Liouville theorems for the sub-linear Lane-Emden equation on the half space

In this article, we study the following Dirichlet problem to the sub-linear Lane-Emden equation {-Delta u=u(p), u(x)>= 0, x is an element of R-+(n), u(x)equivalent to 0, x is an element of partial derivative R-+(n), where n >= 3, 0<p <= 1. By establishing an equivalent integral equation, we give a lower bound of the Kelvin transformation u((sic)). Then, by constructing a new comparison function, we apply the maximum principle based on comparisons and the method of moving planes to obtain that u only depends on x(n). Based on this, we prove the non-existence of non-negative solutions.