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) ( − Δ ) m u ( x ) = u p ( x ) | x | a in ℝ n with n ≥ 3, 1 ≤ m < n 2 , 0 ≤ a < 2 m and p > 1. We establish Liouville theorems in the ranges 1 < p < n + 2 m − 2 a n − 2 m if 0 ≤ a < 2 and 1 < p < +∞ if 2 ≤ a < 2 m for nonnegative classical solutions of equations (0.1), that is, the unique nonnegative solution is u ≡ 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.