Liouville-type theorems for polyhannonic systems in RN

In this note, we consider the polyharmonic system (-Delta)(m) u = v(alpha), (-Delta)(m) v = in R-N with N > 2m and alpha >= 1, beta >= 1, where (-Delta)(m) is the polyharmonic operator. For 1/(alpha + 1) + 1/ (beta + 1) > (N - 2m)/N, we prove the non-existence of non-negative, radial, smooth solutions. For 1 < alpha, beta < (N + 2m)/(N - 2m), we show the non-existence of non-negative smooth solutions. In addition, for either (N - 2m)beta < N/alpha + 2m or (N - 2m)alpha < N/beta + 2m with alpha, beta > 1, we show the non-existence of non-negative smooth solutions for polyharmonic system of inequalities (-Delta)(m) u >= v(alpha), (-Delta)(m) v >= mu(beta). More general, we can prove that all the above results hold for the system (-Delta)(m) u = v(alpha), (-Delta)(n) v = 0 in RN with N > max(2m, 2n) and alpha >= 1, beta >= 1. (c) 2005 Elsevier Inc. All rights reserved.