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 $\mathbb{R}^n, n>1.$ 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.



A basic question about elliptic pde
The answer is **Yes** and is a consequence of Holmgren's uniqueness Theorem. See for instance [Theorem 1.1.4 in this text.](https://webusers.imj-prg.fr/~nicolas.lerner/m2carl.pdf). The ellipticity serves here to ensure that there does not exist a characteristic hypersurface. Applying HUT, you obtain that $u\equiv0$ over the connected component of the complement of the support of $Lu$. In your case, this means $u\equiv0$ away from a finite set, but since $u\in L^\infty$, this is $u=0$.


