I have the two spaces $W_0^{1,p}$ with the norme $$||u||^p=||u||^p_{L^p}+||\nabla u||^p_{L^p}$$ and $$L^{p^*}_{\alpha}=\{ u~\text{measurable}, \int_{\Omega} (|x|^{\alpha} u(x)|)^{p^*} dx<\infty\}$$ equiped with the norm $$||u||_{L^{p^*}_{\alpha}}^{p^*}=\int_{\Omega} (|x|^{\alpha}|u(x)|)^{p^*} dx$$

How to find the condition on $\alpha>0$ such that $W^{1,p}_0$ do not be compactly embeded in $L^{p^*}_{\alpha}$ ?

Where $\Omega\subset\mathbb{R}^N$ is bounded, $N>p$ and $p^*=\frac{Np}{N-p}$

Thank you