On compact embedding of Sobolev space?

Question

Let$\{u_k\}\in W^{1,p}(R^n)$, $p\in[1,n)$, $\sup_k||u_k||_{W^{1,p}(R^n)}<C$, then $\forall r>0$ and $\forall q\in[1,p^*)$,where $p^*=\frac{np}{n-p}$, there exist a subsequence $\{u_{k_i}\}$ of $\{u_k\}$ such that $\{u_{k_i}\}$ is convergence in $L^q(B_r(0))$.

Answer 1

If you know that $W^{1,p}(B_r(0))$ is compactly embedded in $L^q(B_r(0))$ for $q\in[1,p^\star)$ then the result follows, because $\|u_k\|_{1,p}(\mathbb{R}^n)\leq C$ implies that $$\|u_k\|_{1,p}(B_r(0))\leq C$$

You can find the proof of the first statement in the book of Adams or Leoni:

R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

G. Leoni, A First Course in Sobolev Spaces, Graduated Studies in Mathematics; v.105