LIOUVILLE TYPE THEOREMS, A PRIORI ESTIMATES AND EXISTENCE OF SOLUTIONS FOR SUB-CRITICAL ORDER LANE-EMDEN-HARDY EQUATIONS

We study the sub-critical order Lane-Emden-Hardy equations (0.1) (- Delta)(m)u(x) = u(p)(x)/vertical bar x vertical bar(a) in R-n with n >= 3, 1 <= m< n/2, 0 <= a < 2m and p > 1. We establish Liouville theorems in the ranges 1 < p < n+2m-2a/n-2m if 0 <= a < 2 and 1 < p < +infinity if 2 <= a < 2m for nonnegative classical solutions of equations (0.1), that is, the unique nonnegative solution is u equivalent to 0. As an application, we derive a priori estimates and the existence of positive solutions to sub-critical order Lane-Emden equations in bounded domains.