A Liouville type theorem for Lane–Emden systems involving the fractional Laplacian. (28th June 2016) - Quaas, Alexander; Xia, Aliang
Abstract: We establish a Liouville type theorem for the fractional Lane–Emden system: { ( − Δ ) α u = v q i n     R N, ( − Δ ) α v = u p i n     R N, where α ∈ ( 0, 1 ), N > 2 α 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 ( I​R × S + N, where S + N is the half unit sphere in R N + 1 ) based on maximum principles which are obtained by barrier functions and a coupling argument using a fractional Sobolev trace inequality.