If the sequence $u_n\in H^1(R^n)$, $n\geq 3$ and $u_n$ weak converges to $u$ in $H^1(R^n)$, if we can get $u_n$ weak converges to $u$ in $L^2(R^n)$? Furthermore, if we can have $\nabla u_n$ weak converges to $\nabla u$ in $L^2(R^n)$?
Probably, this question is elementary. We have know that under some condition (such as regular bounded) on the domain $\Omega\subset R^n$, any bounded sequences in $H^1(R^n)$ with weak convergence in $H^1(R^n)$ implies that $u_n$ weak converges to $u$ in $L^2(R^n)$.
Is the fact true if $\Omega\subset R^n$, no matter bounded or unbounded?
Thank you very much if you can give me a reply!