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

Abstract

We consider an elliptic differential inequality: $\vert Δu(x) \vert \le C_0(\YYYY^{-γ}\vert u(x)\vert + \YYYY^{-θ}\vert \nabla u(x)\vert)$ in an exterior domain $\R^n \setminus \ooo{U}$, 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 $m\in \{1,...,n\}$, 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 $|x|\to \infty$. 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 $\UUUUU$. The key is a Carleman estimate with typical cut-off arguments.