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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.

MathLearner
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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 $u$.

daw
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  • On the left I get the same norm? And on the right I get t times everything? I don't see where this leads? – MathLearner Oct 17 '18 at 16:29
  • Now let $t\to 0$ or $t\to \infty$. – daw Oct 17 '18 at 18:45
  • So now if $t\to 0$ and the norm of u is positive (say 1) then the inequality doesn't work. But why do we assume that $q≤p^$? I don't understand why the same conclusion would not surface with $p^$. – MathLearner Oct 18 '18 at 05:24
  • The parameters do not play a role, as the right hand side is $|\nabla u|{L^p}$ and not $|u|{W^{1,p}}$. The inequality is false on unbounded domains regardless of parameters $p,q,n$. – daw Oct 18 '18 at 06:34