I have (hopefully) a rather basic question about smooth elliptic partial differential equations.

Let $L$ be a linear elliptic differential operator with polynomial coefficients in $\mathbb{R}^n.$ Let $u\in L^{\infty}(\mathbb{R}^n)$ be such that it has compact support and $L(u)$ is supported on a finite set (as a distribution). Then is $u$ necessarily 0? I (naively) hope that the answer is yes, and perhaps could be proved by using some known asymptotics of the fundamental solution of $L$ near the support of $u.$ Any suggestions or references will be greatly appreciated.