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: {(-Delta)(alpha)u = v(q) in R-N, (-Delta)(alpha)v = u(p) in R-N, where alpha is an element of (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 x S-+(N), where S-+(N) is the half unit sphere in RN+1) based on maximum principles which are obtained by barrier functions and a coupling argument using a fractional Sobolev trace inequality.