A Liouville type theorem for Lane–Emden systems involving the fractional Laplacian. (28th June 2016)

• Record Type: Journal Article
• Authors: Quaas, Alexander; Xia, Aliang
• Is Part Of: Nonlinearity. Volume 29:Number 8(2016:Aug.)
• Page Start: 2279
• Page End: 2297
• Publication Date: 2016-06-28
• Subjects: non-existence -- fractional Lapalcian -- Lane–Emden system; 35B33 -- 35B53 -- 35J45 -- 35S05 -- 35B08 -- 45K05; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/8/2279
• Languages: English
• ISSNs: 0951-7715

Abstract

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.