Compactness in $W^{1,p}_0(B_R)$

Question

I have a sequence of functions $\{u_k\}_k \in u + W^{1,p}_0(B_R, R^N)$, where $B_R \Subset \Omega $ is the ball of radius R and $\Omega \subset R^n$ open and bounded, $u \in W^{1,p}(B_R, R^N)$.

I want to understand why $\{ u_k - u\}_k$ admits a subsequence that converges weakly to a function $u_{\infty} - u.$

Maybe the set $ W^{1,p}_0(B_R, R^N) $ is weakly compact, but why?

Answer 1

The claim is false.

Let $u=0$ and let $v\in W_0^{1,p}(B_R,R^N)$ be a function with $\|v\|>0$. Then define $u_k:=kv$. Then it turns out that there is no weakly convergent subsequence of $\{u_k-u\}_k$ (this is because weakly convergent sequences must be bounded).