Counterexample of Sobolev Embedding Theorem in $W_0^{1,p}$

Question

I am looking for a counterexample of Sobolev Embedding Theorem, i.e. I am seeking for a sobolev function $u\in W_0^{1,p}(\Omega),\,p\in[1,n)$, $\Omega$ is a bounded domain in $\mathbb{R}^n$ such that the inequality: $||u||_{L^q}\leq C||\nabla u||_{L^p}$ does not hold, where $q>p^*:=\frac{np}{n-p}$. A similar question was asked in the past, but with $\mathbb{R}^n$ as a domain and the space used was $W^{1,p}(\Omega)$, see link

Answer 1

You don't need any special function to reach this conclusion.

Any domain contains a ball; so it's enough to consider a ball, which may as well be centered at $0$. Take any function $f$ with nonzero $L^q$ norm supported in this ball. Consider the sequence $f_k(x)=f(kx)$ and use the change of variables $y=kx$ to show that $$ \|f_k\|_{L^q} = k^{-n/q} \|f\|_{L^q},\quad \|\nabla f_k\|_{L^p} = k^{1-n/p} \|f\|_{L^p} $$ The conclusion follows since $-\frac{n}{q} > 1-\frac{n}{p}$.