Under which conditions does
$$\|u\|_{\infty}\leq C\|\Delta u\|_{L_2}+\|u\|_{L_2}$$
hold for some constant $C>0$?
Where $u$ satisfies $\sup_x x^b \partial^a_x u(x) \in L^\infty$ for all multiindices $a,b\in \mathbb{N}_0^n$ and $u\in C^\infty(\mathbb{R}^n)$