If $(u_{n})$ is a sequance in $W^{2, p}_{loc}(R^{N})$ ($1< p < \infty$) such that $u_{n}(x)=0$ if $|x|\geqslant n$, $u_{n} = u_{n+1}$ in $B(0,n)$, $u_{n}\in C_{0}^{2}(B(0,n))$ and $\|u_{n}\|_{W^{2, p}(B(0,m))}\leqslant K(m)$, where $K$ is a constant dependent on $m$. Can we say that $(u_{n})$ has a subsequence $(u_{n_{k}})$ converging weakly to some $u\in W^{2, p}_{loc}(R^{N})$?
What I understand is that if $Q_{1}$ and $Q_{2}$ are two unequal compact subsets of $R^{N}$ then there will be two sequnce $(n_{k})$ and $(m_{k})$ such that $u_{n_{k}}$ converges weakly to some $u_{1}\in W^{2, p}(Q_{1})$ and $u_{m_{k}}$ converges weakly to some $u_{2}\in W^{2, p}(Q_{2})$. And by the conditions, I understand that we can take a common function $u\in W^{2, p}(Q_{1})\cap W^{2, p}(Q_{2})$ and a common sequence $n_{k}$ such that $u_{n_{k}}$ converges weakly to $u$ in $W^{2, p}(Q_{1})$ and in $W^{2, p}(Q_{2})$. But, I don't understand how to get a common function $u$ and a common subsequence $(u_{n_{k}})$ in $W^{2, p}_{loc}(R^{N})$.
The above question came to mind when I was reading about the existence of a viscosity solution of a Hamilton-Jacobi equation with the Dirichlet condition from the book- Generalized Solutions of Hamilton-Jacobi Equations by P.L. Lions (page 99). And I don't even know if my question is right or wrong.
You start from $B(0,1)$ and find a subsequence that converges weakly to some $v_1$ in $W^{2,p}(0,1)$. Then you take that subsequence and consider $B(0,2)$ and find a sub sub sequence that converges weakly to some $v_2$ in $W^{2,p}(B(0,2))$. As you said, $v_1=v_2$ in $B(0,1)$. Inductively, assume that you selected a sub sub sub sub… sequence that converges weakly in $W^{2,p}(0,m)$ to some function $v_m$ and find a further subsequence that converges weakly to some $v_{m+1}$ in $W^{2,p}(0,m+1)$. As before, $v_{m+1}=v_m$ in $B(0,m)$.
Given $x$, find $m$ such that $|x|<m$ and define $u(x)=v_m(x)$. Now you have to use a Cantor diagonal argument similar to the one in the proof of Ascoli Arzela theorem.