Liouville theorems for Neumann problem on ℝⁿ₊

In this paper, we are concerned with the following Lane-Emden equation with Neumann boundary condition on a half space R + n \mathbb R^{n}_+ : { ( − Δ ) u ( x ) = u p ( x ) , a m p ; in R + n , ∂ u ( x ) ∂ r = u q ( x ) , a m p ; on ∂ R + n , \begin{equation*} \begin {cases} (-\Delta )u(x)=u^{p}(x), \qquad &\text { in }\mathbb {R}^{n}_{+}, \\ \displaystyle \frac {\partial u(x)}{\partial r}=u^{q}(x), &\text { on }\partial \mathbb {R}^{n}_{+}, \end{cases} \end{equation*} where u ∈ { C 2 ( R + n