A Liouville type theorem for poly-harmonic Dirichlet problems in a half space

作者：Fang, Yanqin; Chen, Wenxiong^*

来源：Advances in Mathematics, 2012, 229(5): 2835-2867.

DOI：10.1016/j.aim.2012.01.018

• Liouville theorems
• Half space
• Higher order
• Poly-harmonic operators
• Dirichlet problems
• Equivalence
• Integral equations
• Green's functions
• Method of moving planes in integral forms
• Non-existence of solutions

In this paper, we consider the following Dirichlet problem for poly-harmonic operators on a half space R-+(n):
{(-Delta)(m)u = u(p), in R-+(n),
u = partial derivative u/partial derivative x(n) = partial derivative(2)u/partial derivative x(n)(2) = ... = partial derivative(m-1)u/partial derivative x(n)(m-1) = 0, on partial derivative R-+(n). (1)
First, under some very mild growth conditions, we show that problem (1) is equivalent to the integral equation
u(x) = integral(R+n) G(x, y)u(p) dy, (2)
where G(x, y) is the Green's function on the half space.
Then, by combining the method of moving planes in integral forms with some new ideas, we prove that there is no positive solution for integral equation (2) in both subcritical and critical cases. This partially solves an open problem posed by Reichel and Weth (2009) [40]. We also prove non-existence of weak solutions for problem (1).

• 单位
  南京师范大学