A Liouville Type Theorem for Poly-harmonic System with Dirichlet Boundary Conditions in a Half Space

Abstract

Abstract In this paper, we consider the following poly-harmonic system with Dirichlet boundary conditions in a half space ℝn +: where for i = 1, 2. First, we show that, under some mild growth conditions, (0.1) is equivalent to the IE system where 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 = up 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).