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

Abstract

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