A Liouville theorem for a class of fractional systems in R+n

Let 0 < alpha, beta < 2 be any real number. In this paper, we investigate a class of fractional elliptic systems of the form { (-Delta)(alpha/2)u(x) = f (v(x)), (-Delta)(beta/2)v(x) = g(u(x)), x is an element of R-+(n), u,v equivalent to 0, x is not an element of R-+(n). Applying the iteration method and the direct method of moving planes for the fractional Laplacian, without any decay assumption on the solutions at infinity, we prove the Lionville theorem of nonnegative solutions under some natural conditions on f and g. (C) 2017 Elsevier Inc. All rights reserved.