Uniqueness of solutions to an elliptic inequality with rapid decay at infinity

We consider an elliptic differential inequality: $\mathrm{\mid }\mathrm{\Delta }u(x)\mathrm{\mid }\le C_0({\text{YYYY}}^{-\gamma }\mathrm{\mid }u(x)\mathrm{\mid }+{\text{YYYY}}^{-\theta }\mathrm{\mid }\mathrm{\nabla }u(x)\mathrm{\mid })$ in an exterior domain ${\mathbb{R}}^n\setminus \text{ooo}U$, where $U$ is a simply connected bounded domain $U$, $x:=(y,z)\in {\mathbb{R}}^n$ with $y\in {\mathbb{R}}^m$ and $z\in {\mathbb{R}}^{n-m}$ for given $m\in \{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$, and $\gamma ,\theta \in \mathbb{R}$ are constants. We assume that $u(x)$ decays with exponential rate in the $y$-coordinates and polynomial rate in the $z$-coordinates as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. We prove that if decay rates of $u$ satisfy certain conditions related to the constants $\gamma ,\theta \in \mathbb{R}$, then $u\equiv 0$ in $\text{UUUUU}$. The key is a Carleman estimate with typical cut-off arguments.