Here is a fundamental question on PDEs whose answer must be known but is not easy to find:
Let $(Lu)(x) := \sum_{|\alpha|\leq m} c_\alpha(x)\partial_x^\alpha u(x)$ be a partial differential operator with boundedly smooth coefficients $c_\alpha\in C_b^\infty(\Bbb R^n)$. (Then $L:\mathcal D'(\Bbb R^n)\to\mathcal D'(\Bbb R^n)$ is well-defined and continuous.) Assume that $u\in W^{k,p}(\Bbb R^n)$ with $k\in\Bbb N$ and $1\leq p < \infty$ satisfies $Lu=0$. Does there exist for every $\epsilon>0$ such $u_\epsilon\in C^\infty(\Bbb R^n)$ that $\Vert u_\epsilon - u\Vert_{W^{k,p}(\Bbb R^n)}<\epsilon$ and $Lu_\epsilon=0$?
The constant coefficient case (at least with $p=2$) is very easy to obtain using frequency cutting, and my guess is that this idea generalizes. Thanks in advance for a reference or a proof sketch. (My knowledge of $\Psi DO$s is limited, but it would be enlightening to see if their basic theory is applicable here and if this result holds for them as well.)