Non-existence criteria for solutions of the Lane-Emden-Fowler system

By means of a simple proof we show that the system -Delta u(i) = P(i)(vertical bar x vertical bar)f(i)(u(i+1)) and - Delta u(d) = P(d)(vertical bar x vertical bar f(d)(u(1)) for i = (1, d - 1) over bar on R(N), where N > 2, f(i) (i= (1,d) over bar): (0, infinity) -> (0, infinity) are continuous functions bounded in a neighborhood at infinity such that lim(si SE arrow 0)f(i i=(1,d) over bar)(s(i)) = + infinity and p(ii=(1,d) over bar) are positive radial functions which are continuous on R(N), has no positive radial solutions that decay to zero at infinity provided integral(infinity)(0) r Sigma(d)(i=1) p(i)(r)dr = infinity with r := vertical bar x vertical bar. Moreover, a non-existence result for the case N = 2 is obtained. (C) 2011 Elsevier Ltd. All rights reserved.