An open ball $B$ in $\mathbb{R}^{n}$ is given. We are looking for a sequence $(\phi_{j})_{j}$ of $C_{c}^{\infty}(B)$ and a sequence $(K_{j})_{j}$ of compact sets such that:
1) the compact sets $(K_{j})_{j}$ are increasing and their union equals $B$,
2) for each natural number $j$, the laplacian $\Delta \phi_{j}=1$ on $K_{j}$ and its support is in $K_{j+1}$,
3) the sequence $(\phi_{j})_{j}$ satisfies $$\sup_{(x,j)\in B\times\mathbb{N}}|\Delta\phi_{j}(x)|<\infty.$$
My question is: does this problem have a solution?
No, by an almost identical argument to my answer to your previous question. By the divergence theorem,
$$0=\int \Delta\phi_j=\int_{K_j} \Delta\phi_j+\int_{K_{j+1}\setminus K_j} \Delta\phi_j=\operatorname{vol}(K_j)+\int_{K_{j+1}\setminus K_j}\Delta\phi_j,$$ so $$\sup(-\Delta\phi_j)\geq \frac{1}{\operatorname{vol}(K_{j+1}\setminus K_j)}\int_{K_{j+1}\setminus K_j}-\Delta\phi_j = \frac{\operatorname{vol}(K_j)}{\operatorname{vol}(K_{j+1}\setminus K_j)}\to\infty.$$