# On the existence of a function satisfying a certain inequality

Let $$D$$ be an open disk in $$\mathbb{R}^n$$ ($$n\ge 1$$). I am wondering if there is a nonnegative function $$f:D\rightarrow \mathbb{R}$$ with $$f\in H^1(D)\cap L^\infty(D)$$ satisfying $$|\nabla f(x)|\le cf(x)$$ for $$x\in G$$, where $$G$$ is an open layer around the boundary of $$D$$, and $$f(x)=0$$ if $$x$$ is outside of $$G$$. I am trying to find an example at least in dimensions 1 and 2.

No, such a function cannot exist. In one-dimension, the inequality $$|f'(x)|\leq c f(x)$$ implies $$f(x)\leq f(a) e^{cx}$$ (Grönwall's lemma). Thus, if $$f$$ is nonnegative and zero somwhere, it must be zero everywhere. The higher dimensional case can be reduced to the one-dimensional case by restricting $$f$$ to lines.