Is $\Delta C_c^\infty$ a dense subset of $L^p(\mathbb{R}^d)$? I'm struggling to obtain some density result. It is well known that $C^\infty_c(\mathbb{R}^d)$ is dense in $L^p (\mathbb{R}^d)$ for $1\leq p<\infty$. 
It is well known that for $\lambda>0$, $(\lambda -\Delta) C_c^\infty$ is dense in $L^p(\mathbb{R}^d)$ for $1\leq p <\infty$.
Proof of this fact requires maximum principle and Hahn-Banach theorem, and Riesz representation theorem. (Stated in Krylov's Elliptic and Parabolic equation in Sobolev spaces)
I'm wondering whether $\Delta C_c^\infty(\mathbb{R}^d)$  is dense in $L^p (\mathbb{R}^d)$.
I tried by using Newtonial potential, but I fail to obtain the desired result because I found $C^\infty$ function, but I cannot make a sequence of $C^\infty$ functions with compact support. Even I tried cut-off method, I cannot guarantee the fact. 
 A: No for $p=1$, yes for $1<p<\infty$.
If $\phi\in \Delta C^\infty_c$ then $\int\phi=0$; this shows that $\Delta C^\infty_c$ is not dense in $L^1$.
One might think at first that this shows the same thing for other $p$, but it doesn't, because the integral is not a bounded linear functional. Suppose from now on that $1<p<\infty$.
Suppose that $K\in L^1\cap L^\infty$, and for $\delta>0$ define $$K_\delta(x)={\delta^d}K\left( \delta x\right).$$If $\phi\in C^\infty_c$ then $||K_\delta*\phi||_1\le||K||_1||\phi||_1$ and $||K_\delta*\phi||_\infty\le c\delta^d$, hence $$||K_\delta*\phi||_p\to0\quad(\delta\to0).$$
Now say $K=c\chi_{B(0,1)}$, where $c$ is chosen so that $\int K=1$. Say $G$ is the Green's function. Then $$K_\delta*G(x)=G(x)\quad(|x|>1/\delta),$$so if $\phi\in C^\infty_c$ and we set $$\psi_\delta=(\phi-\phi*K_\delta)*G$$then $\psi_\delta\in C^\infty_c$. So $\phi-\phi*K_\delta\in \Delta C^\infty_c$, and if $||f-\phi||_p<\epsilon$ then $||f-(\phi-\phi*K_\delta)||_p<2\epsilon$ for small enough $\delta$.
