $S_{+}^{N}$ is the half unit sphere in ${{\mathbb{R}}^{N+1}}$) based on maximum principles which are obtained by barrier functions and a coupling argument using a fractional Sobolev trace inequality.