In this paper, we study the weighted higher order semilinear equation in an exterior domain \((-\Delta)^{m} u=|x|^{\alpha}g(u) \quad \quad \text{in} \ \mathbb{R}^{N}\setminus B_{R_{0}},\) where N ≥ 1, m ≥ 2 are integers, α > −2m, g is a continuous and nondecreasing function in [0, +∞) and positive in (0, +∞), \(B_{R_{0}}\) is the ball of the radius R0 centered at the origin. We prove that a positive supersolution of the problem which verifies (−Δ)iu > 0 in \(\mathbb{R}^{N}\setminus B_{R_{0}}\) (i=0, …, m−1) exists if and only if N > 2m and \(\int_{0}^{\delta}{g(t)\over t {2(N-m)+\alpha \over N-2m}}{\rm d}t<\infty,\) for some δ > 0. We further obtain some existence and nonexistence results for the positive solution to the Dirichlet problem when g(u) = up with p > 1, by using the Pohozaev identity and an embedding lemma of radial Sobolev spaces.
