LIOUVILLE-TYPE THEOREM FOR HIGH ORDER DEGENERATE LANE-EMDEN SYSTEM

In this paper, we are concerned with the following high order degenerate elliptic system: {(-A)(m)u = v(p) (-A)(m)v = u(q) in R-+(n+1) := {(x, y)vertical bar x is an element of R-n, y > 0}, (1) u >= 0, v >= 0 where the operator A := y partial derivative(2)(y) + a partial derivative(y) + Delta(x), a >= 1 and n + 2a > 2m, m is an element of Z(+), p, q >= 1. We prove the non-existence of positive smooth solutions for 1 < p, q < n+2a+m/n+2a-2m, and classify positive solutions for p = q = n+2a+2m/n+2a-2m, For 1/p+1 + 1/q+1 > n+2a-2m/n+2a, we show the non-existence of positive, ellipse- radial, smooth solutions. Moreover we prove the non-existence of positive smooth solutions for the high order degenerate elliptic system of inequalities (-A)(m)u >= v(p), (-A)(m)v >= u(q), u >= 0, v >= 0, in R-+(n)+1 for either (n+2a-2m)q < n+2a/p + 2m or (n + 2a - 2m)p < n+2a/q + 2m with p, q > 1.