Sobolev embedding counterexample

Question

I am trying to find a counterexample to show that $$W^{1,p}(\mathbb{R ^n}) \nsubseteq C^{0,\alpha}(\mathbb{R^n}) $$

for $p>n$ and $\alpha \le 1 -\frac{n}{p}$.

No clue yet, thanks for your help.

Answer 1

(Summary of comments). Let $u(x) = |x|^b$ on the unit ball, and $0$ outside. Choose $b>0$ appropriately.

The function $u$ is absolutely continuous on every line. Also, $|\nabla u|\in L^p$ whenever $p(b-1) > -n$, so $u\in W^{1,p}$ under this condition. On the other hand, $u\notin C^\alpha$ when $\alpha>b$.

Conclusion: if $\alpha> 1-n/p$, we can choose $b$ to satisfy both $p(b-1) > -n$ and $\alpha>b$, thus obtaining a counterexample.