Yes, such solutions do exist. Let $v$ be the function which is the fundamental solution $|x|^{2-n}$ in $\mathbb{R}^n - B_1$ (with $n \geq 3$) and $1$ in $B_1$, and let $u = \rho \ast v$ for some smooth mollifier $\rho$ supported in $B_{1/4}$.

Then $\Delta u = 0$ outside $B_{2}$ and $u > c(n)$ in $B_{2}$ so $C(x) = \Delta u / u$ is smooth, bounded and vanishes outside $B_2$.

For some $C$ satisfying the decay properties, such solutions do exist. Let $v$ be the function which is the fundamental solution $|x|^{2-n}$ in $\mathbb{R}^n - B_1$ (with $n \geq 3$) and $1$ in $B_1$, and let $u = \rho \ast v$ for some smooth mollifier $\rho$ supported in $B_{1/4}$.

Then $\Delta u = 0$ outside $B_{2}$ and $u > c(n)$ in $B_{2}$ so $C(x) = \Delta u / u$ is smooth, bounded and vanishes outside $B_2$.