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

Abstract

We establish a Liouville type theorem for the fractional Lane–Emden system:{(−Δ)αu=vqin RN,(−Δ)αv=upin RN,where , 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 (, where is the half unit sphere in ) based on maximum principles which are obtained by barrier functions and a coupling argument using a fractional Sobolev trace inequality.