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
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 $L^{p^*}_\alpha (\Omega)$.
