Classification of positive solutions for a static Schrodinger-Maxwell equation with fractional Laplacian

In this paper, we study the fractional-order nonlocal static Schrodinger equation (-Delta)(alpha/2) u = PuP-1 (vertical bar x vertical bar(a-n) * u(P)), u > 0 in R-n, with n >= 3, alpha epsilon (1, n) and p > 1. It can be viewed as an integral system involving the Riesz potentials integral u(x) = root p integral(Rn) u(p-1)(y) upsilon (y) dy/ vertical bar x-y vertical bar(n-alpha) , u > 0 in R-n, upsilon(x) = root p integral(Rn) u(p-1)(y) upsilon (y) dy/ vertical bar x-y vertical bar(n-alpha) , u > 0 in R-n First, the fact p is larger than the Serrin exponent is a necessary condition for the existence of the positive solution. Based on this result, we investigate the classification of the positive solutions. If the system has solutions in L n(p-1)/alpha (Rn), then p must be the critical exponent n, and hence all the positive solutions can be classified as u(x) = v(x) = here c, t are positive constants, and x* epsilon R-n. (C) 2014 Elsevier Ltd. All rights reserved.