Is the Laplacian surjective on $C_0^{\infty}$? Let $M := C_0^{\infty}(\mathbb{R}^n)$ denote the smooth maps with compact support. Then we have a map
$\Delta:M\rightarrow M,\,\, f\mapsto \Delta f$, 
where $\Delta f = \sum_{i=1}^{n} \frac{\partial^2}{\partial x_i^2}f$ is the Laplacian. I am wondering if $\Delta$ is surjective, i.e. if for any $f\in M$ there exists an $F\in M$ with $\Delta F = f$. Is that true?
Thanks for your help! 

It is very far from being surjective. Note that if $f\in C^\infty_0$ and $u$ is any harmonic function in the entire space, then $\int (\Delta f)u=\int f(\Delta u)=0$ (integration by parts or Green). This imposes infinitely many independent restrictions on the functions that can be represented as Laplacians of smooth compactly supported functions in every dimension above $1$ (in dimension $1$ the only harmonic functions are linear).
