A liouville type theorem for poly-harmonic system with dirichlet boundary conditions in a half space

In this paper, we consider the following poly-harmonic system with Dirichlet boundary conditions in a half space Rⁿ₊: { (-Δ)^mu(x) = u^α1 (x)v^β1 (x), x ∈ Rⁿ₊, (-Δ)mv(x) = u^α2 (x)v^β2 (x), x ∈ Rⁿ₊, u = ∂u /∂x_n = ∂²u/ ∂x²_n = · · · = ∂^m-1u ∂x^m-1 _n = 0, x ∈ ∂Rⁿ₊, v = ∂v /∂x_n = ∂²v/ ∂x²_n = · · · = ∂^m-1v ∂x^m-1 _n = 0, x ∈ ∂Rⁿ₊, (0.1) where αi + βi = n+2m n-2m > 2, αi, βi ≥ 1 for i = 1, 2. First, we show that, under some mild growth conditions, (0.1) is equivalent to the IE system { u(x) = ∫ ^Rn_⁺ G⁺_∞(x, y)u^α1 (y)v^β1 (y)dy, v(x) = ∫ ^Rn₊G⁺_∞(x, y)u^α2 (y)v^β2 (y)dy, (0.2) where G⁺_∞(x, y) := c_n /|x - y|^n-2m∫ ^{4x_nny_n |x-y|2} ₀ z_m-1 (z + 1)^n/2 dz is the Green's function in Rn + with the same Dirichlet boundary conditions. Then, inspired by the work [12] of Y. Fang and W. Chen on the Dirichlet problem for (-Δ)^mu = u^p in Rⁿ₊, we use method of moving planes in integral forms to prove the nonexistence of nontrivial nonnegative solutions for IE system (0.2), and as a consequence, we derive the nonexistence of nontrivial nonnegative classical solutions for problem (0.1).