Liouville type theorem for higher order Hénon equations on a half space

In this paper, we are concerned with the higher order Hénon equations with Navier boundary condition on a half space R ₊ ⁿ : (−Δ) ^m u(x)=|x| ^a u ^p (x),u(x)≥0,x∈R ₊ ⁿ ,u=(−Δ)u=⋯=(−Δ) ^m−1 u=0,x∈∂R ₊ ⁿ ,where u∈C ^2m (R ₊ ⁿ )∩C ^2m−2 (R ₊ ⁿ ¯), a≥0, n≥3, 1≤m<[Formula presented] and 1<p<[Formula presented]. We first prove the super poly-harmonic properties and establish the equivalence between (0.1) and the corresponding integral equation. Then, we consider the equivalent integral equation of generalized form, that is, u(x)=∫ _{R + ⁿ} G(x,y)|y| ^a u ^p (y)dywhere G(x,y) denotes the Green's function for (−Δ) ^m on R ₊ ⁿ with Navier or Dirichlet boundary conditions. We establish Liouville theorem for (0.2) via “the method of scaling spheres” in integral forms developed initially in [14] (2018) by Dai and Qin. As a consequence, we obtain the Liouville theorem for (0.1). Extensions to IEs and PDEs with general nonlinearities are also included.