Let $D \subset \mathbb{R}^{N}$ be a bounded sufficiently smooth domain of $f$.
Can we construct a mapping $f : D \to \mathbb{R}$ such that $f$ is concave and $f>0$ in $D\backslash\{a\}$ and $\exists ! a \in D$ such that $f(a) = 0$.
I have tried to construct using $\log$, $\exp$, and even quadratic functions but it seems like the condition of unique existence is very hard to compromise. For example $\log(|y|)$ where $|y| > 1$ in $D$ does not satisfy the uniqueness of $x$ such that $\log|x|=0$.
Any help is pretty much appreciated. Thank you for very much!