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 



Condition to obtain a not compact embedding
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 $L^{p^\*}\_\alpha (\Omega)$.


