I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case.

*Question:*

Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. What is the necessary and sufficient condition $\alpha,\beta$ for which there exists a $u\in C^\infty_{0}(\mathbb{R}^n)$ such that

$$ \|u\|_p=\alpha\text{ and }\|Du\|_p=\beta? $$ What happens if we replace $\mathbb{R}^n$ with an open (not necessarily bounded) set $U\subseteq\mathbb{R}^n$?

Probably, it has something to do with the Poincare Inequality or the first eigenvalue of the domain. But I'm unable to make it precise.

Thank you.

Stefan