The failure of the compactness of the embedding $W^{1, p} (\Omega) \subset L^p (\Omega)$ is local, that is, it can be exhibited in any ball.

An abstract way of seeing the failure of the embedding is to consider a ball $B$ such that $\overline{B} \subset \Omega \setminus \{0\}$. The compactness of the embedding of $W^{1, p} (\Omega)$ in $L^{p^*}_\alpha (\Omega)$ would then imply the compactness of the embedding of $W^{1, p} (B)$ in the unweighted space $L^{p^*} (B)$, in contradiction with the classical theory.

Alternatively, fix a point $a \in \Omega$ and a test function $\varphi \in C^1_c (\mathbb{R}^N)$, and define $$ \varphi_{\lambda} (x) = \lambda^{1-\frac{N}{p}} \varphi \Bigl(\frac{x - a}{\lambda}\Bigr). $$ There exists $\lambda_0>0$ such that if $\lambda \in (0, \lambda_0)$, then $\varphi_\lambda \in C^1_c (\Omega) \subset W^{1,p} (\Omega)$. Moreover the family $(\varphi_\lambda)_{\lambda \in (0, \lambda_0)}$ is bounded in $W^{1, p} (\Omega)$ but is not relatively compact in $L^{p^*}_\alpha (\Omega)$.

The failure of the compactness of the embedding $W^{1, p} (\Omega) \subset L^{p^*} (\Omega)$ is local, that is, it can be exhibited in any ball.

An abstract way of seeing the failure of the embedding is to consider a ball $B$ such that $\overline{B} \subset \Omega \setminus \{0\}$. The compactness of the embedding of $W^{1, p} (\Omega)$ in $L^{p^*}_\alpha (\Omega)$ would then imply the compactness of the embedding of $W^{1, p} (B)$ in the unweighted space $L^{p^*} (B)$, in contradiction with the classical theory.

Alternatively, fix a point $a \in \Omega$ and a test function $\varphi \in C^1_c (\mathbb{R}^N)$, and define $$ \varphi_{\lambda} (x) = \lambda^{1-\frac{N}{p}} \varphi \Bigl(\frac{x - a}{\lambda}\Bigr). $$ There exists $\lambda_0>0$ such that if $\lambda \in (0, \lambda_0)$, then $\varphi_\lambda \in C^1_c (\Omega) \subset W^{1,p} (\Omega)$. Moreover the family $(\varphi_\lambda)_{\lambda \in (0, \lambda_0)}$ is bounded in $W^{1, p} (\Omega)$ but is not relatively compact in $L^{p^*}_\alpha (\Omega)$.