In this paper, we are concerned with the non-critical higher order Lane-Emden-Hardy equationsif 0 ≤ a < 2, and 1 < p < +∞ if 2 ≤ a < 2m. We prove Liouville theorems for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem 1.1), that is, the unique nonnegative solution is u ≡ 0. As an application, we derive a priori estimates and existence of positive solutions to non-critical higher order Lane-Emden equations in bounded domains (Theorem 1.6 and 1.7). The results for critical order Hardy-Hénon equations have been established by Chen, Dai and Qin [5] recently.

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In this paper, we are concerned with the following generalized fully nonlinear nonlocal operators: Fs,m(u(x)) = c N,s m N 2 +s P.V. R N G(u(x) − u(y)) |x − y| N 2 +s K N 2 +s (m|x−y|)dy+m 2s u(x), where s ∈ (0, 1) and mass m > 0. By establishing various maximal principle and using the direct method of moving plane, we prove the monotonicity, symmetry and uniqueness for solutions to fully nonlinear nonlocal equation in unit ball, R N , R N + and a coercive epigraph domain Ω in R N respectively.

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