A test function whose laplacian is positive at the center of a ball

Question

Let $B$ be a ball in $\mathbb{R}^{n}$ centered at a point $x_{0}$. Is there a nonnegative test function $ \phi\in C^{\infty}_{c}(B) $ whose laplacian is positive at the center of the ball: $\Delta\phi(x_{0})>0$? How about a nonnegative test function $ \psi\in C^{\infty}_{c}(B) $ such that $\Delta\psi(x_{0})<0$?

Answer 1

Take any smooth function with the desired behavior at $x_0,$ for example $f(x)=1+c\|x-x_0\|^2$ with $c=\pm 1,$ and multiply by a smooth “cutoff” function that takes the value $1$ in a neighborhood of $x_0$ and is zero outside a radius $r$ from $x_0$ (small enough that $f$ is nonnegative within this distance). The existence of such cutoff functions is standard - they can be constructed using a smooth bump function for example.