Let $\{z\geq 1\}\subset\mathcal{C}^{n}$ be a subset and let $K_{0}\geq 2n+1$ be a positive integer. Given any $\eta_{0}\in(-\delta_{b}/2,\delta_{b}/2)\setminus\{0\}$, if $v\in C^{3K_{0},\alpha}(\{z\geq 1\})$ for and

(5.217)

$$|v|=O(e^{\eta_{0}\cdot z(\bm{x})^{\frac{n}{2}}}),$$

then the Poisson equation

(5.218)

$$\Delta_{g_{\mathcal{C}^{n}}}u=v$$

has a solution $u\in C^{3K_{0}+2,\alpha}(\{z\geq 1\})$ such that for any $\eta>\eta_{0}$

(5.219)

$$|u(\bm{x})|+|\nabla u(\bm{x})|\leq C\cdot e^{\eta\cdot z^{\frac{n}{2}}},$$

as $z(\bm{x})\to+\infty$, where $C>0$ is independent of $\bm{x}\in\mathcal{C}^{n}$.