We know that if $1\leq p < \infty$, the following embbedings of Sobolev spaces in $\mathbb{R}^k$ are true:
- $\text { if } 1 \leq p<k, W^{1, p}\left(\mathbb{R}^{k}\right) \subset L^{q}\left(\mathbb{R}^{k}\right) \text { with } q \in\left[p, p_{*}\right] \text { and } p_{*}=\frac{kp}{k-p}$
- $W^{1, k}\left(\mathbb{R}^{k}\right) \subset L^{q}\left(\mathbb{R}^{k}\right) \text { for every } q \in[k, \infty)$
- $\text { if } p>N, W^{1, p}\left(\mathbb{R}^{k}\right) \subset L^{\infty}\left(\mathbb{R}^{k}\right)$.
Now, I am interested in proving that these emmbedings are not compacts. I am a little lost with this proof, I have found this related quiestion $W^{1,1}(\mathbb{R}^n)$ is not compactly embedded in $L^1(\mathbb{R}^n)$?, but I don't know if the proof which appears there can be generalised to answer my question.