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

作者：Liu, Zhao; Dai, Wei^*

来源：Advanced Nonlinear Studies, 2015, 15(1): 117-134.

DOI：10.1515/ans-2015-0106

• Liouville type theorems
• Equivalence
• Dirichlet problem
• Half space
• Method of moving planes in integral forms
• Rotational symmetry
• Nonexistence

In this paper, we consider the following poly-harmonic system with Dirichlet boundary conditions in a half space R-+(n):
[GRAPHICS]
where alpha(i) + beta(i) = n+2m/n-2m > 2, alpha(i), beta(i) >= 1 for i = 1, 2. First, we show that, under some mild growth conditions, (0.1) is equivalent to the IE system
[GRAPHICS]
where
G(infinity)(+) (x, y) : = c(n)/\x-y\(n-2m) integral(4xnyn/\x-y\2)(0) z(m-1)/(z + 1)(n/2) dz
is the Green's function in R-+(n) with the same Dirichlet boundary conditions. Then, inspired by the work [12] of Y. Fang and W. Chen on the Dirichlet problem for (-Delta)(m) u = u(p) in R-+(n), 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).

• 出版日期2015-2
• 单位北京师范大学