Sobolev inequality cannot hold for all compactly supported smooth functions

Question

I am on a course on Sobolev Spaces and we had this as an exercise:

Let $1\leq p<n$ and $q< p^*$, where $p^*=(pn)/(n-p)$. Show that

$||u||_{L^q(\mathbb{R}^n)}\leq C(q,p,n)||\nabla u||_{L^p(\mathbb{R}^n)}$

cannot hold for all $u\in C_0^\infty(\mathbb{R}^n).$

We were given a hint that if we take some $u\in C_0^\infty(\mathbb{R}^n)$ and then scale the variable inside, then we could show this. But I am at a loss, and don't know how to use the hint.

Answer 1

Hint: For fixed $u \in C_c^\infty$, consider $u_t$ defined by $u_t(x) = u(tx)$ for $t>0$. Then express the norms of $u_t$ in the Sobolev inequality by scaled versions of the norms of $u$.